Category Archives: Computation

Teleportation: Fact and Fiction

When we talk about teleportation, we quickly remember science fiction stories in which both people and artifacts are teleported over great distances instantaneously, overcoming the limitations of relativistic physical laws.

Considering that the theoretical possibility of teleporting quantum information was proposed in the scientific field, Bennett et al [1] (1993), and that it was later experimentally demonstrated to be possible, Bouwmeester et al (1997) [2] and Boschi et al (1998) [3], we can ask the question what is true in this assumption.

For this reason, the aim of this post is to expose the basics of quantum teleportation, analyze its possible practical applications and clarify what is true in the scenarios proposed by science fiction.

Fundamentals of quantum teleportation

Before delving into the fundamentals, it should be clarified that quantum teleportation consists of converting the quantum state of a system into an exact replica of the unknown quantum state of another system with which it is quantum entangled. Therefore, teleportation in no way means the transfer of matter or energy. And as we will see below, teleportation also does not imply the violation of the non-cloning theorem [4] [5].

Thus, the model proposed by Bennet et al [1] is the one shown in the figure below, which is constituted by a set of quantum logic gates that process the states of the three qubits, named A, B and Ancillary. The A qubit corresponds to the system whose state is to be teleported, while the B qubit is the system on which the quantum state of system A is transferred. The ancillary qubit is a qubit necessary to perform the transfer.

Once the three qubits are processed by the logic gates located up to the point indicated by ③ they are quantum entangled [6] [7] [8] [9], in such a way that when a measurement is performed on qubit A and ancillary qubit ④ its state collapses into one of the possible states (|00〉,|01〉,|10〉,|11〉).

From this information, qubit B is processed by a quantum gate U, whose functionality depends on the state obtained from the measurement performed on qubits A and ancillary, according to the following criterion, where I, X, Z are Pauli gates.

|00〉 → U = I.
|01〉 → U = X.
|10〉 → U = Z.
|11〉 → U = XZ.

As a consequence, the state of qubit B corresponds to the original state of qubit A, which in turn is modified as a consequence of the measurement process. This means that once the measurement of qubit A and the ancillary qubit is performed, their state collapses, verifying the non-cloning theorem [4] [5] which establishes the impossibility of creating copies of a quantum state.

From a practical point of view, once the three qubits are entangled, qubit B can be moved to another spatial position, which is constrained by the laws of general relativity, so the velocity of qubit B cannot exceed the speed of light. On the other hand, the measurement result of the A ancillary qubits must be transferred to the location of qubit B by means of a classical information channel, so the information transfer speed cannot exceed the speed of light. The result is that teleportation makes it possible to transfer the state of a quantum particle to another remotely located quantum particle, but this transfer is bound by the laws of general relativity, so it cannot exceed the speed of light.

It is very important to note that in reality the only thing that is transferred between qubit A and qubit B is the information describing the wave function, since what physically constitutes the particles that support the qubit are not teleported. This raises a fundamental question concerning the meaning of teleportation at the level of classical reality, which we will analyze in the context of complex systems consisting of multiple qubits.

But a fundamental aspect in determining the nature of information is the fact that teleportation is based on the transfer of information, which is another indication that information is the support of reality, as we concluded in the post “Reality as an Information Process“.

Quantum teleportation of macroscopic objects

Analogous to the teleportation scenario proposed by Bennett et al [1], it is possible to teleport the quantum state of a complex system consisting of N quantum particles. As shown in the figure below, teleportation from system A to system B requires the use of N ancillary qubits.

This is because the number of combinations of the coefficients a_I of the wave function |ψ_C〉 and their signs is of the order of 2^2N. Thus, when the measurement of the qubits of system A and the auxiliary qubits is performed, 2N classical bits are obtained, which encode 2^2N configurations of the unitary transform U. Thus, the coefficients of the wave function |ψ_C〉 can be rearranged, transforming the wave function of system B into |ψ〉.

Consequently, from the theoretical point of view, the teleportation of complex quantum systems consisting of a large number of particles is possible. However, its practical realization faces the difficulty of maintaining the quantum entanglement of all particles, as a consequence of quantum decoherence [10]. This causes the quantum particles to no longer be entangled as a consequence of the interaction with the environment, which causes the transferred quantum information to contain errors.

Since decoherence effect grows exponentially with the number of particles forming the quantum system, it is evident that the teleportation of N-particle systems is in practice a huge challenge, since the system is composed of 3N particles. The difficulty is even greater if it is considered that in the preparation of the teleportation scenario systems A, B and ancillary qubits will be in the same location. But subsequently system B will have to move to another space-time location in order for the teleportation to make any practical sense. This makes system B under physical conditions that make decoherence much more likely and produce a higher error rate in the transferred quantum state, with respect to the original quantum state of system A.

But suppose that these limitations are overcome in such a way that it is possible in practice to teleport macroscopic objects, even objects of a biological nature. The question arises: what properties of the teleported object are transferred to the receiving object?

In principle, it can be assumed that the properties of the receiving object have the same properties as the original object from the point of view of classical reality, since after the teleportation is completed the receiving object has the same wave function as the teleported object.

In the case of inanimate objects it can be assumed that the classical properties of the receiving object are the same as those of the original object, since its wave function is exactly the same. This must be so since the observables of the object are determined by the wave function. This means that the receiving object will not be distinguishable from the original object, so for all intents and purposes it must be considered the same object. But from this conclusion the question again arises as to what is the nature of reality, since the process of teleportation is based on the transfer of information between the original object and the receiving object. Therefore, it seems obvious that information is a fundamental part of reality.

Another issue is the teleportation of biological objects. In this case the same argument could be used as in the case of non-animate objects. However, in this case it must be considered that in the framework of classical reality decoherence plays a fundamental role, since classical reality emerges as a consequence of the interaction of quantum systems, which observe one another, producing the collapse of their quantum functions, emerging states of classical reality.

This makes the process of entanglement of biological systems necessary in teleportation incompatible with what is defined as life, since this process would inhibit decoherence and therefore the emergence of classical reality. This issue has already been discussed in the posts Reality as an irreducible layered structure and A macroscopic view of Schrodinger’s cat, in which it is made clear that a living being is a set of independent quantum systems, and therefore not entangled among them. Therefore, the process of entanglement of all these systems will require the inhibition of all biological activity, something that will certainly have a profound effect on what is defined as a living being.

Since if teleportation is to be used to move an object to another location, system B must be relocated to that location prior to making measurements on system A and the ancillary system, which is governed by the laws of general relativity. Additionally, once the measurement has been performed, the information must be transferred to the location of system B, which is also limited by general relativity. In short, the teleportation process has no practical advantage over a classical transport process, especially considering that it is also susceptible to possible quantum errors.

Consequently, quantum applications are limited to the implementation of quantum networks and quantum computing systems, the structure of which can be found in the specialized literature [11] [12].

A bit of theory

The functionality of quantum systems is based on tensor calculus and quantum computation [13]. In particular, in order to illustrate the mathematical foundation underpinning quantum teleportation, the figure below shows the functionality of the Hadamard and CNOT logic gates needed to implement quantum teleportation.

Additionally, the following figure shows the functionality of the Pauli gates, necessary to perform the transformation of the wave function of qubit B, once the measurement is performed on the A and auxiliary qubits.

Conclusion

As discussed, quantum teleportation allows the transfer of quantum information between two remote locations by means of particle entanglement. This makes it possible to implement quantum communication and computing systems.

Although for the moment its experimental realization is limited to a very small number of particles, from a theoretical point of view it can be applied to macroscopic objects, which raises the possibility of applying it to transport objects of classical reality, even objects of a biological nature.

However, as has been analyzed, the application of teleportation to macroscopic objects poses a difficulty as a consequence of quantum decoherence, which implies the appearance of errors in the transferred quantum information.

On the other hand, quantum teleportation does not involve overcoming the limitations imposed by the theory of relativity, so the fictitious idea of using quantum teleportation as a means of transferring macroscopic objects at a distance instantaneously is not an option. But in addition, it must be considered that quantum entanglement of biological objects may not be compatible with what is defined as life.

[1]	C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres and W. K. Wootters, “Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels,” Phys. Rev. Lett., vol. 70, pp. 1895-1899, 1993.
[2]	D. Bouwmeester, J.-W. Pan, K. Matte, M. Eibl, H. Weinfurter y A. Heilinger, «Experimental quantum teleportation,» arXiv:1901.11004v1 [quant-ph], 1997.
[3]	D. Boschi, S. Branca, F. De Martini, L. Hardy y S. Popescu, «Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels,» Physical Review Letters, vol. 80, nº 6, pp. 1121-1125, 1998.
[4]	W. K. Wootters y W. H. Zurek, «A Single Quantum Cannot be Cloned,» Nature, vol. 299, pp. 802-803, 1982.
[5]	D. Dieks, «Communication by EPR devices,» Physics Letters, vol. 92A, nº 6, pp. 271-272, 1982.
[6]	E. Schrödinger, «Probability Relations between Separated Systems,» Mathematical Proceedings of the Cambridge Philosophical Society, vol. 32, nº 3, pp. 446-452, 1936.
[7]	A. Einstein, B. Podolsky and N. Rose, “Can Quantum-Mechanical Description of Physical Reality be Considered Complete?,” Physical Review, vol. 47, pp. 777-780, 1935.
[8]	J. S. Bell, «On the Einstein Podolsky Rosen Paradox,» Physics, vol. 1, nº 3, pp. 195-290, 1964.
[9]	A. Aspect, P. Grangier and G. Roger, “Experimental Tests of Realistic Local Theories via Bell’s Theorem,” Phys. Rev. Lett., vol. 47, pp. 460-463, 1981.
[10]	H. D. Zeh, «On the Interpretation of Measurement in Quantum Theory,» Found. Phys., vol. 1, nº 1, pp. 69-76, 1970.
[11]	T. Liu, «The Applications and Challenges of Quantum Teleportation,» Journal of Physics: Conference Series, vol. 1634, nº 1, 2020.
[12]	Z.-H. Yan, J.-L. Qin, Z.-Z. Qin, X.-L. Su, X.-J. Jia, C.-D. Xie y K.-C. Peng, «Generation of non-classical states of light and their application in deterministic quantum teleportation,» Fundamental Research, vol. 1, nº 1, pp. 43-49, 2021.
[13]	M. A. Nielsen and I. L. Chuang, Quantum computation and Quantum Information, Cambridge University Press, 2011.

An interpretation of the collapse of the wave function

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The aim of this post is to hypothesize about the collapse of the wave function based on thermodynamic entropy and computational reversibility. This will be done using arguments based on statistical mechanics, both quantum and classical, and on the theory of computation and the information theory.

In this sense, it is interesting to note that most of the natural processes have a reversible behavior, among which we must highlight the models of gravitation, electromagnetism and quantum physics. In particular, the latter is the basis for all the models of the emerging reality that configure the classical reality (macroscopic reality).

On the contrary, thermodynamic processes have an irreversible behavior, which contrasts with the previous models and poses a contradiction originally proposed by Loschmidt, since they are based on quantum physics, which has a reversible nature. It should also be emphasized that thermodynamic processes are essential to understand the nature of classical reality, since they are present in all macroscopic interactions.

This raises the following question. If the universe as a quantum entity is a reversible system, how is it possible that irreversible behavior exists within it?

This irreversible behavior is materialized in the evolution of thermodynamic entropy, in such a way that the dynamics of thermodynamic systems is determined by an increase of entropy as the system evolves in time. This determines that the complexity of the emerging classical reality grows steadily in time and therefore the amount of information of the classical universe.

To answer this question we will hypothesize how the collapse of the wave function is the mechanism that determines how classical reality emerges from the underlying quantum nature, justifying the increase of entropy and as a consequence the rise in the amount of information.

In order to go deeper into this topic, we will proceed to analyze it from the point of view of the theory of computation and the theory of information, emphasizing the meaning and nature of the concept of entropy. This point of view is fundamental, since quantity of information and entropy are synonyms of the same phenomenon.

Reversible computing

First we must analyze what reversible computation is and how it is implemented. To begin with, it should be emphasized that classical computation has an irreversible nature, which is made clear by a simple example, such as the XOR gate, which constitutes a universal set in classical computation, meaning that with a set of these gates any logical function can be implemented.

This gate performs the logical function X⊕Y from the logical variables X and Y, in such a way that in this process the system loses one bit of information, since the input information corresponds to two bits of information, while the output has only one bit of information. Therefore, once the X⊕Y function has been executed, it is not possible to recover the values of the X and Y variables.

According to Landauer’s principle [1], this loss of information means that the system dissipates energy in the environment, increasing its entropy, so that the loss of one bit of information dissipates a minimum energy k·T·ln2 in the environtment. Where k is Boltzmann’s constant and T is the absolute temperature of the system.

Therefore, for a classic system to be reversible it must verify that it does not lose information, so two conditions must be verified:

The number of input and output bits must be the same.
The relationship between inputs and outputs must be bijective.

The following figure shows the above criteria. But this does not mean that the logic function can be considered a complete set of implementation in a reversible computational context, since the relationship between inputs and outputs is linear and therefore cannot implement nonlinear functions.

It is shown that for this to be possible the number of bits must be n≥3, an example being the Toffoli gate (X,Y,Z)→(X,Y,Z⊕XY) and Fredkin gate (X,Y,Z)→(X, XZ+¬XY,XY+¬XZ), where ¬ is the logical negation.

For this type of gates to form a universal set of quantum computation it is also necessary that they verify the ability to implement nonlinear functions, so according to its truth table the Toffoli gate is not a universal quantum set, unlike the Fredkin gate which is.

One of the reasons for studying universal reversible models of computation, such as the billiard ball model proposed by Fredkin and Toffoli [2], is that they could theoretically lead to real computational systems that consume very low amounts of energy.

But where these models become relevant is in quantum computation, since quantum theory has a reversible nature, which makes it possible to implement reversible algorithms by using reversible logic gates. The reversibility of these algorithms opens up the possibility of reducing the energy dissipated in their execution and approaching the Landauer limit.

Fundamentals of quantum computing

In the case of classical computing a bit of information can take one of the values {0,1}. In contrast, the state of a quantum variable is a superposition of its eigenstates. Thus, for example, the eigenstates of the spin of a particle with respect to some reference axes are {|0〉,|1〉}, so that the state of the particle |Ψ〉 can be in a superposition of the eigenstates |Ψ〉= α|0〉+ β|1〉, α²+ β² = 1. This is what is called a qubit, so that a qubit can simultaneously encode the values {0,1}.

Thus, in a system consisting of n qubits its wave function can be expressed as |Ψ〉 = α₀|00…00〉+α₁|00…01〉+α₂|00…10〉+…+α_N-1|11…11〉, Σ(α_i)² =1, N=2ⁿ, such that the system can encode the N possible combinations of n bits and process them simultaneously, which is an exponential speedup compared to classical computing.

The time evolution of the wave function of a quantum system is determined by a unitary transformation, |Ψ’〉 = U|Ψ〉, such that the transposed conjugate of U is its inverse, U^†U = UU^†= I. Therefore, the process is reversible |Ψ〉 = U^†|Ψ’〉 = U^†U|Ψ〉, keeping the entropy of the system constant throughout the process, so the implementation of quantum computing algorithms must be performed with reversible logic gates. As an example, the inverse function of the Ferdkin gate is itself, as can be easily deduced from its definition.

The evolution of the state of the quantum system continues until it interacts with a measuring device, in what is defined as the quantum measurement, such that the system collapses into one of its possible states |Ψ〉 = |i〉, with probability (α_i)². Without going into further details, this behavior raises a philosophical debate that nevertheless has an empirical confirmation.

Another fundamental feature of quantum reality is particle entanglement, which plays a fundamental role in the implementation of quantum algorithms, quantum cryptography and quantum teleportation.

To understand what particle entanglement means let us first analyze the wave function of two independent quantum particles. Thus, the wave function of a quantum system consisting of two qubits, |Ψ₀〉 = α₀₀|0〉+ α₀₁|1〉, |Ψ₁〉 = α₁₀|0〉+ α₁₁|1〉, can be expressed as:

|Ψ〉= |Ψ₀〉⊗ |Y₁〉= α_00·α₁₀|00〉+α_00·α₁₁|01〉+α_01·α₁₀|10〉+α_01·α₁₁|11〉,

such that both qubits behave as independent systems, since this expression is factorizable in the functions |Ψ₀〉 and |Ψ₁〉. Where ⊗ is the tensor product.

However, quantum theory admits solutions for the system, such as |Ψ〉 = α|00〉+β|11〉, α²+ β² = 1, so if a measurement is performed on one of the qubits, the quantum state of the other collapses instantaneously, regardless of the location of the entangled qubits.

Thus, if one of the qubit collapses in state |0〉 the other qubit collapses also in state |0〉. Conversely, if the qubit collapses into the |1〉 state the other qubit collapses into the |1〉 state as well. This means that the entangled quantum system behaves not as a set of independent qubits, but as a single inseparable quantum system, until the measurement of the system is performed.

This behavior seems to violate the speed limit imposed by the theory of relativity, breaking the principle of locality, which establishes that the state of an object is only influenced by its immediate environment. These inconsistencies gave rise to what is known as the EPR paradox [3], positing that quantum theory was an incomplete theory requiring the existence of hidden local variables in the quantum model.

However, Bell’s theorem [4] proves that quantum physics is incompatible with the existence of local hidden variables. For this purpose, Bell determined what results should be obtained from the measurement of entangled particles, assuming the existence of local hidden variables. This leads to the establishment of a constraint on how the measurement results correlate, known as Bell’s inequalities.

The experimental results obtained by A. Aspect [5] have shown that particle entanglement is a real fact in the world of quantum physics, so that the model of quantum physics is complete and does not require the existence of local hidden variables.

In short, quantum computing is closely linked to the model of quantum physics, based on the concepts of: superposition of states, unitary transformations and quantum measurement. To this we must add particle entanglement, so that a quantum system can be formed by a set of entangled particles, which form a single quantum system.

Based on these concepts, the structure of a quantum computer is as shown in the figure below. Without going into details about the functional structure of each block, the logic gates that constitute the quantum algorithm perform a specific function, for example the product of two variables. In this case, the input qubits would encode all the possible combinations of the input variables, obtaining as a result all the possible products of the input variables, encoded in the superposition of states of the output qubits.

For the information to emerge into the classical world it is necessary to measure the set of output qubits, so that the quantum state randomly collapses into one of its eigenstates, which is embodied in a set of bits that encodes one of the possible outcomes.

But this does not seem to be of practical use. On the one hand, quantum computing involves exponential speedup, by running all products simultaneously. But all this information is lost when measuring quantum information. For this reason, quantum computing requires algorithm design strategies to overcome this problem.

Shor’s factorization algorithm [6] is a clear example of this. In this particular case, the input qubits will encode the number to be factorized, so that the quantum algorithm will simultaneously obtain all the prime divisors of the number. When the quantum measurement is performed, a single factor will be obtained, which will allow the rest of the divisors to be obtained sequentially in polynomial time, which means acceleration with respect to the classical algorithms that require an exponential time.

But fundamental questions arise from all this. It seems obvious that the classical reality emerges from the quantum measurement and, clearly, the information that emerges is only a very small part of the information describing the quantum system. Therefore, one of the questions that arise is: What happens to the information describing the quantum system when performing the measurement? But on the other hand, when performing the measurement information emerges at the classical level, so we must ask: What consequences does this behavior have on the dynamics of the classical universe?

Thermodynamic entropy

The impossibility of directly observing the collapse of the wave function has given rise to various interpretations of quantum mechanics, so that the problem of quantum measurement remains an unsolved mystery [7]. However, we can find some clue if we ask what quantum measurement means and what is its physical foundation.

In this sense, it should be noted that the quantum measurement process is based on the interaction of quantum systems exclusively. The fact that quantum measurement is generally associated with measurement scenarios in an experimental context can give the measurement an anthropic character and, as a consequence, a misperception of the true nature of quantum measurement and of what is defined as a quantum observable.

Therefore, if the quantum measurement involves only quantum systems, the evolution of these systems will be determined by unitary transformations, so that the quantum entropy will remain constant throughout the whole process. But on the other hand, this quantum interaction causes the emergence of information that constitutes classical reality and ultimately produces an increase in classical entropy. Consequently, what is defined as quantum measurement would be nothing more than the emergence of information that conforms classical reality.

The abstract view is clearly shown in practical cases. Thus, for example, from the interaction between atoms that interact with each other emerge the observable properties that determine the properties of the system they form, such as its mechanical properties. However, the quantum system formed by atoms evolves according to the laws of quantum mechanics, keeping the amount of quantum information constant.

Similarly, the interaction between a set of atoms to form a molecule is determined by the laws of quantum mechanics, and therefore by unitary transformations, so that the complexity of the system remains constant at the quantum level. However, at the classical level the resulting system is more complex, emerging new properties that constitute the laws of chemistry and biology.

The question that arises is how it is possible that equations at the microscopic level which are time invariant can lead to a time asymmetry, as shown by the Boltzmann equation of heat diffusion.

Another objection to this behavior, and to a purely mechanical basis for thermodynamics, is due to the fact that every finite system, however complex it may be, must recover its initial state periodically after the so-called recurrence time, as demonstrated by Poincaré [8]. However, by purely statistical analysis it is shown that the probability of a complex thermodynamic system returning to its initial state is practically zero, with recurrence times much longer than the age of the universe itself.

Perhaps the most significant and which clearly highlights the irreversibility of thermodynamic systems is the evolution of the entropy S, which determines the complexity of the system and whose temporal dynamics is increasing, such that the derivative of S is always positive Ṡ > 0. But what is more relevant is that this behavior is demonstrated from the quantum description of the system in what is known as “Pauli’s Master Equation” [9].

This shows that the classical reality emerges from the quantum reality in a natural way, which supports the hypothesis put forward, in such a way that the interaction between quantum systems results in what is called the collapse of the wave function of these systems, emerging the classical reality.

Thermodynamic entropy vs. information theory

The analysis of this behavior from the point of view of information theory confirms this idea. The fact that quantum theory is time-reversible means that the complexity of the system is invariant. In other words, the amount of information describing the quantum system is constant in time. However, the classical reality is subject to an increase of complexity in time determined by the evolution of thermodynamic entropy, which means that the amount of information of the classical system is increasing with time.

If we assume that classical reality is a closed system, this poses a contradiction since in such a system information cannot grow over time. Thus, in a reversible computing system the amount of information remains unchanged, while in a non-reversible computing system the amount of information decreases as the execution progresses. Consequently, classical reality cannot be considered as an isolated system, so the entropy increase must be produced by an underlying reality that injects information in a sustained way.

In short, this analysis is consistent with the results obtained from quantum physics, by means of the “Pauli’s Master Equation”, which shows that the entropy growth of classical reality is obtained from its quantum nature.

It is important to note that the thermodynamic entropy can be expressed as a function of the probability of the microstates as S = – k Σ(p_i ln p_i), where k is the Boltzmann constant and which matches the amount of information in a system, if the physical dimensions are chosen such that k = 1. Therefore, it seems clear that the thermodynamic entropy represents the amount of information that emerges from the quantum reality.

But there remains the problem of understanding the physical process by which quantum information emerges into the classical reality layer¹. It should be noted that the analysis to obtain the classical entropy from the quantum state of the system is purely mathematical and does not provide physical criteria on the nature of the process. Something similar happens with the analysis of the system from the point of view of classical statistical mechanics [10], where the entropy of the system is obtained from the microstates of the system (generalized coordinates qi and generalized momentum pi), so it does not provide physical criteria to understand this behavior either.

The inflationary universe

The expansion of the universe [11] is another example of how the entropy of the universe is growing steadily since its beginning, suggesting that the classical universe is an open system. But, unlike thermodynamics, in this case the physical structure involved is the vacuum.

It is important to emphasize that historically physical models integrate the vacuum as a purely mathematical structure of space-time in which physical phenomena occur, so that conceptually it is nothing more than a reference frame. This means that in classical models, the vacuum or space-time is not explicitly considered as a physical entity, as is the case with other physical concepts.

The development of the theory of relativity is the first model in which it is recognized, at least implicitly, that the vacuum must be a complex physical structure. While it continues to be treated as a reference frame, two aspects clearly highlight this complexity: the interaction between space-time and momentum-energy, and its relativistic nature.

Experiments such as the Casimir effect [12] or the Lamb effect show the complexity of the vacuum, so that quantum mechanics attributes to the basic state of electromagnetic radiation zero-point electric field fluctuations that pervade empty space at all frequencies. Similarly, the Higgs field suggests that it permeates all of space, such that particles interacting with it acquire mass.But ultimately there is no model that defines spacetime beyond a simple abstract reference frame.

However, it seems obvious that the vacuum must be a physical entity, since physical phenomena occur within it and, above all, its size and complexity grow systematically. This means that its entropy grows as a function of time, so the system must be open, there being a source that injects information in a sustained manner. The current theory assumes that dark energy is the cause of inflation [13], although its existence and nature is still a hypothesis.

Conclusions

From the previous analysis it is deduced that the entropy increase of the classical systems emerges from the quantum reality, which produces a sustained increase of the information of the classical reality. For this purpose different points of view have been used, such as classical and quantum thermodynamic criteria, and mathematical criteria such as classical and quantum computation theory and information theory.

The results obtained by these procedures are concordant, allowing verification of the hypothesis that classical reality emerges in a sustained manner from quantum interaction, providing insight into what is meant by the collapse of the wave function.

What remains a mystery is how this occurs, for while the entropy increase is demonstrated from the quantum state of the system, this analysis does not provide physical criteria for how this occurs.

Evidently, this must be produced by the quantum interaction of the particles involved, so that the collapse of their wave function is a source of information at the classical level. However, it is necessary to confirm this behavior in different scenarios since, for example, in a system in equilibrium there is no increase in entropy and yet there is still a quantum interaction between the particles.

Another factor that must necessarily intervene in this behavior is the vacuum, since the growth of entropy is also determined by variations in the dimensions of the system, which is also evident in the case of the inflationary universe. However, the lack of a model of the physical vacuum describing its true nature makes it difficult to establish hypotheses to explain its possible influence on the sustained increase of entropy.

In conclusion, the increase of information produced by the expansion of the universe is an observable fact that is not yet justified by a physical model. On the contrary, the increase of information determined by entropy is a phenomenon that emerges from quantum reality and that is justified by the model of quantum physics and that, as has been proposed in this essay, would be produced by the collapse of the wave function.

Appendix

¹ The irreversibility of the system is obtained from the quantum density matrix:

ρ(t)= ∑_i p_i |i〉〈i|

Being |i〉 the eigenstates of the Hamiltonian ℌ₀, such that the general Hamiltonian is ℌ=ℌ₀+V, where the perturbation V is the cause of the state transitions. Thus for example, in an ideal gas ℌ₀, could be the kinetic energy and V the interaction as a consequence of the collision of the atoms of the gas.

Consequently, “Pauli’s Master Equation” takes into consideration the interaction of particles with each other and their relation to the volume of the system, but in an abstract way. Thus, the interaction of two particles has a quantum nature, exchanging energy by means of bosons, something that is hidden in the mathematical development.

Similarly, gas particles interact with the vacuum, this interaction being fundamental, as is evident in the expansion of the gas shown in the figure. However, the quantum nature of this interaction is hidden in the model. Moreover, it is also not possible to establish what this interaction is like, beyond its motion, since we lack a vacuum model that allows this analysis.

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[11]	A. H. Guth, The Inflationary Universe, Perseus, 1997.
[12]	H. B. G. Casimir, «On the Attraction Between Two Perfectly Conducting Plates,» Indag. Math. , vol. 10, p. 261–263., 1948.
[13]	P. J. E. Peebles y B. Ratra, «The cosmological constant and dark energy,» Reviews of Modern Physics, vol. 75, nº 2, p. 559–606, 2003.

The unreasonable effectiveness of mathematics

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In the post “What is the nature of mathematics“, the dilemma of whether mathematics is discovered or invented by humans has been exposed, but so far no convincing evidence has been provided in either direction.

A more profound way of approaching the issue is as posed by Eugene P. Wigner [1], asking about the unreasonable effectiveness of mathematics in the natural sciences.

According to Roger Penrose this poses three mysteries [2] [3], identifying three distinct “worlds”: the world of our conscious perception, the physical world and the Platonic world of mathematical forms. Thus:

The world of physical reality seems to obey laws that actually reside in the world of mathematical forms.
The perceiving minds themselves – the realm of our conscious perception – have managed to emerge from the physical world.
Those same minds have been able to access the mathematical world by discovering, or creating, and articulating a capital of mathematical forms and concepts.

The effectiveness of mathematics has two different aspects. An active one in which physicists develop mathematical models that allow them to accurately describe the behavior of physical phenomena, but also to make predictions about them, which is a striking fact.

Even more extraordinary, however, is the passive aspect of mathematics, such that the concepts that mathematicians explore in an abstract way end up being the solutions to problems firmly rooted in physical reality.

But this view of mathematics has detractors especially outside the field of physics, in areas where mathematics does not seem to have this behavior. Thus, the neurobiologist Jean-Pierre Changeux notes [4], “Asserting the physical reality of mathematical objects on the same level as the natural phenomena studied in biology raises, in my opinion, a considerable epistemological problem. How can an internal physical state of our brain represent another physical state external to it?”

Obviously, it seems that analyzing the problem using case studies from different areas of knowledge does not allow us to establish formal arguments to reach a conclusion about the nature of mathematics. For this reason, an abstract method must be sought to overcome these difficulties. In this sense, Information Theory (IT) [5], Algorithmic Information Theory (AIT) [6] and Theory of Computation (TC) [7] can be tools of analysis that help to solve the problem.

What do we understand by mathematics?

The question may seem obvious, but mathematics is structured in multiple areas: algebra, logic, calculus, etc., and the truth is that when we refer to the success of mathematics in the field of physics, it underlies the idea of physical theories supported by mathematical models: quantum physics, electromagnetism, general relativity, etc.

However, when these mathematical models are applied in other areas they do not seem to have the same effectiveness, for example in biology, sociology or finance, which seems to contradict the experience in the field of physics.

For this reason, a fundamental question is to analyze how these models work and what are the causes that hinder their application outside the field of physics. To do this, let us imagine any of the successful models of physics, such as the theory of gravitation, electromagnetism, quantum physics or general relativity. These models are based on a set of equations defined in mathematical language, which determine the laws that control the described phenomenon, which admit analytical solutions that describe the dynamics of the system. Thus, for example, a body subjected to a central attractive force describes a trajectory defined by a conic.

This functionality is a powerful analysis tool, since it allows to analyze systems under hypothetical conditions and to reach conclusions that can be later verified experimentally. But beware! This success scenario masks a reality that often goes unnoticed, since generally the scenarios in which the model admits an analytical solution are very limited. Thus, the gravitational model does not admit an analytical solution when the number of bodies is n>=3 [8], except in very specific cases such as the so-called Lagrange points. Moreover, the system has a very sensitive behavior to the initial conditions, so that small variations in these conditions can produce large deviations in the long term.

This is a fundamental characteristic of nonlinear systems and, although the system is governed by deterministic laws, its behavior is chaotic. Without going into details that are beyond the scope of this analysis, this is the general behavior of the cosmos and everything that happens in it.

One case that can be considered extraordinary is the quantum model which, according to the Schrödinger equation or the Heisenberg matrix model, is a linear and reversible model. However, the information that emerges from quantum reality is stochastic in nature.

In short, the models that describe physical reality only have an analytical solution in very particular cases. For complex scenarios, particular solutions to the problem can be obtained by numerical series, but the general solution of any mathematical proposition is obtained by the Turing Machine (TM) [9].

This model can be represented in an abstract form by the concatenation of three mathematical objects〈xyz〉(bit sequences) which, when executed in a Turing machine TM(〈xyz〉), determine the solution. Thus, for example, in the case of electromagnetism, the object z will correspond to the description of the boundary conditions of the system, y to the definition of Maxwell’s equations and x to the formal definition of the mathematical calculus. TM is the Turing machine defined by a finite set of states. Therefore, the problem is reduced to the treatment of a set of bits〈xyz〉 according to axiomatic rules defined in TM, and that in the optimal case can be reduced to a machine with three states (plus the HALT state) and two symbols (bit).

Nature as a Turing machine

And here we return to the starting point. How is it possible that reality can be represented by a set of bits and a small number of axiomatic rules?

Prior to the development of IT, the concept of information had no formal meaning, as evidenced by its classic dictionary definition. In fact, until communication technologies began to develop, words such as “send” referred exclusively to material objects.

However, everything that happens in the universe is interaction and transfer, and in the case of humans the most elaborate medium for this interaction is natural language, which we consider to be the most important milestone on which cultural development is based. It is perhaps for this reason that in the debate about whether mathematics is invented or discovered, natural language is used as an argument.

But TC shows that natural language is not formal, not being defined on axiomatic grounds, so that arguments based on it may be of questionable validity. And it is here that IT and TC provide a broad view on the problem posed.

In a physical system each of the component particles has physical properties and a state, in such a way that when it interacts with the environment it modifies its state according to its properties, its state and the external physical interaction. This interaction process is reciprocal and as a consequence of the whole set of interactions the system develops a temporal dynamics.

Thus, for example, the dynamics of a particle is determined by the curvature of space-time which indicates to the particle how it should move and this in turn interacts with space-time, modifying its curvature.

In short, a system has a description that is distributed in each of the parts that make up the system. Thus, the system could be described in several different ways:

As a set of TMs interacting with each other.
As a TM describing the total system.
As a TM partially describing the global behavior, showing emergent properties of the system.

The fundamental conclusion is that the system is a Turing machine. Therefore, the question is not whether the mathematics is discovered or invented or to ask ourselves how it is possible for mathematics to be so effective in describing the system. The question is how it is possible for an intelligent entity – natural or artificial – to reach this conclusion and even to be able to deduce the axiomatic laws that control the system.

The justification must be based on the fact that it is nature that imposes the functionality and not the intelligent entities that are part of nature. Nature is capable of developing any computable functionality, so that among other functionalities, learning and recognition of behavioral patterns is a basic functionality of nature. In this way, nature develops a complex dynamic from which physical behavior, biology, living beings, and intelligent entities emerge.

As a consequence, nature has created structures that are able to identify its own patterns of behavior, such as physical laws, and ultimately identify nature as a Universal Turing Machine (UTM). This is what makes physical interaction consistent at all levels. Thus, in the above case of the ability of living beings to establish a spatio-temporal map, this allows them to interact with the environment; otherwise their existence would not be possible. Obviously this map corresponds to a Euclidean space, but if the living being in question were able to move at speeds close to light, the map learned would correspond to the one described by relativity.

A view beyond physics

While TC, IT and AIT are the theoretical support that allows sustaining this view of nature, advances in computer technology and AI are a source of inspiration, showing how reality can be described as a structured sequence of bits. This in turn enables functions such as pattern extraction and recognition, complexity determination and machine learning.

Despite this, fundamental questions remain to be answered, in particular what happens in those cases where mathematics does not seem to have the same success as in the case of physics, such as biology, economics or sociology.

Many of the arguments used against the previous view are based on the fact that the description of reality in mathematical terms, or rather, in terms of computational concepts does not seem to fit, or at least not precisely, in areas of knowledge beyond physics. However, it is necessary to recognize that very significant advances have been made in areas such as biology and economics.

Thus, knowledge of biology shows that the chemistry of life is structured in several overlapping languages:

The language of nucleic acids, consisting of an alphabet of 4 symbols that encodes the structure of DNA and RNA.
The amino acid language, consisting of an alphabet of 64 symbols that encodes proteins. The transcription process for protein synthesis is carried out by means of a concordance between both languages.
The language of the intergenic regions of the genome. Their functionality is still to be clarified, but everything seems to indicate that they are responsible for the control of protein production in different parts of the body, through the activation of molecular switches.

On the other hand, protein structure prediction by deep learning techniques is a solid evidence that associates biology to TC [10]. To emphasize also that biology as an information process must verify the laws of logic, in particular the recursion theorem [11], so DNA replication must be performed at least in two phases by independent processes.

In the case of economics there have been relevant advances since the 80’s of the twentieth century, with the development of computational finance [12]. But as a paradigmatic example we will focus on the financial markets, which should serve to test in an environment far from physics the hypothesis that nature has the behavior of a Turing machine.

Basically, financial markets are a space, which can be physical or virtual, through which financial assets are exchanged between economic agents and in which the prices of such assets are defined.

A financial market is governed by the law of supply and demand. In other words, when an economic agent wants something at a certain price, he can only buy it at that price if there is another agent willing to sell him that something at that price.

Traditionally, economic agents were individuals but, with the development of complex computer applications, these applications now also act as economic agents, both supervised and unsupervised, giving rise to different types of investment strategies.

This system can be modeled by a Turing machine that emulates all the economic agents involved, or as a set of Turing machines interacting with each other, each of which emulates an economic agent.

The definition of this model requires implementing the axiomatic rules of the market, as well as the functionality of each of the economic agents, which allow them to determine the purchase or sale prices at which they are willing to negotiate. This is where the problem lies, since this depends on very diverse and complex factors, such as the availability of information on the securities traded, the agent’s psychology and many other factors such as contingencies or speculative strategies.

In brief, this makes emulation of the system impossible in practice. It should be noted, however, that brokers and automated applications can gain a competitive advantage by identifying global patterns, or even by insider trading, although this practice is punishable by law in suitably regulated markets.

The question that can be raised is whether this impossibility of precise emulation invalidates the hypothesis put forward. If we return to the case study of Newtonian gravitation, determined by the central attractive force, it can be observed that, although functionally different, it shares a fundamental characteristic that makes emulation of the system impossible in practice and that is present in all scenarios.

If we intend to emulate the case of the solar system we must determine the position, velocity and angular momentum of all celestial bodies involved, sun, planets, dwarf planets, planetoids, satellites, as well as the rest of the bodies located in the system, such as the asteroid belt, the Kuiper belt and the Oort cloud, as well as the dispersed mass and energy. In addition, the shape and structure of solid, liquid and gaseous bodies must be determined. It will also be necessary to consider the effects of collisions that modify the structure of the resulting bodies. Finally, it will be necessary to consider physicochemical activity, such as geological, biological and radiation phenomena, since they modify the structure and dynamics of the bodies and are subject to quantum phenomena, which is another source of uncertainty. And yet the model is not adequate, since it is necessary to apply a relativistic model.

This makes accurate emulation impossible in practice, as demonstrated by the continuous corrections in the ephemerides of GPS satellites, or the adjustments of space travel trajectories, where the journey to Pluto by NASA’s New Horizons spacecraft is a paradigmatic case.

Conclusions

From the previous analysis it can be hypothesized that the universe is an axiomatic system governed by laws that determine a dynamic that is a consequence of the interaction and transference of the entities that compose it.

As a consequence of the interaction and transfer phenomena, the system itself can partially and approximately emulate its own behavior, which gives rise to learning processes and finally gives rise to life and intelligence. This makes it possible for living beings to interact in a complex way with the environment and for intelligent entities to observe reality and establish models of this reality.

This gave rise to abstract representations such as natural language and mathematics. With the development of IT [5] it is concluded that all objects can be represented by a set of bits, which can be processed by axiomatic rules [7] and which optimally encoded determine the complexity of the object, defined as Kolmogorov complexity [6].

The development of TC establishes that these models can be defined as a TM, so that in the limit it can be hypothesized that the universe is equivalent to a Turing machine and that the limits of reality can go beyond the universe itself, in what is defined as multiverse and that it would be equivalent to a UTM. Esta concordancia entre un universo y una TM permite plantear la hipótesis de que el universo no es más que información procesada por reglas axiomáticas.

Therefore, from the observation of natural phenomena we can extract the laws of behavior that constitute the abstract models (axioms), as well as the information necessary to describe the cases of reality (information). Since this representation is made on a physical reality, its representation will always be approximate, so that only the universe can emulate itself. Since the universe is consistent, models only corroborate this fact. But reciprocally, the equivalence between the universe and a TM implies that the deductions made from consistent models must be satisfied by reality.

However, everything seems to indicate that this way of perceiving reality is distorted by the senses, since at the level of classical reality what we observe are the consequences of the processes that occur at this functional level, appearing concepts such as mass, energy, inertia.

But when we explore the layers that support classical reality, this perception disappears, since our senses do not have the direct capability for its observation, in such a way that what emerges is nothing more than a model of axiomatic rules that process information, and the physical sensory conception disappears. This would justify the difficulty to understand the foundations of reality.

It is sometimes speculated that reality may be nothing more than a complex simulation, but this poses a problem, since in such a case a support for its execution would be necessary, implying the existence of an underlying reality necessary to support such a simulation [13].

There are two aspects that have not been dealt with and that are of transcendental importance for the understanding of the universe. The first concerns irreversibility in the layer of classical reality. According to the AIT, the amount of information in a TM remains constant, so the irreversibility of thermodynamic systems is an indication that these systems are open, since they do not verify this property, an aspect to which physics must provide an answer.

The second is related to the non-cloning theorem. Quantum systems are reversible and, according to the non-cloning theorem, it is not possible to make exact copies of the unknown quantum state of a particle. But according to the recursion theorem, at least two independent processes are necessary to make a copy. This would mean that in the quantum layer it is not possible to have at least two independent processes to copy such a quantum state. An alternative explanation would be that these quantum states have a non-computable complexity.

Finally, it should be noted that the question of whether mathematics was invented or discovered by humans is flawed by an anthropic view of the universe, which considers humans as a central part of it. But it must be concluded that humans are a part of the universe, as are all the entities that make up the universe, particularly mathematics.

References

[1]	E. P. Wigner, “The unreasonable effectiveness of mathematics in the natural sciences.,” Communications on Pure and Applied Mathematics, vol. 13, no. 1, pp. 1-14, 1960.
[2]	R. Penrose, The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics, Oxford: Oxford University Press, 1989.
[3]	R. Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe, London: Jonathan Cape, 2004.
[4]	J.-P. Changeux and A. Connes, Conversations on Mind, Matter, and Mathematics, Princeton N. J.: Princeton University Press, 1995.
[5]	C. E. Shannon, “A Mathematical Theory of Communication,” The Bell System Technical Journal, vol. 27, pp. 379-423, 1948.
[6]	P. Günwald and P. Vitányi, “Shannon Information and Kolmogorov Complexity,” arXiv:cs/0410002v1 [cs:IT], 2008.
[7]	M. Sipser, Introduction to the Theory of Computation, Course Technology, 2012.
[8]	H. Poincaré, New Methods of Celestial Mechanics, Springer, 1992.
[9]	A. M. Turing, “On computable numbers, with an application to the Entscheidungsproblem.,” Proceedings, London Mathematical Society, pp. 230-265, 1936.
[10]	A. W. Senior, R. Evans and e. al., “Improved protein structure prediction using potentials from deep learning,” Nature, vol. 577, pp. 706-710, Jan 2020.
[11]	S. Kleene, “On Notation for ordinal numbers,” J. Symbolic Logic, no. 3, p. 150–155, 1938.
[12]	A. Savine, Modern Computational Finance: AAD and Parallel Simulations, Wiley, 2018.
[13]	N. Bostrom, “Are We Living in a Computer Simulation?,” The Philosophical Quarterly, vol. 53, no. 211, p. 243–255, April 2003.

What is the nature of mathematics?

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The ability of mathematics to describe the behavior of nature, particularly in the field of physics, is a surprising fact, especially when one considers that mathematics is an abstract entity created by the human mind and disconnected from physical reality. But if mathematics is an entity created by humans, how is this precise correspondence possible?

Throughout centuries this has been a topic of debate, focusing on two opposing ideas: Is mathematics invented or discovered by humans?

This question has divided the scientific community: philosophers, physicists, logicians, cognitive scientists and linguists, and it can be said that not only is there no consensus, but generally positions are totally opposed. Mario Livio in the essay “Is God a Mathematician? [1] describes in a broad and precise way the historical events on the subject, from Greek philosophers to our days.

The aim of this post is to analyze this dilemma, introducing new analysis tools such as Information Theory (IT) [2], Algorithmic Information Theory (AIT) [3] and Computer Theory (CT) [4], without forgetting the perspective that shows the new knowledge about Artificial Intelligence (AI).

In this post we will make a brief review of the current state of the issue, without entering into its historical development, trying to identify the difficulties that hinder its resolution, for in subsequent posts to analyze the problem from a different perspective to the conventional, using the logical tools that offer us the above theories.

Currents of thought: invented or discovered?

In a very simplified way, it can be said that at present the position that mathematics is discovered by humans is headed by Max Tegmark, who states in “Our Mathematical Universe” [5] that the universe is a purely mathematical entity, which would justify that mathematics describes reality with precision, but that reality itself is a mathematical entity.

On the other extreme, there is a large group of scientists, including cognitive scientists and biologists who, based on the fact of the brain’s capabilities, maintain that mathematics is an entity invented by humans.

Max Tegmark: Our Mathematical Universe

In both cases, there are no arguments that would tip the balance towards one of the hypotheses. Thus, in Max Tegmark’s case he maintains that the definitive theory (Theory of Everything) cannot include concepts such as “subatomic particles”, “vibrating strings”, “space-time deformation” or other man-made constructs. Therefore, the only possible description of the cosmos implies only abstract concepts and relations between them, which for him constitute the operative definition of mathematics.

This reasoning assumes that the cosmos has a nature completely independent of human perception, and its behavior is governed exclusively by such abstract concepts. This view of the cosmos seems to be correct insofar as it eliminates any anthropic view of the universe, in which humans are only a part of it. However, it does not justify that physical laws and abstract mathematical concepts are the same entity.

In the case of those who maintain that mathematics is an entity invented by humans, the arguments do not usually have a formal structure and it could be said that in many cases they correspond more to a personal position and sentiment. An exception is the position maintained by biologists and cognitive scientists, in which the arguments are based on the creative capacity of the human brain and which would justify that mathematics is an entity created by humans.

For these, mathematics does not really differ from natural language, so mathematics would be no more than another language. Thus, the conception of mathematics would be nothing more than the idealization and abstraction of elements of the physical world. However, this approach presents several difficulties to be able to conclude that mathematics is an entity invented by humans.

On the one hand, it does not provide formal criteria for its demonstration. But it also presupposes that the ability to learn is an attribute exclusive to humans. This is a crucial point, which will be addressed in later posts. In addition, natural language is used as a central concept, without taking into account that any interaction, no matter what its nature, is carried out through language, as shown by the TC [4], which is a theory of language.

Consequently, it can be concluded that neither current of thought presents conclusive arguments about what the nature of mathematics is. For this reason, it seems necessary to analyze from new points of view what is the cause for this, since physical reality and mathematics seem intimately linked.

Mathematics as a discovered entity

In the case that considers mathematics the very essence of the cosmos, and therefore that mathematics is an entity discovered by humans, the argument is the equivalence of mathematical models with physical behavior. But for this argument to be conclusive, the Theory of Everything should be developed, in which the physical entities would be strictly of a mathematical nature. This means that reality would be supported by a set of axioms and the information describing the model, the state and the dynamics of the system.

This means a dematerialization of physics, something that somehow seems to be happening as the development of the deeper structures of physics proceeds. Thus, the particles of the standard model are nothing more than abstract entities with observable properties. This could be the key, and there is a hint in Landauer’s principle [6], which establishes an equivalence between information and energy.

But solving the problem by physical means or, to be more precise, by contrasting mathematical models with reality presents a fundamental difficulty. In general, mathematical models describe the functionality of a certain context or layer of reality, and all of them have a common characteristic, in such a way that these models are irreducible and disconnected from the underlying layers. Therefore, the deepest functional layer should be unraveled, which from the point of view of AIT and TC is a non-computable problem.

Mathematics as an invented entity

The current of opinion in favor of mathematics being an entity invented by humans is based on natural language and on the brain’s ability to learn, imagine and create.

But this argument has two fundamental weaknesses. On the one hand, it does not provide formal arguments to conclusively demonstrate the hypothesis that mathematics is an invented entity. On the other hand, it attributes properties to the human brain that are a general characteristic of the cosmos.

The Hippocampus: A paradigmatic example of the dilemma discovered or invented

To clarify this last point, let us take as an example the invention of whole numbers by humans, which is usually used to support this view. Let us now imagine an animal interacting with the environment. Therefore, it has to interpret spacetime accurately as a basic means of survival. Obviously, the animal must have learned or invented the space-time map, something much more complex than natural numbers.

Moreover, nature has provided or invented the hippocampus [7], a neuronal structure specialized in acquiring long-term information that forms a complex convolution, forming a recurrent neuronal network, very suitable for the treatment of the space-time map and for the resolution of trajectories. And of course this structure is physical and encoded in the genome of higher animals. The question is: Is this structure discovered or invented by nature?

Regarding the use of language as an argument, it should be noted that language is the means of interaction in nature at all functional levels. Thus, biology is a language, the interaction between particles is formally a language, although this point requires a deeper analysis for its justification. In particular, natural language is in fact a non-formal language, so it is not an axiomatic language, which makes it inconsistent.

Finally, in relation to the learning capability attributed to the brain, this is a fundamental characteristic of nature, as demonstrated by mathematical models of learning and evidenced in an incipient manner by AI.

Another way of approaching the question about the nature of mathematics is through Wigner’s enigma [8], in which he asks about the inexplicable effectiveness of mathematics. But this topic and the topics opened before will be dealt with and expanded in later posts.

References

[1]	M. Livio, Is God a Mathematician?, New York: Simon & Schuster Paperbacks, 2009.
[2]	C. E. Shannon, «A Mathematical Theory of Communication,» The Bell System Technical Journal, vol. 27, pp. 379-423, 1948.
[3]	P. Günwald and P. Vitányi, “Shannon Information and Kolmogorov Complexity,” arXiv:cs/0410002v1 [cs:IT], 2008.
[4]	M. Sipser, Introduction to the Theory of Computation, Course Technology, 2012.
[5]	M. Tegmark, Our Mathematical Universe: My Quest For The Ultimate Nature Of Reality, Knopf Doubleday Publishing Group, 2014.
[6]	R. Landauer, «Irreversibility and Heat Generation in Computing Process,» IBM J. Res. Dev., vol. 5, pp. 183-191, 1961.
[7]	S. Jacobson y E. M. Marcus, Neuroanatomy for the Neuroscientist, Springer, 2008.
[8]	E. P. Wigner, «The unreasonable effectiveness of mathematics in the natural sciences.,» Communications on Pure and Applied Mathematics, vol. 13, nº 1, pp. 1-14, 1960.

COVID-19: What makes this pandemic different?

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Zoonosis, or the jump from an animal virus to humans, has the characteristics of a contingent event. In principle, this leap can be limited by sanitary control of domestic animal species and by regulation of trade, contact and consumption of wild species. However, given the complexity of modern society and the close contact between humans at a global level, the probability of a virus jump to humans is not an avoidable event, so zoonosis can be considered a contingent phenomenon.

This situation has been clearly shown in recent times with the appearance of MERS (MERS-Cov), SARS (SARS-Cov) and recently the COVID-19 (SARS-Cov-2). This propagation is fundamentally motivated by globalization, although the factors are multiple and complex, such as health controls and the structure of livestock farms. But the list is long, and we can also mention the expansion of other viral diseases due to climate change, such as Zika, Chikungunya or Dengue.

The question that arises in this scenario is: What factors influence the magnitude and speed of the spread of a pandemic? Thus, in the cases mentioned above, a very significant difference in the behavior and spread of infection can be seen. Except in the case of COVID-19, the spread has been limited and outbreaks have been localized and isolated, avoiding a global spread.

In contrast, the situation has been completely different with CoVID-19. Thus, its rapid expansion has caught societies unfamiliar with this type of problem unawares, so that health systems have been overwhelmed and without appropriate protocols for the treatment of the infection. On the other hand, authorities unaware of the magnitude of the problem, and ignorant of the minimum precautions to prevent the spread of the virus, seem to have made a series of chained errors, typical of catastrophic processes, such as economic bankruptcies and air accidents.

The long-term impact is still very difficult to assess, as it has triggered a vicious circle of events affecting fundamental activities of modern society.

In particular, the impact on health services will leave a deep imprint, with extension to areas that in principle are not directly related to the COVID-19, such as the psychological and psychiatric effects derived from the perception of danger and social confinement. But even more important is the detraction of resources in other health activities, having reduced the flow of daily health activity, so it is foreseeable a future increase in morbidity and mortality rates of other diseases, especially cancer.

To all this must be added the deterioration of economic activity, with reductions in GDP of up to two figures, which will trigger an increase in poverty, especially in the most disadvantaged segments of the population. And since the economic factor is the transmission belt of human activity, it is easy to imagine a perfect storm scenario.

Pandemic Influencing Factors COVID-19

But let’s return to the question that has been raised, about the singularity of SARS-Cov-2, so that its expansion has been unstoppable and that we are now facing a second wave.

To unravel this question we can analyze what the mathematical models of expansion of an infection show us, starting with the classic SIR model. This type of model allows us to determine the rates of infection (β) and recovery (γ), as well as the basic reproduction rate (R0=β/γ) from the observed morbidity.

The origin of the SIR models (Susceptible, Infectious, and Recovered) goes back to the beginning of the 20th century, proposed by Kermack and McKendrick in 1927. The advantage of these models is that they are based on a system of differential equations, which can be solved analytically and therefore suitable for resolution at the time they were proposed.

However, these types of models are basic and do not facilitate considerations of geographical distribution, mobility, probability of infection, clinical status, temporal development of each of the phases of the infection, age, sex, social distance, protection, tracking and testing strategies. On the other hand, the classic SIR model has a deductive structure, exclusively. This means that from the morbidity data it is possible to determine the basic reproduction rate exclusively, hiding fundamental parameters in the pandemic process, as will be justified below.

To contrast this idea, it is necessary to propose new approaches to the simulation of the pandemic process, as is the case of the study proposed in “A model of the spread of Covid-19” and in its implementation. In this case, the model is a discrete SIR structure, in which individuals go through an infection and recovery process with realistic states, in addition to including all the parameters for defining the scenario mentioned above, that is, probability of infection, geographical distribution of the population, mobility, etc. This allows an accurate simulation of the pandemic and, despite its complexity, its structure is very suitable for implementation with existing computational means.

The first conclusion drawn from the simulations of the initial phase of the pandemic was the need to consider the existence of a very significant asymptomatic population. Thus, in the classical model it is possible to obtain a rapid expansion of the pandemic simply by considering high values of the infection rate (β).

On the contrary, in the discrete model the application of existing data did not justify the observed data, unless there was a very significant asymptomatic population that hid the true magnitude of the spread of the infection. The symptomatic population in the early stages of the pandemic should be considered to be small. This, together with the data on spread through different geographical areas and the possible probability of infection, produced temporary results of much slower expansion that did not even trigger the priming of the model.

In summary, the result of the simulations led to totally inconsistent scenarios, until a high population of asymptomatic people was included, from which the model began to behave according to the observed data. At present, there are already more precise statistics that confirm this behavior that, in the group of infected people, get to establish that 80% are asymptomatic, 15% are symptomatic that require some type of medical attention by means of treatment or hospital admission and, the rest, 5% that require from basic level life support to advanced life support.

These figures help explain the virulence of a pandemic, which is strongly regulated by the percentage of asymptomatic individuals. This behavior justifies the enormous difference between the behaviors of different types of viruses. Thus, if a virus has a high morbidity it is easy to track and isolate, since the infectious cases do not remain hidden. On the contrary, a virus with low morbidity keeps hidden the individuals who are vectors of the disease, since they belong to the group of asymptomatic people. Unlike the viruses mentioned above, COVID-19 is a paradigmatic example of this scenario, with the added bonus that it is a virus that has demonstrated a great capacity for contagion.

This behavior has meant that when the pandemic has shown its face there was already a huge group of individual vectors. And this has probably been the origin of a chain of events with serious health, economic and social consequences.

The mechanisms of expansion and containment of the pandemic

In retrospect, the apparent low incidence in the first few weeks suggested that the risk of a pandemic was low and not very virulent. Obviously, an observation clearly distorted by the concealment of the problem caused by the asymptomatic nature of the majority of those infected.

This possibly also conditioned the response to their containment. The inadequate management of the threat by governments and institutions, the lack of protection resources and the message transmitted to the population ended up materializing the pandemic.

In this context, there is one aspect that calls for deep attention. A disease with a high infectious capacity requires a very effective means of transmission and since the first symptoms were of pulmonary type it should have been concluded that the airway was the main means of transmission. However, much emphasis was placed on direct physical contact and social distance. The minimization of the effect of aerosols, which are very active in closed spaces, as is now being recognized, is remarkable.

Another seemingly insignificant nuance related to the behavior of the pandemic under protective measures should also be noted. This is related to the modeling of the pandemic. The classical SIR model assumes that the infection rate (β) and recovery rate (γ) are uniquely proportional to the sizes of the populations in the different States. However, this is an approach that masks the underlying statistical process, and in the case of the recovery is also a conceptual flaw. This assumption determines the structure of the differential equations of the model, imposing a general solution of exponential type that is not necessarily the real one.

By the way, the exponential functions introduce a phase delay, which produces the effect that the recovery of an individual occurs in pieces, for example, first the head and then the legs!

But the reality is that the process of infection is a totally stochastic process that is a function of the probability of contagion determined by the capacity of the virus, the susceptibility of the individual, the interaction between infected and susceptible individuals, the geographical distribution, mobility, etc. In short, this process has a Gaussian nature.

As will later be justified, this Gaussian process is masked by the overlap of infection in different geographical areas, so they are only visible in separate local outbreaks, as a result of effective containment. An example of this can be found in the case of South Korea, represented in the figure below.

In the case of recovery, the process corresponds to a stochastic delay line and therefore Gaussian, since it only depends on the temporary parameters of recovery imposed by the virus, the response of the individual and the healing treatments. Therefore, the recovery process is totally independent for each individual.

The result is that the general solution of the discrete SIR model is Gaussian and therefore responds to a quadratic exponential function, unlike the order one exponential functions of the classical SIR model. This makes the protection measures much more effective than those exposed by the conventional models. So they must be considered a fundamental element to determine the strategy for the containment of the pandemic.

The point is that once a pandemic is evident, containment and confinement measures must be put in place. It is at this point that COVID-19 poses a challenge of great complexity, as a result of the large proportion of asymptomatic individuals, who are the main contributors to the spread of infection.

A radical solution to the problem requires strict confinement of the entire population for a period no less than the latency period of the virus in an infected person. To be effective, this measure must be accompanied by protective measures in the family or close environment, as well as extensive screening campaigns. This strategy has shown its effectiveness in some Asian countries.

In reality, early prophylaxis and containment is the only measure to effectively contain the pandemic, as the model output for different dates of containment shows. Interestingly, the dispersion of the curves in the model’s priming areas is a consequence of the stochastic nature of the model.

But the late implementation of this measure, when the number of people infected in hiding was already very high, together with the lack of a culture of prophylaxis against pandemics in Western countries has meant that these measures have been ineffective and very damaging.

In this regard, it should be noted that the position of the governments has been lukewarm and in most cases totally erratic, which has contributed to the fact that the confinement measures have been followed very laxly by the population.

Here it is important to note that in the absence of effective action, governments have based their distraction strategy on the availability of a vaccine, which is clearly not a short-term solution.

As a consequence of the ineffectiveness of the measure, the period of confinement has been excessively prolonged, with restrictions being lifted once morbidity and mortality statistics were lowered. The result is that, since the virus is widespread in the population, new waves of infection have inevitably occurred.

This is another important aspect in interpreting the pandemic’s spread figures. According to the classic SIR model, everything seems to indicate that in the progression of the figures, a peak of infections should be expected, which should decrease exponentially. Throughout the first months, those responsible for the control of the pandemic have been looking for this peak, as well as the flattening of the integration curve of the total cases. Something expected but never seemed to come.

The explanation for this phenomenon is quite simple. The spread of the pandemic is not subject to infection of a closed group of individuals, as the classical SIR model assumes. Rather, the spread of the virus is a function of geographic areas with specific population density and the mobility of individuals between them. The result is that the curves that describe the pandemic are a complex superposition of the results of this whole conglomerate, as shown by the curve of deaths in Spain, on the dates indicated.

The result is that the process can be spread out over time, so that the dynamics of the curves are a complex overlap of outbreaks that evolve according to multiple factors, such as population density and mobility, protective measures, etc.

This indicates that the concepts of pandemic spread need to be thoroughly reviewed. This should not be surprising if we consider that throughout history there have been no reliable data that have allowed contrasting their behavior.

Evolution of morbidity and mortality

Another interesting aspect is the study of the evolution of morbidity and mortality of SARS-Cov-2. For this purpose, case records can be used, especially now that data from a second wave of infection are beginning to be available, as shown in the figure below.

In view of these data a premature conclusion could be drawn, assuring that the virus is affecting the population with greater virulence, increasing morbidity, but on the other hand it could also be said that mortality is decreasing dramatically.

But nothing could be further from reality if we consider the procedure for obtaining data on diagnosed cases. Thus, it can be seen that the magnitude of the curve of diagnosed cases in the second phase is greater than in the first phase, indicating greater morbidity. However, in the first phase the diagnosis was mainly of a symptomatic type, given the lack of resources for testing. On the contrary, in the second phase the diagnosis was made in a symptomatic way and by means of tests, PCR and serology.

This has only brought to light the magnitude of the group of asymptomatic infected, which were hidden in the first phase. Therefore, we cannot speak of a greater morbidity. On the contrary, if we look at the slope of evolution of the curve, it is smoother, indicating that the probability of infection is being much lower than that shown in the month of March. This is a clear indication that the protective measures are effective. And they would be even more so if the discipline were greater and the messages would converge on this measure, instead of creating confusion and uncertainty.

If the slopes of the case curves are compared, it is clear that the expansion of the pandemic in the first phase was very abrupt, as a result of the existence of a multitude of asymptomatic vectors and the absolute lack of prevention measures. In the second phase, the slope is gentler, attributable to the prevention measures. The comparison of these slopes is by a factor of approximately 4.

However, it is possible that without prevention measures the second phase could be much more aggressive. This is true considering that it is very possible that the number of vectors of infection at present is much higher than in the first phase, since the pandemic is much more widespread. Therefore the spread factor could have been much higher in the second phase, as a consequence of this parameter.

In terms of mortality, the ratio deceased/diagnosed seems to have dropped dramatically, which would lead to say that the lethality of the virus has dropped. Thus at the peak of the first phase its value was approximately 0.1, while in the second phase it has a value of approximately 0.01, that is, an order of magnitude lower.

But considering that in the figures of diagnosed in the first phase the asymptomatic were hidden, both ratios are not comparable. Obviously, the term corresponding to the asymptomatic would allow us to explain this apparent decrease, although we must also consider that the real mortality has decreased as a result of improved treatment protocols.

Consequently, it is not possible to draw consequences on the evolution of the lethality of the virus, but what is certain is that the magnitudes of mortality are decreasing for two reasons. One is virtual one, such as the availability of more reliable figures of infected people, and the other is real, as a result of improved treatment protocols.

Strategies for the future

At present, it seems clear that the spread of the virus is a consolidated fact, so the only possible strategy in the short and medium term is to limit its impact. In the long term, the availability of a vaccine could finally eradicate the disease, although the possibility of the disease becoming endemic or recurrent will also have to be considered.

For this reason, and considering the implications of the pandemic on human activity of all kinds, future plans must be based on a strategy of optimization, so as to minimize the impact on the general health of the population and on the economy. This is because increased poverty may have a greater impact than the pandemic itself.

Under this point of view and considering the aspects analyzed above, the strategy should be based on the following points:

Strict protection and prophylaxis measures: masks, cleaning, ventilation, social distance in all areas.
Protection of the segments of the population at risk.
Maintain as much as possible the economic and daily activities.
Social awareness: Voluntary declaration and isolation in case of infection. Compliance with regulations without the need for coercive measures.
Implementing an organizational structure for mass testing, tracking and isolation of infected.

It is important to note that, as experience is demonstrating, aggressive containment measures are not adequate to prevent successive waves of infection and are generally highly ineffective, producing distrust and rejection, which is a brake on fighting the pandemic.

Another interesting aspect is that the implementation of the previous points does not correspond to strictly health-related projects, but rather to resource management and control projects. For this reason, the activities aimed at fighting the pandemic must be ad hoc projects, since the pandemic is an eventual event, to which specific efforts must be devoted.

Directing the effort through organizations such as the health system itself will only result in a destructuring of the organization and a dispersion of resources, a task for which it has not been created nor does it have the profile to do so.

Covid-19: Interpretation of data

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In view of the expansion of the Covid-19 in different countries, and taking as a reference the model of spreading exposed in the previous post, it is possible to make an interpretation of the data, in order to solve some doubts and contradictions raised in different forums.

But before starting this analysis, it is important to highlight an outstanding feature of the Covid-19 expansion shown by the model. In general, the modeling of infectious processes usually focuses on the infection rate of individuals, leaving temporal aspects such as incubation or latency periods of the pathogens in the background. This is justified as a consequence of the fact that their influence is generally unnoticed, besides introducing difficulties in the analytical study of the models.

However, in the case of Covid-19 its rapid expansion makes the effect of time parameters evident, putting health systems in critical situations and making it difficult to interpret the data that emerge as the pandemic spreads.

In this sense, the outstanding characteristics of the Covid-19 are:

The high capacity of infection.
The capacity of infection of individuals in the incubation phase.
The capacity of infection of asymptomatic individuals.

This makes the number of possible asymptomatic cases very high, presenting a great difficulty in diagnosis, as a result of the lack of resources caused by the novelty and rapid spread of the virus.

For this reason, the model has been developed taking into account the temporal parameters of the spread of the infection, which requires a numerical model, since the analytical solution is very complex and possibly without a purely analytical solution.

As a result, the model has a distinctive feature compared to conventional models, which is shown in the figure below.

This consists in that it is necessary to distinguish groups of asymptomatic and symptomatic individuals, since they present a temporal evolution delayed in time. As a consequence, the same happens with the curves of hospitalized and ICU individuals.

This allows clarifying some aspects linked to the real evolution of the virus. For example, in relation to the declaration of the exceptional measures in Italy and Spain, a substantial improvement in the contention of the pandemic was expected, something that still seems distant. The reason for this behavior is that the contention measures have been taken on the basis of the evolution of the curve of symptomatic individuals, ignoring the fact that there was already a very important population of asymptomatic individuals.

As can be seen in the graphs, the measurements should have been taken at least three weeks in advance, that is, according to the evolution curve of asymptomatic individuals. But in order to make this decision correctly, this data should have been available, something that was completely impossible, as a result of the lack of a test campaign on the population.

This situation is supported by the example of China, which although the spread of the virus could not be contained at an early stage, containment measures were taken several weeks earlier, on a comparative time scale.

The data from Germany are also very significant, exhibiting a much lower mortality rate than Italy and Spain. Although this raises a question about the capacity of infection in this country, it is actually easy to explain. In Italy and Spain, testing for Covid-19 infection is beginning. However, in Germany these tests have been carried out for several weeks at a rate of several hundred thousand per week. In contrast, the numbers of individuals diagnosed in Italy and Spain should be reviewed in the future.

This explains the lower mortality rate for a large number of infected individuals. This also has a decisive advantage, since early diagnosis allows for the isolation of infected individuals, reducing the possibility of infection of other individuals, which ultimately will result in a lower mortality rate.

Therefore, a quick conclusion can be made that can be summarized in the following points:

Measures to isolate the population are necessary but ineffective when taken at an advanced stage of the pandemic.
Early detection of infection is a totally decisive aspect in the contention of the pandemic and above all in the reduction of the mortality rate.

A model of the spread of Covid-19

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The reason for addressing this issue is twofold. On the one hand, Covid-19 is the most important challenge for humanity at the moment, but on the other hand the process of expansion of the virus is an example of how nature establishes models based on information processing.

The analysis of the dynamics of the virus expansion and its consequences will be based on a model implemented in Python, which for those who are interested can be downloaded, being able to make the changes that are considered appropriate to analyze different scenarios.

The model

The model is based on a structure of 14 states and 20 parameters, which determine the probabilities and the temporal dynamics of transitions between states. It is important to note that in the model the only vectors for virus spread are the “symptomatic” and “asymptomatic” states. The model also establishes parameters for the mobility of individuals and the rate of infection.

Some simplifications have been made to the model. Thus, it assumes that the geographical distribution of the population is homogeneous, which has contributed to a significant reduction in computational effort. In principle, this may seem to be a major limitation, but we will see that it is not an obstacle to drawing overall conclusions. The following figure represents in a simplified way the state diagram of the model. The conditions that establish the transitions can be consulted in the model.

The parameters have been adjusted according to experience gained from the progression of the virus, so information is limited and should be subject to further review. In any case, it seems clear that the virus has a high efficiency in infiltrating the cells to perform the copying process, so the viral load required for the infection seems to be small. This presupposes a high rate of infection, so it is also assumed that a significant part of the population will be infected.

Scenarios for the spread of the virus can be listed in the following sections:

Early action measures to confine the spread of the virus
Uncontrolled spread of the virus.
Exceptional measures to limit the propagation of virus.

The first scenario is not going to be analyzed as this is not the case in the current situation. This scenario can be analyzed by modifying the parameters of the model.

Therefore, the scenarios of interest are those of uncontrolled propagation and exceptional measures, as these represent the current state of the pandemic.

The natural evolution

The model dynamics for the case of uncontrolled propagation are shown in the figure below. It can be seen that the most important vectors in the propagation of the virus are asymptomatic individuals, for three fundamental reasons. The first is the broad impact of the virus on the population. The second is determined by the fact that it only produces a symptomatic picture in a limited fraction of the population. The third is directly related to the practical limitations in diagnosing asymptomatic individuals, as a consequence of the novelty and rapid spread of Covid-19.

For this reason, it seems clear that the extraordinary measures to contain the virus must be aimed at drastically limiting contact between humans. This is what has surely advised the possible suspension of academic activities, which includes the child and youth population, not because they are a risk group but because they are the most active population in the spread of the virus.

The other characteristic of the spreading dynamics is the abrupt temporary growth of those affected by the virus, until it reaches the whole population, initiating a rapid recovery, but condemning the groups at risk to be admitted to the Intensive Care Unit (ICU) and probably to death.

This will pose an acute problem in health systems, and an increase in collateral cases can be expected, which could easily surpass the direct cases produced by Covid-19. This makes it advisable to take extraordinary measures, but at the same time, the effectiveness of these measures is in doubt, since their rapid expansion may reduce the effectiveness of these measures, leading to late decision-making.

Present situation

This scenario is depicted in the following figures where quarantine is decreed for a large part of the population, restricting the movement of the propagation vectors. To confirm the above, two scenarios have been modeled. The first, in which the decision of extraordinary measures has been taken before the curve of diagnosed symptoms begins to grow, which in the figure occurs around day 40 from patient zero. The second in whom the decision has been taken a few days later, when the curve of diagnosed symptoms is clearly increasing, around day 65 from patient zero.

These two scenarios clearly indicate that it is more than possible that measures have been taken late and that the pandemic is following its natural course, due to the delay between the infected and symptomatic patient curves. Consequently, it seems that the containment measures will not be as effective as expected, and considering that economic factors will possibly have very profound consequences in the long and medium term for the well-being of society, alternative solutions should be considered.

It is interesting to note how the declaration of special measures modifies the temporal behavior of the pandemic. But once these have not been taken at an early stage of the virus’ emergence, the consequences are profound.

What can be expected

Obviously, the most appropriate solution would be to find remedies to cure the disease, which is being actively worked on, but which has a developmental period that may exceed those established by the dynamics of the pandemic.

However, since the groups at risk, the impact and the magnitude of these are known, a possible alternative solution would be:

Quarantine these groups, keeping them totally isolated from the virus and implementing care services to make this isolation effective until the pandemic subsides, or effective treatment is found.
Implement hospitals dedicated exclusively to the treatment of Covid-19.
For the rest of the population not included in the risk groups, continue with normal activity, allowing the pandemic to spread (something that already seems to be an inevitable possibility). However, strict prophylactic and safety measures must be taken.

This strategy has undeniable advantages. Firstly, it would reduce the pressure on the health system, preventing the collapse of normal system activity and leading to a faster recovery. Secondly, it would reduce the problems of treasury and cash management of states, which can lead to an unprecedented crisis, the consequences of which will certainly be more serious than the pandemic itself.

Finally, an important aspect of the model remains to be analyzed, such as its limitation for modeling a non-homogeneous distribution of the population. This section is easy to solve if we consider that it works correctly for cities. Thus, in order to model the case of a wider geographical extension, one only has to model the particular cases of each city or community with a time lag as the extension of the pandemic itself is showing.

One aspect, namely the duration of the extraordinary measures, remains to be determined. If it is considered that the viral load to infect an individual is small, it is possible that the remnants at the end of the quarantine period may reactivate the disease, in those individuals who have not yet been exposed to the virus or who have not been immunized. This is especially important considering that cured people may continue to be infected for another 15 days.

Perception of complexity

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In previous posts, the nature of reality and its complexity has been approached from the point of view of Information Theory. However, it is interesting to make this analysis from the point of view of human perception and thus obtain a more intuitive view.

Obviously, making an exhaustive analysis of reality from this perspective is complex due to the diversity of the organs of perception and the physiological and neurological aspects that develop over them. In this sense, we could explain how the information perceived is processed, depending on each of the organs of perception. Especially the auditory and visual systems, as these are more culturally relevant. Thus, in the post dedicated to color perception it has been described how the physical parameters of light are encoded by the photoreceptor cells of the retina.

However, in this post the approach will consist of analyzing in an abstract way how knowledge influences the interpretation of information, in such a way that previous experience can lead the analysis in a certain direction. This behavior establishes a priori assumptions or conditions that limit the analysis of information in all its extension and that, as a consequence, prevent to obtain certain answers or solutions. Overcoming these obstacles, despite the conditioning posed by previous experience, is what is known as lateral thinking.

To begin with, let’s consider the case of series math puzzles in which a sequence of numbers, characters, or graphics is presented, asking how the sequence continues. For example, given the sequence “IIIIIIIVVV”, we are asked to determine which the next character is. If the Roman culture had not developed, it could be said that the next character is “V”, or also that the sequence has been made by little scribblers. But this is not the case, so the brain begins to engineer determining that the characters can be Roman and that the sequence is that of the numbers “1,2,3,…”. Consequently, the next character must be “I”.

In this way, it can be seen how the knowledge acquired conditions the interpretation of the information perceived by the senses. But from this example another conclusion can be drawn, consisting of the ordering of information as a sign of intelligence. To expose this idea in a formal way let’s consider a numerical sequence, for example the Fibonacci series “0,1,2,3,5,8,…”. Similarly to the previous case, the following number should be 13, so that the general term can be expressed as f_n=f_n-1+f_n-₂. However, we can define another discrete mathematical function that takes the values “0,1,2,3,5,8” for n = 0,1,2,3,4,5, but differs for the rest of the values of n belonging to the natural numbers, as shown in the following figure. In fact, with this criterion it is possible to define an infinite number of functions that meet this criterion.

The question, therefore, is: What is so special about the Fibonacci series in relation to the set of functions that meet the condition defined above?

Here we can make the argument already used in the case of the Roman number series. So that mathematical training leads to identifying the series of numbers as belonging to the Fibonacci series. But this poses a contradiction, since any of the functions that meet the same criterion could have been identified. To clear up this contradiction, Algorithmic Information Theory (AIT) should be used again.

Firstly, it should be stressed that culturally the game of riddles implicitly involves following logical rules and that, therefore, the answer is free from arbitrariness. Thus, in the case of number series the game consists of determining a rule that justifies the result. If we now try to identify a simple mathematical series that determines the sequence “0,1,2,3,5,8,…” we see that the expression f_n=f_n-1+f_n-2 fulfills these requirements. In fact, it is possible that this is the simplest within this type of expressions. The rest are either complex, arbitrary or simple expressions that follow different rules from the implicit rules of the puzzle.

From the AIT point of view, the solution that contains the minimum information and can therefore be expressed most readily will be the most likely response that the brain will give in identifying a pattern determined by a stimulus. In the example above, the description of the predictable solution will be the one composed of:

A Turing machine.
The information to code the calculus rules.
The information to code the analytical expression of the simplest solution. In the example shown it corresponds to the expression of the Fibonacci series.

Obviously, there are solutions of similar or even less complexity, such as the one performed by a Turing machine that periodically generates the sequence “0,1,2,3,5,8”. But in most cases the solutions will have a more complex description, so that, according to the AIT, in most cases their most compact description will be the sequence itself, which cannot be compressed or expressed analytically.

For example, it is easy to check that the function:

generates for integer values of n the sequence “0,1,1,2,3,5,8,0,-62,-279,…”, so it could be said that the quantities following the proposed series are “…,0,-62,-279,… Obviously, the complexity of this sequence is higher than that of the Fibonacci series, as a result of the complexity of the description of the function and the operations to be performed.

Similarly, we can try to define other algorithms that generate the proposed sequence, which will grow in complexity. This shows the possibility of interpreting the information from different points of view that go beyond the obvious solutions, which are conditioned by previous experiences.

If, in addition to all the above, it is considered that, according to Landauer’s principle, information complexity is associated with greater energy consumption, the resolution of complex problems not only requires a greater computational effort, but also a greater energy effort.

This may explain the feeling of satisfaction produced when a certain problem is solved, and the tendency to engage in relaxing activities that are characterized by simplicity or monotony. Conversely, the lack of response to a problem produces frustration and restlessness.

This is in contrast to the idea that is generally held about intelligence. Thus, the ability to solve problems such as the ones described above is considered a sign of intelligence. But on the contrary, the search for more complex interpretations does not seem to have this status. Something similar occurs with the concept of entropy, which is generally interpreted as disorder or chaos and yet from the point of view of information it is a measure of the amount of information.

Another aspect that should be highlighted is the fact that the cognitive process is supported by the processing of information and, therefore, subject to the rules of mathematical logic, whose nature is irrefutable. This nuance is important, since emphasis is generally placed on the physical and biological mechanisms that support the cognitive processes, which may eventually be assigned a spiritual or esoteric nature.

Therefore, it can be concluded that the cognitive process is subject to the nature and structure of information processing and that from the formal point of view of the Theory of Computability it corresponds to a Turing machine. In such a way that nature has created a processing structure based on the physics of emerging reality – classical reality -, materialized in a neural network, which interprets the information coded by the perception senses, according to the algorithmic established by previous experience. As a consequence, the system performs two fundamental functions, as shown in the figure:

Interact with the environment, producing a response to the input stimuli.
Enhance the ability to interpret, acquiring new skills -algorithmic- as a result of the learning capacity provided by the neural network.

But the truth is that the input stimuli are conditioned by the sensory organs, which constitute a first filter of information and therefore they condition the perception of reality. The question that can be raised is: What impact does this filtering have on the perception of reality?

Reality as an information process

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The purpose of physics is the description and interpretation of physical reality based on observation. To this end, mathematics has been a fundamental tool to formalize this reality through models, which in turn have allowed predictions to be made that have subsequently been experimentally verified. This creates an astonishing connection between reality and abstract logic that makes suspect the existence of a deep relationship beyond its conceptual definition. In fact, the ability of mathematics to accurately describe physical processes can lead us to think that reality is nothing more than a manifestation of a mathematical world.

But perhaps it is necessary to define in greater detail what we mean by this. Usually, when we refer to mathematics we think of concepts such as theorems or equations. However, we can have another view of mathematics as an information processing system, in which the above concepts can be interpreted as a compact expression of the behavior of the system, as shown by the algorithmic information theory [1].

In this way, physical laws determine how the information that describes the system is processed, establishing a space-time dynamic. As a consequence, a parallelism is established between the physical system and the computational system that, from an abstract point of view, are equivalent. This equivalence is somewhat astonishing, since in principle we assume that both systems belong to totally different fields of knowledge.

But apart from this fact, we can ask what consequences can be drawn from this equivalence. In particular, computability theory [2] and information theory [3] [1] provide criteria for determining the computational reversibility and complexity of a system [4]. In particular:

In a reversible computing system (RCS) the amount of information remains constant throughout the dynamics of the system.
In a non-reversible computational system (NRCS) the amount of information never increases along the dynamics of the system.
The complexity of the system corresponds to the most compact expression of the system, called Kolmogorov complexity and is an absolute measure.

It is important to note that in an NRCS system information is not lost, but is explicitly discarded. This means that there is no fundamental reason why such information should not be maintained, as the complexity of an RCS system remains constant. In practice, the implementation of computer systems is non-reversible in order to optimize resources, as a consequence of the technological limitations for its implementation. In fact, the energy currently needed for its implementation is much higher than that established by the Landauer principle [5].

If we focus on the analysis of reversible physical systems, such as quantum mechanics, relativity, Newtonian mechanics or electromagnetism, we can observe invariant physical magnitudes that are a consequence of computational reversibility. These are determined by unitary mathematical processes, which mean that every process has an inverse process [6]. But the difficulties in understanding reality from the point of view of mathematical logic seem to arise immediately, with thermodynamics and quantum measurement being paradigmatic examples.

In the case of quantum measurement, the state of the system before the measurement is made is in a superposition of states, so that when the measurement is made the state collapses in one of the possible states in which the system was [7]. This means that the quantum measurement scenario corresponds to that of a non-reversible computational system, in which the information in the system decreases when the superposition of states disappears, making the system non-reversible as a consequence of the loss of information.

This implies that physical reality systematically loses information, which poses two fundamental contradictions. The first is the fact that quantum mechanics is a reversible theory and that observable reality is based on it. The second is that this loss of information contradicts the systematic increase of classical entropy, which in turn poses a deeper contradiction, since in classical reality there is a spontaneous increase of information, as a consequence of the increase of entropy.

The solution to the first contradiction is relatively simple if we eliminate the anthropic vision of reality. In general, the process of quantum measurement introduces the concept of observer, which creates a certain degree of subjectivity that is very important to clarify, as it can lead to misinterpretations. In this process there are two clearly separated layers of reality, the quantum layer and the classical layer, which have already been addressed in previous posts. The realization of quantum measurement involves two quantum systems, one that we define as the system to be measured and another that corresponds to the measurement system, which can be considered as a quantum observer, and both have a quantum nature. As a result of this interaction, classical information emerges, where the classical observer is located, who can be identified e.g. with a physicist in a laboratory.

Now consider that the measurement is structured in two blocks, one the quantum system under observation and the other the measurement system that includes the quantum observer and the classical observer. In this case it is being interpreted that the quantum system under measurement is an open quantum system that loses quantum information in the measurement process and that as a result a lesser amount of classical information emerges. In short, this scenario offers a negative balance of information.

But, on the contrary, in the quantum reality layer the interaction of two quantum systems takes place which, it can be said, mutually observe each other according to unitary operators, so that the system is closed producing an exchange of information with a null balance of information. As a result of this interaction, the classical layer emerges. But then there seems to be a positive balance of information, as classical information emerges from this process. But what really happens is that the emerging information, which constitutes the classical layer, is simply a simplified view of the quantum layer. For this reason we can say that the classical layer is an emerging reality.

So, it can be said that the quantum layer is formed by subsystems that interact with each other in a unitary way, constituting a closed system in which the information and, therefore, the complexity of the system is invariant. As a consequence of these interactions, the classical layer emerges as an irreducible reality of the quantum layer.

As for the contradiction produced by the increase in entropy, the reasons justifying this behavior seem more subtle. However, a first clue may lie in the fact that this increase occurs only in the classical layer. It must also be considered that, according to the algorithmic information theory, the complexity of a system, and therefore the amount of information that describes the system, is the set formed by the processed information and the information necessary to describe the processor itself.

A physical scenario that can illustrate this situation is the case of the big bang [8], in which it is considered that the entropy of the system in its beginning was small or even null. This is so because the microwave background radiation shows a fairly homogeneous pattern, so the amount of information for its description and, therefore, its entropy is small. But if we create a computational model of this scenario, it is evident that the complexity of the system has increased in a formidable way, which is incompatible from the logical point of view. This indicates that in the model not only the information is incomplete, but also the description of the processes that govern it. But what physical evidence do we have to show that this is so?

Perhaps the clearest sample of this is cosmic inflation [9], so that the space-time metric changes with time, so that the spatial dimensions grow with time. To explain this behavior the existence of dark energy has been postulated as the engine of this process [10], which in a physical form recognizes the gaps revealed by mathematical logic. Perhaps one aspect that is not usually paid attention is the interaction between vacuum and photons, which produces a loss of energy in photons as space-time expands. This loss supposes a decrease of information that necessarily must be transferred to space-time.

This situation causes the vacuum, which in the context of classical physics is nothing more than an abstract metric, to become a fundamental physical piece of enormous complexity. Aspects that contribute to this conception of vacuum are the entanglement of quantum particles [11], decoherence and zero point energy [12].

From all of the above, a hypothesis can be made as to what the structure of reality is from a computational point of view, as shown in the following figure. If we assume that the quantum layer is a unitary and closed structure, its complexity will remain constant. But the functionality and complexity of this remains hidden from observation and it is only possible to model it through an inductive process based on experimentation, which has led to the definition of physical models, in such a way that these models allow us to describe classical reality. As a consequence, the quantum layer shows a reality that constitutes the classical layer and that is a partial vision and, according to the theoretical and experimental results, extremely reduced of the underlying reality and that makes the classical reality an irreducible reality.

The fundamental question that can be raised in this model is whether the complexity of the classical layer is constant or whether it can vary over time, since it is only bound by the laws of the underlying layer and is a partial and irreducible view of that functional layer. But for the classical layer to be invariant, it must be closed and therefore its computational description must be closed, which is not verified since it is subject to the quantum layer. Consequently, the complexity of the classical layer may change over time.

Consequently, the question arises as to whether there is any mechanism in the quantum layer that justifies the fluctuation of the complexity of the classical layer. Obviously one of the causes is quantum decoherence, which makes information observable in the classical layer. Similarly, cosmic inflation produces an increase in complexity, as space-time grows. On the contrary, attractive forces tend to reduce complexity, so gravity would be the most prominent factor.

From the observation of classical reality we can answer that currently its entropy tends to grow, as a consequence of the fact that decoherence and inflation are predominant causes. However, one can imagine recession scenarios, such as a big crunch scenario in which entropy decreased. Therefore, the entropy trend may be a consequence of the dynamic state of the system.

In summary, it can be said that the amount of information in the quantum layer remains constant, as a consequence of its unitary nature. On the contrary, the amount of information in the classical layer is determined by the amount of information that emerges from the quantum layer. Therefore, the challenge is to determine precisely the mechanisms that determine the dynamics of this process. Additionally, it is possible to analyze specific scenarios that generally correspond to the field of thermodynamics. Other interesting scenarios may be quantum in nature, such as the one proposed by Hugh Everett on the Many-Worlds Interpretation (MWI).

Bibliography

[1]	P. Günwald and P. Vitányi, “Shannon Information and Kolmogorov Complexity,” arXiv:cs/0410002v1 [cs:IT], 2008.
[2]	M. Sipser, Introduction to the Theory of Computation, Course Technology, 2012.
[3]	C. E. Shannon, “A Mathematical Theory of Communication,” vol. 27, pp. 379-423, 623-656, 1948.
[4]	M. A. Nielsen and I. L. Chuang, Quantum computation and Quantum Information, Cambridge University Press, 2011.
[5]	R. Landauer, «Irreversibility and Heat Generation in Computing Process,» IBM J. Res. Dev., vol. 5, pp. 183-191, 1961.
[6]	J. Sakurai y J. Napolitano, Modern Quantum Mechanics, Cambridge University Press, 2017.
[7]	G. Auletta, Foundations and Interpretation of Quantum Mechanics, World Scientific, 2001.
[8]	A. H. Guth, The Inflationary Universe, Perseus, 1997.
[9]	A. Liddle, An Introduction to Modern Cosmology, Wiley, 2003.
[10]	P. J. E. Peebles and Bharat Ratra, “The cosmological constant and dark energy,” arXiv:astro-ph/0207347, 2003.
[11]	A. Aspect, P. Grangier and G. Roger, “Experimental Tests of Realistic Local Theories via Bell’s Theorem,” Phys. Rev. Lett., vol. 47, pp. 460-463, 1981.
[12]	H. B. G. Casimir and D. Polder, “The Influence of Retardation on the London-van der Waals Forces,” Phys. Rev., vol. 73, no. 4, pp. 360-372, 1948.

On the complexity of PI (π)

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Introduction

There is no doubt that since the origins of geometry humans have been seduced by the number π. Thus, one of its fundamental characteristics is that it determines the relationship between the length of a circumference and its radius. But this does not stop here, since this constant appears systematically in mathematical and scientific models that describe the behavior of nature. In fact, it is so popular that it is the only number that has its own commemorative day. The great fascination around π has raised speculations about the information encoded in its figures and above all has unleashed an endless race for its determination, having calculated several tens of billions of figures to date.

Formally, the classification of real numbers is done according to the rules of calculus. In this way, Cantor showed that real numbers can be classified as countable infinities and uncountable infinities, what are commonly called rational and irrational. Rational numbers are those that can be expressed as a quotient of two whole numbers. While irrational numbers cannot be expressed this way. These in turn are classified as algebraic numbers and transcendent numbers. The former correspond to the non-rational roots of the algebraic equations, that is, the roots of polynomials. On the contrary, transcendent numbers are solutions of transcendent equations, that is, non-polynomial, such as exponential and trigonometric functions.

Georg Cantor. Co-creator of Set Theory

Without going into greater detail, what should catch our attention is that this classification of numbers is based on positional rules, in which each figure has a hierarchical value. But what happens if the numbers are treated as ordered sequences of bits, in which the position is not a value attribute. In this case, the Algorithmic Information Theory (AIT) allows to establish a measure of the information contained in a finite sequence of bits, and in general of any mathematical object, and that therefore is defined in the domain of natural numbers.

What does the AIT tell us?

This measure is based on the concept of Kolmogorov complexity (KC). So that, the Kolmogorov complexity K(x) of a finite object x is defined as the length of the shortest effective binary description of x. Where the term “effective description” connects the Kolmogorov complexity with the Theory of Computation, so that K(x) would correspond to the length of the shortest program that prints x and enters the halt state. To be precise, the formal definition of K(x) is:

K(x) = min_p,i{K(i) + l(p):T_i (p) = x } + O(1)

Where T_i(p) is the Turing machine (TM) i that executes p and prints x, l(p) is the length of p, and K(i) is the complexity of T_i. Therefore, object p is a compressed representation of object x, relative to T_i, since x can be retrieved from p by the decoding process defined by T_i, so it is defined as meaningful information. The rest is considered as meaningless, redundant, accidental or noise (meaningless information). The term O(1) indicates that K(i) is a recursive function and in general it is non-computable, although by definition it is machine independent, and whose result has the same order of magnitude in each one of the implementations. In this sense, Gödel’s incompleteness theorems, Turing machine and Kolmogorov complexity lead to the same conclusion about undecidability, revealing the existence of non-computable functions.

KC shows that information can be compressed, but does not establish any general procedure for its implementation, which is only possible for certain sequences. In effect, from the definition of KC it is demonstrated that this is an intrinsic property of bitstreams, in such a way that there are sequences that cannot be compressed. Thus, the number of n-bit sequences that can be encoded by m bits is less than 2^m, so the fraction of n-bit sequences with K(x) ≥ n-k is less than 2^-k. If the n-bit possible sequences are considered, each one of them will have a probability of occurrence equal to 2^-n, so the probability that the complexity of a sequence is K(x) ≥ n-k is equal to or greater than (1-2^-k). In short, most bit sequences cannot be compressed beyond their own size, showing a high complexity as they do not present any type of pattern. Applied to the field of physics, this behavior justifies the ergodic hypothesis. As a consequence, this means that most of the problems cannot be solved analytically, since they can only be represented by themselves and as a consequence they cannot be described in a compact way by means of formal rules.

It could be thought that the complexity of a sequence can be reduced at will, by applying a coding criterion that modifies the sequence into a less complex sequence. In general, this only increases the complexity, since in the calculation of K(x) we would have to add the complexity of the coding algorithm that makes it grow as n². Finally, add that the KC is applicable to any mathematical object, integers, sets, functions, and it is demonstrated that, as the complexity of the mathematical object grows, K(x) is equivalent to the entropy H defined in the context of Information Theory. The advantage of AIT is that it performs a semantic treatment of information, being an axiomatic process, so it does not require having a priori any type of alphabet to perform the measurement of information.

What can be said about the complexity of π?

According to its definition, KC cannot be applied to irrational numbers, since in this case the Turing machine does not reach the halt state, and as we know these numbers have an infinite number of digits. In other words, and to be formally correct, the Turing machine is only defined in the field of natural numbers (it must be noted that their cardinality is the same as that of the rationals), while irrational numbers have a cardinality greater than that of rational numbers. This means that KC and the equivalent entropy H of irrational numbers are undecidable and therefore non-computable.

To overcome this difficulty we can consider an irrational number X as the concatenation of a sequence of bits composed of a rational number x and a residue δx, so that in numerical terms X=x+δx, but in terms of information X={x,δx}. As a consequence, δx is an irrational number δx→0, and therefore δx is a sequence of bits with an undecidable KC and hence non-computable. In this way, it can be expressed:

K(X) = K(x)+K(δx)

The complexity of X can be assimilated to the complexity of x. A priori this approach may seem surprising and inadmissible, since the term K(δx) is neglected when in fact it has an undecidable complexity. But this is similar to the approximation made in the calculation of the entropy of a continuous variable or to the renormalization process used in physics, in order to circumvent the complexity of the underlying processes that remain hidden from observable reality.

Consequently, the sequence p, which runs the Turing machine i to get x, will be composed of the concatenation of:

The sequence of bits that encode the rules of calculus in the Turing machine i.
The bitstream that encodes the compressed expression of x, for example a given numerical series of x.
The length of the sequence x that is to be decoded and that determines when the Turing machine should reach the halt state, for example a googol (10¹⁰⁰).

In short, it can be concluded that the complexity K(x) of known irrational numbers, e.g. √2, π, e,…, is limited. For this reason, the challenge must be to obtain the optimum expression of K(x) and not the figures that encode these numbers, since according to what has been said, their uncompressed expression, or the development of their figures, has a high degree of redundancy (meaningless information).

What in theory is a surprising and questionable fact is in practice an irrefutable fact, since the complexity of δx will always remain hidden, since it is undecidable and therefore non-computable.

Another important conclusion is that it provides a criterion for classifying irrational numbers into two groups: representable and non-representable. The former correspond to irrational numbers that can be represented by mathematical expressions, which would be the compressed expression of these numbers. While non-representable numbers would correspond to irrational numbers that could only be expressed by themselves and are therefore undecidable. In short, the cardinality of representable irrational numbers is that of natural numbers. It should be noted that the previous classification criterion is applicable to any mathematical object.

On the other hand, it is evident that mathematics, and calculus in particular, de facto accepts the criteria established to define the complexity K(x). This may go unnoticed because, traditionally in this context, numbers are analyzed from the perspective of positional coding, in such a way that the non-representable residue is filtered out through the concept of limit, in such a way that δx→0. However, when it comes to evaluating the informative complexity of a mathematical object, it may be required to apply a renormalization procedure.