key: cord-0517700-cqg3dyus
authors: Pomorski, Krzysztof
title: Equivalence between classical epidemic model and non-dissipative and dissipative quantum tight-binding model
date: 2020-12-17
journal: nan
DOI: nan
sha: 583d2f6a68fd5cfc3731a184dd6d74c56c9acf4b
doc_id: 517700
cord_uid: cqg3dyus

The equivalence between classical epidemic model and nondissipative and dissipative quantum tight-binding model is derived. Classical epidemic model can reproduce the quantum entanglement emerging in the case of electrostatically coupled qubits described by von-Neumann entropy both in non-dissipative and dissipative case. The obtained results shows that quantum mechanical phenomena might be almost entirely simulated by classical statistical model. It includes the quantum like entanglement and superposition of states. Therefore coupled epidemic models expressed by classical systems in terms of classical physics can be the base for possible incorporation of quantum technologies and in particular for quantum like computation and quantum like communication. The classical density matrix is derived and described by the equation of motion in terms of anticommutator. Existence of Rabi like oscillations is pointed in classical epidemic model. Furthermore the existence of Aharonov-Bohm effect in quantum systems can also be reproduced by the classical epidemic model. Every quantum system made from quantum dots and described by simplistic tight-binding model by use of position-based qubits can be effectively described by classical model with very specific structure of S matrix that has twice bigger size as it is the case of quantum matrix Hamiltonian. Obtained results partly question fundamental and unique character of quantum mechanics and are placing ontology of quantum mechanics much in the framework of classical statistical physics what can bring motivation for emergence of other fundamental theories bringing suggestion that quantum mechanical is only effective and phenomenological but not fundamental picture of reality.

Epidemic model can model sickness propagation and various phenomena in sociology, physics and biology. The most basic form of epidemic model relies on co-dependence of two probabilities of occurrence of state 1 and 2 that can be identified at the state of being healthy and sick as it is being depicted in Fig.1 . It is expressed in the compact way in the following way:

(s 11 (t) |1 1| + s 22 (t) |2 2| + s 12 (t) |2 1| + s 21 (t) |1 2|)(p 1 (t) |1 + p 2 (t) |2 ) = d dt (p 1 |1 + p 2 |2 ) = = d dt

Quite naturally such system evolves in natural statistical environment before the measurement is done. Once the measurement is done the statistical system state is changed from undetermined and spanned by two probabilities into the case of being either with probability p 1 = 1 or p 1 = 0 so it corresponds to two projections: P →1 = |1 1| = 1 0 0 0 ,P →2 = |2 2| = 0 0 0 1 ,P →1 +P →2 =Î = 1 0 0 1 P →1 |ψ classical = |1 = |ψ 1 a f ter , 0 0 0 1 ,P →2 |ψ classical = |2 = |ψ 2 a f ter ,

where occurrence of |ψ1 a f ter and |ψ2 a f ter occurs with probability p 1 (t measurement ) and p 2 (t measurement ).

We notice that matrixŜŜ = s 11 (t) s 12 (t) s 21 (t) s 22 (t) 

It shall be underlined that necessary condition for identification the superposition of two classical eigenstates is expressed by (− (s 11 

We have superposition of states with 2 statistical ensembles occurring with probabilities p I (t) and p II (t) that are encoded in probabilities p 1 (t) and p 2 (t) that are directly observable. We can extract probabilities p I (t) and p II (t) from |ψ(t) classical in the following way p 1 (t), p 2 (t) (11) and p II (t) = 1 n E 2 (t) ψ E 2 | |ψ(t) classical = 1 − 

Probabilities p I (t) and p II (t) will describe the occupancy of energy levels E 1 and E 2 in real time domain of epidemic simplistic model. We have the same superposition of two eigenergies as in the case of quantum tight-binding model. The same reasoning can be conducted for N-th state classical epidemic model expressed as

In principle we can also introduce weak measurement procedure that will be partly omitted in this work. In very real way if we have the population of N individuals possibly infected with COVID we can inspect N 1 individuals, where N 1 < N and we can introduce some corrections to

are probabilities obtained by testing N 1 individuals what could correspond to weak measurement conducted on assemble of N individuals. Let us consider the state of the system before measurement and its natural evolution Such set of equations has two analytical solutions for probabilities p 1 (t) and p 2 (t) expressed as exp t t0 s 11 (t)dt 

with

and

Sum of probabilities is not normalized. However physical significance between ratio p 1 (t) and p 2 that is expressed by ratio

It means that Rabi oscillations or more precisely change of occupancy among levels is naturally build in classical epidemic model. Still superposition of two states is mainatianed so the analogy of classical epidemic model to quantum tight-binding model is deep.

We consider the case of p I (t) = constant 1 and p II (t) = constant II . We have time-dependent parameters s 11 , s 22 , s 12 and s 21 and we obtain the following equations of motion

We obtain the set of 2 equations

Consequently we obtain

and it implies Let us consider the equations of motion in the following way:

This equation is equivalent to the set of 2 coupled ordinary differential equations given as

,

and can be rewritten to be as

It will lead to further simplification that can be written as

We can write it in the compact form as

The solution is analytical given as

and can be written aŝ

where

and

G 1,2 (t,t 0 ) = 2g 12 (t,t 0 )e g 11 (t,t 0 )+g 22 (t,t 0 ) 2 sinh 1 2 (g 11 (t,t 0 ) − g 22 (t,t 0 )) 2 + 4g 12 (t,t 0 )g 21 (t,t 0 ) (g 11 (t,t 0 ) − g 22 (t,t 0 )) 2 + 4g 12 (t,t 0 )g 21 (t,t 0 ) ,

G 2,1 (t,t 0 ) = 2g 21 (t,t 0 )e g 11 (t,t 0 )+g 22 (t,t 0 ) 2 sinh 1 2 (g 11 (t,t 0 ) − g 22 (t,t 0 )) 2 + 4g 12 (t,t 0 )g 21 (t,t 0 ) (g 11 (t,t 0 ) − g 22 (t,t 0 )) 2 + 4g 12 (t,t 0 )g 21 (t,t 0 ) (50)

It is now easy to generalize our considerations for 2 coupled statistical ensembles as corresponding to cities with flight traffic. We have 

We make the analytic simplifications by assumption of two symmetric systems A and B interacting interacting in asymmetric way as

We have 4 eigenstates

and |V 3 (t) and |V 4 (t) are physically justifiable in framework of epidemic model that can be used for classical entanglement. We have 4 projectors corresponding to the measurement of p 1A , p 2A , p 1B and p 2B represented as matriceŝ

Measurement conducted on system A also brings the change of system B state what is due to presence of non-diagonal matrix elements in system evolution equations (generalized epidemic model). It is therefore analogical to measurement of quantum entangled state. After the measurements we obtain the following classical states

We are going to reformulate the description of 2 classical noninteracting systems A and B. We havê

and we have the

where |x kA |x lB are time-independent and k and l are 1 or 2 and

) are describing probabilities for 4 different states of statistic finite machine. Similar situation occurs in the case of Schroedinger equation written for 2 non-interacting systems, but instead of probabilities we have square root of probabilities times phase factor. In general case of interaction between A and B systems in classical epidemic model we havê

The presented analytical approach can be applied for system A with 4 distinct states as well as for system B with 4 distinct states since isolated system A(B) with 4 states can be described by evolution matrix 4 by 4 that has 4 analytical eigenvalues and eigenstates and becomes non-analytical for 5 and more distinct states due to fact that roots of polynomial of higher order than 4 becomes non-analytical and becomes numerical with some limited exceptions. We can write the matrix

Let us be motivated by work on single electron devices by [19] , [1] , [20] , [5] and [7] . Instead of probabilities it will be useful to operate with square root of probabilities as they are present in quantum mechanics and in Schroedinger or Dirac equation. Since

we can rewrite the epidemic equation as

This equation is equivalent to

and we set t 1 = iht so one has dt 1 = ihdt and t = t 1 ih so we have

The following notation is introduced:

,

and one obtains

Let us start from quantum mechanical perspective

and we obtain the set of 4 equations

that can be rewritten

and that can be translated into

so one can write

what can be written asÂ

what implies

Furthermore we can obtain

The last set of equations is equivalent to

B. Case of 2 coupled 2 level classical stochastic finite state machines Instead of probabilities it will be useful to operate with square root of probabilities as they are present in quantum mechanics and in Schroedinger or Dirac equation. Since 

and we have

In very real way we obtain the following classical physics Hamiltonian and we obtain 

At this stage we can identify analytical solutions given as

and it implies that 

. We recognize analogies with Schroedinger equation that are occurring only in case of real valued wave functions. We can introduce the object similar to quantum density matrix that is classical density matrix. We havê

We can obtain the analytical solution for such system in quite straighforward waŷ

We can compute the reduced density matrix of subsystem A and B in standard way and we can establish von-Neumann entropy relation.

By introducing t 1 = iht we have t 1 ih = t and it leads to

It leads us towards analytical solution

Dealing in similar fashion we can obtain quantum density matrix given as

It is quite straightforward to establish the reduced density matrix of system A and B and hence obtain von-Neumann entropy with time that characterizes the quantum like entropy in time in classical epidemic model in 2 coupled systems. We can extract density of matrix for system A given as

and for system B given as

. Those density matrices needs to be normalized so finally we obtain

and for system B given as

. In such case we can apply von-Neumman entropy measure to characterize the density matrix of classical state so we have S A = Tr(ρ A log(ρ A )) and S B = Tr(ρ B log(ρ B )). In our general case S A = S B , while in case of two quantum systems with mutual interaction, but isolated from external environment we have S A = S B . We conduct reverse reasoning and show that any quantum system describing by tight binding model can be modeled by stochastic finite state machine. This proves the fact that classical electronics can model quantum system made from singleelectron devices as in CMOS quantum computer. This reasoning can be conduced for N coupled quantum dots implementing CMOS quantum computer. It has therefore its meaning in quantum machine learning implemented in CMOS quantum computer.

We are considering two position-based qubits interacting by means of electrostatic Coulomb force as depicted in Fig.2 . We define the Hamiltonian of 2 interacting Wannier qubits in the simplistic form given by tight-binding model. We havê

and for quantum state of the form

and with γ 1 (t) = p I (t)e iΘ 1 (t) , .., γ IV (t) = p IV (t)e iΘ IV (t) . We have the following evolution of non-disspative quantum state of isolated system in tight-binding model given as

After reshaping we have

[2] [3] V(x) of SEL and final form of classical epidemic model mimicking the quantum tight-binding model in the form as

We obtain d dt

and equivalently we have − p I (t)sin(Θ I (t)) + p II (t)cos(Θ II (t)) − p II (t)sin(Θ II (t)) + p III (t)cos(Θ III (t)) − p III (t)sin(Θ III (t)) + p IV (t)cos(Θ IV (t)) − p IV (t)sin(Θ IV (t))

p I (t)(cos(Θ I (t))) 2 p I (t)(sin(Θ I (t))) 2 p II (t)(cos(Θ II (t))) 2 p II (t)(sin(Θ II (t))) 2 p III (t)(cos(Θ III (t))) 2 p III (t)(sin(Θ III (t))) 2 p IV (t)(cos(Θ IV (t))) 2 p IV (t)(sin(Θ IV (t))) 2 

Im(E p1A + E p1B +Re(E p1A + E p1B + q 2 d 1A−1B ) (t sim(2 B →1 B ) ) +(t sr(2 B →1 B ) ) (t sim(2 A →1 A ) ) (t sr(2 A →1 A ) ) 0 0 −Re(E p1A + E p1B + q 2 d 1A−1B ) Im(E p1A + E p1B −(t sr(2 B →1 B ) ) (t sim(2 B →1 B ) ) (−t sr(2 A →1 A ) ) (t sim(2 A →1 A ) ) 0 0 

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and in compact form we have d dt[t s(2 B →1 B ) ] e i(Θ II (t)−Θ I (t)) p I (t) p II (t) [t s(2 A →1 A ) ] e i(Θ III (t)−Θ I (t)) p I (t) p III (t) 0 t s(1 B →2 B ) e i(Θ I (t)−Θ II (t)) p II (t) p I (t)

Let us start from equations of motion for 2 electrostatically coupled position based qubits as by [16] expressed by minimalistic tight-binding model and we havep I (t)cos(Θ I (t)) + i p I (t)sin(Θ I (t)) p II (t)cos(Θ II (t)) + i p II (t)sin(Θ II (t)) p III (t)cos(Θ III (t)) + i p III (t)sin(Θ III (t)) p IV (t)cos(Θ IV (t)) + i p IV (t)sin(Θ IV (t))and equivalently we can writeThe established relation can be written in the compact form asThe S matrix from classical epidemic model can be written aŝand can be written in relation to 2 coupled single electron devices asˆS Re(p I (t)) Im(p I (t)) (t sim(2 B →1 B ) ) Re(p I (t)) Re(p II (t)) +(t sr(2 B →1 B ) ) Re(p I (t)) Im(p II (t)) (t sim(2 A →1 A ) ) Re(p I (t)) Re(p III (t)) (t sr(2 A →1 A ) ) Re(p I (t)) Im(p III (t)) 0

Im(E p1A + E p1B ) −(t sr(2 B →1 B ) ) Im(p I (t)) Re(p II (t)) (t sim(2 B →1 B ) ) Im(p I (t)) Im(p II (t)) (−t sr(2 A →1 A ) ) Im(p I (t)) Re(p III (t)) (t sim(2 A →1 A ) ) Im(p I (t)) Im(p III (t)) 0and matrixŜ 2 is given aŝRe(p II (t)) Im(p II (t)) Re(p III (t)) Im(p III (t)) Re(p IV (t)) Im(p IV (t))Re(p II (t)) Im(p II (t)) Re(p III (t)) Im(p III (t)) Re(p IV (t)) Im(p IV (t))with Re(p I (t)) = cos(Θ I (t)) 2 p I (t), (115) Im(p I (t)) = sin(Θ I (t)) 2 p I (t), (116) Re(p II (t)) = cos(Θ II (t)) 2 p II (t), (117) Im(p II (t)) = sin(Θ II (t)) 2 p II (t), (118) Re(p III (t)) = cos(Θ III (t)) 2 p III (t),(119) Im(p III (t)) = sin(Θ III (t)) 2 p III (t),(120) Re(p IV (t)) = cos(Θ IV (t)) 2 p IV (t),(121) Im(p IV (t)) = sin(Θ IV (t)) 2 p IV (t).(122)After conducted considerations it becomes explictly visible that usage of magnetic Aharonov-Bohm effect will have impact on the occupancy of nodes). Let us assume that two interacting qubits are symmetric and aligned along x axes. In such case we need to renormalize phases dynamics with time sowhere δ L is the diameter each of 4 quantum dots and e is elementary charge of electron. Such situation was described by means of Schroedinger formalism in [18] . Furthermore the noninvasive detection of charge movement by single electron devices [26] can also be also mapped to epidemic model. Various analytical solutions of tight-binding model that can be used in epidemic model can be found in [28] , [29] , [13] , [30] .Here we have assumed that there is possible escape of electron(s) from the system of quantum dots to outside environment by tunneling from the system of 2 coupled quantum dots what can be reflected in imaginary values of E p1A , E p2A , E p1B , E p2B . Furthermore we can also assume that electron is being injected to the structure of 2 coupled quantum dots what also corresponds to non-zero imaginary value of E p1A , E p2A , E p1B , E p2B and is denoted as Im(E p1A ), Im(E p2A ), Im(E p1B ), Im(E p2B ). Presence of electrons at positions (1A,1B),(1A,2B), (2A,1B), (2A,2B) is reflected by probabilities p 1 (t) + p 2 (t), p 3 (t) + p 4 (t), p 5 (t) + p 6 (t), p 7 (t) + p 8 (t). Dependence of evolution of quantum phases Θ I (t),Θ II (t),Θ III (t),Θ IV (t) is encoded by square of tangents of phases given by tan(Θ I (t)) 2 = p 2 (t) p 1 (t) , tan(Θ II (t)) 2 = p 4 (t) p 3 (t) , tan(Θ III (t)) 2 = p 6 (t) p 5 (t) , tan(Θ IV (t)) 2 = p 8 (t) p 7 (t) . If tight-binding model is time-independent the evolution of phase can be tracked in unique way by classical epidemic model. Furthermore matrixŜ(t) of classical epidemic model can be determined in unique way, since all its eigenstates and eigenvectors can be determined analytically. This is not the case of bigger matrices than 4 by 4. Furthermore one can incorporate the noise in case of 2 coupled position based qubits with use of delta functions and such situation can also be mapped to classical epidemic model. We notice that for the case of coupled position based qubits system expressed by matrix N by N and N possible states we have mapping to classical epidemic model with obtained matrixŜ that has dimension 2N by 2N and it described evolution of 2N classical states in the framework of stochastic finite state machine.

The obtained results shows that quantum mechanical phenomena might be almost entirely simulated by classical statistical model. It includes the quantum like entanglement and superposition of states. Therefore coupled epidemic models expressed by classical systems in terms of classical physics can be the base for possible incorporation of quantum technologies and in particular for quantum like computation and quantum like communication. In the conduced computations Wolfram software was used [23] . All work presented at [25] , [13] can be expressed by classical epidemic model. It is expected that time crystals can be also described in the given framework [14] , [15] . It is open issue to what extent we can parameterize various condensed matter phenomena by stochastic finite state machine [21] .