Violation of Bell-like inequalities with spatial Boltzmann path ensemble: Ising model?

[Duda Jarek]

Quantum mechanics is equivalent with Feynman path ensemble, which after Wick rotation becomes Boltzmann path ensemble, which can be normalized into stochastic process as maximal entropy random walk MERW.

But Boltzmann path ensemble has also spatial realization: 1D Ising model and its generalizations: Boltzmann distribution among spatial sequences of spins or some more complicated objects.

For E_uv energy of interaction between u and v neighboring spins or something more general, define M_uv = exp(-beta E_uv) as transition matrix and find its dominant eigenvalue/vector: M psi = lambda psi for maximal |lambda|. Now it is easy to find (e.g. derived here) that probability distribution of one and two neighboring values inside such sequence are:

Pr(u) = (psi_u)^2
Pr(u,v) = psi_u (M_uv / lambda) psi_v

The former resembles QM Born rule, the latter TSVF – the two ending psi come from propagators from both infinities as M^p ~ lambda^p psi psi^T for unique dominant eigenvalue thanks to Frobenius-Perron theorem. We nicely see this Born rule coming from symmetry here: spatial in Ising, time in MERW.

Having Ising-like models as spatial realization of Boltzmann path integrals getting Born rule from symmetry, maybe we could construct Bell violation example with it?

Here is MERW construction (page 9 here) for violation of Mermin’s Pr(A=B) + Pr(A=C) + Pr(B=C) >= 1 inequality for 3 binary variables ABC, intuitively “tossing 3 coins, at least 2 are equal” (e.g. here is QM violation )



From Ising perspective, we need 1D lattice of 3 spins with constraints – allowing neighbors only accordingly to blue edges in above diagram, or some other e.g. just forbidding |000> and |111>.

Measurement of AB spins is defect in this lattice as above – fixing only the measured values. Assuming uniform probability distribution among all possible sequences, the red boxes have correspondingly 1/10, 4/10, 4/10, 1/10 probabilities – leading to Pr(A=B) + Pr(A=C) + Pr(B=C) = 0.6 violation.

Could this kind of spin lattice construction be realized?

What types of constraints/interaction in spin lattices can be realized?

While Ising-like models provide spatial realization of Boltzmann path integrals, is there spatial realization of Feynman path integrals?

Edited January 15, 2020 by Duda Jarek
[math] [/math] latex does not work (?)

[Duda Jarek]

Another paper with Bell violation using Ising model: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.123.170604

[Duda Jarek]

Construction of Bell violation can be extended further to quantum-like computers realized as Ising model: e.g. by somehow printing on a surface conditions for solving a given problem by Ising's: Boltzmann distribution among possible sequences.

Such Wick-rotated quantum gates seem a bit weaker computationally, but spatial realization allows to fix amplitudes from both directions: left and right, what seems(?) to allow to quickly solve NP-complete problems like 3-SAT (end of https://arxiv.org/pdf/1912.13300  ) :

Edited January 22, 2020 by Duda Jarek