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

• January 15th, 2020, 04:11 AM
Jarek Duda
Violation of Bell-like inequalities with spatial Boltzmann path ensemble: Ising model?
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):

https://i.stack.imgur.com/MvGba.png

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?

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Update: Born-like formulas from symmetry in Ising model (Boltzmann sequence ensemble): Pr(i)=(ψ_i)^2 where one amplitude ("hidden variable") comes from left, second from right:

https://i.imgur.com/SSFN1BG.png
• January 17th, 2020, 03:41 AM
Jarek Duda
Another paper with Bell violation using Ising model: https://journals.aps.org/prl/abstrac...ett.123.170604
• January 21st, 2020, 11:53 PM
Jarek Duda
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 ):

https://i.imgur.com/ls3KAFi.png
• February 16th, 2020, 03:21 AM
Jarek Duda
Let me ask a related question: do Feynman path integrals satisfy Bell locality assumption?

There are generally two basic ways to solve physics models:

1) Asymmetric, e.g. Euler-Lagrange equation in CM, Schrödinger equation in QM
2) Symmetric, e.g. the least action principle in CM, Feynman path integrals in QM, Feynman diagrams in QFT.

Having solution found with 1) or 2), we can transform it into the second, but generally solutions originally found using 1) or 2) seem to have a bit different properties - for example regarding "hidden variables" in Bell theorem.
The asymmetric ones 1) like Schrödinger equation usually satisfy assumptions used to derive Bell inequality, which is violated by physics - what is seen as contradiction of local realistic "hidden variables" models. Does it also concern the symmetric ones 2)?

We successfully use classical field theories like electromagnetism or general relativity, which assume existence of objective state of their field - how does this field differ from local realistic "hidden variables"?

Wanting to resolve this issue, there are e.g. trials to undermine the locality assumption by proposing faster-than-light communication, but these classical field theories don't allow for that.

So I would like to ask about another way to dissatisfy Bell's locality assumption: there is general belief that physics is CPT-symmetric, so maybe it solves its equations in symmetric ways 2) like through Feynman path integrals?

Good intuitions for solving in symmetric way provides Ising model, where asking about probability distribution inside such Boltzmann sequence ensemble, we mathematically get Pr(u)=(psi_u)^2, where one amplitude comes from left, second from right, such Born rule allows for Bell-violation construction. Instead of single "hidden variable", due to symmetry we have two: from both directions.

From perspective of e.g. general relativity, we usually solve it through Einstein's equation, which is symmetric - spacetime is kind of "4D jello" there, satisfying this this local condition for intrinsic curvature. It seems tough (?) to solve it in asymmetric way like through Euler-Lagrange, what would require to "unroll" spacetime.

Assuming physics solves its equations in symmetric way, e.g. QM with Feynman path integrals instead of Schrödinger equation, do Bell's assumptions hold - are local realistic "hidden variables" still disproven?