Ising model as Boltzmann path ensemble–Born rule, Bell viol.

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Ising model as Boltzmann path ensemble–Born rule, Bell viol.

Postby Jarek » Mon Jan 06, 2020 10:38 pm

Ising-like models as spatial Boltzmann path ensembles – with Born rule, TSVF … Bell violation?

Quantum mechanics is equivalent with Feynman path ensembles, which after Wick rotation becomes Boltzmann path ensemble, which can be seen as MERW ( https://en.wikipedia.org/wiki/Maximal_E ... andom_Walk ): diffusion repaired to agree with maximal entropy principle and having QM-like localization property: e.g. predicting rho~sin^2 stationary density for [0,1], providing intuition for this Born rule square as direct consequence of time symmetry etc.

While such Boltzmann ensembles of paths along time are nonintuitive due to their time symmetry, their spatial analogues are non-controversial basic tools of condensed matter physics: Ising model and its generalizations.

Imagine such infinite 1D lattice of spins or some more general objects (e.g. patterns for 2D Ising: https://arxiv.org/pdf/1912.13300 ), with E_uv energy of interaction between u and v neighboring generalized spins. Define M_uv = exp(-beta E_uv) as transition matrix and find its dominant eigenvalue/vector: M psi = lambda psi for maximal |lambda|.

Now assuming Boltzmann distribution among such infinite sequences (in space for Ising-like models, in time for MERW), one can easily find that probability of one and two neighboring values is:
Pr(u) = (psi_u)^2
Pr(u,v) = psi_u (M_uv / lambda) psi_v

The former resembles 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 in https://arxiv.org/pdf/0910.2724 ) 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”:

Image

From Ising perspective, we need 1D lattice of 3 spins with constraints here – allowing neighbors only accordingly to blue edges in above diagram, or some other e.g. 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?
Jarek
 
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Re: Ising model as Boltzmann path ensemble–Born rule, Bell v

Postby Jarek » Thu Jan 16, 2020 5:11 am

Another paper about Bell violation with Ising model: https://journals.aps.org/prl/abstract/1 ... 123.170604
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Re: Ising model as Boltzmann path ensemble–Born rule, Bell v

Postby Jarek » Tue Jan 21, 2020 10:21 pm

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 ):

Image
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