localmodels

The modules bilocal.py and triangle.py provide tools to find explicit local models in two network topologies: the bilocal scenario [1] and the triangle scenario with no inputs [2].

See the two examples below to learn how use the modules.

Bilocal scenario

This example tries to find a local model for the (I,J) distribution found in figure 3 of reference [1] in the bilocal scenario with binary inputs and outputs.

import numpy as np
from bilocal import *

a = np.arange(0,2)
b = np.arange(0,2)
c = np.arange(0,2)
x = np.arange(0,2)
y = np.arange(0,2)
z = np.arange(0,2)
a, b, c, x, y, z = np.meshgrid(a,b,c,x,y,z,indexing='ij')
I = 0
J = 1
pI = 1/8*(1+(y==0)*(-1)**(a+b+c))
pJ = 1/8*(1+(y==1)*(-1)**(x+z+a+b+c))
p0 = 1/8
p = I*pI + J*pJ + (1-I-J)*p0
solution = bilocal(p, Ma=2, Mb=2, Mc=2, ma=2, mb=2, mc=2, c_lambda=2, c_mu=2)
p_lambda, p_mu, p_a, p_b, p_c = model(solution.x, c_lambda=2, c_mu=2)

After running the code above, the probability distributions of the hidden variables should be in the variables p_lambda, p_mu, and the response functions of Alice, Bob and Charles should be in the variables p_a, p_b, p_c. The indexing of these variables works as follows:

p_lambda[i] is the probability that lambda assumes the value i.

p_mu[i] is the probability that mu assumes the value i.

p_a[i,j,k] is the probability that Alice outputs value i, given that she receives x=j and lambda=k.

p_b[i,j,k,l] is the probability that Bob outputs value i, given that he receives y=j, lambda=k and mu=l.

p_c[i,j,k] is the probability that Charles outputs value i, given that he receives z=j and mu=k.

Triangle scenario with no inputs

For the triangle scenario, the class TrilocalModel implements a bunch of cool functionalities such as model optimization, relabelling of hidden variables or outputs, exchange of parties and hidden variable cardinality reduction. If you are interested in using all that, I recommend you read the docstrings for the class methods.

For a quick introduction the following example tries to find a local model for the GHZ distribution mixed with a uniform distribution with visibility v = 0.33 in the triangle scenario with no inputs and binary outputs. The hidden variable cardinalities are (3, 2, 2).

import numpy as np
from triangle import TrilocalModel

v = 0.33
pGHZ = np.zeros((2, 2, 2))
pGHZ[0, 0, 0] = pGHZ[1, 1, 1] = 1/2
p0 = 1/8
p = v*pGHZ + (1-v)*p0
model = TrilocalModel.uniform(c_alpha=3, c_beta=2, c_gamma=2, ma=2, mb=2, mc=2)
model = model.optimize(p, number_of_trials=10)
print(model)

After running the code above, the probability distributions of the hidden variables should be in the attributes model.p_alpha, model.p_beta, model.p_gamma, and the response functions of Alice, Bob and Charles should be in the attributes model.p_a, model.p_b, model.p_c. The indexing of these attributes works as follows:

p_alpha[i] is the probability that alpha assumes the value i.

p_beta[i] is the probability that beta assumes the value i.

p_gamma[i] is the probability that gamma assumes the value i.

p_a[i,j,k] is the probability that Alice outputs value i, given that she receives beta=j and gamma=k.

p_b[i,j,k] is the probability that Bob outputs value i, given that he receives gamma=j, alpha=k.

p_c[i,j,k] is the probability that Charles outputs value i, given that he receives alpha=j and beta=k.

References:

[1]: BRANCIARD, C.; ROSSET, D.; GISIN, N.; PIRONIO, S. Bilocal versus nonbilocal correlations in entanglement-swapping experiments. Physical Review A, APS, v. 85, n. 3, p. 032119, 2012.

[2]: RENOU, M.-O.; BÄUMER, E.; BOREIRI, S.; BRUNNER, N.; GISIN, N.; BEIGI, S. Genuine quantum nonlocality in the triangle network. Physical review letters, APS, v. 123, n. 14, p. 140401, 2019.