Source code for toqito.nonlocal_games.extended_nonlocal_game

"""Two-player extended nonlocal game."""
from __future__ import annotations
from collections import defaultdict

import cvxpy
import numpy as np

from toqito.matrix_ops import tensor
from toqito.random import random_unitary
from toqito.helper import update_odometer, npa_constraints


[docs]class ExtendedNonlocalGame: r""" Create two-player extended nonlocal game object. *Extended nonlocal games* are a superset of nonlocal games in which the players share a tripartite state with the referee. In such games, the winning conditions for Alice and Bob may depend on outcomes of measurements made by the referee, on its part of the shared quantum state, in addition to Alice and Bob's answers to the questions sent by the referee. Extended nonlocal games were initially defined in [JMRW16]_ and more information on these games can be found in [Russo17]_. References ========== .. [JMRW16] Nathaniel Johnston, Rajat Mittal, Vincent Russo, John Watrous "Extended non-local games and monogamy-of-entanglement games", Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472.2189 (2016), https://arxiv.org/abs/1510.02083 .. [Russo17] Vincent Russo "Extended nonlocal games" https://arxiv.org/abs/1704.07375 """ def __init__(self, prob_mat: np.ndarray, pred_mat: np.ndarray, reps: int = 1) -> None: """ Construct extended nonlocal game object. :param prob_mat: A matrix whose (x, y)-entry gives the probability that the referee will give Alice the value `x` and Bob the value `y`. :param pred_mat: :param reps: Number of parallel repetitions to perform. """ if reps == 1: self.prob_mat = prob_mat self.pred_mat = pred_mat self.reps = reps else: ( dim_x, dim_y, num_alice_out, num_bob_out, num_alice_in, num_bob_in, ) = pred_mat.shape self.prob_mat = tensor(prob_mat, reps) pred_mat2 = np.zeros( ( dim_x**reps, dim_y**reps, num_alice_out**reps, num_bob_out**reps, num_alice_in**reps, num_bob_in**reps, ) ) i_ind = np.zeros(reps, dtype=int) j_ind = np.zeros(reps, dtype=int) for i in range(num_alice_in**reps): for j in range(num_bob_in**reps): to_tensor = np.empty([reps, dim_x, dim_y, num_alice_out, num_bob_out]) for k in range(reps - 1, -1, -1): to_tensor[k] = pred_mat[:, :, :, :, i_ind[k], j_ind[k]] pred_mat2[:, :, :, :, i, j] = tensor(to_tensor) j_ind = update_odometer(j_ind, num_bob_in * np.ones(reps)) i_ind = update_odometer(i_ind, num_alice_in * np.ones(reps)) self.pred_mat = pred_mat2 self.reps = reps
[docs] def unentangled_value(self) -> float: r""" Calculate the unentangled value of an extended nonlocal game. The *unentangled value* of an extended nonlocal game is the supremum value for Alice and Bob's winning probability in the game over all unentangled strategies. Due to convexity and compactness, it is possible to calculate the unentangled extended nonlocal game by: .. math:: \omega(G) = \max_{f, g} \lVert \sum_{(x,y) \in \Sigma_A \times \Sigma_B} \pi(x,y) V(f(x), g(y)|x, y) \rVert where the maximum is over all functions :math:`f : \Sigma_A \rightarrow \Gamma_A` and :math:`g : \Sigma_B \rightarrow \Gamma_B`. :return: The unentangled value of the extended nonlocal game. """ dim_x, dim_y, alice_out, bob_out, alice_in, bob_in = self.pred_mat.shape max_unent_val = float("-inf") for a_out in range(alice_out): for b_out in range(bob_out): p_win = np.zeros([dim_x, dim_y], dtype=complex) for x_in in range(alice_in): for y_in in range(bob_in): p_win += ( self.prob_mat[x_in, y_in] * self.pred_mat[:, :, a_out, b_out, x_in, y_in] ) rho = cvxpy.Variable((dim_x, dim_y), hermitian=True) objective = cvxpy.Maximize(cvxpy.real(cvxpy.trace(p_win.conj().T @ rho))) constraints = [cvxpy.trace(rho) == 1, rho >> 0] problem = cvxpy.Problem(objective, constraints) unent_val = problem.solve() max_unent_val = max(max_unent_val, unent_val) return max_unent_val
[docs] def nonsignaling_value(self) -> float: r""" Calculate the non-signaling value of an extended nonlocal game. The *non-signaling value* of an extended nonlocal game is the supremum value of the winning probability of the game taken over all non-signaling strategies for Alice and Bob. A *non-signaling strategy* for an extended nonlocal game consists of a function .. math:: K : \Gamma_A \times \Gamma_B \times \Sigma_A \times \Sigma_B \rightarrow \text{Pos}(\mathcal{R}) such that .. math:: \sum_{a \in \Gamma_A} K(a,b|x,y) = \rho_b^y \quad \text{and} \quad \sum_{b \in \Gamma_B} K(a,b|x,y) = \sigma_a^x, for all :math:`x \in \Sigma_A` and :math:`y \in \Sigma_B` where :math:`\{\rho_b^y : y \in \Sigma_A, \ b \in \Gamma_B\}` and :math:`\{\sigma_a^x : x \in \Sigma_A, \ a \in \Gamma_B\}` are collections of operators satisfying .. math:: \sum_{a \in \Gamma_A} \rho_b^y = \tau = \sum_{b \in \Gamma_B} \sigma_a^x, for every choice of :math:`x \in \Sigma_A` and :math:`y \in \Sigma_B` where :math:`\tau \in \text{D}(\mathcal{R})` is a density operator. :return: The non-signaling value of the extended nonlocal game. """ dim_x, dim_y, alice_out, bob_out, alice_in, bob_in = self.pred_mat.shape constraints = [] # The cvxpy package does not support optimizing over more than # 2-dimensional objects. To overcome this, we use a dictionary to index # between the questions and answers, while the cvxpy variables held at # this positions are `dim_x`-by-`dim_y` cvxpy Variable objects. # Define K(a,b|x,y) variable. k_var = defaultdict(cvxpy.Variable) for a_out in range(alice_out): for b_out in range(bob_out): for x_in in range(alice_in): for y_in in range(bob_in): k_var[a_out, b_out, x_in, y_in] = cvxpy.Variable( (dim_x, dim_y), hermitian=True ) constraints.append(k_var[a_out, b_out, x_in, y_in] >> 0) # Define \sigma_a^x variable. sigma = defaultdict(cvxpy.Variable) for a_out in range(alice_out): for x_in in range(alice_in): sigma[a_out, x_in] = cvxpy.Variable((dim_x, dim_y), hermitian=True) # Define \rho_b^y variable. rho = defaultdict(cvxpy.Variable) for b_out in range(bob_out): for y_in in range(bob_in): rho[b_out, y_in] = cvxpy.Variable((dim_x, dim_y), hermitian=True) # Define \tau density operator variable. tau = cvxpy.Variable((dim_x, dim_y), hermitian=True) p_win = 0 for a_out in range(alice_out): for b_out in range(bob_out): for x_in in range(alice_in): for y_in in range(bob_in): p_win += self.prob_mat[x_in, y_in] * cvxpy.trace( self.pred_mat[:, :, a_out, b_out, x_in, y_in].conj().T
[docs] @ k_var[a_out, b_out, x_in, y_in] ) objective = cvxpy.Maximize(cvxpy.real(p_win)) # The following constraints enforce the so-called non-signaling # constraints. # Enforce that: # \sum_{b \in \Gamma_B} K(a,b|x,y) = \sigma_a^x for x_in in range(alice_in): for y_in in range(bob_in): for a_out in range(alice_out): b_sum = 0 for b_out in range(bob_out): b_sum += k_var[a_out, b_out, x_in, y_in] constraints.append(b_sum == sigma[a_out, x_in]) # Enforce non-signaling constraints on Alice marginal: # \sum_{a \in \Gamma_A} K(a,b|x,y) = \rho_b^y for x_in in range(alice_in): for y_in in range(bob_in): for b_out in range(bob_out): a_sum = 0 for a_out in range(alice_out): a_sum += k_var[a_out, b_out, x_in, y_in] constraints.append(a_sum == rho[b_out, y_in]) # Enforce non-signaling constraints on Bob marginal: # \sum_{a \in \Gamma_A} \sigma_a^x = \tau for x_in in range(alice_in): sig_a_sum = 0 for a_out in range(alice_out): sig_a_sum += sigma[a_out, x_in] constraints.append(sig_a_sum == tau) # Enforce that: # \sum_{b \in \Gamma_B} \rho_b^y = \tau for y_in in range(bob_in): rho_b_sum = 0 for b_out in range(bob_out): rho_b_sum += rho[b_out, y_in] constraints.append(rho_b_sum == tau) # Enforce that tau is a density operator. constraints.append(cvxpy.trace(tau) == 1) constraints.append(tau >> 0) problem = cvxpy.Problem(objective, constraints) ns_val = problem.solve() return ns_val
def quantum_value_lower_bound(self, iters: int = 5, tol: float = 10e-6) -> float: r""" Calculate lower bound on the quantum value of an extended nonlocal game. Test :return: The quantum value of the extended nonlocal game. """ # Get number of inputs and outputs for Bob's measurements. _, _, _, num_outputs_bob, _, num_inputs_bob = self.pred_mat.shape best_lower_bound = float("-inf") for _ in range(iters): # Generate a set of random POVMs for Bob. These measurements serve as a # rough starting point for the alternating projection algorithm. bob_povms = defaultdict(int) for y_ques in range(num_inputs_bob): random_mat = random_unitary(num_outputs_bob) for b_ans in range(num_outputs_bob): random_mat_trans = random_mat[:, b_ans].conj().T.reshape(-1, 1) bob_povms[y_ques, b_ans] = random_mat[:, b_ans] * random_mat_trans # Run the alternating projection algorithm between the two SDPs. it_diff = 1 prev_win = -1 best = float("-inf") while it_diff > tol: # Optimize over Alice's measurement operators while fixing Bob's. # If this is the first iteration, then the previously randomly # generated operators in the outer loop are Bob's. Otherwise, Bob's # operators come from running the next SDP. rho, lower_bound = self.__optimize_alice(bob_povms) bob_povms, lower_bound = self.__optimize_bob(rho) it_diff = lower_bound - prev_win prev_win = lower_bound # As the SDPs keep alternating, check if the winning probability # becomes any higher. If so, replace with new best. best = max(best, lower_bound) best_lower_bound = max(best, best_lower_bound) return best_lower_bound
def __optimize_alice(self, bob_povms) -> tuple[dict, float]: """Fix Bob's measurements and optimize over Alice's measurements.""" # Get number of inputs and outputs. ( dim, _, num_outputs_alice, num_outputs_bob, num_inputs_alice, num_inputs_bob, ) = self.pred_mat.shape # The cvxpy package does not support optimizing over 4-dimensional objects. # To overcome this, we use a dictionary to index between the questions and # answers, while the cvxpy variables held at this positions are # `dim`-by-`dim` cvxpy variables. rho = defaultdict(cvxpy.Variable) for x_ques in range(num_inputs_alice): for a_ans in range(num_outputs_alice): rho[x_ques, a_ans] = cvxpy.Variable( (dim * num_outputs_bob, dim * num_outputs_bob), hermitian=True ) tau = cvxpy.Variable((dim * num_outputs_bob, dim * num_outputs_bob), hermitian=True) win = 0 for x_ques in range(num_inputs_alice): for y_ques in range(num_inputs_bob): for a_ans in range(num_outputs_alice): for b_ans in range(num_outputs_bob): if isinstance(bob_povms[y_ques, b_ans], np.ndarray): win += self.prob_mat[x_ques, y_ques] * cvxpy.trace( ( np.kron( self.pred_mat[:, :, a_ans, b_ans, x_ques, y_ques], bob_povms[y_ques, b_ans], ) ) .conj() .T @ rho[x_ques, a_ans] ) if isinstance( bob_povms[y_ques, b_ans], cvxpy.expressions.variable.Variable, ): win += self.prob_mat[x_ques, y_ques] * cvxpy.trace( ( np.kron( self.pred_mat[:, :, a_ans, b_ans, x_ques, y_ques], bob_povms[y_ques, b_ans].value, ) ) .conj() .T @ rho[x_ques, a_ans] ) objective = cvxpy.Maximize(cvxpy.real(win)) constraints = [] # Sum over "a" for all "x" for Alice's measurements. for x_ques in range(num_inputs_alice): rho_sum_a = 0 for a_ans in range(num_outputs_alice): rho_sum_a += rho[x_ques, a_ans] constraints.append(rho[x_ques, a_ans] >> 0) constraints.append(rho_sum_a == tau) constraints.append(cvxpy.trace(tau) == 1) constraints.append(tau >> 0) problem = cvxpy.Problem(objective, constraints) lower_bound = problem.solve() return rho, lower_bound def __optimize_bob(self, rho) -> tuple[dict, float]: """Fix Alice's measurements and optimize over Bob's measurements.""" # Get number of inputs and outputs. ( dim, _, num_outputs_alice, num_outputs_bob, num_inputs_alice, num_inputs_bob, ) = self.pred_mat.shape # The cvxpy package does not support optimizing over 4-dimensional objects. # To overcome this, we use a dictionary to index between the questions and # answers, while the cvxpy variables held at this positions are # `dim`-by-`dim` cvxpy variables. bob_povms = defaultdict(cvxpy.Variable) for y_ques in range(num_inputs_bob): for b_ans in range(num_outputs_bob): bob_povms[y_ques, b_ans] = cvxpy.Variable((dim, dim), hermitian=True) win = 0 for x_ques in range(num_inputs_alice): for y_ques in range(num_inputs_bob): for a_ans in range(num_outputs_alice): for b_ans in range(num_outputs_bob): win += self.prob_mat[x_ques, y_ques] * cvxpy.trace( ( cvxpy.kron( self.pred_mat[:, :, a_ans, b_ans, x_ques, y_ques], bob_povms[y_ques, b_ans], ) )
[docs] @ rho[x_ques, a_ans].value ) objective = cvxpy.Maximize(cvxpy.real(win)) constraints = [] # Sum over "b" for all "y" for Bob's measurements. for y_ques in range(num_inputs_bob): bob_sum_b = 0 for b_ans in range(num_outputs_bob): bob_sum_b += bob_povms[y_ques, b_ans] constraints.append(bob_povms[y_ques, b_ans] >> 0) constraints.append(bob_sum_b == np.identity(num_outputs_bob)) problem = cvxpy.Problem(objective, constraints) lower_bound = problem.solve() return bob_povms, lower_bound def commuting_measurement_value_upper_bound(self, k: int | str = 1) -> float: """ Compute an upper bound on the commuting measurement value of an extended nonlocal game. This function calculates an upper bound on the commuting measurement value by using k-levels of the NPA hierarchy [NPA]_. The NPA hierarchy is a uniform family of semidefinite programs that converges to the commuting measurement value of any extended nonlocal game. You can determine the level of the hierarchy by a positive integer or a string of a form like '1+ab+aab', which indicates that an intermediate level of the hierarchy should be used, where this example uses all products of one measurement, all products of one Alice and one Bob measurement, and all products of two Alice and one Bob measurements. References ========== .. [NPA] Miguel Navascues, Stefano Pironio, Antonio Acin, "A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations." https://arxiv.org/abs/0803.4290 :param k: The level of the NPA hierarchy to use (default=1). :return: The upper bound on the commuting strategy value of an extended nonlocal game. """ referee_dim, _, alice_out, bob_out, alice_in, bob_in = self.pred_mat.shape mat = defaultdict(cvxpy.Variable) for x_in in range(alice_in): for y_in in range(bob_in): mat[x_in, y_in] = cvxpy.Variable( (alice_out * referee_dim, bob_out * referee_dim), name=f"K(a, b | {x_in}, {y_in})", hermitian=True, ) p_win = cvxpy.Constant(0) for a_out in range(alice_out): for b_out in range(bob_out): for x_in in range(alice_in): for y_in in range(bob_in): p_win += self.prob_mat[x_in, y_in] * cvxpy.trace( self.pred_mat[:, :, a_out, b_out, x_in, y_in].conj().T @ mat[x_in, y_in][ a_out * referee_dim : (a_out + 1) * referee_dim, b_out * referee_dim : (b_out + 1) * referee_dim, ] ) npa = npa_constraints(mat, k, referee_dim) objective = cvxpy.Maximize(cvxpy.real(p_win)) problem = cvxpy.Problem(objective, npa) cs_val = problem.solve() return cs_val