:py:mod:`nonlocal_games.nonlocal_game` ====================================== .. py:module:: nonlocal_games.nonlocal_game .. autoapi-nested-parse:: Two-player nonlocal game. Module Contents --------------- Classes ~~~~~~~ .. autoapisummary:: nonlocal_games.nonlocal_game.NonlocalGame .. py:class:: NonlocalGame(prob_mat, pred_mat, reps = 1) Create two-player nonlocal game object. *Nonlocal games* are a mathematical framework that abstractly models a physical system. This game is played between two players, Alice and Bob, who are not allowed to communicate with each other once the game has started and who play cooperative against an adversary referred to as the referee. The nonlocal game framework was originally introduced in :cite:`Cleve_2010_Consequences`. A tutorial is available in the documentation. For more info, see :ref:`ref-label-nl-games-tutorial`. .. rubric:: References .. bibliography:: :filter: docname in docnames .. py:method:: from_bcs_game(constraints, reps = 1) :classmethod: Construct nonlocal game object from a binary constraint system game. :raises ValueError: At least one constraint needs to be supplied. :param constraints: A list of m matrices corresponding to the `m` constraints in the BCS game. Each constraint matrix is an `n`-dimensional matrix of shape `2 x 2 x ... x 2`. The `(i, j, k, ...)`-th entry is 1 if and only if the contraint is satisfied given the values v_1 = i, v_2 = j, v_3 = k, ..., and otherwise 0. :param reps: Number of parallel repetitions to perform. Default is 1. :return: An instance of a nonlocal game object. .. py:method:: classical_value() Compute the classical value of the nonlocal game. This function has been adapted from the QETLAB package. :return: A value between [0, 1] representing the classical value. .. py:method:: quantum_value_lower_bound(dim = 2, iters = 5, tol = 1e-05) Compute a lower bound on the quantum value of a nonlocal game :cite:`Liang_2007_Bounds`. Calculates a lower bound on the maximum value that the specified nonlocal game can take on in quantum mechanical settings where Alice and Bob each have access to `dim`-dimensional quantum system. This function works by starting with a randomly-generated POVM for Bob, and then optimizing Alice's POVM and the shared entangled state. Then Alice's POVM and the entangled state are fixed, and Bob's POVM is optimized. And so on, back and forth between Alice and Bob until convergence is reached. Note that the algorithm is not guaranteed to obtain the optimal local bound and can get stuck in local minimum values. The alleviate this, the `iter` parameter allows one to run the algorithm some pre-specified number of times and keep the highest value obtained. The algorithm is based on the alternating projections algorithm as it can be applied to Bell inequalities as shown in :cite:`Liang_2007_Bounds`. The alternating projection algorithm has also been referred to as the "see-saw" algorithm as it goes back and forth between the following two semidefinite programs: .. math:: \begin{equation} \begin{aligned} \textbf{SDP-1:} \quad & \\ \text{maximize:} \quad & \sum_{(x,y \in \Sigma)} \pi(x,y) \sum_{(a,b) \in \Gamma} V(a,b|x,y) \langle B_b^y, A_a^x \rangle \\ \text{subject to:} \quad & \sum_{a \in \Gamma_{\mathsf{A}}}= \tau, \qquad \qquad \forall x \in \Sigma_{\mathsf{A}}, \\ \quad & A_a^x \in \text{Pos}(\mathcal{A}), \qquad \forall x \in \Sigma_{\mathsf{A}}, \ \forall a \in \Gamma_{\mathsf{A}}, \\ & \tau \in \text{D}(\mathcal{A}). \end{aligned} \end{equation} .. math:: \begin{equation} \begin{aligned} \textbf{SDP-2:} \quad & \\ \text{maximize:} \quad & \sum_{(x,y \in \Sigma)} \pi(x,y) \sum_{(a,b) \in \Gamma} V(a,b|x,y) \langle B_b^y, A_a^x \rangle \\ \text{subject to:} \quad & \sum_{b \in \Gamma_{\mathsf{B}}}= \mathbb{I}, \qquad \qquad \forall y \in \Sigma_{\mathsf{B}}, \\ \quad & B_b^y \in \text{Pos}(\mathcal{B}), \qquad \forall y \in \Sigma_{\mathsf{B}}, \ \forall b \in \Gamma_{\mathsf{B}}. \end{aligned} \end{equation} .. rubric:: Examples The CHSH game The CHSH game is a two-player nonlocal game with the following probability distribution and question and answer sets. .. math:: \begin{equation} \begin{aligned} \pi(x,y) = \frac{1}{4}, \qquad (x,y) \in \Sigma_A \times \Sigma_B, \qquad \text{and} \qquad (a, b) \in \Gamma_A \times \Gamma_B, \end{aligned} \end{equation} where .. math:: \begin{equation} \Sigma_A = \{0, 1\}, \quad \Sigma_B = \{0, 1\}, \quad \Gamma_A = \{0,1\}, \quad \text{and} \quad \Gamma_B = \{0, 1\}. \end{equation} Alice and Bob win the CHSH game if and only if the following equation is satisfied. .. math:: \begin{equation} a \oplus b = x \land y. \end{equation} Recall that :math:`\oplus` refers to the XOR operation. The optimal quantum value of CHSH is :math:`\cos(\pi/8)^2 \approx 0.8536` where the optimal classical value is :math:`3/4`. >>> import numpy as np >>> from toqito.nonlocal_games.nonlocal_game import NonlocalGame >>> >>> dim = 2 >>> num_alice_inputs, num_alice_outputs = 2, 2 >>> num_bob_inputs, num_bob_outputs = 2, 2 >>> prob_mat = np.array([[1 / 4, 1 / 4], [1 / 4, 1 / 4]]) >>> pred_mat = np.zeros((num_alice_outputs, num_bob_outputs, num_alice_inputs, num_bob_inputs)) >>> >>> for a_alice in range(num_alice_outputs): ... for b_bob in range(num_bob_outputs): ... for x_alice in range(num_alice_inputs): ... for y_bob in range(num_bob_inputs): ... if np.mod(a_alice + b_bob + x_alice * y_bob, dim) == 0: ... pred_mat[a_alice, b_bob, x_alice, y_bob] = 1 >>> >>> chsh = NonlocalGame(prob_mat, pred_mat) >>> chsh.quantum_value_lower_bound() # doctest: +SKIP 0.85 .. rubric:: References .. bibliography:: :filter: docname in docnames :param dim: The dimension of the quantum system that Alice and Bob have access to (default = 2). :param iters: The number of times to run the alternating projection algorithm. :param tol: The tolerance before quitting out of the alternating projection semidefinite program. :return: The lower bound on the quantum value of a nonlocal game. .. py:method:: __optimize_alice(dim, bob_povms) Fix Bob's measurements and optimize over Alice's measurements. .. py:method:: __optimize_bob(dim, alice_povms) Fix Alice's measurements and optimize over Bob's measurements. .. py:method:: nonsignaling_value() Compute the non-signaling value of the nonlocal game. :return: A value between [0, 1] representing the non-signaling value. .. py:method:: commuting_measurement_value_upper_bound(k = 1) Compute an upper bound on the commuting measurement value of the nonlocal game. This function calculates an upper bound on the commuting measurement value by using k-levels of the NPA hierarchy :cite:`Navascues_2008_AConvergent`. The NPA hierarchy is a uniform family of semidefinite programs that converges to the commuting measurement value of any 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. .. rubric:: References .. bibliography:: :filter: docname in docnames :param k: The level of the NPA hierarchy to use (default=1). :return: The upper bound on the commuting strategy value of a nonlocal game.