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nonlocal_games.extended_nonlocal_game¶
Two-player extended nonlocal game.
Module Contents¶
Classes¶
Create two-player extended nonlocal game object. |
- class nonlocal_games.extended_nonlocal_game.ExtendedNonlocalGame(prob_mat, pred_mat, reps=1)¶
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 [2] and more information on these games can be found in [3].
An example demonstration is available as a tutorial in the documentation. Go to Example: The BB84 extended nonlocal game.
References
[1]Miguel Navascués, Stefano Pironio, and Antonio Acín. A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations. New Journal of Physics, 10(7):073013, Jul 2008. URL: http://dx.doi.org/10.1088/1367-2630/10/7/073013, doi:10.1088/1367-2630/10/7/073013.
[2]Nathaniel Johnston, Rajat Mittal, Vincent Russo, and John Watrous. Extended non-local games and monogamy-of-entanglement games. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472(2189):20160003, May 2016. URL: https://arxiv.org/abs/1510.02083.
[3]Vincent Russo. Extended nonlocal games. 2017. arXiv:1704.07375.
- Parameters:
prob_mat (numpy.ndarray)
pred_mat (numpy.ndarray)
reps (int)
- unentangled_value()¶
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:
\[\omega(G) = \max_{f, g} \lVert \sum_{(x,y) \in \Sigma_A \times \Sigma_B} \pi(x,y) V(f(x), g(y)|x, y) \rVert\]where the maximum is over all functions \(f : \Sigma_A \rightarrow \Gamma_A\) and \(g : \Sigma_B \rightarrow \Gamma_B\).
- Returns:
The unentangled value of the extended nonlocal game.
- Return type:
float
- nonsignaling_value()¶
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
\[K : \Gamma_A \times \Gamma_B \times \Sigma_A \times \Sigma_B \rightarrow \text{Pos}(\mathcal{R})\]such that
\[\sum_{a \in \Gamma_A} K(a,b|x,y) = \rho_b^y \quad \text{and} \quad \sum_{b \in \Gamma_B} K(a,b|x,y) = \sigma_a^x,\]for all \(x \in \Sigma_A\) and \(y \in \Sigma_B\) where \(\{\rho_b^y : y \in \Sigma_A, \ b \in \Gamma_B\}\) and \(\{\sigma_a^x : x \in \Sigma_A, \ a \in \Gamma_B\}\) are collections of operators satisfying
\[\sum_{a \in \Gamma_A} \rho_b^y = \tau = \sum_{b \in \Gamma_B} \sigma_a^x,\]for every choice of \(x \in \Sigma_A\) and \(y \in \Sigma_B\) where \(\tau \in \text{D}(\mathcal{R})\) is a density operator.
- Returns:
The non-signaling value of the extended nonlocal game.
- Return type:
float
- quantum_value_lower_bound(iters=5, tol=1e-05)¶
Calculate lower bound on the quantum value of an extended nonlocal game.
Test
- Returns:
The quantum value of the extended nonlocal game.
- Parameters:
iters (int)
tol (float)
- Return type:
float
- __optimize_alice(bob_povms)¶
Fix Bob’s measurements and optimize over Alice’s measurements.
- Return type:
tuple[dict, float]
- __optimize_bob(rho)¶
Fix Alice’s measurements and optimize over Bob’s measurements.
- Return type:
tuple[dict, float]
- commuting_measurement_value_upper_bound(k=1)¶
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 [1]. 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
[1]Miguel Navascués, Stefano Pironio, and Antonio Acín. A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations. New Journal of Physics, 10(7):073013, Jul 2008. URL: http://dx.doi.org/10.1088/1367-2630/10/7/073013, doi:10.1088/1367-2630/10/7/073013.
[2]Nathaniel Johnston, Rajat Mittal, Vincent Russo, and John Watrous. Extended non-local games and monogamy-of-entanglement games. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472(2189):20160003, May 2016. URL: https://arxiv.org/abs/1510.02083.
[3]Vincent Russo. Extended nonlocal games. 2017. arXiv:1704.07375.
- Parameters:
k (int | str) – The level of the NPA hierarchy to use (default=1).
- Returns:
The upper bound on the commuting strategy value of an extended nonlocal game.
- Return type:
float