nonlocal_games.quantum_hedging

Semidefinite programs for obtaining values of quantum hedging scenarios.

Module Contents

Classes

QuantumHedging

Calculate optimal winning probabilities for hedging scenarios.

class nonlocal_games.quantum_hedging.QuantumHedging(q_a, num_reps)

Calculate optimal winning probabilities for hedging scenarios.

Calculate the maximal and minimal winning probabilities for quantum hedging to occur in certain two-party scenarios [1, 2].

Examples

This example illustrates the initial example of perfect hedging when Alice and Bob play two repetitions of the game where Alice prepares the maximally entangled state:

\[u = \frac{1}{\sqrt{2}}|00\rangle + \frac{1}{\sqrt{2}}|11\rangle,\]

and Alice applies the measurement operator defined by vector

\[v = \cos(\pi/8)|00\rangle + \sin(\pi/8)|11\rangle.\]

As was illustrated in [2], the hedging value of the above scenario is \(\cos(\pi/8)^2 \approx 0.8536\)

>>> from numpy import kron, cos, sin, pi, sqrt, isclose
>>> from toqito.states import basis
>>> from toqito.nonlocal_games.quantum_hedging import QuantumHedging
>>>
>>> e_0, e_1 = basis(2, 0), basis(2, 1)
>>> e_00, e_01 = kron(e_0, e_0), kron(e_0, e_1)
>>> e_10, e_11 = kron(e_1, e_0), kron(e_1, e_1)
>>>
>>> alpha = 1 / sqrt(2)
>>> theta = pi / 8
>>> w_var = alpha * cos(theta) * e_00 + sqrt(1 - alpha ** 2) * sin(theta) * e_11
>>>
>>> l_1 = -alpha * sin(theta) * e_00 + sqrt(1 - alpha ** 2) * cos(theta) * e_11
>>> l_2 = alpha * sin(theta) * e_10
>>> l_3 = sqrt(1 - alpha ** 2) * cos(theta) * e_01
>>>
>>> q_1 = w_var * w_var.conj().T
>>> q_0 = l_1 * l_1.conj().T + l_2 * l_2.conj().T + l_3 * l_3.conj().T
>>> molina_watrous = QuantumHedging(q_0, 1)
>>>
>>> # cos(pi/8)**2 \approx 0.8536
>>> '%.2f' % molina_watrous.max_prob_outcome_a_primal()
'0.85'

Note

You do not need to use ‘%.2f’ % when you use this function.

We use this to format our output such that doctest compares the calculated output to the expected output upto two decimal points only. The accuracy of the solvers can calculate the float output to a certain amount of precision such that the value deviates after a few digits of accuracy.

References

[1]

Srinivasan Arunachalam, Abel Molina, and Vincent Russo. Quantum hedging in two-round prover-verifier interactions. 2017. arXiv:1310.7954.

[2] (1,2)

Abel Molina and John Watrous. Hedging bets with correlated quantum strategies. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 468(2145):2614–2629, Apr 2012. URL: https://arxiv.org/abs/1104.1140.

Parameters:

• q_a (numpy.ndarray)

• num_reps (int)

max_prob_outcome_a_primal()

Compute the maximal probability for calculating outcome “a”.

The primal problem for the maximal probability of “a” is given as:

\[\begin{split}\begin{equation} \begin{aligned} \text{maximize:} \quad & \langle Q_{a_1} \otimes \ldots \otimes Q_{a_n}, X \rangle \\ \text{subject to:} \quad & \text{Tr}_{\mathcal{Y}_1 \otimes \ldots \otimes \mathcal{Y}_n}(X) = I_{\mathcal{X}_1 \otimes \ldots \otimes \mathcal{X}_n},\\ & X \in \text{Pos}(\mathcal{Y}_1 \otimes \mathcal{X}_1 \otimes \ldots \otimes \mathcal{Y}_n \otimes \mathcal{X}_n) \end{aligned} \end{equation}\end{split}\]

Returns:

The optimal maximal probability for obtaining outcome “a”.

Return type:

float

max_prob_outcome_a_dual()

Compute the maximal probability for calculating outcome “a”.

The dual problem for the maximal probability of “a” is given as:

\[\begin{split}\begin{equation} \begin{aligned} \text{minimize:} \quad & \text{Tr}(Y) \\ \text{subject to:} \quad & \pi \left(I_{\mathcal{Y}_1 \otimes \ldots \otimes \mathcal{Y}_n} \otimes Y \right) \pi^* \geq Q_{a_1} \otimes \ldots \otimes Q_{a_n}, \\ & Y \in \text{Herm} \left(\mathcal{X} \otimes \ldots \otimes \mathcal{X}_n \right) \end{aligned} \end{equation}\end{split}\]

Returns:

The optimal maximal probability for obtaining outcome “a”.

Return type:

float

min_prob_outcome_a_primal()

Compute the minimal probability for calculating outcome “a”.

The primal problem for the minimal probability of “a” is given as:

\[\begin{split}\begin{equation} \begin{aligned} \text{minimize:} \quad & \langle Q_{a_1} \otimes \ldots \otimes Q_{a_n}, X \rangle \\ \text{subject to:} \quad & \text{Tr}_{\mathcal{Y}_1 \otimes \ldots \otimes \mathcal{Y}_n}(X) = I_{\mathcal{X}_1 \otimes \ldots \otimes \mathcal{X}_n},\\ & X \in \text{Pos}(\mathcal{Y}_1 \otimes \mathcal{X}_1 \otimes \ldots \otimes \mathcal{Y}_n \otimes \mathcal{X}_n) \end{aligned} \end{equation}\end{split}\]

Returns:

The optimal minimal probability for obtaining outcome “a”.

Return type:

float

min_prob_outcome_a_dual()

Compute the minimal probability for calculating outcome “a”.

The dual problem for the minimal probability of “a” is given as:

\[\begin{split}\begin{equation} \begin{aligned} \text{maximize:} \quad & \text{Tr}(Y) \\ \text{subject to:} \quad & \pi \left(I_{\mathcal{Y}_1 \otimes \ldots \otimes \mathcal{Y}_n} \otimes Y \right) \pi^* \leq Q_{a_1} \otimes \ldots \otimes Q_{a_n}, \\ & Y \in \text{Herm} \left(\mathcal{X} \otimes \ldots \otimes \mathcal{X}_n \right) \end{aligned} \end{equation}\end{split}\]

Returns:

The optimal minimal probability for obtaining outcome “a”.

Return type:

float