Calculate optimal winning probabilities for hedging scenarios.
Calculate the maximal and minimal winning probabilities for quantum
hedging to occur in certain two-party scenarios [@Arunachalam_2017_QuantumHedging, Molina_2012_Hedging].
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 1, the hedging value of the above scenario is
\(\cos(\pi/8)^2 \approx 0.8536\)
import numpy as np
from toqito.states import basis
from numpy import kron, cos, sin, pi, sqrt, isclose
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
print(np.around(molina_watrous.max_prob_outcome_a_primal(), decimals=2))
This example demonstrates strong duality with matching primal and dual values, as can be seen below:
import numpy as np
from toqito.states import basis
from numpy import kron, cos, sin, pi, sqrt, isclose
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)
print(np.around(molina_watrous.max_prob_outcome_a_dual(), decimals=2))
and
import numpy as np
from toqito.states import basis
from numpy import kron, cos, sin, pi, sqrt, isclose
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)
print(np.around(molina_watrous.min_prob_outcome_a_primal(), decimals=2))
import numpy as np
from toqito.states import basis
from numpy import kron, cos, sin, pi, sqrt, isclose
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)
print(np.around(molina_watrous.min_prob_outcome_a_dual(), decimals=2))
References
1 Molina, Abel and Watrous, John. Hedging bets with correlated quantum strategies. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. vol. 468(2145). (2012). link.
Initialize the variables for semidefinite program.
Parameters:
-
q_a
(ndarray)
–
-
num_reps
(int)
–
The number of parallel repetitions.
Source code in toqito/nonlocal_games/quantum_hedging.py
| def __init__(self, q_a: np.ndarray, num_reps: int) -> None:
"""Initialize the variables for semidefinite program.
Args:
q_a: The fixed SDP variable.
num_reps: The number of parallel repetitions.
"""
self._q_a = q_a
self._num_reps = num_reps
self._sys = list(range(0, 2 * self._num_reps - 1, 2))
self._dim = 2 * np.ones((1, 2 * self._num_reps)).astype(int).flatten()
self._dim = self._dim.tolist()
# For the dual problem, the following unitary operator is used to
# permute the subsystems of Alice and Bob which is defined by the
# action:
# π(y1 ⊗ y2 ⊗ x1 ⊗ x2) = y1 ⊗ x1 ⊗ y2 ⊗ x2
# for all y1 ∈ Y1, y2 ∈ Y2, x1 ∈ X1, x2 ∈ X2.).
l_1 = list(range(self._num_reps))
l_2 = list(range(self._num_reps, self._num_reps**2))
if self._num_reps == 1:
self._pperm = np.array([1])
else:
perm = [*sum(zip(l_1, l_2), ())]
self._pperm = permutation_operator(2, perm)
|
max_prob_outcome_a_primal() -> float
Compute the maximal probability for calculating outcome "a".
The primal problem for the maximal probability of "a" is given as:
\[
\begin{equation}
\begin{aligned}
\text{maximize:} \quad & \langle Q_{a_1} \otimes \ldots
\otimes Q_{a_n}, X \rangle \\
\text{subject to:} \quad & \text{Tr}_{\mathcal{Y}_1 \otimes
\ldots \otimes \mathcal{Y}_n}(X) =
I_{\mathcal{X}_1 \otimes \ldots
\otimes \mathcal{X}_n},\\
& X \in \text{Pos}(\mathcal{Y}_1
\otimes \mathcal{X}_1 \otimes \ldots
\otimes \mathcal{Y}_n \otimes
\mathcal{X}_n)
\end{aligned}
\end{equation}
\]
Returns:
-
float
–
The optimal maximal probability for obtaining outcome "a".
Source code in toqito/nonlocal_games/quantum_hedging.py
| def max_prob_outcome_a_primal(self) -> float:
r"""Compute the maximal probability for calculating outcome "a".
The primal problem for the maximal probability of "a" is given as:
\[
\begin{equation}
\begin{aligned}
\text{maximize:} \quad & \langle Q_{a_1} \otimes \ldots
\otimes Q_{a_n}, X \rangle \\
\text{subject to:} \quad & \text{Tr}_{\mathcal{Y}_1 \otimes
\ldots \otimes \mathcal{Y}_n}(X) =
I_{\mathcal{X}_1 \otimes \ldots
\otimes \mathcal{X}_n},\\
& X \in \text{Pos}(\mathcal{Y}_1
\otimes \mathcal{X}_1 \otimes \ldots
\otimes \mathcal{Y}_n \otimes
\mathcal{X}_n)
\end{aligned}
\end{equation}
\]
Returns:
The optimal maximal probability for obtaining outcome "a".
"""
x_var = cvxpy.Variable((4**self._num_reps, 4**self._num_reps), hermitian=True)
objective = cvxpy.Maximize(cvxpy.real(cvxpy.trace(self._q_a.conj().T @ x_var)))
constraints = [partial_trace(x_var, self._sys, self._dim) == np.identity(2**self._num_reps), x_var >> 0]
problem = cvxpy.Problem(objective, constraints)
return problem.solve(verbose=False)
|
max_prob_outcome_a_dual() -> float
Compute the maximal probability for calculating outcome "a".
The dual problem for the maximal probability of "a" is given as:
\[
\begin{equation}
\begin{aligned}
\text{minimize:} \quad & \text{Tr}(Y) \\
\text{subject to:} \quad & \pi \left(I_{\mathcal{Y}_1
\otimes \ldots \otimes \mathcal{Y}_n} \otimes Y \right)
\pi^* \geq Q_{a_1} \otimes \ldots \otimes Q_{a_n}, \\
& Y \in \text{Herm} \left(\mathcal{X} \otimes \ldots \otimes
\mathcal{X}_n \right)
\end{aligned}
\end{equation}
\]
Returns:
-
float
–
The optimal maximal probability for obtaining outcome "a".
Source code in toqito/nonlocal_games/quantum_hedging.py
| def max_prob_outcome_a_dual(self) -> float:
r"""Compute the maximal probability for calculating outcome "a".
The dual problem for the maximal probability of "a" is given as:
\[
\begin{equation}
\begin{aligned}
\text{minimize:} \quad & \text{Tr}(Y) \\
\text{subject to:} \quad & \pi \left(I_{\mathcal{Y}_1
\otimes \ldots \otimes \mathcal{Y}_n} \otimes Y \right)
\pi^* \geq Q_{a_1} \otimes \ldots \otimes Q_{a_n}, \\
& Y \in \text{Herm} \left(\mathcal{X} \otimes \ldots \otimes
\mathcal{X}_n \right)
\end{aligned}
\end{equation}
\]
Returns:
The optimal maximal probability for obtaining outcome "a".
"""
y_var = cvxpy.Variable((2**self._num_reps, 2**self._num_reps), hermitian=True)
objective = cvxpy.Minimize(cvxpy.trace(cvxpy.real(y_var)))
kron_var = cvxpy.kron(np.eye(2**self._num_reps), y_var)
if self._num_reps == 1:
u_var = cvxpy.multiply(cvxpy.multiply(self._pperm, kron_var), self._pperm.conj().T)
constraints = [cvxpy.real(u_var) >> self._q_a]
else:
constraints = [cvxpy.real(self._pperm @ kron_var @ self._pperm.conj().T) >> self._q_a]
problem = cvxpy.Problem(objective, constraints)
return problem.solve(verbose=False)
|
min_prob_outcome_a_primal() -> float
Compute the minimal probability for calculating outcome "a".
The primal problem for the minimal probability of "a" is given as:
\[
\begin{equation}
\begin{aligned}
\text{minimize:} \quad & \langle Q_{a_1} \otimes \ldots
\otimes Q_{a_n}, X \rangle \\
\text{subject to:} \quad & \text{Tr}_{\mathcal{Y}_1 \otimes
\ldots \otimes \mathcal{Y}_n}(X) =
I_{\mathcal{X}_1 \otimes \ldots
\otimes \mathcal{X}_n},\\
& X \in \text{Pos}(\mathcal{Y}_1
\otimes \mathcal{X}_1 \otimes \ldots
\otimes \mathcal{Y}_n \otimes
\mathcal{X}_n)
\end{aligned}
\end{equation}
\]
Returns:
-
float
–
The optimal minimal probability for obtaining outcome "a".
Source code in toqito/nonlocal_games/quantum_hedging.py
| def min_prob_outcome_a_primal(self) -> float:
r"""Compute the minimal probability for calculating outcome "a".
The primal problem for the minimal probability of "a" is given as:
\[
\begin{equation}
\begin{aligned}
\text{minimize:} \quad & \langle Q_{a_1} \otimes \ldots
\otimes Q_{a_n}, X \rangle \\
\text{subject to:} \quad & \text{Tr}_{\mathcal{Y}_1 \otimes
\ldots \otimes \mathcal{Y}_n}(X) =
I_{\mathcal{X}_1 \otimes \ldots
\otimes \mathcal{X}_n},\\
& X \in \text{Pos}(\mathcal{Y}_1
\otimes \mathcal{X}_1 \otimes \ldots
\otimes \mathcal{Y}_n \otimes
\mathcal{X}_n)
\end{aligned}
\end{equation}
\]
Returns:
The optimal minimal probability for obtaining outcome "a".
"""
x_var = cvxpy.Variable((4**self._num_reps, 4**self._num_reps), hermitian=True)
objective = cvxpy.Minimize(cvxpy.real(cvxpy.trace(self._q_a.conj().T @ x_var)))
constraints = [partial_trace(x_var, self._sys, self._dim) == np.identity(2**self._num_reps), x_var >> 0]
problem = cvxpy.Problem(objective, constraints)
return problem.solve(verbose=False)
|
min_prob_outcome_a_dual() -> float
Compute the minimal probability for calculating outcome "a".
The dual problem for the minimal probability of "a" is given as:
\[
\begin{equation}
\begin{aligned}
\text{maximize:} \quad & \text{Tr}(Y) \\
\text{subject to:} \quad & \pi \left(I_{\mathcal{Y}_1
\otimes \ldots \otimes \mathcal{Y}_n} \otimes Y \right)
\pi^* \leq Q_{a_1} \otimes \ldots \otimes Q_{a_n}, \\
& Y \in \text{Herm} \left(\mathcal{X} \otimes \ldots \otimes
\mathcal{X}_n \right)
\end{aligned}
\end{equation}
\]
Returns:
-
float
–
The optimal minimal probability for obtaining outcome "a".
Source code in toqito/nonlocal_games/quantum_hedging.py
| def min_prob_outcome_a_dual(self) -> float:
r"""Compute the minimal probability for calculating outcome "a".
The dual problem for the minimal probability of "a" is given as:
\[
\begin{equation}
\begin{aligned}
\text{maximize:} \quad & \text{Tr}(Y) \\
\text{subject to:} \quad & \pi \left(I_{\mathcal{Y}_1
\otimes \ldots \otimes \mathcal{Y}_n} \otimes Y \right)
\pi^* \leq Q_{a_1} \otimes \ldots \otimes Q_{a_n}, \\
& Y \in \text{Herm} \left(\mathcal{X} \otimes \ldots \otimes
\mathcal{X}_n \right)
\end{aligned}
\end{equation}
\]
Returns:
The optimal minimal probability for obtaining outcome "a".
"""
y_var = cvxpy.Variable((2**self._num_reps, 2**self._num_reps), hermitian=True)
objective = cvxpy.Maximize(cvxpy.trace(cvxpy.real(y_var)))
kron_var = cvxpy.kron(np.eye(2**self._num_reps), y_var)
if self._num_reps == 1:
u_var = cvxpy.multiply(cvxpy.multiply(self._pperm, kron_var), self._pperm.conj().T)
constraints = [cvxpy.real(u_var) << self._q_a]
else:
constraints = [cvxpy.real(self._pperm @ kron_var @ self._pperm.conj().T) << self._q_a]
problem = cvxpy.Problem(objective, constraints)
return problem.solve(verbose=False)
|