toqito.state_opt.ppt_distinguishability

Calculates the probability of PPT state distinguishability when done optimally.

Module Contents

toqito.state_opt.ppt_distinguishability.ppt_distinguishability(vectors, subsystems, dimensions, probs=None, strategy='min_error', solver='cvxopt', primal_dual='dual')[source]

Compute probability of optimally distinguishing a state via PPT measurements [@Cosentino_2013_PPT].

Implements the semidefinite program (SDP) whose optimal value is equal to the maximum probability of perfectly distinguishing orthogonal maximally entangled states using any PPT measurement; a measurement whose operators are positive under partial transpose. This SDP was explicitly provided in [@Cosentino_2013_PPT].

One can specify the distinguishability method using the dist_method argument.

For dist_method = “min_error”, this is the default method that yields the probability of distinguishing quantum states via PPT measurements that minimize the probability of error.

For dist_method = “unambig”, Alice and Bob never provide an incorrect answer, although it is possible that their answer is inconclusive.

For more background, see the state_distinguishability example in the quantum states gallery.

!!! Note

This function supports both pure states (vectors) and mixed states (density matrices). The PPT constraints are applied to the measurement operators to restrict the class of allowed measurements.

Examples

Consider the following Bell states:

[
begin{equation}

begin{aligned} |psi_0 rangle = frac{|00rangle + |11rangle}{sqrt{2}}, &quad |psi_1 rangle = frac{|01rangle + |10rangle}{sqrt{2}}, \ |psi_2 rangle = frac{|01rangle - |10rangle}{sqrt{2}}, &quad |psi_3 rangle = frac{|00rangle - |11rangle}{sqrt{2}}. end{aligned}

end{equation}

]

It was illustrated in [@Yu_2012_Four] that for the following set of states

[
begin{equation}

begin{aligned} rho_1^{(2)} &= |psi_0 rangle | psi_0 rangle langle psi_0 | langle psi_0 |, quad rho_2^{(2)} &= |psi_1 rangle | psi_3 rangle langle psi_1 | langle psi_3 |, \ rho_3^{(2)} &= |psi_2 rangle | psi_3 rangle langle psi_2 | langle psi_3 |, quad rho_4^{(2)} &= |psi_3 rangle | psi_3 rangle langle psi_3 | langle psi_3 |, \ end{aligned}

end{equation}

]

that the optimal probability of distinguishing via a PPT measurement should yield (7/8 approx 0.875) as was proved in [@Yu_2012_Four].

```python exec=”1” source=”above” import numpy as np from toqito.states import bell from toqito.state_opt import ppt_distinguishability

# Bell vectors: psi_0 = bell(0) psi_1 = bell(2) psi_2 = bell(3) psi_3 = bell(1)

# YDY vectors from [@Yu_2012_Four]. x_1 = np.kron(psi_0, psi_0) x_2 = np.kron(psi_1, psi_3) x_3 = np.kron(psi_2, psi_3) x_4 = np.kron(psi_3, psi_3)

# YDY density matrices. rho_1 = x_1 @ x_1.conj().T rho_2 = x_2 @ x_2.conj().T rho_3 = x_3 @ x_3.conj().T rho_4 = x_4 @ x_4.conj().T

states = [rho_1, rho_2, rho_3, rho_4] probs = [1 / 4, 1 / 4, 1 / 4, 1 / 4]

opt_val, _ = ppt_distinguishability(vectors=states, probs=probs, dimensions=[2, 2, 2, 2], subsystems=[0, 2])

print(f”Optimal value: {opt_val:.3f}”) ```

Parameters:

  • vectors (list[numpy.ndarray]) – A list of states provided as either matrices or vectors.

  • probs (list[float] | None) – Respective list of probabilities each state is selected.

  • subsystems (list[int]) – A list of integers that correspond to the complex Euclidean space dimensions.

  • dimensions (list[int]) – A list of integers that correspond to the dimensions of the subsystems.

  • strategy (str) – The method of distinguishing states.

  • solver (str) – The SDP solver to use.

  • primal_dual (str) – Option for the optimization problem.

Returns:

The optimal probability with which the states can be distinguished via PPT measurements.

Return type:

float