toqito.state_opt.ppt_distinguishability

toqito.state_opt.ppt_distinguishability(states, probs=None, dist_method='min-error', strategy=False)[source]

Compute probability of optimally distinguishing a state via PPT measurements [COS13].

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 [COS13].

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 = "unambiguous", Alice and Bob never provide an incorrect answer, although it is possible that their answer is inconclusive.

Examples

Consider the following Bell states:

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

It was illustrated in [YDY12] that for the following set of states

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

that the optimal probability of distinguishing via a PPT measurement should yield \(7/8 \approx 0.875\) as was proved in [YDY12].

>>> 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 [YDY12]_.
>>> 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]
>>> ppt_distinguishability(states, probs)
0.875

References

[COS13] (1,2)

Cosentino, Alessandro. “Positive-partial-transpose-indistinguishable states via semidefinite programming.” Physical Review A 87.1 (2013): 012321. https://arxiv.org/abs/1205.1031

[YDY12] (1,2)

Yu, Nengkun, Runyao Duan, and Mingsheng Ying. “Four locally indistinguishable ququad-ququad orthogonal maximally entangled states.” Physical review letters 109.2 (2012): 020506. https://arxiv.org/abs/1107.3224

Parameters:

  • states – A list of states provided as either matrices or vectors.

  • dist_method – The method of distinguishing states.

  • probs – Respective list of probabilities each state is selected.

  • dist_method – Method of distinguishing to use.

  • strategy – Returns strategy if True and does not otherwise.

Returns:

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