state_metrics.fidelity_of_separability

Add function for fidelity of separability as defined in [2].

The constraints for this function are positive partial transpose (PPT) & k-extendible states.

Module Contents

Functions

fidelity_of_separability(input_state_rho, ...[, k, ...])

Define the first benchmark introduced in Appendix H of [2].
state_metrics.fidelity_of_separability.fidelity_of_separability(input_state_rho, input_state_rho_dims, k=1, verbosity_option=0, solver_option='cvxopt')

Define the first benchmark introduced in Appendix H of [2].

If you would like to instead use the benchmark introduced in Appendix I, go to toqito.channel_metrics.fidelity_of_separability.

In [2] a variational quantum algorithm (VQA) is introduced to test the separability of a general bipartite state. The algorithm utilizes quantum steering between two separated systems such that the separability of the state is quantified.

Due to the limitations of currently available quantum computers, two optimization semidefinite programs (SDP) benchmarks were introduced to maximize the fidelity of separability subject to some state constraints (Positive Partial Transpose (PPT), symmetric extensions (k-extendibility ) [3] ) This function approximites the fidelity of separability by maximizing over PPT states & k-extendible states i.e. an optimization problem over states [1].

The following expression (Equation (H2) from [2] ) defines the constraints for approxiamting

\(\sqrt{\widetilde{F}_s^1}(\rho_{AB}) {:}=\)

\[\begin{split}\begin{multline} \max_{\substack{X_{AB} \in\mathcal{L}(\mathcal{H}_{AB}),\\\sigma_{AB^{k}}\geq0}} \left\{\begin{array} [c]{c} \operatorname{Re}[\operatorname{Tr}[X_{AB}]]:\\% \begin{bmatrix} \rho_{AB} & X_{AB}\\ X_{AB}^{\dagger} & \sigma_{AB_{1}}% \end{bmatrix} \geq0,\\ \operatorname{Tr}[\sigma_{AB^{k}}]=1,\\ \sigma_{AB^{k}}=\mathcal{P}_{B^{k}}(\sigma_{AB^{k}}),\\ T_{B_{1\cdots j}}(\sigma_{AB_{1\cdots j}})\geq 0 \quad \forall j\leq k \end{array}\right\} \end{multline}\end{split}\]

\(\sqrt{\widetilde{F}_s^1}(\rho_{AB})\) is the quantity to be approximated but this function returns \(\widetilde{F}_s^1(\rho_{AB})\).

\(\operatorname{Re}[\operatorname{Tr}[X_{AB}]]\) is the maximization problem subject to PPT & k-extendibile state constraints.

Here, \(\mathcal{L}(\mathcal{H}_{AB})\) is the space of linear operators over space \(\mathcal{H}_{AB}\).

\(\sigma_{AB^{k}}\) is a k-extension of \(\rho_{AB}\).

\(\mathcal{P}_{B^{k}}\) is the permutation operator among systems \(B_1, B_2, \ldots , B_{k}\) which has no effect on the k-extended state \(\sigma_{AB^{k}}\).

The other constraints are due to the PPT condition [4].

Examples

Let’s consider a density matrix of a state that we know is pure and separable; \(|00 \rangle = |0 \rangle \otimes |0 \rangle\).

The expected approximation of fidelity of separability is the maximum value possible i.e. very close to 1.

\[\rho_{AB} = |00 \rangle \langle 00|\]
>>> from toqito.state_metrics import fidelity_of_separability
>>> from toqito.matrix_ops import tensor
>>> from toqito.states import basis
>>> state = tensor(basis(2, 0), basis(2, 0))
>>> rho = state @ state.conj().T
>>> expected_value = fidelity_of_separability(rho, [2, 2])
>>> '%.2f' % expected_value
'1.00'

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]

John Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. doi:10.1017/9781316848142.

[2] (1,2,3,4,5)

Aby Philip, Soorya Rethinasamy, Vincent Russo, and Mark M. Wilde. Schrödinger as a quantum programmer: estimating entanglement via steering. 2023. arXiv:2303.07911.

[3]

Patrick Hayden, Kevin Milner, and Mark M. Wilde. Two-message quantum interactive proofs and the quantum separability problem. In 2013 IEEE Conference on Computational Complexity. IEEE, Jun 2013. URL: http://dx.doi.org/10.1109/CCC.2013.24, doi:10.1109/ccc.2013.24.

[4]

Asher Peres. Separability criterion for density matrices. Physical Review Letters, 77(8):1413–1415, Aug 1996. URL: http://dx.doi.org/10.1103/PhysRevLett.77.1413, doi:10.1103/physrevlett.77.1413.

Parameters:
Raises:

  • AssertionError – If the provided dimensions are not for a bipartite density matrix.

  • ValueError – If the matrix is not a density matrix (square matrix that is PSD with trace 1).

  • ValueError – the input state is entangled.

  • ValueError – the input state is a mixed state.

Returns:

Optimized value of the SDP when maximized over a set of linear operators subject to some constraints.

Return type:

float