channel_metrics.fidelity_of_separability

Add functions for channel fidelity of Separability as defined in [2].

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

Functions

fidelity_of_separability(psi, psi_dims[, k, ...])

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

Module Contents

channel_metrics.fidelity_of_separability.fidelity_of_separability(psi, psi_dims, k=1, verbosity_option=0, solver_option='cvxopt')

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

If you would like to instead use the benchmark introduced in Appendix H, go to toqito.state_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]). Entangled states do not have k-symmetric extensions. If an extension exists, it cannot be assumed directly that the state is separable. This function approximites the fidelity of separability by maximizing over PPT channels & k-extendible entanglement breaking channels i.e. an optimization problem over channels [1] .

The following discussion (Equation (I4) from [2] ) defines the constraints for approximating \(\widetilde{F}_s^2(\rho_{AB})\) in \(\frac{1}{2}(1+\widetilde{F}_s^2(\rho_{AB}))\).

\[\operatorname{Tr}[ \Pi_{A^{\prime}A}^{\operatorname{sym}} \operatorname{Tr}_{R}[ T_R(\psi_{RAB})\Gamma^{\mathcal{E}^{k}}_{RA^{\prime}_1}]]\]

Above expression defines the maximization problem subject to PPT & k-extendibile channel constraints over \(\max_{\Gamma^{\mathcal{E}^{k}}_{RA^{\prime k}}\geq 0}\)

The constraint expressions are listed below:

\[\operatorname{Tr}_{A^{\prime k}}[\Gamma^{\mathcal{E}^{k}}_{RA^{\prime k}}]=I_R\]

\(\Gamma^{\mathcal{E}^{k}}_{RA^{\prime}}\) is Choi operator of entanglement breaking channel \(\mathcal{E}^{k}\).

\[\Gamma^{\mathcal{E}^{k}}_{RA^{\prime k}}= \mathcal{P}_{A^{\prime k}}(\Gamma^{\mathcal{E}^{k}}_{RA^{\prime k}})\]

\(\mathcal{P}_{A^{\prime k}}\) is the permutation operator over k-extensions \(A^{\prime k}\).

\[T_{A^{\prime}_{1\cdots j}}(\Gamma^{\mathcal{E}^{k}_{RA^{\prime k}}}) \geq 0 \quad \forall j\leq k\]

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

Examples

Let’s consider a density matrix of a state that we know is pure & separable. \(|000 \rangle = |0 \rangle \otimes |0 \rangle \otimes |0 \rangle\).

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

\[\rho_{AB} = |000 \rangle \langle 000|\]
>>> import numpy as np
>>> 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
>>> np.around(fidelity_of_separability(rho, [2, 2]), decimals=2)
np.float64(1.0)

References

[1]

John Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. URL: https://johnwatrous.com/wp-content/uploads/TQI.pdf, 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 tripartite 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