state_props.has_symmetric_extension

Determine whether there exists a symmetric extension for a given quantum state.

Functions

has_symmetric_extension(rho[, level, dim, ppt, tol])

Determine whether there exists a symmetric extension for a given quantum state.

Module Contents

state_props.has_symmetric_extension.has_symmetric_extension(rho, level=2, dim=None, ppt=True, tol=0.0001)

Determine whether there exists a symmetric extension for a given quantum state.

For more information, see [2].

Determining whether an operator possesses a symmetric extension at some level level can be used as a check to determine if the operator is entangled or not.

This function was adapted from QETLAB.

Examples

2-qubit symmetric extension:

In [1], it was shown that a 2-qubit state \(\rho_{AB}\) has a symmetric extension if and only if

\[\text{Tr}(\rho_B^2) \geq \text{Tr}(\rho_{AB}^2) - 4 \sqrt{\text{det}(\rho_{AB})}.\]

This closed-form equation is much quicker to check than running the semidefinite program.

>>> import numpy as np
>>> from toqito.state_props import has_symmetric_extension
>>> from toqito.channels import partial_trace
>>> rho = np.array([[1, 0, 0, -1],
...                 [0, 1, 1/2, 0],
...                 [0, 1/2, 1, 0],
...                 [-1, 0, 0, 1]])
>>> # Show the closed-form equation holds
>>> np.trace(np.linalg.matrix_power(partial_trace(rho, 1), 2)) >= np.trace(rho**2) - 4 * np.sqrt(np.linalg.det(rho))
np.True_
>>> # Now show that the `has_symmetric_extension` function recognizes this case.
>>> has_symmetric_extension(rho)
True

Higher qubit systems:

Consider a density operator corresponding to one of the Bell states.

\[\begin{split}\rho = \frac{1}{2} \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \end{pmatrix}\end{split}\]

To make this state over more than just two qubits, let’s construct the following state

\[\sigma = \rho \otimes \rho.\]

As the state \(\sigma\) is entangled, there should not exist a symmetric extension at some level. We see this being the case for a relatively low level of the hierarchy.

>>> import numpy as np
>>> from toqito.states import bell
>>> from toqito.state_props import has_symmetric_extension
>>>
>>> rho = bell(0) @ bell(0).conj().T
>>> sigma = np.kron(rho, rho)
>>> has_symmetric_extension(sigma)
False

References

[1]

Jianxin Chen, Zhengfeng Ji, David Kribs, Norbert Lütkenhaus, and Bei Zeng. Symmetric extension of two-qubit states. Physical Review A, Sep 2014. URL: http://dx.doi.org/10.1103/PhysRevA.90.032318, doi:10.1103/physreva.90.032318.

[2]

A. C. Doherty, Pablo A. Parrilo, and Federico M. Spedalieri. Distinguishing separable and entangled states. Physical Review Letters, Apr 2002. URL: http://dx.doi.org/10.1103/PhysRevLett.88.187904, doi:10.1103/physrevlett.88.187904.

Raises:

ValueError – If dimension does not evenly divide matrix length.

Parameters:

  • rho (numpy.ndarray) – A matrix or vector.

  • level (int) – Level of the hierarchy to compute.

  • dim (numpy.ndarray | int) – The default has both subsystems of equal dimension.

  • ppt (bool) – If True, this enforces that the symmetric extension must be PPT.

  • tol (float) – Tolerance when determining whether a symmetric extension exists.

Returns:

True if mat has a symmetric extension; False otherwise.

Return type:

bool