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 [1].

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 [2], 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.matrix_ops 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

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