state_props.is_separable

Checks if a quantum state is a separable state.

Functions

is_separable(state[, dim, level, tol])

Determine if a given state (given as a density matrix) is a separable state [1].

Module Contents

state_props.is_separable.is_separable(state, dim=None, level=2, tol=1e-08)

Determine if a given state (given as a density matrix) is a separable state [1].

Examples

Consider the following separable (by construction) state:

\[\rho = \rho_1 \otimes \rho_2. \rho_1 = \frac{1}{2} \left( |0 \rangle \langle 0| + |0 \rangle \langle 1| + |1 \rangle \langle 0| + |1 \rangle \langle 1| \right) \rho_2 = \frac{1}{2} \left( |0 \rangle \langle 0| + |1 \rangle \langle 1| \right)\]

The resulting density matrix will be:

\[\begin{split}\rho = \frac{1}{4} \begin{pmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{pmatrix} \in \text{D}(\mathcal{X}).\end{split}\]

We provide the input as a density matrix \(\rho\).

On the other hand, a random density matrix will be an entangled state (a separable state).

>>> import numpy as np
>>> from toqito.rand import random_density_matrix
>>> from toqito.state_props import is_separable
>>> rho_separable = np.array([[1, 0, 1, 0],
...                           [0, 1, 0, 1],
...                           [1, 0, 1, 0],
...                           [0, 1, 0, 1]])
>>> is_separable(rho_separable)
True

>>> rho_not_separable = np.array([[ 0.13407875+0.j        , -0.08263926-0.17760437j,
...    -0.0135111 -0.12352182j,  0.0368423 -0.05563985j],
...   [-0.08263926+0.17760437j,  0.53338542+0.j        ,
...     0.19782968-0.04549732j,  0.11287093+0.17024249j],
...   [-0.0135111 +0.12352182j,  0.19782968+0.04549732j,
...     0.21254612+0.j        , -0.00875865+0.11144344j],
...   [ 0.0368423 +0.05563985j,  0.11287093-0.17024249j,
...    -0.00875865-0.11144344j,  0.11998971+0.j        ]])
>>> is_separable(rho_not_separable)
False

References

[1] (1,2)

Wikipedia. Separable state. URL: https://en.wikipedia.org/wiki/Separable_state.

Raises:

ValueError – If dimension is not specified.

Parameters:

  • state (numpy.ndarray) – The matrix to check.

  • dim (None | int | list[int]) – The dimension of the input.

  • level (int) – The level up to which to search for the symmetric extensions.

  • tol (float) – Numerical tolerance used.

Returns:

True if rho is separabale and False otherwise.

Return type:

bool