state_props.is_separable

Check if state is separable.

Module Contents

Functions

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

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

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 denisty matrix \(\rho\).

On the other hand, a random density matrix will be an entangled state (a separable state). >>> 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]]) >>> rho_random = random_density_matrix(4) >>> is_separable(rho_separable) True >>> is_separable(rho_random) False

References

[1] (1,2)

Wikipedia. Separable state. 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