matrix_props.is_k_incoherent¶

Checks if the matrix is $k$-incoherent.

Functions¶

is_k_incoherent(mat, k[, tol])

Determine whether a quantum state is k-incoherent [1].

Module Contents¶

matrix_props.is_k_incoherent.is_k_incoherent(mat, k, tol=1e-15)¶

Determine whether a quantum state is k-incoherent [1].

For a positive integers, $k$ and $n$, the matrix $X \in \text{Pos}(\mathbb{C}^n)$ is called $k$-incoherent if there exists a positive integer $m$, a set $S = \{|\psi_0\rangle, |\psi_1\rangle,\ldots, |\psi_{m-1}\rangle\} \subset \mathbb{C}^n$ with the property that each $|\psi_i\rangle$ has at most $k$ non-zero entries, and real scalars $c_0, c_1, \ldots, c_{m-1} \geq 0$ for which

\[X = \sum_{j=0}^{m-1} c_j |psi_j\rangle \langle \psi_j|.\]

This function checks if the provided density matrix mat is k-incoherent. It returns True if mat is k-incoherent and False if mat is not.

The function first handles trivial cases. Then it computes the comparison matrix (via matrices.comparison.comparison()) and performs further tests based on the trace of $mat^2$ and a dephasing channel. If no decision is reached, the function recurses by checking incoherence for k-1. Finally, if still indeterminate, an SDP is formulated to decide incoherence.

Examples

If $n = 3$ and $k = 2$, then the following matrix is $2$-incoherent:

import numpy as np
from toqito.matrix_props import is_k_incoherent
mat = np.array([[2, 1, 2],
            [1, 2, -1],
            [2, -1, 5]])
is_k_incoherent(mat, 2)

True