state_props.l1_norm_coherence

Computes the l1-norm of coherence of a quantum state.

Functions

l1_norm_coherence(rho)

Compute the l1-norm of coherence of a quantum state [1].

Module Contents

state_props.l1_norm_coherence.l1_norm_coherence(rho)

Compute the l1-norm of coherence of a quantum state [1].

The \(\ell_1\)-norm of coherence of a quantum state \(\rho\) is defined as

\[C_{\ell_1}(\rho) = \sum_{i \not= j} \left|\rho_{i,j}\right|,\]

where \(\rho_{i,j}\) is the \((i,j)^{th}\)-entry of \(\rho\) in the standard basis.

The \(\ell_1\)-norm of coherence is the sum of the absolute values of the sum of the absolute values of the off-diagonal entries of the density matrix rho in the standard basis.

This function was adapted from QETLAB.

Examples

The largest possible value of the \(\ell_1\)-norm of coherence on \(d\)-dimensional states is \(d-1\), and is attained exactly by the “maximally coherent states”: pure states whose entries all have the same absolute value.

from toqito.state_props import l1_norm_coherence
import numpy as np
# Maximally coherent state.
v = np.ones((3,1))/np.sqrt(3)
l1_norm_coherence(v)

np.float64(2.000000000000001)

References

[1] (1,2)

Swapan Rana, Preeti Parashar, Andreas Winter, and Maciej Lewenstein. Logarithmic coherence: operational interpretation of $\ensuremath \ell _1$-norm coherence. Phys. Rev. A, 96:052336, Nov 2017. URL: https://link.aps.org/doi/10.1103/PhysRevA.96.052336, doi:10.1103/PhysRevA.96.052336.

Parameters:

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

Returns:

The l1-norm coherence of rho.

Return type:

float