state_props.l1_norm_coherence

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

Module Contents

Functions

l1_norm_coherence(rho)

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

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)
>>> '%.1f' % l1_norm_coherence(v)
'2.0'

Note

You do not need to use ‘%.1f’ % when you use this function.

We use this to format our output such that doctest compares the calculated output to the expected output upto two decimal points only. The accuracy of the solvers can calculate the float output to a certain amount of precision such that the value deviates after a few digits of accuracy.

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