toqito.state_props.l1_norm_coherence

toqito.state_props.l1_norm_coherence(rho)

Compute the l1-norm of coherence of a quantum state [RPWL17].

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)
2

References

[RPWL17]

“Logarithmic coherence: Operational interpretation of 1-norm coherence” Rana, Swapan, Parashar, Preeti, Winter, Andreas, Lewenstein, Maciej Physical Review A 96.5 (2017): 052336. https://arxiv.org/pdf/1612.09234.pdf

Parameters:

rho – A matrix or vector.

Returns:

The l1-norm coherence of rho.