L1NormCoherence

L1NormCoherence is a function that computes the ℓ1-norm of coherence of a quantum state $\rho$, defined as follows:


 * $$C_{\ell_1}(\rho) := \sum_{i \neq j} |\rho_{ij}|,$$

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

Syntax

 * L1C = L1NormCoherence(RHO)

Argument descriptions

 * RHO: A state (either pure or mixed) to have its ℓ1-norm of coherence computed.

Pure states or mixed states
If $|v\rangle$ is a pure state then its ℓ1-norm of coherence is computed from the density matrix $|v\rangle\langle v|$:

Maximally coherent states
The largest possible value of the ℓ1-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.

Can be used within CVX
The ℓ1-norm of coherence is a convex function and can be used in the same way as any other convex function within CVX. Thus you can minimize the ℓ1-norm of coherence or use the ℓ1-norm of coherence in constraints of CVX optimization problems. For example, the following code minimizes the ℓ1-norm of coherence over all density matrices that are within a trace distance of $1/2$ from the maximally coherent state $|v\rangle = (1,1,1,1,1)/\sqrt{5}$: