TraceDistanceCoherence

TraceDistanceCoherence is a function that computes the trace distance of coherence of a quantum state $\rho$, defined as follows:


 * $$C_{\text{tr}}(\rho) := \min_{D \in \mathcal{I}}\big\{ \|\rho - D\|_{\text{tr}}\big\},$$

where $\mathcal{I}$ is the set of incoherent quantum states (i.e., the set of density matrices that are diagonal in the standard basis).

Syntax

 * TDC = TraceDistanceCoherence(RHO)
 * [TDC,D] = TraceDistanceCoherence(RHO)

Input arguments

 * RHO: A state (either pure or mixed) to have its trace distance of coherence computed.

Output arguments

 * TDC: The trace distance of coherence of RHO.
 * D: A vector such that diag(D) is the closest incoherent state to RHO</tt>.

Maximally coherent states
The largest possible value of the trace distance of coherence on $d$-dimensional states is $2 - 2/d$, and is attained exactly by the "maximally coherent states": pure states whose entries all have the same absolute value.