Estimating coherence with respect to general quantum measurements

TL;DR

This paper extends the concept of quantum coherence to general POVMs, analyzing properties of various coherence measures, deriving bounds, and connecting them to quantum state discrimination and uncertainty relations.

Contribution

It introduces new bounds and relations for coherence measures with respect to general POVMs, unifying and extending previous coherence concepts.

Findings

Upper bounds for $C_{l_{1}}$ coherence measure.

Minimal error probability linked to $C_{T,1/2}$ coherence.

Derived uncertainty relations involving $C_{r}$.

Abstract

The conventional coherence is defined with respect to a fixed orthonormal basis, i.e., to a von Neumann measurement. Recently, generalized quantum coherence with respect to general positive operator-valued measurements (POVMs) has been presented. Several well-defined coherence measures, such as the relative entropy of coherence $C_{r}$, the $l_{1}$ norm of coherence $C_{l_{1}}$ and the coherence $C_{T,\alpha }$ based on Tsallis relative entropy with respect to general POVMs have been obtained. In this work, we investigate the properties of $C_{r}$, $l_{1}$ and $C_{T,\alpha }$. We estimate the upper bounds of $C_{l_{1}}$; we show that the minimal error probability of the least square measurement state discrimination is given by $C_{T,1/2}$; we derive the uncertainty relations given by $C_{r}$, and calculate the average values of $C_{r}$, $C_{T,\alpha }$ and $C_{l_{1}}$ over random pure quantum states. All these results include the corresponding results of the conventional coherence as special cases.