Estimating coherence with respect to general quantum measurements

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.

8 pages, 2 figures

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Cited by in corpus (3)

• Detecting coherence with respect to general quantum measurements
• Complementarity relations for design-structured POVMs in terms of generalized entropies of order $\alpha \in (0,2)$
• Bounds on positive operator-valued measure based coherence of superposition