Family of coherence measures and duality between quantum coherence and path distinguishability

Abstract

Coherence measures and their operational interpretations lay the cornerstone of coherence theory. In this paper, we introduce a class of coherence measures with α-affinity, say α-affinity of coherence for α ∈ (0, 1). Furthermore, we obtain the analytic formulae for these coherence measures and study their corresponding convex roof extension. We provide an operational interpretation for 1/2-affinity of coherence by showing that it is equal to the error probability to discrimination a set of pure states with the least square measurement. Employing this relationship we regain the optimal measurement for equiprobable quantum state discrimination. Moreover, we compare these coherence quantifiers, and establish a complementarity relation between 1/2-affinity of coherence and path distinguishability for some special cases. *

Coherence measures with respect to general quantum measurements

“…We remark that when the POVM E = {E j } n j=1 is a rank-one projective measurement E = {|j j|} d j=1 , both the Theorem 3 and Theorem 4 reduce to the corresponding conclusions in the theory of conventional coherence [15].…”

“…When E is a rank-one projective measurement, C T,α (ρ, E) returns to the standard coherence measure C r (ρ, {|j j|} d j=1 ) proposed in Ref. [14,15]. A quantifier of standard coherence measure based on Tsallis entropy was proposed in Ref.…”

“…When the POVM E is a rank-one projective measurement E = {|j j|} d j=1 , the definitions of POVM incoherent states and POVM incoherent channels, as well as the conditions (P1) to (P4) all reduce to the cases of the conventional coherence theory, for which various kinds of coherence measures have been proposed together with their operational interpretations and applications [1,[7][8][9][10][11][12][13][14][15][16][17][18][19][20]. However, less is known about the POVM coherence measures.…”

Quantifying coherence with respect to general measurements via quantum Fisher information

“…So we expect that some of these standard coherence measures can be generalized to be the corresponding POVM coherence measures. In fact, several standard coherence measures have been properly generalized to be POVM coherence measures, such as the relative entropy of POVM coherence [1,5,7], robustness of POVM coherence [6,8,9], l 1 norm of POVM coherence [1,6,10], and the POVM coherence based on the Tsallis entropy [10][11][12][13][14]. Also, there is a physical interpretation for the relative entropy of POVM coherence * Electronic address: xxujianwei@nwafu.edu.cn [6].…”

$l_{1}$ norm of coherence is not equal to its convex roof quantifier

“…with p ∈ [0, 1], ρ 1 , ρ 2 any states. Many coherence measures have been proposed, such as the relative entropy of coherence C r [1], l 1 norm of coherence C l1 [1,5], coherence based on Tsallis relative entropy C T,α [6][7][8][9], robustness of coherence C rob [10,11], geometric coherence [12] C g , modified trace norm of coherence C ′ tr [4], coherence weight [13,14]. Among them, C r and C l1 are widely used, they have the explicit expressions as…”