Polynomial measure of coherence

Abstract

Coherence, the superposition of orthogonal quantum states, is indispensable in various quantum processes. Inspired by the polynomial invariant for classifying and quantifying entanglement, we first define polynomial coherence measure and systematically investigate its properties. Except for the qubit case, we show that there is no polynomial coherence measure satisfying the criterion that its value takes zero if and only if for incoherent states. Then, we release this strict criterion and obtain a necessary condition for polynomial coherence measure. Furthermore, we give a typical example of polynomial coherence measure for pure states and extend it to mixed states via a convex-roof construction. Analytical formula of our convex-roof polynomial coherence measure is obtained for symmetric states which are invariant under arbitrary basis permutation. Consequently, for general mixed states, we give a lower bound of our coherence measure.

One-shot coherence distillation with catalysts

“…where {P i } form the incoherent permutation operator group on H N . After twirling, every label of the output state is symmetric, so the output state can without loss of generality be written as [26],…”

“…There are different definitions of free operations such as maximal incoherent operations [15] (MIO), incoherent operations [17] (IO), dephasing-covariant incoherent operations [19,20] (DIO), strictly incoherent operations [21,22] (SIO), and physically implementable incoherent operations [19] (PIO), showing differences in their operational ability and physical relevance. Furthermore, * zhaoqithu10@gmail.com different measures of coherence of a state are defined, such as the l 1 -norm, the relative entropy of coherence [17], the coherence of formation [15,23,24], the robustness of coherence [25], polynomial measure of coherence [26], etc. The operational meanings of some measures are further uncovered [21,23,27,28].…”

No-go theorems for deterministic purification and probabilistic enhancement of coherence

“…With these notions, the essential conditions of coherence measures have been established [11,12]. Based on these conditions, a number of legitimate coherence measures also have been proposed [21][22][23][24][25][26][27][28][29][30][31], which can be used to study the role of coherence in many physical contexts quantitatively [32][33][34][35][36][37][38]. Many studies have shown that the amount of quantum coherence is directly related to the success or failure of some quantum information processing tasks, such as quantum phase discrimination [21], quantum algorithm [39,40], and the secrete key rate in quantum key distribution [41].…”

The Tightness of Multipartite Coherence from Spectrum Estimation

“…Based on the general framework, several coherence measures have been proposed, such as the l 1 norm of coherence, the relative entropy of coherence [12], the geometric measure of coherence [17], the robustness of coherence [19,20], some convex roof quantifiers of coherence [21][22][23][24][25], and others [26][27][28][29][30][31][32]. These coherence measures make it possible to quantify the role of coherence in different quantum information processing tasks, especially in the multipartite scenario, such as quantum state merging [33], coherence of assistance [34], incoherent teleportation [35], coherence localization [36], and anti-noise quantum metrology [37].…”