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arXiv:2002.03823 · Physical Review A

l1 -norm coherence of assistance

Ming‐Jing Zhao, Teng Ma, Quan Quan, Heng Fan, Rajesh Pereira

Published Jul 15, 2019• 33 citations
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Abstract

We introduce and study the ${l}_{1}$-norm coherence of assistance both theoretically and operationally. We first provide an upper bound for the ${l}_{1}$-norm coherence of assistance and show a necessary and sufficient condition for the saturation of the upper bound. For two- and three-dimensional quantum states, the analytical expression of the ${l}_{1}$-norm coherence of assistance is given. Operationally, the mixed quantum coherence can always be increased with the help of another party's local measurement and one way classical communication since the ${l}_{1}$-norm coherence of assistance, as well as the relative-entropy coherence of assistance, are shown to be strictly larger than the original coherence. The relation between the ${l}_{1}$-norm coherence of assistance and entanglement is revealed. Finally, a comparison between the ${l}_{1}$-norm coherence of assistance and the relative-entropy coherence of assistance is made.

DOI
10.1103/physreva.100.012315

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We introduce and study the l1{l}_{1}l1​-norm coherence of assistance both theoretically and operationally. We first provide an upper bound for the l1{l}_{1}l1​-norm coherence of assistance and show a necessary and sufficient condition for the saturation of the upper bound. For two- and three-dimensional quantum states, the analytical expression of the l1{l}_{1}l1​-norm coherence of assistance is given. Operationally, the mixed quantum coherence can always be increased with the help of another party's local measurement and one way classical communication since the l1{l}_{1}l1​-norm coherence of assistance, as well as the relative-entropy coherence of assistance, are shown to be strictly larger than the original coherence. The relation between the l1{l}_{1}l1​-norm coherence of assistance and entanglement is revealed. Finally, a comparison between the l1{l}_{1}l1​-norm coherence of assistance and the relative-entropy coherence of assistance is made.
Coherence and entanglement are fundamental concepts in resource theory. The coherence (entanglement) of assistance is the coherence (entanglement) that can be extracted assisted by another party with local measurement and classical communication. We introduce and study the general coherence of assistance. First, in terms of real symmetric concave functions on the probability simplex, the coherence of assistance and the entanglement of assistance are shown to be in one-to-one correspondence. We then introduce two classes of quantum states: the assisted maximally coherent states and the assisted maximally entangled states. They can be transformed into maximally coherent or entangled pure states with the help of another party using local measurement and classical communication. We give necessary conditions for states to be assisted maximally coherent or assisted maximally entangled. Based on these, a unified framework between coherence and entanglement including coherence (entanglement) measures, coherence (entanglement) of assistance, coherence (entanglement) resources is proposed. Then we show that the coherence of assistance as well as entanglement of assistance are strictly larger than the coherence of convex roof and entanglement of convex roof for all full rank density matrices. So all full rank quantum states are distillable in the assisted coherence distillation.
We study the relation between the coherence of assistance and the regularized coherence of assistance introduced by Chitambar et al. [E. Chitambar et al., Phys. Rev. Lett. 116, 070402 (2016)]. The necessary and sufficient conditions that these two quantities coincide are provided. Detailed examples are analyzed and the optimal pure state decompositions such that the coherence of assistance equals the regularized coherence of assistance are derived. Moreover, we present the protocol for obtaining the maximal relative entropy coherence, assisted by another party under local measurement and one-way communication in one copy setting.
Various measures have been suggested recently for quantifying the coherence of a quantum state with respect to a given basis. We first use two of these, the l1{l}_{1}l1​-norm and relative entropy measures, to investigate tradeoffs between the coherences of mutually unbiased bases. Results include relations between coherence, uncertainty, and purity; tight general bounds restricting the coherences of mutually unbiased bases; and an exact complementarity relation for qubit coherences. We further define the average coherence of a quantum state. For the l1{l}_{1}l1​-norm measure this is related to a natural ``coherence radius'' for the state and leads to a conjecture for an l2{l}_{2}l2​-norm measure of coherence. For relative entropy the average coherence is determined by the difference between the von Neumann entropy and the quantum subentropy of the state and leads to upper bounds for the latter quantity. Finally, we point out that the relative entropy of coherence is a special case of G-asymmetry, which immediately yields several operational interpretations in contexts as diverse as frame alignment, quantum communication, and metrology, and suggests generalizing the property of quantum coherence to arbitrary groups of physical transformations.
We show that the distillable coherence---which is equal to the relative entropy of coherence---is, up to a constant factor, always bounded by the ${\ensuremath{\ell}}_{1}$-norm measure of coherence (defined as the sum of absolute values of off diagonals). Thus the latter plays a similar role as logarithmic negativity plays in entanglement theory and this is the best operational interpretation from a resource-theoretic viewpoint. Consequently the two measures are intimately connected to another operational measure, the robustness of coherence. We find also relationships between these measures, which are tight for general states, and the tightest possible for pure and qubit states. For a given robustness, we construct a state having minimum distillable coherence.
In the task of assisted coherence distillation via the set of operations X, where X is either local incoherent operations and classical communication (LICC), local quantum-incoherent operations and classical communication (LQICC), separable incoherent operations (SI), or separable quantum incoherent operations (SQI), two parties, namely Alice and Bob, share many copies of a bipartite joint state. The aim of the process is to generate the maximal possible coherence on the subsystem of Bob. In this paper, we investigate the assisted coherence distillation of some special mixed states, the states with vanished basis-dependent discord and Werner states. We show that all the four sets of operations are equivalent for assisted coherence distillation, whenever Alice and Bob share one of those mixed quantum states. Moreover, we prove that the assisted coherence distillation of the former can reach the upper bound, namely QI relative entropy, while that of the latter can not. Meanwhile, we also present a sufficient condition such that the assistance of Alice via the set of operations X can not help Bob improve his distillable coherence, and this condition is that the state shared by Alice and Bob has vanished basis-dependent discord.