Alternative framework for quantifying coherence

Abstract

We propose an alternative framework for quantifying coherence. The framework is based on a natural property of coherence, the additivity of coherence for subspace-independent states, which is described by an operationindependent equality rather than operation-dependent inequalities and therefore applicable to various physical contexts. Our framework is compatible with all the known results on coherence measures but much more flexible and convenient for applications, and by using it many open questions can be resolved.Quantum coherence is a fundamental feature of quantum mechanics, describing the capability of a quantum state to exhibit quantum interference phenomena. It is an essential ingredient in quantum information processing [1], and plays a central role in emergent fields, such as quantum metrology [2][3][4], nanoscale thermodynamics [5][6][7][8][9][10][11], and quantum biology [12][13][14][15][16]. Although the theory of quantum coherence is historically well developed in quantum optics [17][18][19], it is only in recent years that the quantification of coherence has attracted a growing interest [20][21][22][23][24] due to the development of quantum information science.By following the approach that has been established for entanglement resource [25,26], Baumgratz et al. proposed a seminal framework for quantifying coherence as a resource in Ref. [22]. The framework comprises four conditions, of which the first two are based on the notions of free states and free operations in the resource theories, while the third and fourth conditions are two constraints imposed on coherence measures. Based on this framework, a number of coherence measures, such as the relative entropy of coherence, the l 1 norm of coherence, and the coherence of formation [20,22,27,28], have been put forward. With the coherence measures, various properties of quantum coherence, such as the relations between quantum coherence and quantum correlations [29][30][31][32][33], the freezing phenomenon of coherence [34,35], and the distillation of coherence [28,36], were investigated. Hereafter, we refer to the framework proposed by Baumgratz et al. as the BCP framework for simplicity.Although the BCP framework has been widely used as an approach to coherence measures, there are arguments against the necessity of its last two conditions [29,37], and researchers have different opinions on the definition of free operations. Besides the incoherent operations defined in the BCP framework, there have been many different suggestions on the definition of free operations, such as maximally incoherent operations [20], translationally invariant operations [23], and others [38][39][40]. These arguments against the conditions and free operations imply that the frameworks for quantifying coherence are not unique. There can be other frameworks different from the BCP framework. For instance, the framework proposed by Marvian and Spekkens in Ref. [23], called the MS framework for simplicity, is based on the translationally invariant operations, and it compris...

One-shot entanglement distillation beyond local operations and classical communication

“…This quantity, considered first in the resource theory of coherence as the modified trace distance [36], generalises the commonly employed trace distance measure  r s -sÎ   min 1 . The reason why  ( ) T 1 is a more suitable measure of entanglement than the trace distance itself is the fact that, contrary to  ( ) T 1 , the trace distance does not satisfy strong monotonicity under LOCC [36,37] (i.e. the requirement that a measure M obeys…”

Flag additivity in quantum resource theories

“…When they are applied to coherence measures, they give the results in Ref. [17] as a special case. Compared with the strong monotonicity and the convexity, the flag supadditivity and flag subadditivity are much easier to prove or disprove, because they do not involve the Kraus operators and the structure of flagged states is much simpler than the mixing of ensembles.…”

“…It is difficult to prove whether or not C tr satisfies the strong monotonicity by examining it directly [25,26]. However, one can easily prove that C tr violates the strong monotonicity by examining the flag additivity [17].…”

“…In general, a resource measure must satisfy the nonnegativity and the monotonicity. Other useful properties, such as the strong monotonicity, the convexity, and the additivity, may also be introduced in different physical and mathematical contexts [11][12][13][14][15][16][17][18][19][20][21][22].…”

Generalized-mean Cramér-Rao bounds for multiparameter quantum metrology

“…Furthermore, we show that the f -mean QFIs have monotonicity under quantum operations and thus can be considered as information-theoretic quantities. We demonstrate that they are useful in quantum resource theory, e.g., in quantifying asymmetry [27] and coherence [28][29][30][31][32][33][34] as quantum resources.…”