Maximum Relative Entropy of Coherence: An Operational Coherence Measure

One of the central problems in the study of quantum resource theories is to provide a given resource with an operational meaning, characterizing physical tasks in which the resource can give an explicit advantage over all resourceless states. We show that this can always be accomplished for all convex resource theories. We establish in particular that any resource state enables an advantage in a channel discrimination task, allowing for a strictly greater success probability than any state without the given resource. Furthermore, we find that the generalized robustness measure serves as an exact quantifier for the maximal advantage enabled by the given resource state in a class of subchannel discrimination problems, providing a universal operational interpretation to this fundamental resource quantifier. We also consider a wider range of subchannel discrimination tasks and show that the generalized robustness still serves as the operational advantage quantifier for several well-known theories such as entanglement, coherence, and magic.Introduction. -A rigorous understanding of quantum resources has been one of the ultimate goals in quantum information science. In addition to the apparent theoretical interest, it also has high relevance to burgeoning quantum information technologies such as quantum communication [1,2], quantum cryptography [3,4], and quantum computation [5,6].Quantum resource theories [7] have recently attracted much attention as powerful tools which offer formal frameworks dealing with quantification and manipulation of intrinsic resources associated with quantum systems. One could consider different theories depending on the relevant physical constraints, and indeed various resource theories have been proposed and analyzed, such as entanglement [8,9], coherence [10][11][12], asymmetry [13,14], quantum thermodynamics [15,16], non-Markovianity [17], magic [18,19], and non-Gaussianity [20][21][22]. Although these resource theories provide deeper insights into their specific physical settings,they do not tell us much about how to understand the individual properties and results in a unified fashion. In particular, despite the generality of the resource theoretical framework, only a small number of results reported in the literature are applicable to wide classes of general quantum resource theories [23][24][25][26][27][28][29]. In this work, we add a fundamental item to this list with regard to one of the central questions asked in the study of resource theories: the operational characterization of quantum states and the resources they possess.An essential building block of a resource theory is the set of free states. It is the set of states that are considered "easy to prepare" in that theory, and any state outside of this set is called a resource state. A common and intuitive assumption is that the set of free states should be convex and closed. Convexity reflects a natural attribute in many physical settings, i.e. the fact that losing information about which free state was prepared, hence resulti...

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“…One can easily verify that, for instance, coherence theory satisfies this condition, which recovers the result in [49,50,56].…”

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confidence: 66%

Operational Advantage of Quantum Resources in Subchannel Discrimination

et al. 2019

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“…One important pillar of the field is the study on quantification of coherence. Since the seminal work [1] on defining a good coherence measure in terms of the resource theory, quantum coherence has been widely studied and applied to many quantum information processing [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

“…For instance, the relative entropy coherence can be understood as the optimal rate for distilling a maximally coherent state from given states [8]. The max-relative entropy of coherence can exactly characterize the subchannel discrimination problems such that the coherent state allows for a higher probability of successfully discriminating subchannels than that of all incoherent states [20]. The robustness of coherence quantifies the advantage enabled by a quantum state in a phase discrimination task [13].…”

Section: Introductionmentioning

confidence: 99%

Triangle-like inequalities related to coherence and entanglement negativity

Jin

Li‐Jost

Fei

2018

Quantum Inf Process

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Quantum coherence and entanglement are two key features in quantum mechanics and play important roles in quantum information processing and quantum computation. We provide a general triangle-like inequality satisfied by the l 1 -norm measure of coherence for convex combination of arbitrary n pure states of a quantum state ρ. Furthermore, we present triangle-like inequality for the convex-roof extended negativity for any states of rank 2, which gives a positive answer to a conjecture raised in [Phys. Rev. A 96, 062308 (2017)]. Detailed examples are given to illustrate the relations characterized by the triangle-like inequalities.

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“…Note that the relative entropy is not a proper distance because it does not satisfy symmetric and triangle inequality, but the relative entropy of coherence [9] is also viewed as a distance-based coherence quantifiers, and there are many interesting properties [9,11,24,30,34]. Next we will discuss separately the ǫ generalization of relative entropy of coherence.…”

Section: Distance-based Coherent Measuresmentioning

confidence: 99%

“…This quantity has some interesting properties in the manipulation of coherence. It provided an upper bound of oneshot coherence distillation, and in the asymptotic limit the smoothed minimum relative entropy of coherence is equivalent to the relative entropy of coherence [30]. Then the next definition formalizes the notion of the one-shot coherence distillation under the fixed distance D and any coherence measure C.…”

Section: Measure Of Coherencementioning

confidence: 99%

Epsilon-smooth measure of coherence

Yuwen

2019

Phys. Rev. A

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In this paper, by minimizing the coherence quantifiers over all states in an ǫ ball around a given state, we define a generalized smooth quantifier, called the ǫ-smooth measure of coherence. We use it to estimate the difference between the expected state and the actually prepared state and quantify quantum coherence contained in an actually prepared state, and it can been interpreted as the minimal coherence guaranteed to present in an ǫ ball around given quantum state. We find that the ǫ-smooth measure of any coherence monotone is still a coherence monotone, but it does not satisfy monotonicity on average under incoherent operations. We show the ǫ-smooth measure of coherence is continuous even if the original coherence quantifier is not. We also study the ǫsmooth measure of distance-based coherence quantifiers, and some interesting properties are given. Moreover, we discuss the dual form of the ǫ-smooth measure of coherence by maximizing over all states in an ǫ ball around the given state and show that the dual ǫ-smooth measure of coherence provides an upper bound of one-shot coherence distillation.PACS numbers:

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Maximum Relative Entropy of Coherence: An Operational Coherence Measure

Cited by 194 publications

References 51 publications

Operational Advantage of Quantum Resources in Subchannel Discrimination

Operational Advantage of Quantum Resources in Subchannel Discrimination

Triangle-like inequalities related to coherence and entanglement negativity

Epsilon-smooth measure of coherence

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