Quantum majorization and a complete set of entropic conditions for quantum thermodynamics

Gour, Gilad; Jennings, David; Buscemi, Francesco; Duan, Runyao; Marvian, Iman

doi:10.1038/s41467-018-06261-7

Cited by 140 publications

(148 citation statements)

References 97 publications

(184 reference statements)

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“…entanglement, coherence, thermal non-equilibrium), allowing one to obtain nontrivial bounds for optimal resource manipulation in specific contexts. Our results also complement the studies on the complete set of monotones [26,35,77,78], which provide the necessary and sufficient conditions for state transformations between two states under free operations. A complete set of monotones generally consists of infinite number of resource monotones [23], which makes the computation impractical.…”

Section: Discussionsupporting

confidence: 73%

One-Shot Operational Quantum Resource Theory

2019

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A fundamental approach for the characterization and quantification of all kinds of resources is to study the conversion between different resource objects under certain constraints. Here we analyze, from a resource-non-specific standpoint, the optimal efficiency of resource formation and distillation tasks with only a single copy of the given quantum state, thereby establishing a unified framework of one-shot quantum resource manipulation. We find general bounds on the optimal rates characterized by resource measures based on the smooth max/min-relative entropies and hypothesis testing relative entropy, as well as the free robustness measure, providing them with general operational meanings in terms of optimal state conversion. Our results encompass a wide class of resource theories via the theory-dependent coefficients we introduce, and the discussions are solidified by important examples, such as entanglement, coherence, superposition, magic states, asymmetry, and thermal

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Section: Discussionsupporting

confidence: 73%

One-Shot Operational Quantum Resource Theory

2019

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“…We call a (possibly infinite) family of monotones a complete set of monotones if it fully characterizes the necessary and sufficient conditions for the existence of a free transformation. Such sets of monotones were discussed in several specific settings [25,26,32,[121][122][123][124][125][126][127][128][129][130][131], but no set of general conditions with a clear operational meaning was previously known to provide a comprehensive characterization of transformations in general resource theories.…”

Section: Complete Sets Of Monotonesmentioning

confidence: 99%

General Resource Theories in Quantum Mechanics and Beyond: Operational Characterization via Discrimination Tasks

2019

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We establish an operational characterization of general convex resource theories -describing the resource content of not only states, but also measurements and channels, both within quantum mechanics and in general probabilistic theories (GPTs) -in the context of state and channel discrimination. We find that discrimination tasks provide a unified operational description for quantification and manipulation of resources by showing that the family of robustness measures can be understood as the maximum advantage provided by any physical resource in several different discrimination tasks, as well as establishing that such discrimination problems can fully characterize the allowed transformations within the given resource theory.Specifically, we introduce a quantifier of resourcefulness of a measurement in any GPT, the generalized robustness of measurement, and show that it admits an operational interpretation as the maximum advantage that a given measurement provides over resourceless measurements in all state discrimination tasks. In the special case of quantum mechanics, we connect discrimination problems with single-shot information theory by showing that the generalized robustness of any measurement can be alternatively understood as the maximal increase in one-shot accessible information when compared to free measurements. We introduce two different approaches to quantifying the resource content of a physical channel based on the generalized robustness measures, and show that they quantify the maximum advantage that a resourceful channel can provide in several classes of state and channel discrimination tasks. Furthermore, we endow another measure of resourcefulness of states, the standard robustness, with an operational meaning in general GPTs as the exact quantifier of the maximum advantage that a state can provide in binary channel discrimination tasks. Finally, we establish that several classes of channel and state discrimination tasks form complete families of monotones fully characterizing the transformations of states and measurements, respectively, under general classes of free operations. Our results establish a fundamental connection between the operational tasks of discrimination and core concepts of resource theories -the geometric quantification of resources and resource manipulation -valid for all physical theories beyond quantum mechanics with no additional assumptions about the structure of the GPT

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“…As a consequence of Blackwell's equivalence theorem [5], also the more general case of dichotomies is completely characterized by a finite set of simple inequalities, which directly reduce to those of Hardy, Littlewood and Polya if q 1 and q 2 are both taken to be uniform. Also in this more general scenario, a relative Lorenz curve can be associated to each dichotomy, and the preorder visualized accordingly [45].In relation to quantum information sciences, while the preorder of majorization has found early applications in entanglement theory [39], the notion of relative majorization have started being employed only more recently, especially due to its applications in quantum thermodynamics [24,6,7,45,9,21] and generalized resource theories [13]. In the quantum setting, the objects of comparison are quantum dichotomies, namely, pairs of density matrices.Let us consider two arbitrary finite-dimensional quantum dichotomies (ρ 1 , σ 1 ) and (ρ 2 , σ 2 ).…”

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confidence: 99%

An information-theoretic treatment of quantum dichotomies

Buscemi

Sutter

Tomamichel

2019

Quantum

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Given two pairs of quantum states, we want to decide if there exists a quantum channel that transforms one pair into the other. The theory of quantum statistical comparison and quantum relative majorization provides necessary and sufficient conditions for such a transformation to exist, but such conditions are typically difficult to check in practice. Here, by building upon work by Matsumoto, we relax the problem by allowing for small errors in one of the transformations. In this way, a simple sufficient condition can be formulated in terms of one-shot relative entropies of the two pairs. In the asymptotic setting where we consider sequences of state pairs, under some mild convergence conditions, this implies that the quantum relative entropy is the only relevant quantity deciding when a pairwise state transformation is possible. More precisely, if the relative entropy of the initial state pair is strictly larger compared to the relative entropy of the target state pair, then a transformation with exponentially vanishing error is possible. On the other hand, if the relative entropy of the target state is strictly larger, then any such transformation will have an error converging exponentially to one. As an immediate consequence, we show that the rate at which pairs of states can be transformed into each other is given by the ratio of their relative entropies. We discuss applications to the resource theories of athermality and coherence. * marco.tomamichel@uts.edu.au 1 A transformation is said to be bistochastic if it transforms probability distributions to probability distributions, while keeping the uniform distribution fixed.The majorization preorder is particularly relevant and useful because of a famous result by Hardy, Littlewood, and Polya, according to which the relation p 1 p 2 can be expressed in terms of a finite set of inequalities [22] of the form f i ( p 1 ) ≥ f i ( p 2 ), intuitively capturing the idea that p 1 is "less uniform" than p 2 . Such inequalities can be conveniently visualized by plotting the Lorenz curve of p 1 versus that of p 2 [32].As it involves the comparison of two probability distributions relative to a third one (i.e., the uniform distribution), the majorization preorder is naturally generalized by considering two pairs of probability distributions, that is, two dichotomies ( p 1 , q 1 ) and ( p 2 , q 2 ), where now q 1 and q 2 are arbitrary distributions. One then writes ( p 1 , q 1 ) ( p 2 , q 2 ) whenever there exists a stochastic transformation simultaneously mapping p 1 to p 2 and q 1 to q 2 . As a consequence of Blackwell's equivalence theorem [5], also the more general case of dichotomies is completely characterized by a finite set of simple inequalities, which directly reduce to those of Hardy, Littlewood and Polya if q 1 and q 2 are both taken to be uniform. Also in this more general scenario, a relative Lorenz curve can be associated to each dichotomy, and the preorder visualized accordingly [45].In relation to quantum information sciences, while the preorder of majoriz...

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Quantum majorization and a complete set of entropic conditions for quantum thermodynamics

Cited by 140 publications

References 97 publications

One-Shot Operational Quantum Resource Theory

One-Shot Operational Quantum Resource Theory

General Resource Theories in Quantum Mechanics and Beyond: Operational Characterization via Discrimination Tasks

An information-theoretic treatment of quantum dichotomies

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