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arXiv:1708.00381 · Physical Review Letters

Quantifying Resources in General Resource Theory with Catalysts

Anurag Anshu, Min-Hsiu Hsieh, Rahul Jain

Published Nov 9, 2018• 66 citations
arXivHTMLPDF

Abstract

A question that is commonly asked in all areas of physics is how a certain property of a physical system can be used to achieve useful tasks and how to quantify the amount of such a property in a meaningful way. We answer this question by showing that, in a general resource-theoretic framework that allows the use of free states as catalysts, the amount of "resources" contained in a given state, in the asymptotic scenario, is equal to the regularized relative entropy of a resource of that state. While we need to place a few assumptions on our resource-theoretical framework, it is still sufficiently general, and its special cases include quantum resource theories of entanglement, coherence, asymmetry, athermality, nonuniformity, and purity. As a by-product, our result also implies that the amount of noise one has to inject locally to erase all the entanglement contained in an entangled state is equal to the regularized relative entropy of entanglement.

DOI
10.1103/physrevlett.121.190504

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A question that is commonly asked in all areas of physics is how a certain property of a physical system can be used to achieve useful tasks and how to quantify the amount of such a property in a meaningful way. We answer this question by showing that, in a general resource-theoretic framework that allows the use of free states as catalysts, the amount of "resources" contained in a given state, in the asymptotic scenario, is equal to the regularized relative entropy of a resource of that state. While we need to place a few assumptions on our resource-theoretical framework, it is still sufficiently general, and its special cases include quantum resource theories of entanglement, coherence, asymmetry, athermality, nonuniformity, and purity. As a by-product, our result also implies that the amount of noise one has to inject locally to erase all the entanglement contained in an entangled state is equal to the regularized relative entropy of entanglement.
In recent years it has been recognized that properties of physical systems such as entanglement, athermality, and asymmetry, can be viewed as resources for important tasks in quantum information, thermodynamics, and other areas of physics. This recognition was followed by the development of specific quantum resource theories (QRTs), such as entanglement theory, determining how quantum states that cannot be prepared under certain restrictions may be manipulated and used to circumvent the restrictions. Here we discuss the general structure of QRTs, and show that under a few assumptions (such as convexity of the set of free states), a QRT is asymptotically reversible if its set of allowed operations is maximal, that is, if the allowed operations are the set of all operations that do not generate (asymptotically) a resource. In this case, the asymptotic conversion rate is given in terms of the regularized relative entropy of a resource which is the unique measure or quantifier of the resource in the asymptotic limit of many copies of the state. This measure also equals the smoothed version of the logarithmic robustness of the resource.
We review the basic idea behind resource theories, where we quantify quantum resources by specifying a restricted class of operations. This divides the state space into various sets, including states which are free (because they can be created under the class of operations), and those which are a resource (because they cannot be). One can quantify the worth of the resource by the relative entropy distance to the set of free states, and under certain conditions, this is a unique measure which quantifies the rate of state to state transitions. The framework includes entanglement, asymmetry and purity theory. It also includes thermodynamics, which is a hybrid resource theory combining purity theory and asymmetry. Another hybrid resource theory which merges purity theory and entanglement can be used to study quantumness of correlations and discord, and we present quantumness in this more general framework of resource theories.
We review the basic idea behind resource theories, where we quantify quantum resources by specifying a restricted class of operations. This divides the state space into various sets, including states which are free (because they can be created under the class of operations), and those which are a resource (because they cannot be). One can quantify the worth of the resource by the relative entropy distance to the set of free states, and under certain conditions, this is a unique measure which quantifies the rate of state to state transitions. The framework includes entanglement, asymmetry and purity theory. It also includes thermodynamics, which is a hybrid resource theory combining purity theory and asymmetry. Another hybrid resource theory which merges purity theory and entanglement can be used to study quantumness of correlations and discord, and we present quantumness in this more general framework of resource theories.
Quantum physics currently provides our deepest description of nature. To understand precisely how it differs from classical physics is not only of foundational interest, but also important to advance technology: the advent of quantum information has led to the insight that certain quantum phenomena, e.g., entanglement and coherence, are resources which enable us to solve practical tasks that are unsolvable within the realm of classical physics. But how can we quantify quantum resources and how do we decide which resource is relevant for a specific operational advantage in quantum cryptography, quantum computation, or quantum metrology? How can we employ these resources optimally? Addressing these questions is the motivation for the development of quantum resource theories, a mathematically rigorous framework that allows us to investigate a specific quantum phenomenon as a resource. Formally, such theories emerge from restrictions which are imposed in addition to the laws of quantum mechanics. A prominent example is the restriction to local operations and classical communication, which singles out entanglement as a resource. Typically, such theories describe the value of quantum states, i.e., static resources. An important question in such theories concerns their structure as determined by the restriction under consideration, which is analyzed for coherence theory on qubits in this thesis. Only recently, so-called dynamical resource theories that quantify the value of operations were proposed, which are advantageous for several reasons: they include resource theories of states as special cases, allow us to quantify additional quantum properties such as the detection or usage of static resources, which will be a main topic of this thesis, and are more natural from an operational perspective. But are different types of dynamical resources present in quantum operations related? For the case of dynamical coherence and entanglement, it will be shown that there is not only a quantitative, but also a qualitative connection.
The ideas of thermodynamics have proved fruitful in the setting of quantum information theory, in particular the notion that when the allowed transformations of a system are restricted, certain states of the system become useful resources with which one can prepare previously inaccessible states. The theory of entanglement is perhaps the best-known and most well-understood resource theory in this sense. Here, we return to the basic questions of thermodynamics using the formalism of resource theories developed in quantum information theory and show that the free energy of thermodynamics emerges naturally from the resource theory of energy-preserving transformations. Specifically, the free energy quantifies the amount of useful work which can be extracted from asymptotically many copies of a quantum system when using only reversible energy-preserving transformations and a thermal bath at fixed temperature. The free energy also quantifies the rate at which resource states can be reversibly interconverted asymptotically, provided that a sublinear amount of coherent superposition over energy levels is available, a situation analogous to the sublinear amount of classical communication required for entanglement dilution.
We show that the minimal rate of noise needed to catalytically erase the entanglement in a bipartite quantum state is given by the regularized relative entropy of entanglement. This offers a solution to the central open question raised in [Groisman et al., Phys. Rev. A 72, 032317 (2005)PLRAAN1050-294710.1103/PhysRevA.72.032317] and complements their main result that the minimal rate of noise needed to erase all correlations is given by the quantum mutual information. We extend our discussion to the tripartite setting where we show that an asymptotic rate of noise given by the regularized relative entropy of recovery is sufficient to catalytically transform the state to a locally recoverable version of the state.