Reversible Framework for Quantum Resource Theories

Abstract

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. Classical and quantum information theories can be viewed as examples of theories of interconversions among different resources [1]. These resources are classified as being quantum or classical, dynamic or static, noisy or noiseless, and therefore enable a plethora of quantum information processing tasks [2,3]. For example, quantum teleportation can be viewed as a resource interconversion task in which one entangled bit (a quantum static noiseless resource) is transformed by local operations and classical communication (LOCC) into a single use of a quantum channel (a quantum dynamic noiseless resource) [4]. Just as the restriction of LOCC leads to the theory of entanglement [5], in general, every restriction on quantum operations defines a resource theory, determining how quantum states that cannot be prepared under the restriction may be manipulated and used to circumvent the restriction.The scope of quantum resource theories (QRTs) goes far beyond quantum information science. In recent years a lot of work has been done formulating QRTs in different areas of physics, such as the resource theory of athermality in quantum thermodynamics [6][7][8][9][10][11], the resource theory of asymmetry [12,13] (which led to generalizations of important theorems in physics such as Noether's theorem [13]), the resource theory of non-Gaussianity in quantum optics [14,15], the resource theory of stabilizer computation in quantum computing [16], noncontextuality in the foundations of quantum physics [17], and more recently it was suggested that non-Markovian evolution can be formulated as a resource theory [18]. In addition, tools and ideas from quantum resource theories have been applied in many-body physics (see, e.g., Ref. [19] and references therein), and even for a universal formulation of the uncertainty principle [20]. Furthermore, very recently an abstract formulation using concepts from categor...

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