Operational Resource Theory of Coherence

Abstract

We establish an operational theory of coherence (or of superposition) in quantum systems, by focusing on the optimal rate of performance of certain tasks. Namely, we introduce the two basic concepts -"coherence distillation" and "coherence cost" in the processing quantum states under so-called incoherent operations [Baumgratz/Cramer/Plenio, Phys. Rev. Lett. 113:140401 (2014)]. We then show that in the asymptotic limit of many copies of a state, both are given by simple singleletter formulas: the distillable coherence is given by the relative entropy of coherence (in other words, we give the relative entropy of coherence its operational interpretation), and the coherence cost by the coherence of formation, which is an optimization over convex decompositions of the state. An immediate corollary is that there exists no bound coherent state in the sense that one would need to consume coherence to create the state but no coherence could be distilled from it. Further we demonstrate that the coherence theory is generically an irreversible theory by a simple criterion that completely characterizes all reversible states.PACS numbers: 03.65. Aa, 03.67.Mn Introduction.-The universality of the superposition principle is the fundamental non-classical characteristic of quantum mechanics: Given any configuration space X, its elements x label an orthogonal basis |x of a Hilbert space, and we have all superpositions x ψ x |x as the possible states of the system. In particular, we could choose a completely different orthonormal basis as an equally valid computational basis, in which to express the superpositions. However, often a basis is distinguished, be it the eigenbasis of an observable or of the system's Hamiltonian, so that conservation laws or even superselection rules apply. In such a case, the eigenstates |x are distinguished as "simple" and superpositions are "complex". Indeed, in the presence of conservation laws, structured superpositions of eigenstates can serve as so-called "reference frames", which are resources to overcome the conservation laws [1][2][3]. Based on this idea,Åberg [4] and more recently Baumgratz et al. [5] have proposed to consider any non-trivial superposition as a resource, and to create a theory in which computational basis states and their probabilistic mixtures are for free (or worthless), and operations preserving these "incoherent" states are free as well. This suggests that coherence theory can be regarded as a resource theory.Let us briefly recall the general structure of a quantum resource theory (QRT) and basic questions that be asked in a QRT through entanglement theory (ET), which is a well-known QRT. For our purposes, a QRT has three ingredients: (1) free states (separable state in ET), (2) resource states (entangled state in ET), (3) the restricted or free operations (LOCC in ET). A necessary requirement for a consistent QRT is that no resource state can be created from any free state under any free operation. The QRT is then the study of interconversion between resource states u...

Measures and applications of quantum correlations

“…The incoherent operations may be, for instance, those defined in [43], which admit an operator sum representation where all Kraus operators map the set of incoherent states into itself, or a more restricted set such as the so-called covariant operations, which have been defined in the context of the resource theory of asymmetry [69]; we defer the reader to recent studies where merits and setbacks of various proposals for incoherent operations are discussed [70,44,71,45,72,73]. We list a sample of coherence measures C η of relevance in various resource theoretic contexts in Table 5.…”

“…To be more precise, let us recall some basic terminology used in the characterisation of quantum coherence, see e.g. [42,43,44,45].…”

“…Measures of quantum coherence have been studied quite extensively in the last few years, see e.g. [42,69,43,48,49,71,44,45,46,82,83,70,47] and the very recent review [84]. While several connections between coherence, entanglement, and QCs have been pointed out quite recently, including quantitative interconversion schemes [85,49,86,46,47] somehow analogous to the entanglement activation framework [36,35], a consistent definition of QCs quantifiers in terms of coherence has not been reported in full generality (to our knowledge), and will be introduced here.…”

“…In entanglement theory, for example, LOCC are singled out by the physical paradigm of distant laboratories, and form a strict subset of the maximal set of operations preserving separable states [68] (as the latter also contain certain nonlocal operations such as the swap gate), while in the resource theories of quantum coherence there are currently several alternative proposals for free operations, with different characterisations, and a still ongoing debate on their physical consistency [69,43,70,44,71,45,72,73].…”

“…Similarly, the set I {|a A ,|b B } AB of incoherent-incoherent (or incoherent on A and B) states is defined as [48,49,44] I…”