We introduce and study the task of assisted coherence distillation. This task arises naturally in bipartite systems where both parties work together to generate the maximal possible coherence on one of the subsystems. Only incoherent operations are allowed on the target system while general local quantum operations are permitted on the other, an operational paradigm that we call local quantum-incoherent operations and classical communication (LQICC). We show that the asymptotic rate of assisted coherence distillation for pure states is equal to the coherence of assistance, an analog of the entanglement of assistance, whose properties we characterize. Our findings imply a novel interpretation of the von Neumann entropy: it quantifies the maximum amount of extra quantum coherence a system can gain when receiving assistance from a collaborative party. Our results are generalized to coherence localization in a multipartite setting and possible applications are discussed.PACS numbers: 03.65. Aa, 03.67.Mn Introduction. Quantum coherence represents a basic feature of quantum systems that is not present in the classical world. Recently, researchers have begun developing a resource-theoretic framework for understanding quantum coherence [1][2][3][4][5][6][7][8][9]. In this setting, coherence is regarded as a precious resource that cannot be generated or increased under a restricted class of operations known as incoherent operations [2,3]. A resource-theoretic treatment of coherence is physically motivated, in part, by certain processes in biology [10][11][12], transport theory [2,13,14], and thermodynamics [7,15,16], for which the presence of quantum coherence plays an important role.In this paper, we consider the task of assisted coherence distillation. It involves (at least) two parties, Alice (A) and Bob (B), who share one or many copies of some bipartite state ρ AB . Their goal is to maximize the quantum coherence of Bob's system by Alice performing arbitrary quantum operations on her subsystem, while Bob is restricted to just incoherent operations on his. The duo is further allowed to communicate classically with one another. Overall, we refer to the allowed set of operations in this protocol as Local Quantum-Incoherent operations and Classical Communication (LQICC). As we will show, the operational LQICC setting reveals fundamental properties about the quantum coherence accessible to Bob. In particular, the von Neumann entropy of his state, S (ρ B ), quantifies precisely how much extra coherence can be generated in Bob's subsystem using LQICC than when no communication is allowed between him and any correlated party.Alice and Bob's objective here is analogous to the task of assisted entanglement distillation. In the latter, entanglement is shared between three parties, A, B, C, and the goal is for B and C to obtain maximal bipartite entanglement when all parties use (unrestricted) Local Operations and Classical Communication (LOCC). The corresponding maximal entanglement that can be generated between B and C is ...
We show that trace distance measure of coherence is a strong monotone for all qubit and, so called, $X$ states. An expression for the trace distance coherence for all pure states and a semi definite program for arbitrary states is provided. We also explore the relation between $l_1$-norm and relative entropy based measures of coherence, and give a sharp inequality connecting the two. In addition, it is shown that both $l_p$-norm- and Schatten-$p$-norm-based measures violate the (strong) monotonicity for all $p\in(1,\infty)$.Comment: 7 pages, 1 figure; published versio
The search for a simple description of fundamental physical processes is an important part of quantum theory. One example for such an abstraction can be found in the distance lab paradigm: if two separated parties are connected via a classical channel, it is notoriously difficult to characterize all possible operations these parties can perform. This class of operations is widely known as local operations and classical communication (LOCC). Surprisingly, the situation becomes comparably simple if the more general class of separable operations is considered, a finding which has been extensively used in quantum information theory for many years. Here, we propose a related approach for the resource theory of quantum coherence, where two distant parties can only perform measurements which do not create coherence and can communicate their outcomes via a classical channel. We call this class local incoherent operations and classical communication (LICC). While the characterization of this class is also difficult in general, we show that the larger class of separable incoherent operations (SI) has a simple mathematical form, yet still preserving the main features of LICC. We demonstrate the relevance of our approach by applying it to three different tasks: assisted coherence distillation, quantum teleportation, and single-shot quantum state merging. We expect that the results obtained in this work also transfer to other concepts of coherence which are discussed in recent literature. The approach presented here opens new ways to study the resource theory of coherence in distributed scenarios. arXiv:1509.07456v3 [quant-ph]
Understanding the resource consumption in distributed scenarios is one of the main goals of quantum information theory. A prominent example for such a scenario is the task of quantum state merging where two parties aim to merge their parts of a tripartite quantum state. In standard quantum state merging, entanglement is considered as an expensive resource, while local quantum operations can be performed at no additional cost. However, recent developments show that some local operations could be more expensive than others: it is reasonable to distinguish between local incoherent operations and local operations which can create coherence. This idea leads us to the task of incoherent quantum state merging, where one of the parties has free access to local incoherent operations only. In this case the resources of the process are quantified by pairs of entanglement and coherence. Here, we develop tools for studying this process, and apply them to several relevant scenarios. While quantum state merging can lead to a gain of entanglement, our results imply that no merging procedure can gain entanglement and coherence at the same time. We also provide a general lower bound on the entanglement-coherence sum, and show that the bound is tight for all pure states. Our results also lead to an incoherent version of Schumacher compression: in this case the compression rate is equal to the von Neumann entropy of the diagonal elements of the corresponding quantum state.Introduction. While coherence has long been known in classical physics as a fundamental waves property [1], in quantum mechanics coherent superposition is elevated to a universal principle governing all processes. Indeed, the fact that all matter exhibits wave behavior was first understood by de Broglie [2], which became the basis of the now standard formulation of quantum mechanics in Schrödinger's wave equation [3]. The universality of the superposition principle, i.e. the tenet that any two valid states of a system can be superposed to form a new valid state, marks a radical departure from classical physics. It is at the heart of the many counterintuitive features of quantum theory, perhaps most famously in Schrödinger's Gedankenexperiment of the cat [4]. Quantum entanglement can be considered as a particular manifestation of coherence, and both of these nonclassical phenomena have led to extensive debates in the early days of quantum mechanics [5,6].While the study of the resource theory of entanglement has a long tradition [7,8], the resource theory of quantum coherence has been formulated only recently [9,10], although other attempts in this direction have been also presented earlier [11][12][13][14][15][16]. The basis of any resource theory are free states, these are states which can be created at no cost. In entanglement theory, these are all separable states. In coherence theory these are incoherent states [9], i.e., states which are diagonal in a fixed basis |i . The second important ingredient of any resource theory are free operations, i.e., operations...
Quantum coherence is an essential feature of quantum mechanics which is responsible for the departure between classical and quantum world. The recently established resource theory of quantum coherence studies possible quantum technological applications of quantum coherence, and limitations which arise if one is lacking the ability to establish superpositions. An important open problem in this context is a simple characterization for incoherent operations, constituted by all possible transformations allowed within the resource theory of coherence. In this work, we contribute to such a characterization by proving several upper bounds on the maximum number of incoherent Kraus operators in a general incoherent operation. For a single qubit, we show that the number of incoherent Kraus operators is not more than 5, and it remains an open question if this number can be reduced to 4. The presented results are also relevant for quantum thermodynamics, as we demonstrate by introducing the class of Gibbs-preserving strictly incoherent operations, and solving the corresponding mixedstate conversion problem for a single qubit.Quantum resource theories [1, 2] provide a strong framework for studying fundamental properties of quantum systems and their applications for quantum technology. The basis of any quantum resource theory is the definition of free states and free operations. Free states are quantum states which can be prepared at no additional cost, while free operations capture those physical transformations which can be implemented without consumption of resources. Having identified these two main features, one can study the basic properties of the corresponding theory, such as possibility of state conversion, resource distillation, and quantification. An important example is the resource theory of entanglement, where free states are separable states, and free operations are local operations and classical communication [3,4].In the resource theory of quantum coherence [5-9], free states are identified as incoherent statesi.e., states which are diagonal in a fixed specified basis {|i }. The choice of this basis depends on the particular problem under study, and in many relevant scenarios such a basis is naturally singled out by the unavoidable decoherence [10]. The definition of free operations within the theory of coherence is not unique, and several approaches have been discussed in the literature, based on different physical (or mathematical) considerations [8]. Two important frameworks are known as incoherent [6] and strictly incoherent operations [7,11], which will be denoted by IO and SIO, respectively. The characterizing feature of IO is the fact that they admit an incoherent Kraus decomposition, i.e., they can be written as [6] where each of the Kraus operators K j cannot create coherence individually, K j |m ∼ |n for suitable integers n and m. This approach is motivated by the fact that any quantum operation can be interpreted as a selective measurement in which outcome j occurs with probability p j = Tr[K j ρK ...
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