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arXiv:1710.04565 · Open Systems & Information Dynamics

On GKLS Dynamics for Local Operations and Classical Communication

Lajos Diósi, Antoine Tilloy

Published Dec 1, 2017• 8 citations
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Abstract

We define a time continuous version of the concept of “local operations and classical communication” (LOCC), ubiquitous in quantum information theory. It allows us to construct GKLS master equations for particle systems that have (1) an arbitrary pair potential, and (2) local decoherence terms, but that do not entangle the constituents. The local decoherence terms take a particularly simple form if a principle of least decoherence is applied.

DOI
10.1142/s1230161217400200

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We define a time continuous version of the concept of “local operations and classical communication” (LOCC), ubiquitous in quantum information theory. It allows us to construct GKLS master equations for particle systems that have (1) an arbitrary pair potential, and (2) local decoherence terms, but that do not entangle the constituents. The local decoherence terms take a particularly simple form if a principle of least decoherence is applied.
Observations of quantum systems carried out by finite observers who subsequently communicate their results using classical data structures can be described as “local operations, classical communication” (LOCC) observations. The implementation of LOCC observations by the Hamiltonian dynamics prescribed by minimal quantum mechanics is investigated. It is shown that LOCC observations cannot be described using decoherence considerations alone, but rather require the a priori stipulation of a positive operator-valued measure (POVM) about which communicating observers agree. It is also shown that the transfer of classical information from system to observer can be described in terms of system-observer entanglement, raising the possibility that an apparatus implementing an appropriate POVM can reveal the entangled system-observer states that implement LOCC observations.
We study the finite dimensional C∗C^{*}C∗-algebras and their representation theory. The physical description of quantum local operations and classical communication (LOCC) and its schematics are presented. Focusing on the mathematical description of one-way LOCC, we give detailed analysis of recently derived operator relations in quantum information theory. We also show how functional analytic tools such as operatorx systems, operator algebras, and Hilbert C∗C^{*}C∗-modules all naturally emerge in this setting. We make use of these structures to derive some key results in one-way LOCC. Perfect distinguishability of one-way LOCC versus arbitrary quantum operations is analyzed. It turns out that they are equivalent for several families of operators that appear jointly in matrix and operator theory and quantum information theory. The main results of this work are contained in the paper \\citep{comfort}.
In the absence of quantum channels, local operations on subsystems and classical communication between parties (LOCC) constitute the most general protocols available on spatially separated quantum systems. Every LOCC protocol implements a separable quantum measurement, but it is known that there exist separable measurements that cannot be implemented by LOCC. A longstanding problem in quantum information theory is to understand the difference between LOCC and the full set of separable measurements. Toward this end, we show in this paper how to construct an LOCC protocol to implement an arbitrary separable measurement whenever such a protocol exists. In addition, given a measurement that cannot be implemented by LOCC within some fixed maximum number of rounds, the method shows explicitly that this is the case.
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 apply these results to the task of assisted coherence distillation, where the aim is to locally extract maximally coherent states. For all pure states and certain mixed states we observe a collapse of all classes: the distillation rate is always the same. It remains open if the distillation rate can be different for general mixed states, and if the intersection of LOCC and SI is equivalent to LICC. We also introduce the task of incoherent teleportation and show that perfect incoherent teleportation of one qubit is possible with one singlet and two bits of classical communication. The approach presented here opens new ways to study the resource theory of coherence in distributed scenarios.
In this work (multipartite) entanglement, discord, and coherence are unified as different aspects of a single underlying resource theory defined through simple and operationally meaningful elemental operations. This is achieved by revisiting the resource theory defining entanglement, local operations, and classical communication (LOCC), placing the focus on the underlying quantum nature of the communication channels. Taking the natural elemental operations in the resulting generalization of LOCC yields a resource theory that singles out coherence in the wire connecting the spatially separated systems as an operationally useful resource. The approach naturally allows us to consider a reduced setting as well, namely, the one with only the wire connected to a single quantum system, which leads to discordlike resources. The general form of free operations in this latter setting is derived and presented as a closed form. We discuss in what sense the present approach defines a resource theory of quantum discord and in which situations such an interpretation is sound—and why in general discord is not a resource. This unified and operationally meaningful approach makes transparent many features of entanglement that in LOCC might seem surprising, such as the possibility to use a particle to entangle two parties, without it ever being entangled with either of them, or that there exist different forms of multipartite entanglement.