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Register a new userGeneral approach to quantum channel impossibility by local operations and classical communication
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Scott M. Cohen
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We describe a general approach to proving the impossibility of implementing a quantum channel by local operations and classical communication (LOCC), even with an infinite number of rounds, and find that this can often be demonstrated by solving a set of linear equations. The method also allows one to design an LOCC protocol to implement the channel whenever such a protocol exists in any finite number of rounds. Perhaps surprisingly, the computational expense for analyzing LOCC channels is not much greater than that for LOCC measurements. We apply the method to several examples, two of which provide numerical evidence that the set of quantum channels that are not LOCC is not closed and that there exist channels that can be implemented by LOCC either in one round or in three rounds that are on the boundary of the set of all LOCC channels. Although every LOCC protocol must implement a separable quantum channel, it is a very difficult task to determine whether or not a given channel is separable. Fortunately, prior knowledge that the channel is separable is not required for application of our method.
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We give a conceptually simple necessary condition such that a separable quantum operation can be implemented by local operations on subsystems and classical communication between parties (LOCC), a condition which follows from a novel approach to understanding LOCC. This necessary condition holds for any number of parties and any finite number of rounds of communication and as such, also provides a completely general sufficient condition that a given separable operation cannot be exactly implemented by LOCC. Furthermore, it demonstrates an extremely strong difference between separable operations and LOCC, in that there exist examples of the former for which the condition is extensively violated. More precisely, the violation by separable operations of our necessary condition for LOCC grows without limit as the number of parties increases.
Given a protocol ${cal P}$ that implements multipartite quantum channel ${cal E}$ by repeated rounds of local operations and classical communication (LOCC), we construct an alternate LOCC protocol for ${cal E}$ in no more rounds than ${cal P}$ and no more than a fixed, constant number of outcomes for each local measurement, the same constant number for every party and every round. We then obtain another upper bound on the number of outcomes that, under certain conditions, improves on the first. The latter bound shows that for LOCC channels that are extreme points of the convex set of all quantum channels, the parties can restrict the number of outcomes in their individual local measurements to no more than the square of their local Hilbert space dimension, $d_alpha$, suggesting a possible link between the required resources for LOCC and the convex structure of the set of all quantum channels. Our bounds on the number of outcomes indicating the need for only constant resources per round, independent of the number of rounds $r$ including when that number is infinite, are a stark contrast to the exponential $r$-dependence in the only previously published bound of which we are aware. If a lower bound is known on the number of product operators needed to represent the channel, we obtain a lower bound on the number of rounds required to implement the given channel by LOCC. Finally, we show that when the quantum channel is not required but only that a given task be implemented deterministically, then no more than $d_alpha^2$ outcomes are needed for each local measurement by party $alpha$.
Recent advances have lead towards first prototypes of a quantum internet in which entanglement is distributed by sources producing bipartite entangled states with high fidelities. This raises the question which states can be generated in quantum networks based on bipartite sources using local operations and classical communication. In this work we study state transformations under finite rounds of local operations and classical communication in networks based on maximally entangled two-qubit states. We first derive the symmetries for arbitrary network structures as these determine which transformations are possible. Then we show that contrary to tree graphs for which it has already been shown that any state within the same entanglement class can be reached there exist states which can be reached probabilistically but not deterministically if the network contains a cycle. Furthermore, we provide a systematic way to determine states which are not reachable in networks consisting of a cycle. Moreover, we provide a complete characterization of the states which can be reached in a cycle network with a protocol where each party measures only once and each step of the protocol results in a deterministic transformation. Finally, we present an example which cannot be reached with such a simple protocol.
We study the task of entanglement distillation in the one-shot setting under different classes of quantum operations which extend the set of local operations and classical communication (LOCC). Establishing a general formalism which allows for a straightforward comparison of their exact achievable performance, we relate the fidelity of distillation under these classes of operations with a family of entanglement monotones and the rates of distillation with a class of smoothed entropic quantities based on the hypothesis testing relative entropy. We then characterise exactly the one-shot distillable entanglement of several classes of quantum states and reveal many simplifications in their manipulation. We show in particular that the $varepsilon$-error one-shot distillable entanglement of any pure state is the same under all sets of operations ranging from one-way LOCC to separability-preserving operations or operations preserving the set of states with positive partial transpose, and can be computed exactly as a quadratically constrained linear program. We establish similar operational equivalences in the distillation of isotropic and maximally correlated states, reducing the computation of the relevant quantities to linear or semidefinite programs. We also show that all considered sets of operations achieve the same performance in environment-assisted entanglement distillation from any state.
We consider an infinite class of unambiguous quantum state discrimination problems on multipartite systems, described by Hilbert space $cal{H}$, of any number of parties. Restricting consideration to measurements that act only on $cal{H}$, we find the optimal global measurement for each element of this class, achieving the maximum possible success probability of $1/2$ in all cases. This measurement turns out to be both separable and unique, and by our recently discovered necessary condition for local quantum operations and classical communication (LOCC), it is easily shown to be impossible by any finite-round LOCC protocol. We also show that, quite generally, if the input state is restricted to lie in $cal{H}$, then any LOCC measurement on an enlarged Hilbert space is effectively identical to an LOCC measurement on $cal{H}$. Therefore, our necessary condition for LOCC demonstrates directly that a higher success probability is attainable for each of these problems using general separable measurements as compared to that which is possible with any finite-round LOCC protocol.
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