Abstract
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 a 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.
Highlights
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
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
One may ask what characterizes a quantum channel’s ability to create and/or increase entanglement between the subsystems upon which it acts. One significant such characterization is that the entanglement cannot increase when the channel can be simulated by local quantum operations and classical communication (LOCC), which are the only operations that can be implemented by spatially separated parties who lack the means to bring their subsystems together in a single laboratory
Summary
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. We show that the method of [47] for measurements is readily extended to the case of multipartite quantum channels, allowing one to design an LOCC protocol to implement a given channel whenever this is possible in a finite number of rounds, with a computational effort for the case of channels that is never more than a quadratic increase over that needed for measurements.
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