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Canadian Journal of Physics

Deterministic entanglement of assistance in quantum networks

Barry C. Sanders, Gilad Gour, DAVID A. MEYER

Published Jan 15, 2006• 0 citations
Publisher

Abstract

We present a powerful theorem for tripartite remote entanglement distribution protocols, which provides an operational interpretation of concurrence as a type of entanglement capacity, and we establish that concurrence of assistance, which we show is an entanglement monotone, identifies capabilities of and limitations to producing pure bipartite entangled states from pure tripartite entangled states. In addition, we show that, if concurrence of assistance for the pure tripartite state is at least as large as the concurrence of the desired pure bipartite state, then the former may be transformed to the latter via local operations and classical communication, and we calculate the maximum probability for this transformation when this condition is not met.PACS Nos.: 03.67.Mn, 03.67.Hk, 03.65.Ud

DOI
10.1139/p06-019

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We present a powerful theorem for tripartite remote entanglement distribution protocols, which provides an operational interpretation of concurrence as a type of entanglement capacity, and we establish that concurrence of assistance, which we show is an entanglement monotone, identifies capabilities of and limitations to producing pure bipartite entangled states from pure tripartite entangled states. In addition, we show that, if concurrence of assistance for the pure tripartite state is at least as large as the concurrence of the desired pure bipartite state, then the former may be transformed to the latter via local operations and classical communication, and we calculate the maximum probability for this transformation when this condition is not met.PACS Nos.: 03.67.Mn, 03.67.Hk, 03.65.Ud
Certain quantum-information tasks require entanglement of assistance, namely, a reduction of a tripartite entangled state to a bipartite entangled state via local measurements. We establish that concurrence of assistance (COA) identifies capabilities of and limitations to producing pure bipartite entangled states from pure tripartite entangled states and prove that COA is an entanglement monotone for $(2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}n)$-dimensional pure states. Moreover, if the COA for the pure tripartite state is at least as large as the concurrence of the desired pure bipartite state, then the former may be transformed to the latter via local operations and classical communication, and we calculate the maximum probability for this transformation when this condition is not met.
We consider the maximum bipartite entanglement that can be distilled from a single copy of a multipartite mixed entangled state, where we focus mostly on $(d\ifmmode\times\else\texttimes\fi{}d\ifmmode\times\else\texttimes\fi{}n)$-dimensional tripartite mixed states. We show that this assisted entanglement, when measured in terms of the generalized concurrence (called the GGG concurrence), is (tightly) bounded by an entanglement monotone, which we call the GGG concurrence of assistance. The GGG concurrence is one of the possible generalizations of the concurrence to higher dimensions, and for pure bipartite states it measures the geometric mean of the Schmidt numbers. For a large (nontrivial) class of $(d\ifmmode\times\else\texttimes\fi{}d)$-dimensional mixed states, we are able to generalize Wootters' formula for the concurrence into lower and upper bounds on the GGG concurrence. Moreover, we have found an explicit formula for the GGG concurrence of assistance that generalizes the expression for the concurrence of assistance for a large class of $(d\ifmmode\times\else\texttimes\fi{}d\ifmmode\times\else\texttimes\fi{}n)$-dimensional tripartite pure states.
We extend the definition of concurrence into a family of entanglement monotones, which we call concurrence monotones. We discuss their properties and advantages as computational manageable measures of entanglement, and show that for pure bipartite states all measures of entanglement can be written as functions of the concurrence monotones. We then show that the concurrence monotones provide bounds on quantum information tasks. As an example, we discuss their applications to remote entanglement distributions (RED) such as entanglement swapping and remote preparation of bipartite entangled states (RPBES). We prove a powerful theorem which states what kind of (possibly mixed) bipartite states or distributions of bipartite states cannot be remotely prepared. The theorem establishes an upper bound on the amount of GGG-concurrence (one member in the concurrence family) that can be created between two single-qudit nodes of quantum networks by means of tripartite RED. For pure bipartite states the bound on the GGG-concurrence can always be saturated by RPBES.
The entanglement of assistance quantifies the entanglement that can be generated between two parties, Alice and Bob, given assistance from a third party, Charlie, when the three share a tripartite state and where the assistance consists of Charlie initially performing a measurement on his share and communicating the result to Alice and Bob through a one-way classical channel. We argue that if this quantity is to be considered an operational measure of entanglement, then it must be understood to be a tripartite rather than a bipartite measure. We compare it with a distinct tripartite measure that quantifies the entanglement that can be generated between Alice and Bob when they are allowed to make use of a two-way classical channel with Charlie. We show that the latter quantity, which we call the entanglement of collaboration, can be greater than the entanglement of assistance. This demonstrates that the entanglement of assistance (considered as a tripartite measure of entanglement), and its multipartite generalizations such as the localizable entanglement, are not entanglement monotones, thereby undermining their operational significance.