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IEEE International Symposium on Information Theory (ISIT)

A tight lower bound on the classical communication cost of entanglement dilution

Aram W. Harrow, Hoi‐Kwong Lo

Published Jan 1, 2003• 8 citations
Publisher

Abstract

In this paper, we will discuss the resource of entanglement specifically, bipartite pure state entanglement. This sort of entanglement was proposed as asymptotically fungible (and thus a resource) by BBPS algorithm.
DOI
10.1109/isit.2003.1228446

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Entanglement bits or ``ebits'' have been proposed as a quantitative measure of a fundamental resource in quantum information processing. It is, thus, important to show that the same number of ebits in different forms are interconvertible in the asymptotic limit. Here we draw attention to a very important but hitherto unnoticed aspect of entanglement manipulation---the classical communication cost. We construct an explicit procedure which demonstrates that for bipartite pure states, in the asymptotic limit, entanglement can be concentrated or diluted with vanishing classical communication cost. Entanglement of bipartite pure states is thus established as a truly interconvertible resource.
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The quantum entanglement is the necessary resource,which could realize some quantum tasks.It is very important to study the algorithm for the entanglement transformation between the quantum states determinately or probability.An algorithm first is presented that transforms one pure bipartite quantum state into a mixed bipartite by Local Operations and Classical Communication(LOCC),this algorithm is much simpler than the original algorithm.The algorithm for the transformation between two pure bipartite states with catalysis by LOCC is then discussed.
The theory of the asymptotic manipulation of pure bipartite quantum systems can be considered completely understood: the rates at which bipartite entangled states can be asymptotically transformed into each other are fully determined by a single number each, the respective entanglement entropy. In the multipartite setting, similar questions of the optimally achievable rates of transforming one pure state into another are notoriously open. This seems particularly unfortunate in the light of the revived interest in such questions due to the perspective of experimentally realizing multipartite quantum networks. In this Letter, we report substantial progress by deriving simple upper and lower bounds on the rates that can be achieved in asymptotic multipartite entanglement transformations. These bounds are based on ideas of entanglement combing and state merging. We identify cases where the bounds coincide and hence provide the exact rates. As an example, we bound rates at which resource states for the cryptographic scheme of quantum secret sharing can be distilled from arbitrary pure tripartite quantum states. This result provides further scope for quantum internet applications, supplying tools to study the implementation of multipartite protocols over quantum networks.
We derive lower limits on the inefficiency and classical communication costs of dilution between two-term bipartite pure states that are partially entangled. We first calculate explicit relations between the allowable error and classical communication costs of entanglement dilution using a previously described protocol, then consider a two-stage dilution from singlets with this protocol followed by some unknown protocol for conversion between partially entangled states. Applying overall lower bounds on classical communication and inefficiency to this two-stage protocol, we derive bounds for the unknown protocol. In addition we derive analogous (but looser) bounds for general pure states.