Suppose two distant observers, Alice and Bob, share some form of entanglement - quantum correlations - in some bipartite pure quantum states. They may apply local operations and classical communication to convert one form of entanglement to another. Since entanglement is regarded as a resource in quantum information processing, it is an important question to ask how much classical communication, which is also a resource, is needed in the inter-conversion process of entanglement. In this paper, we address this important question in the many-copy case. The inter-conversion process of entanglement is usually divided into two types: concentrating the entanglement from many partially entangled states into a smaller number of maximally entangled states (i.e., singlets) and the reverse process of diluting singlets into partially entangled states. It is known that entanglement concentration requires no classical communication, but the best prior art result for diluting to N copies of a partially entangled state requires an amount of communication on the order of /spl radic/N. Our main result is to prove that this prior art result is optimal up to a constant factor; any procedure for approximately creating N partially entangled states from singlets requires /spl Omega/(/spl radic/N) bits of classical communication. Previously not even a constant bound was known for approximate entanglement transformations. We also prove a lower bound on the inefficiency of the process: to dilute singlets to N copies of a partially entangled state, the entropy of entanglement must decrease by /spl Omega/(/spl radic/N). Moreover, we introduce two new tools - /spl delta/-significant subspaces and the standard form protocol reduction in entanglement manipulations. We hope that these two new tools will be useful in other work in quantum information theory.
We study the amount of communication needed for two parties to transform some given joint pure state into another one, either exactly or with some fidelity. Specifically, we present a method to lower bound this communication cost even when the amount of entanglement does not increase. Moreover, the bound applies even if the initial state is supplemented with unlimited entanglement in the form of EPR (Einstein-Podolsky-Rosen) pairs and the communication is allowed to be quantum mechanical. We then apply the method to the determination of the communication cost of asymptotic entanglement concentration and dilution. While concentration is known to require no communication whatsoever, the best known protocol for dilution, discovered by H.-K. Lo and S. Popescu [Phys. Rev. Lett. 83, 1459 (1999)], requires exchange of a number of bits that is of the order of the square root of the number of EPR pairs. Here we prove a matching lower bound of the same asymptotic order, demonstrating the optimality of the Lo-Popescu protocol up to a constant factor and establishing the existence of a fundamental asymmetry between the concentration and dilution tasks. We also discuss states for which the minimal communication cost is proportional to their entanglement, such as the states recently introduced in the context of ``embezzling entanglement'' (W. van Dam and P. Hayden, e-print quant-ph/0201041).
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.
We study the following task: Alice is given a classical description of a rank-
k projector
P on
Cd, and Alice and Bob want to prepare the quantum state
P/k on Bob's side using shared entanglement and classical communication. The general form of this task is known as remote state preparation (RSP). We give nearly-matching lower and upper bounds for the entanglement cost and communication cost for RSP of the states
P/k. Ours are the first nearly matching upper and lower bounds for RSP of mixed states, and in the special case of pure states, our lower bound outperforms the best previously known lower bound. Our results show that any pure entangled state that can be used to do RSP of these states with
o(d) bits of communication, can distill
logd ebits of entanglement, and conversely, any state that can distill
logd ebits of entanglement can be used to do RSP of these states efficiently. As applications of our results, we rederive a previously-known incompressibility result for states of the form
P/k, and give a new entanglement-assisted communication protocol for the equality function that uses
21logn+O(1) many ebits, and
O(1) communication.
This article will examine states that superpose different amounts of entanglement and protocols that run in superposition but generate or consume different amounts of entanglement. In both cases we find a uniquely quantum difficulty: entanglement cannot be conditionally discarded without either using communication or causing decoherence. I will first describe the problem of entanglement spread in states and operations, as well as some methods of dealing with it. Then I'll describe three applications to problems that at first glance appear to be quite different: first, a reinterpretation of the old observation that creating n partially entangled states from singlets requires theta(sqrt(n)) communication, but cannot itself be used to communicate; second, a new lower bound technique for communication complexity; third, an explanation of how to extend the quantum reverse Shannon theorem from tensor power sources to general sources.