The entanglement-assisted classical capacity of a noisy quantum channel (C E ) is the amount of information per channel use that can be sent over the channel in the limit of many uses of the channel, assuming that the sender and receiver have access to the resource of shared quantum entanglement, which may be used up by the communication protocol. We show that the capacity C E is given by an expression parallel to that for the capacity of a purely classical channel: i.e., the maximum, over channel inputs ρ, of the entropy of the channel input plus the entropy of the channel output minus their joint entropy, the latter being defined as the entropy of an entangled purification of ρ after half of it has passed through the channel. We calculate entanglement-assisted capacities for two interesting quantum channels, the qubit amplitude damping channel and the bosonic channel with amplification/attenuation and Gaussian noise. We discuss how many independent parameters are required to completely characterize the asymptotic behavior of a general quantum channel, alone or in the presence of ancillary resources such as prior entanglement. In the classical analog of entanglement assisted communication-communication over a discrete memoryless channel (DMC) between parties who share prior random information-we show that one parameter is sufficient, i.e., that in the presence of prior shared random information, all DMC's of equal capacity can simulate one another with unit asymptotic efficiency.

Prior entanglement between sender and receiver, which exactly doubles the classical capacity of a noiseless quantum channel, can increase the classical capacity of some noisy quantum channels by an arbitrarily large constant factor depending on the channel, relative to the best known classical capacity achievable without entanglement. The enhancement factor is greatest for very noisy channels, with positive classical capacity but zero quantum capacity. Although such quantum channels can be simulated classically, no violation of causality is implied, because the simulation requires at least as much forward classical communication as the entanglement-assisted classical capacity of the channel being simulated. We obtain exact expressions for the entanglement-assisted capacity of depolarizing and erasure channels in d dimensions.Comment: 4 pages including 3 figures. Eq 4 restored to be an equality, not merely the lower bound claimed in v4; reference added; one figure removed to save spac

Hoping to simplify the classification of pure entangled states of multi (m)-partite quantum systems, we study exactly and asymptotically (in n) reversible transformations among n th tensor powers of such states (ie n copies of the state shared among the same m parties) under local quantum operations and classical communication (LOCC). For exact transformations, we show that two states whose marginal one-party entropies agree are either locally-unitarily (LU) equivalent or else LOCCincomparable. In particular we show that two tripartite Greenberger-Horne-Zeilinger (GHZ) states are LOCC-incomparable to three bipartite Einstein-Podolsky-Rosen (EPR) states symmetrically shared among the three parties. Asymptotic transformations yield a simpler classification than exact transformations; for example, they allow all pure bipartite states to be characterized by a single parameter-their partial entropy-which may be interpreted as the number of EPR pairs asymptotically interconvertible to the state in question by LOCC transformations. We show that m-partite pure states having an m-way Schmidt decomposition are similarly parameterizable, with the partial entropy across any nontrivial partition representing the number of standard "Cat" states | 0 ⊗m +| 1 ⊗m asymptotically interconvertible to the state in question. For general m-partite states, partial entropies across different partitions need not be equal, and since partial entropies are conserved by asymptotically reversible LOCC operations, a multicomponent entanglement measure is needed, with each scalar component representing a different kind of entanglement, not asymptotically interconvertible to the other kinds. In particular we show that the m=4 Cat state is not isentropic to, and therefore not asymptotically interconvertible to, any combination of bipartite and tripartite states shared among the four parties. Thus, although the m=4 cat state can be prepared from bipartite EPR states, the preparation process is necessarily irreversible, and remains so even asymptotically. For each number of parties m we define a minimal reversible entanglement generating set (MREGS) as a set of states of minimal cardinality sufficient to generate all m-partite pure states by asymptotically reversible LOCC transformations. Partial entropy arguments provide lower bounds on the size of the MREGS, but for m > 2 we know no upper bounds. We briefly consider several generalizations of LOCC transformations, including transformations with some probability of failure, transformations with the catalytic assistance of states other than the states we are trying to transform, and asymptotic LOCC transformations supplemented by a negligible (o(n)) amount of quantum communication.

We exhibit a two-parameter family of bipartite mixed states ρ bc , in a d ⊗ d Hilbert space, which are negative under partial transposition (NPT), but for which we conjecture that no maximally entangled pure states in 2 ⊗ 2 can be distilled by local quantum operations and classical communication (LQ+CC). Evidence for this undistillability is provided by the result that, for certain states in this family, we cannot extract entanglement from any arbitrarily large number of copies of ρ bc using a projection on 2 ⊗ 2. These states are canonical NPT states in the sense that any bipartite mixed state in any dimension with NPT can be reduced by LQ+CC operations to an NPT state of the ρ bc form. We show that the main question about the distillability of mixed states can be formulated as an open mathematical question about the properties of composed positive linear maps.

Abstract. The newfound importance of "entanglement as a resource" in quantum computation and quantum communication behooves us to quantify it in as many distinct ways as possible. Here we explore a new quantification of entanglement of a general (mixed) quantum state for a bipartite system, which we name entanglement of assistance. We show it to be the maximum of the average entanglement over all ensembles consistent with the density matrix describing the bipartite state. With the help of lower and upper bounds we calculate entanglement of assistance for a few cases and use these results to show the surprising property of superadditivity. We believe that this may throw some light on the question of additivity of entanglement of formation.