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arXiv:quant-ph/9907021 · Physical Review Letters

Classical Information and Distillable Entanglement

Jens Eisert, Timo Felbinger, P. Papadopoulos, Martin Bodo Plenio, Martin Wilkens

Published Feb 14, 2000• 62 citations
arXivHTMLPDF

Abstract

We establish a quantitative connection between the amount of lost classical information about a quantum state and the concomitant loss of entanglement. Using methods that have been developed for the optimal purification of mixed states, we find a class of mixed states with known distillable entanglement. These results can be used to determine the quantum capacity of a quantum channel which randomizes the order of transmitted signals.

DOI
10.1103/physrevlett.84.1611

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We establish a quantitative connection between the amount of lost classical information about a quantum state and the concomitant loss of entanglement. Using methods that have been developed for the optimal purification of mixed states, we find a class of mixed states with known distillable entanglement. These results can be used to determine the quantum capacity of a quantum channel which randomizes the order of transmitted signals.
Based on a unified approach to all kinds of quantum capacities we show that the rate of quantum information transmission is bounded by the maximal attainable rate of coherent information. Moreover, we show that if for any bipartite state the one-way distillable entanglement is no less than coherent information, then one obtains Shannon-like formulas for all the capacities. The inequality also implies that the decrease of distillable entanglement due to the mixing process does not exceed that of the corresponding loss of information about a system.
Distillation and purification of entanglement is an important field of quantum information both in theory and in experiment. This paper presents some main conclusions in this field, and discusses mainly the distillation of entanglement from finite copies of a mixed state. By using the conceptions of quasiseparable state and distillable subspace, we show a necessary condition and a sufficient and necessary condition of the distillability from finite copies of a mixed state. We also get the most efficient protocols of entanglement distillation for the case of finite copies. Finally we consider the generalization of these results to the case of infinite copies of a mixed state.
Entanglement is an essential resource for quantum communication and quantum computation, similar to shared random bits in the classical world. Entanglement distillation extracts nearly-perfect entanglement from imperfect entangled state. The classical communication complexity of these protocols is the minimal amount of classical information that needs to be exchanged for the conversion. In this paper, we focus on the communication complexity of protocols that operate with incomplete information, i.e., where the inputs are mixed states and/or prepared adversarially. We consider three models of imperfect entanglement, namely, the bounded measurement model, the depolarization model, and the fidelity model. We describe these models as well as the motivations for studying them. For the bounded measurement model and the depolarization model, we prove tight and almost-tight bounds on the output quality of non-interactive protocols. For the fidelity model, we prove a lower bound that matches the upper bound given by Ambainis et al., and thus completely characterizes communication complexity of entanglement distillation protocols for this model. Our result also suggests the optimality of the BB84 protocol in terms of communication complexity. We emphasize that although some of the results appear intuitively straightforward, their proofs are not. In fact, two novel techniques are developed for proving these results. We believe that these techniques are of independent interests, too.
Abstract We consider the problem of invariance of distillable entanglement D and quantum capacities Q under erasure of information about single copy of quantum state or channel respectively. We argue that any 2 ⊗N two-way distillable state is still two-way distillable after erasure of single copy information. For some known distillation protocols the obtained two-way distillation rate is the same as if Alice and Bob knew the state from the very beginning. The isomorphism between quantum states and quantum channels is also investigated. In particular it is pointed out that any transmission rate down the channel is equal to distillation rate with formal LOCC-like superoperator that uses in general nonphysical Alice actions. This allows to we prove that if given channel Λ has nonzero capacity (Q → or Q ⟺) then the corresponding quantum state ϱ(Λ) has nonzero distillable entanglement (D → or D ⟺). Follwoing the latter arguments are provided that any channel mapping single qubit into N level system allows for reliable two-way transmission after erasure of information about single copy. Some open problems are discussed.
Abstract Entanglement is an essential resource for quantum com-munication and quantum computation, similar to shared random bits in the classical world. Entanglement distil-lation extracts nearly-perfect entanglement from imperfect entangled state. The classical communication complexity ofthese protocols is the minimal amount of classical information that needs to be exchanged for the conversion. In thispaper, we focus on the communication complexity of protocols that operate with incomplete information, i.e., wherethe inputs are mixed states and/or prepared adversarially. We consider three models of imperfect entanglement,namely, the bounded measurement model, the depolarization model, and the fidelity model. We describe there mod-els as well as the motivations for studying them. For the bounded measurement model and the depolarization model,we prove tight and almost-tight bounds on the output quality of non-interactive protocols. For the fidelity model weprove a lower bound that matches the upper bound given by Ambainis et al., and thus completely characterizes commu-nication complexity of entanglement distillation protocols for this model. Our result also suggests the optimality ofthe BB84 protocol in terms of communication complexity. We emphasize that although some of the results appearintuitively straightforward, their proofs are not. In fact, two novel techniques are developed for proving these results.We believe that these techniques are of independent interests, too.
An upper limit is given to the amount of quantum information that can be transmitted reliably down a noisy, decoherent quantum channel using the high-probability states of quantum sources. A class of quantum error-correcting codes is presented that allows the information transmitted to attain this limit. The result is a quantum analog of Shannon's bound and code for the noisy classical channel [C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (University of Illinois Press, Chicago, 1948)].