Mixed-state entanglement and quantum error correction

Phys. Rev. A 54, 3824 – Published 1 November 1996
Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters

Abstract

Entanglement purification protocols (EPPs) and quantum error-correcting codes (QECCs) provide two ways of protecting quantum states from interaction with the environment. In an EPP, perfectly entangled pure states are extracted, with some yield D, from a mixed state M shared by two parties; with a QECC, an arbitrary quantum state |ξ〉 can be transmitted at some rate Q through a noisy channel χ without degradation. We prove that an EPP involving one-way classical communication and acting on mixed state M^(χ) (obtained by sharing halves of Einstein-Podolsky-Rosen pairs through a channel χ) yields a QECC on χ with rate Q=D, and vice versa. We compare the amount of entanglement E(M) required to prepare a mixed state M by local actions with the amounts D1(M) and D2(M) that can be locally distilled from it by EPPs using one- and two-way classical communication, respectively, and give an exact expression for E(M) when M is Bell diagonal. While EPPs require classical communication, QECCs do not, and we prove Q is not increased by adding one-way classical communication. However, both D and Q can be increased by adding two-way communication. We show that certain noisy quantum channels, for example a 50% depolarizing channel, can be used for reliable transmission of quantum states if two-way communication is available, but cannot be used if only one-way communication is available. We exhibit a family of codes based on universal hashing able to achieve an asymptotic Q (or D) of 1-S for simple noise models, where S is the error entropy. We also obtain a specific, simple 5-bit single-error-correcting quantum block code. We prove that iff a QECC results in high fidelity for the case of no error then the QECC can be recast into a form where the encoder is the matrix inverse of the decoder. © 1996 The American Physical Society.

DOI: http://dx.doi.org/10.1103/PhysRevA.54.3824

  • Received 23 April 1996
  • Published in the issue dated November 1996

© 1996 The American Physical Society

Authors & Affiliations

Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters

  • IBM Research Division, Yorktown Heights, New York 10598
  • Physics Department, University of California at Los Angeles, Los Angeles, California 90024
  • Physics Department, Williams College, Williamstown, Massachusetts 01267

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