Mixed State Entanglement and Quantum Error Correction

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

doi:10.1103/PhysRevA.54.3824

paper

Mixed State Entanglement and Quantum Error Correction

arXiv:quant-ph/9604024 · doi:10.1103/PhysRevA.54.3824

Abstract

Entanglement purification protocols (EPP) and quantum error-correcting codes (QECC) 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 arbi- trary 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 EPR 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 $D_{1} (M)$ and $D_{2} (M)$ 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 communica- tion, 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 com- munication. 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 com- munication is available. We exhibit a family of codes based on universal hash- ing able toachieve 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 {\em iff} a QECC results in high fidelity for the case of no error the QECC can be recast into a form where the encoder is the matrix inverse of the decoder.

Resubmission with various corrections and expansions. See also http://vesta.physics.ucla.edu/~smolin/ for related papers and information. 82 pages latex including 19 postscript figures included using psfig macros

Mixed State Entanglement and Quantum Error Correction

Abstract

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