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      Mixed State Entanglement and Quantum Error Correction


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          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 \(|\xi\rangle\) can be transmitted at some rate Q through a noisy channel \(\chi\) without degradation. We prove that an EPP involving one- way classical communication and acting on mixed state \(\hat{M}(\chi)\) (obtained by sharing halves of EPR pairs through a channel \(\chi\)) yields a QECC on \(\chi\) 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.

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          23 April 1996
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          Resubmission with various corrections and expansions. See also for related papers and information. 82 pages latex including 19 postscript figures included using psfig macros


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