MIXED-STATE ENTANGLEMENT AND QUANTUM ERROR-CORRECTION

Entanglement purification protocols (EPPs) and quantum error-correctin g codes (QECCs) provide two ways of protecting quantum states from int eraction with the environment. In an EPP, perfectly entangled pure sta tes are extracted, with some yield D, from a mixed state IM shared by two parties; with a QECC, an arbitrary quantum state \xi] can be trans mitted at some rate Q through a noisy channel chi without degradation. We prove that an EPP involving one-way classical communication and ac ting on mixed state M(chi) (obtained by sharing halves of Einstein-Pod olsky-Rosen pairs through a channel chi) yields a QECC on chi with rat e Q=D, and vice versa. We compare the amount of entanglement E(M) requ ired to prepare a mixed slate 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 classi cal communication, QECCs do not, and we prove Q is not increased by ad ding one-way classical communication. However, both D and Q can be inc reased by adding two-way communication. We show that certain noisy qua ntum channels, for example a 50% depolarizing channel, can be used for reliable transmission of quantum states if two-way communication is a vailable, but cannot be used if only one-way communication is availabl e. We exhibit a family of codes based on universal hashing able to ach ieve 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-err or-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 int o a form where the encoder is the matrix inverse of the decoder.