Mixed-state entanglement and quantum error correction.

Entanglement purification protocols (EPP) and quantum errorcorrecting 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 bipartite mixed state M ; 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 M̂(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 D1(M) andD2(M) that can be locally distilled from it by EPPs using oneand two-way classical communication respectively, and give an exact expression for E(M) whenM is Bell-diagonal. While EPPs require classical communication, quantum channel coding does 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-errorcorrecting quantum block code. We prove that iff a QECC results in perfect fidelity for the case of the no-error error syndrome the QECC can be recast into a form where the encoder is the matrix inverse of the decoder.