Mixed State Entanglement and Quantum Error Correction

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 a yield $D$ from a bipartite mixed state $M$; in a QECC, an arbitrary 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 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 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 {\em iff} a QECC results in perfect 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.