Mixed State Entanglement and Quantum Error Correction
(August 1996)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 $|\xi\rangle$ 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 $M(\chi)$ (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 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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