Approximate Quantum Error Correction
and Michael D. Westmoreland
Received January 16, 2002; accepted March 26, 2002
The errors that arise in a quantum channel can be corrected perfectly if and only if the
channel does not decrease the coherent information of the input state. We show that,
if the loss of coherent information is small, then approximate quantum error cor-
rection is possible.
KEY WORDS: quantum error correction.
PACS: 03.67.H; 03.65.U.
1. PERFECT QUANTUM ERROR CORRECTION
The problem of quantum information transfer via a channel can be cast as
the problem of sending quantum entanglement using the channel.
R and Q are subsystems of a composite quantum system initially in a pure
entangled state jÉ
i. This state has a Schmidt decomposition
where the ftjk
ig and fjk
ig are orthonormal sets of R and Q states,
System Q is transmitted from the sender to the receiver via a noisy
channel while R remains isolated at the sender’s end. In the noisy case, the
evolution of Q involves a unitary interaction with an environment system E
(initially in some state j0
i), leading to a ﬁnal joint state
1570-0755/02/0400-0005/0 # 2002 Plenum Publishing Corporation
Department of Physics, Kenyon College, Gambier, Ohio 43022.
Department of Mathematical Sciences, Denison University, Granville, Ohio 43023.
Quantum Information Processing, Vol. 1, Nos. 1/2, April 2002 (# 2002)