Entanglement-Assisted Quantum Error Correction
Mark Wilde
USC
Pre-shared entanglement between a sender and receiver enhances
the capacity of noisy quantum channel for sending quantum
information [1]. This capacity is the "entanglement-assisted
capacity" of a noisy quantum channel for an asymptotically large
number of independent uses of the channel. These asymptotic
arguments of quantum Shannon theory are useful for determining
capacity, but give no explicit construction of error correcting
codes for approaching capacity other than random coding
arguments.
Brun, Devetak, and Hsieh [2] developed the entanglement-assisted
stabilizer formalism for quantum error correction as an extension
of the standard stabilizer theory. The entanglement-assisted
stabilizer formalism is useful for constructing practical quantum
error correcting codes when a finite number of channel uses are
available and when the sender and receiver share entanglement.
The standard stabilizer theory restricts a quantum error
correcting code to be a simultaneous eigenstate of a commuting
set of Pauli operators. This restriction limits the classical
block codes which are useful for quantum error correction. The
classical codes must be self-orthogonal or dual-containing.
Preshared entanglement lifts this restriction of
self-orthogonality so that sender and receiver can use an
arbitrary classical code for correction of quantum error.
I first briefly review the stabilizer theory of quantum error
correction. I then discuss the entanglement-assisted stabilizer
formalism by means of the example in [2]. I give an algorithm
which automatically determines a Clifford encoding circuit for
the entanglement-assisted code and that also determines the
minimal number of ebits needed. I finally discuss the recent
extension of this theory to continuous-variable quantum
information [3] and hint at new directions for this theory that
lie in quantum convolutional coding for error correction and
entanglement distillation [4].
[1] Bennett et al., Phys. Rev. Lett. 83, 3081 - 3084 (1999).
[2] Brun, Devetak, Hsieh, Science 314, pp. 436 - 439 (2006).
[3] Wilde, Krovi, Brun, arXiv:0705.4314 (2007).
[4] Wilde, Krovi, Brun, in preparation (2007).