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).