On Subsystem Codes Beating the Hamming or Singleton Bound

Details On Subsystem Codes Beating the Hamming or Singleton Bound On Subsystem Codes Beating the Hamming or Singleton Bound

2007-03-22

arxiv.org/abs/quant-ph/0703213v1

Physics

Quantum Physics

18 pages more densely packed than classically possible

Scientific paper

10.1098/rspa.2007.0028

Subsystem codes are a generalization of noiseless subsystems, decoherence free subspaces, and quantum error-correcting codes. We prove a Singleton bound for GF(q)-linear subsystem codes. It follows that no subsystem code over a prime field can beat the Singleton bound. On the other hand, we show the remarkable fact that there exist impure subsystem codes beating the Hamming bound. A number of open problems concern the comparison in performance of stabilizer and subsystem codes. One of the open problems suggested by Poulin's work asks whether a subsystem code can use fewer syndrome measurements than an optimal MDS stabilizer code while encoding the same number of qudits and having the same distance. We prove that linear subsystem codes cannot offer such an improvement under complete decoding.

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