On the Structure of Additive Quantum Codes and the Existence of Nonadditive Codes

Details On the Structure of Additive Quantum Codes and the Existence of Nonadditive Codes On the Structure of Additive Quantum Codes and the Existence of Nonadditive Codes

1997-10-09

arxiv.org/abs/quant-ph/9710031v1

Physics

Quantum Physics

21 pages

Scientific paper

We first present a useful characterization of additive (stabilizer) quantum error-correcting codes. Then we present several examples of We first present a useful characterization of additive (stabilizer) quantum error--correcting codes. Then we present several examples of nonadditive codes. We show that there exist infinitely many non-trivial nonadditive codes with different minimum distances, and high rates. In fact, we show that nonadditive codes that correct t errors can reach the asymptotic rate R=1-2H(2t/n), where H(x) is the binary entropy function. Finally, we introduce the notion of strongly nonadditive codes (i.e., quantum codes with the following property: the trivial code consisting of the entire Hilbert space is the only additive code that is equivalent to any code containing the given code), and provide a construction for an ((11,2,3)) strongly nonadditive code.

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