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IEEE International Symposium on Information Theory (ISIT)

On upper bound of error probability by quantum reliability function

Yuki Ishida, Kentaro Kato, Tsuyoshi Sasaki Usuda

Published Nov 20, 2005• 0 citations
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In this paper, the sphere packing bound of Fano, Shannon, Gal- lager and Berlekamp and the zero-rate bound of Berlekamp are extended to general classical-quantum channels. The upper bound for the reliability func- tion obtained from the sphere packing coincides at high rates, for the case of pure state channels, with a lower bound derived by Burnashev and Holevo (21). Thus, for pure state channels, the reliability function at high rates is now ex- actly determined. For the general case, the obtained upper bound expression at high rates was conjectured to represent also a lower bound to the reliability function, but a complete proof has not been obtained yet. Finally, the obtained zero-rate upper bound to the reliability function of a general classical-quantum channel with no zero-error capacity coincides with a lower bound obtained by Holevo, thus determining the exact expression.
We derive both upper and lower bounds on the decoding error probability of maximum-likelihood (ML) decoded low-density parity-check (LDPC) codes. The results hold for any binary-input symmetric-output channel. Our results indicate that for various appropriately chosen ensembles of LDPC codes, reliable communication is possible up to channel capacity. However, the ensemble averaged decoding error probability decreases polynomially, and not exponentially. The lower and upper bounds coincide asymptotically, thus showing the tightness of the bounds. However, for ensembles with suitably chosen parameters, the error probability of almost all codes is exponentially decreasing, with an error exponent that can be set arbitrarily close to the standard random coding exponent.
From last fourteen years the work on undetected error probability for quantum codes has been silent. The undetected error probability has been discussed by Ashikhmin [3] in which it was proved that the average probability of undetected error for a given code is essentially given by a function of its weight enumerators. In this paper, new upper bounds on undetected error probability for quantum codes used for error detection on depolarization channel are given. It has also been established that the probability of undetected errors for quantum codes over depolarization channel do satisfy the upper bound analogous to classical codes.