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arXiv:quant-ph/0202002 · Physical Review A

Exponents of quantum fixed-length pure-state source coding

Masahito Hayashi

Published Sep 27, 2002• 48 citations
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Abstract

We derive the optimal exponent of the error probability of the quantum fixed-length pure-state source coding in both cases of blind coding and visible coding. The optimal exponent is universally attained by Jozsa et al.'s [Phys. Rev. Lett. 81, 1714 (1998)] universal code. In the direct part, a group representation theoretical type method is essential. In the converse part, Nielsen and Kempe's [Phys. Rev. Lett. 86, 5184 (2001)] lemma is essential.

DOI
10.1103/physreva.66.032321

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We derive the optimal exponent of the error probability of the quantum fixed-length pure-state source coding in both cases of blind coding and visible coding. The optimal exponent is universally attained by Jozsa et al.'s [Phys. Rev. Lett. 81, 1714 (1998)] universal code. In the direct part, a group representation theoretical type method is essential. In the converse part, Nielsen and Kempe's [Phys. Rev. Lett. 86, 5184 (2001)] lemma is essential.
The simplest example of a quantum information source with memory is a mixed source, which emits signals entirely from one of two memoryless quantum sources with given a prioria~prioria priori probabilities. Considering a mixed source consisting of a general one-parameter family of memoryless sources, we derive the second-order asymptotic rate for fixed-length visible source coding. Furthermore, we specialize our main result to a mixed source consisting of two memoryless sources. Our results provide the first example of the second-order asymptotics for a quantum information-processing task employing a resource with memory. For the case of a classical mixed source (using a finite alphabet), our results reduce to those obtained by Nomura and Han. To prove the achievability part of our main result, we introduce universal quantum source codes achieving the second-order asymptotic rates. These are obtained by an extension of Hayashi’s construction of their classical counterparts.
The simplest example of a quantum information source with memory is a mixed source, which emits signals entirely from one of two memoryless quantum sources with given a priori probabilities. Considering a mixed source consisting of a general one-parameter family of memoryless sources, we derive the second-order asymptotic rate for fixed-length visible source coding. Furthermore, we specialize our main result to a mixed source consisting of two memoryless sources. Our results provide the first example of the second-order asymptotics for a quantum information-processing task employing a resource with memory. For the case of a classical mixed source (using a finite alphabet), our results reduce to those obtained by Nomura and Han. To prove the achievability part of our main result, we introduce universal quantum source codes achieving the second-order asymptotic rates. These are obtained by an extension of Hayashi's construction of their classical counterparts.
We establish exponential strong converse theorems for blind quantum data compression of finite-dimensional mixed-state sources, with and without entanglement assistance. Blind compression is a fundamental quantum information processing task in which the sender aims to transmit quantum states without direct access to the classical label. Strong converse is a notable aspect of data compression: the accuracy of the compression abruptly drops down to zero once the compression rate goes below a threshold. Strong converses for data compression have been established for pure-state sources; however, for mixed-state sources, the optimal compression rate is determined by a structural decomposition of quantum states and can change substantially under small perturbations. This sensitivity makes it difficult to quantitatively analyze the trade-off between compression rate and error. Despite recent progress, whether a strong converse holds at the optimal rate has remained unresolved in general. In this work, we resolve this question affirmatively. For unassisted compression, we show that at every rate below the optimum, the accuracy decays exponentially with the block length for general mixed-state sources without restrictions on protocols. We also prove exponential strong converse for blind compression with unlimited entanglement assistance at its optimal rate. Combining the two strong converse bounds yields exponential decay at every compression-entanglement rate pair outside the complete achievable rate region. Our proof introduces a new overlap quantity based on the structure of quantum states; this quantity connects preservation of the source information to the dimension of the transmitted system. We expect that this approach provides useful tools for converse analyses of other quantum information-processing tasks involving mixed states.
The simplest example of a quantum information source with memory is a mixed source which emits signals entirely from either one of two memoryless quantum sources with given a priori probabilities. For such a source, we derive the second order asymptotic rates for fixedlength (visible) source coding. This provides the first example of second order asymptotics for a quantum information-processing task employing a resource with memory. For the case of a classical mixed source (using a finite alphabet), our results reduce to those obtained by Nomura and Han [21]. Moreover, the known optimal asymptotic rate and strong converse rate for fixed-length source coding for an arbitrary sequence of mixed sources (derived in the classical case by Han [11] and in the quantum case in [4]) are directly obtainable from our results.