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IEEE International Symposium on Information Theory (ISIT)

Simple construction of quantum universal variable-length source coding

Masahito Hayashi, Koji Matsumoto

Published Jan 1, 2003• 4 citations
Publisher

Abstract

This paper describes the construction of a quantum universal variable-length source code in which, independent of information source, both of the average error and the probability that the coding rate is greater than the entropy rate H(/spl rho/~/sub p/), tend to 0. If H(/spl rho/~/sub p/) is estimated, and compress the coding rate to the admissible rate H(/spl rho/~/sub p/) with a probability close to 1. However, when we perform a naive measurement for the estimation of H(/spl rho/~/sub p/), the input state is demolished. By smearing the measurement, we successfully treat the trade-off between the estimation of H(/spl rho/~/sub p/) and the nondemolition of the input state. Our protocol can be used not only for the Schumacher's compression but also for the compression of entangled states.

DOI
10.1109/isit.2003.1228476

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This paper describes the construction of a quantum universal variable-length source code in which, independent of information source, both of the average error and the probability that the coding rate is greater than the entropy rate H(/spl rho//sub p/), tend to 0. If H(/spl rho//sub p/) is estimated, and compress the coding rate to the admissible rate H(/spl rho//sub p/) with a probability close to 1. However, when we perform a naive measurement for the estimation of H(/spl rho//sub p/), the input state is demolished. By smearing the measurement, we successfully treat the trade-off between the estimation of H(/spl rho/~/sub p/) and the nondemolition of the input state. Our protocol can be used not only for the Schumacher's compression but also for the compression of entangled states.
We simply construct a quantum universal variable-length source code in which, independent of information source, both of the average error and the probability that the coding rate is greater than the entropy rate H(\overline{\rho}_p), tend to 0. If H(\overline{\rho}_p) is estimated, we can compress the coding rate to the admissible rate H(\overline{\rho}_p) with a probability close to 1. However, when we perform a naive measurement for the estimation of H(\overline{\rho}_p), the input state is demolished. By smearing the measurement, we successfully treat the trade-off between the estimation of H(\overline{\rho}_p) and the non-demolition of the input state. Our protocol can be used not only for the Schumacher's scheme but also for the compression of entangled states.
We simply construct a quantum universal variable-length source code in which, independent of information source, both of the average error and the probability that the coding rate is greater than the entropy rate H(rhop)H(rho_p)H(rhop​), tend to 0. If H(rhop)H(rho_p)H(rhop​) is estimated, we can compress the coding rate to the admissible rate H(rhop)H(rho_p)H(rhop​) with a probability close to 1. However, when we perform a naive measurement for the estimation of H(rhop)H(rho_p)H(rhop​), the input state is demolished. By smearing the measurement, we successfully treat the trade-off between the estimation of H(rhop)H(rho_p)H(rhop​) and the non-demolition of the input state. Our protocol can be used not only for the Schumacher's scheme but also for the compression of entangled states.
We describe a method for lossless quantum compression if the output of the information source is not known. We compute the best possible compression rate, minimizing the expected base length of the output quantum bit string (the base length of a quantum string is the maximal length in the superposition). This complements work by Schumacher and Westmoreland who calculated the corresponding rate for minimizing the output's average length. Our compressed code words are prefix-free indeterminate-length quantum bit strings which can be concatenated in the case of multiple sources. Therefore, we generalize the known theory of prefix-free quantum codes to the case where strings have indeterminate length. Moreover, we describe a communication model which allows the lossless transmission of the compressed code words. The benefit of compression is then the reduction of transmission errors in the presence of noise.
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We describe a universal information–compression scheme that compresses any pure quantum independent identically distributed (i.i.d.) source asymptotically to its von Neumann entropy, with no prior knowledge of the structure of the source. We introduce a diagonalization procedure that enables any classical compression algorithm to be used in a quantum context. Our scheme is then based on the corresponding quantum translation of the classical Lempel–Ziv algorithm. Our methods lead to a conceptually simple way of estimating the entropy of a source in terms of the measurement of an associated length parameter while maintaining high fidelity for long blocks. As a byproduct we also estimate the eigenbasis of the source. Since our scheme is based on the Lempel–Ziv method, it can potentially be applied also to non–i.i.d. sources that satisfy some further regularity conditions.