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arXiv:quant-ph/0009073

Lossless quantum coding in many-letter spaces

Kim Joris Boström

Published Jan 1, 2008• 2 citations
arXivHTMLPDF

Abstract

Based on the concept of many-letter theory a general characterization of quantum codes using the Kraus representation is given. Compression codes are defined by their property of decreasing the average information content of a given a priori message ensemble. Lossless quantum codes, in contrast to lossy codes, provide the retrieval of the original input states with perfect fidelity. A general lossless coding scheme is given that translates between two quantum alphabets. It is shown that this scheme is never compressive. Furthermore, a lossless quantum coding scheme, analog to the classical Huffman scheme but different from the Braunstein scheme, is implemented, which provides optimal compression. Motivated by the concept of lossless quantum compression, an observable is defined that measures the amount of compressible quantum information contained in a particular message with respect to a given \\emph{a priori} message ensemble. The average of this observable yields the von Neumann entropy, which is finally established as the amount of compressible quantum information of any message ensemble, confirming its commonly assigned meaning.

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Physical Review A•Sep 1, 1996•497 citations
Based on the concept of many-letter theory a general characterization of quantum codes using the Kraus representation is given. Compression codes are defined by their property of decreasing the average information content of a given a priori message ensemble. Lossless quantum codes, in contrast to lossy codes, provide the retrieval of the original input states with perfect fidelity. A general lossless coding scheme is given that translates between two quantum alphabets. It is shown that this scheme is never compressive. Furthermore, a lossless quantum coding scheme, analog to the classical Huffman scheme but different from the Braunstein scheme, is implemented, which provides optimal compression. Motivated by the concept of lossless quantum compression, an observable is defined that measures the amount of compressible quantum information contained in a particular message with respect to a given \emph{a priori} message ensemble. The average of this observable yields the von Neumann entropy, which is finally established as the amount of compressible quantum information of any message ensemble, confirming its commonly assigned meaning.
In order to compress quantum messages without loss of information it is necessary to allow the length of the encoded messages to vary. We develop a general framework for variable-length quantum messages in close analogy to the classical case and show that lossless compression is only possible if the message to be compressed is known to the sender. The lossless compression of an ensemble of messages is bounded from below by its von-Neumann entropy. We show that it is possible to reduce the number of qbits passing through a quantum channel even below the von Neumann entropy by adding a classical side channel. We give an explicit communication protocol that realizes lossless and instantaneous quantum data compression and apply it to a simple example. This protocol can be used for both online quantum communication and storage of quantum data.
After K. Bostro/spl uml/m and T. Felbinger observed that lossless quantum data compression does not exist unless decoders know the lengths of codewords, they introduced a classical noiseless channel to inform the decoder of a quantum source about the lengths of codewords. In this paper we analyze their codes and present: 1) a sufficient and necessary condition for the existence of such codes for given lists of lengths of codes; 2) a characterization of the optimal compression rate for their codes. However our main contribution is a more efficient way to use the classical channel. We propose a more general coding scheme. It turned out that the optimal compression can always be achieved by a code obtained by this scheme. A von Neumann entropy lower bound to rates of our codes and a necessary and sufficient condition to achieve the bound are obtained. The gap between this lower bound and the compression rates is also well analyzed. For a special family of quantum sources we provide a sharper lower bound in terms of Shannon entropy. Finally, we propose some problems for further research.
Based on the problem of quantum data compression in a lossless way, we present here an operational interpretation for the family of quantum Rényi entropies. In order to do this, we appeal to a very general quantum encoding scheme that satisfies a quantum version of the Kraft-McMillan inequality. Then, in the standard situation, where one is intended to minimize the usual average length of the quantum codewords, we recover the known results, namely that the von Neumann entropy of the source bounds the average length of the optimal codes. Otherwise, we show that by invoking an exponential average length, related to an exponential penalization over large codewords, the quantum Rényi entropies arise as the natural quantities relating the optimal encoding schemes with the source description, playing an analogous role to that of von Neumann entropy.
Based on the problem of quantum data compression in a lossless way, we present here an operational interpretation for the family of quantum Rényi entropies. In order to do this, we appeal to a very general quantum encoding scheme that satisfies a quantum version of the Kraft-McMillan inequality. Then, in the standard situation, where one is intended to minimize the usual average length of the quantum codewords, we recover the known results, namely that the von Neumann entropy of the source bounds the average length of the optimal codes. Otherwise, we show that by invoking an exponential average length, related to an exponential penalization over large codewords, the quantum Rényi entropies arise as the natural quantities relating the optimal encoding schemes with the source description, playing an analogous role to that of von Neumann entropy.
This chapter describes the principle of compression in quantum communication channels. The underlying concept is that it is possible to convey “faithfully” a quantum message with a large number of qubits, while transmitting a compressed version of this message with a reduced number of qubits through the channel. Beyond the mere notion of fidelity , which characterizes the quality of quantum message transmission, the description brings the new concept of typicality in the space defined by all possible “quantum codewords.” The theorem of Schumacher's quantum compression states that for a qubit source with von Neumann entropy S , the message compression factor R has S − ε for the lower bound, where ε is any nonnegative parameter that can be made arbitrarily small for sufficiently long messages (hence, R ≈ S is the best possible compression factor). An original graphical and numerical illustration of the effect of Schumacher's quantum compression and the evolution of the typical quantum-codeword subspace with increasing message length is provided. Quantum data compression and fidelity In this chapter, we have reached the stage where it is possible to start addressing the issues that are central to information theory, namely, “How efficiently can we code information in a quantum communication channel?” both in terms of economy of means – the concept of data compression – and accuracy of transmission – the concept of message integrity or minimal data error , referred to here as fidelity .
We consider the transmission of classical information over a quantum channel. The channel is defined by an ``alphabet'' of quantum states, e.g., certain photon polarizations, together with a specified set of probabilities with which these states must be sent. If the receiver is restricted to making separate measurements on the received ``letter'' states, then the Kholevo theorem implies that the amount of information transmitted per letter cannot be greater than the von Neumann entropy H of the letter ensemble. In fact the actual amount of transmitted information will usually be significantly less than H. We show, however, that if the sender uses a block coding scheme consisting of a choice of code words that respects the a priori probabilities of the letter states, and the receiver distinguishes whole words rather than individual letters, then the information transmitted per letter can be made arbitrarily close to H and never exceeds H. This provides a precise information-theoretic interpretation of von Neumann entropy in quantum mechanics. We apply this result to ``superdense'' coding, and we consider its extension to noisy channels. \textcopyright{} 1996 The American Physical Society.