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arXiv:quant-ph/9607010

Data Compression for Quantum Code

Kwok Yip Szeto

Published Jul 12, 1996• 1 citation
arXivHTMLPDF

Abstract

The general scheme of data compression using the quantum noiseless coding theorem of Schumacher is dicussed for general quantum sources. When the Hilbert space of the quantum source is decomposable into orthogonal subspaces, one can first perform classical data compression before performing the quantum data compression for individual subspaces. For minimizing the resource in quantum coding, a general parameterization of the dimensions in the Jozsa-Schumacher quantum data compression scheme is presented.

DOI
10.48550/arxiv.quant-ph/9607010

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The general scheme of data compression using the quantum noiseless coding theorem of Schumacher is dicussed for general quantum sources. When the Hilbert space of the quantum source is decomposable into orthogonal subspaces, one can first perform classical data compression before performing the quantum data compression for individual subspaces. For minimizing the resource in quantum coding, a general parameterization of the dimensions in the Jozsa-Schumacher quantum data compression scheme is presented.
Schumacher and Westmoreland [Phys. Rev. A 64, 42304 (2001)] have established a quantum analog of a well-known classical information theory result on a role of relative entropy as a measure of nonoptimality in (classical) data compression. In this paper, we provide an alternative simple and constructive proof of this result by constructing quantum compression codes (schemes) from classical data compression codes. Moreover, as the quantum data compression or coding task can be effectively reduced to a (quasi)classical one, we show that relevant results from classical information theory and data compression become applicable and therefore can be extended to the quantum domain.
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 .
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.
One of the fundamental tasks in classical information theory is the compression of information. Given access to many uses of a noiseless classical channel, what is the best that a sender and receiver can make of this resource for compressed data transmission? Shannon's compression theorem demonstrates that the Shannon entropy is the fundamental limit for the compression rate in the IID setting (recall the development in Section 13.4). That is, if one compresses at a rate above the Shannon entropy, then it is possible to recover the compressed data perfectly in the asymptotic limit, and otherwise, it is not possible to do so. This theorem establishes the prominent role of the entropy in Shannon's theory of information. In the quantum world, it very well could be that one day a sender and a receiver would have many uses of a noiseless quantum channel available, and the sender could use this resource to transmit compressed quantum information. But what exactly does this mean in the quantum setting? A simple model of a quantum information source is an ensemble of quantum states { pX(x) , ∣ψ x ⟩}, i.e., the source outputs the state ∣ψ x ⟩ with probability pX(x) , and the states {∣ψ x ⟩} do not necessarily have to form an orthonormal basis. Let us suppose for the moment that the classical data x is available as well, even though this might not necessarily be the case in practice. A naive strategy for compressing this quantum information source would be to ignore the quantum states coming out, handle the classical data instead, and exploit Shannon's compression protocol from Section 13.4.
An explicit algorithm for performing Schumacher's noiseless compression of quantum bits is given. This algorithm is based on a combinatorial expression for a particular bijection among binary strings. The algorithm, which adheres to the rules of reversible programming, is expressed in a high-level pseudocode language. It is implemented using O(n3{\mathit{n}}^{3}n3) two- and three-bit primitive reversible operations, where n is the length of the qubit strings to be compressed. Also, the algorithm makes use of O(n) auxiliary qubits. Space-saving techniques based on those proposed by Bennett are developed which reduce this workspace to O(\ensuremath{\surd}n) while maintaining a running time of O(n3{\mathit{n}}^{3}n3) (albeit with a larger constant). This latter algorithm is of interest because it has a slightly smaller time-space product, which is considered to be the relevant figure of merit for efficiency in some physical models. \textcopyright{} 1996 The American Physical Society.
We consider the most general (finite-dimensional) quantum mechanical information source, which is given by a quantum system A that is correlated with a reference system R. The task is to compress A in such a way as to reproduce the joint source state ρARat the decoder with asymptotically high fidelity. This includes Schumacher's original quantum source coding problem of a pure state ensemble and that of a single pure entangled state, as well as general mixed state ensembles. Here, we determine the optimal compression rate (in qubits per source system) in terms of the Koashi-Imoto decomposition of the source into a classical, a quantum, and a redundant part. The same decomposition yields the optimal rate in the presence of unlimited entanglement between compressor and decoder, and indeed the full region of feasible qubitebit rate pairs. Full version at arXiv:1912.08506 [1].