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arXiv:quant-ph/9511019 · Journal of Mathematical Physics

Quantum sources and a quantum coding theorem

Christopher King, Andrezj Leśniewski

Published Jan 1, 1998• 8 citations
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Abstract

We define a large class of quantum sources and prove a quantum analog of the asymptotic equipartition property. Our proof relies on using local measurements on the quantum source to obtain an associated classical source. The classical source provides an upper bound for the dimension of the relevant subspace of the quantum source, via the Shannon–McMillan noiseless coding theorem. Along the way we derive a bound for the von Neumann entropy of the quantum source in terms of the Shannon entropy of the classical source, and we provide a definition of ergodicity of the quantum source. Several explicit models of quantum sources are also presented.

DOI
10.1063/1.532309

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We define a large class of quantum sources and prove a quantum analog of the asymptotic equipartition property. Our proof relies on using local measurements on the quantum source to obtain an associated classical source. The classical source provides an upper bound for the dimension of the relevant subspace of the quantum source, via the Shannon–McMillan noiseless coding theorem. Along the way we derive a bound for the von Neumann entropy of the quantum source in terms of the Shannon entropy of the classical source, and we provide a definition of ergodicity of the quantum source. Several explicit models of quantum sources are also presented.
Abstract. We define a large class of quantum sources and prove a quantum analog of the asymptotic equipartition property. Our proof relies on using local measurements on the quantum source to obtain an associated classical source. The classical source provides an upper bound for the dimension of the relevant subspace of the quantum source, via the Shannon-McMillan noiseless coding theorem. Along the way we derive a bound for the von Neumann entropy of the quantum source in terms of the Shannon entropy of the classical source, and we provide a definition of ergodicity of the quantum source. Several explicit models of quantum sources are also presented. 1 Supported in part by the National Science Foundation under grant DMS-9424344C. KING and A. LESNIEWSKI
The information content of a source is defined in terms of the minimum number of bits needed to store the output of the source in a perfectly recoverable way. A similar definition can be given in the case of quantum sources, with qubits replacing bits. In the mentioned cases the information content can be quantified through Shannon's and von Neumann's entropy, respectively. Here we extend the definition of information content to operational probabilistic theories, and prove relevant properties such as the subadditivity and the relation between purity and information content of a state. We prove the consistency of the present notion of information content when applied to the classical and the quantum case. Finally, the relation with one of the notions of entropy that can be introduced in general probabilistic theories, the maximum accessible information, is given in terms of a lower bound.
We investigate the recently introduced notion of smooth Renyi entropy for the case of ergodic information sources, thereby generalizing previous work which concentrated mainly on i.i.d. information sources. We will actually consider ergodic quantum information sources, of which ergodic classical information sources are a special case. We prove that the average smooth Renyi entropy rate will approach the entropy rate of a stationary, ergodic source, which is equal to the Shannon entropy rate for a classical source and the von Neumann entropy rate for a quantum source.
In this paper, we consider ergodic causal classical-quantum channels (cq-channels) which additionally have a decaying input memory. In the first part, we develop some structural properties of ergodic cq-channels and provide equivalent conditions for ergodicity. In the second part, we prove the coding theorem with weak converse for causal ergodic cq-channels with decaying input memory. Our proof is based on the possibility to introduce a joint input–output state for the cq-channels and an application of the Shannon–McMillan theorem for ergodic quantum states. In the last part of the paper, it is shown how this result implies a coding theorem for the classical capacity of a class of causal ergodic quantum channels.
Consider a source of pure quantum states with density matrix ρ and von Neumann entropy S . It is known that the quantum information of the source may be faithfully compressed to 5 qubits/signal (asymptotically for long sequences) and that this limit is optimal if we demand high fidelity for long strings of signals. The usual method for achieving this compression, based on the typical subspace of the source, requires knowledge of the density matrix. We describe a new “universal” compression scheme which does not even require knowledge of ρ. More precisely, suppose that the source is known to have von Neumann entropy less than some given bound S 0 but is otherwise completely unspecified. Our method will faithfully compress the quantum information of any such source to S 0 qubits/signal. It is known that S is always less than or equal to the Shannon entropy H of the prior distribution of the source. According to Shannon’s theorem, H provides the limit for compression of classical signals, so one may ask the question: what is the physical origin of the extra compression available in the quantum case compared to the classical case (where the prior probabilities are the same)? One may suggest that this effect is a consequence of possible non-orthogonality of signal states. However we demonstrate that it is possible to make each pair of signal states of a source more parallel (keeping the prior probabilities the same) while increasing the von Neumann entropy. This casts doubt on the conventional wisdom that overlap of quantum states is a measure of their (non-)distinguishability. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.