Quantum entropy-typical subspace and universal data compression

Quantum data compression is one of the most fundamental tasks in quantum information theory[1, 2]. Schumacher first provided a tight bound (equal to the von Neumann entropy of the source) to which quantum information may be compressed[3]. From then on, many compression schemes have been proposed[4–13]. In this paper we will consider the universal quantum data compression in the case that we only know the entropy of the source does not exceed some given value h. In classical information, an explicit example of such compression is a scheme based on the theory of types developed by Csiszar and Körner[14]. They showed that the data can be compressed to h bits/siganl by encoding all the sequences x for which H(px) ≤ h + ε(called CK sequence), where px is the type of x n and H(·) is the Shannon entropy function. For quantum information, an analogous theory was established[10] by Jozsa et al. They extended the classical CK sequence to a quantum subspace Ξ(B) for a given basis B, and then to Υ which is the span of Ξ(B) as B ranges over all bases. They proved that the dimension of Υ is up to some polynomial multiple of dimΞ(B), so the compression rate h is achievable asymptotically. We note that, their proof is based on the CK sequence set, so a natural question arises: Is the CK set essential to the proof? Or can it be replaced by a smaller set? In this paper, we give the answer. It will be shown that, if we replace the CK set with the entropy-typical set {xn : |H(px)− h| ≤ ε}, the proof still holds. This result is based on the quantum entropy-typical subspace theory which reveals that any ρ with entropy≤ h can be preserved by the entropy-typical subspace with entropy= h. Before presenting our main results, we begin with describing some basic concepts which will be used later. Let χ = {1, 2, ..., d} be a alphabet with d symbols. We use p = (p(1), p(2), ..., p(d)) to denote a probability distribution on χ, where p(a) is the probability of the symbol a. Let X1, X2, ..., Xn be a sequence of n symbols from χ. We will use the notation x to denote a sequence x1, x2, ..., xn. The type px of x n is the relative proportion of occurrences of each symbol in χ, i.e. px(a) = N(a|xn)/n for all