Data Compression Limit for an Information Source of
and Yuri Suhov
Received July 11, 2002; accepted October 4, 2002
A system of interacting qubits can be viewed as a non-i.i.d quantum information source.
A possible model of such a source is provided by a quantum spin system, in which spin-1/
2 particles located at sites of a lattice interact with each other. We establish the limit for
the compression of information from such a source and show that asymptotically it is
given by the von Neumann entropy rate. Our result can be viewed as a quantum ana-
logue of Shannon’s noiseless coding theorem for a class of non-i.i.d. quantum informa-
tion sources. From the probabilistic point of view it is an analog of the Shannon–Mc-
Millan–Breiman theorem considered as a cornerstone of modern Information Theory.
KEY WORDS: von Neumann entropy; data compression; Gibbs ensemble;
quantum Pirogov Sinai theory.
PACS: 03.67-a; 03.67.Lx
In this paper we study the issue of compression of information from a
particular class of quantum information sources, formed by systems of
interacting qubits. Our aim is to quantify the minimal physical resources
necessary to store the output from such a source or to transmit it through a
noiseless channel. We shall use the words message, signal and output from a
source interchangeably. The parameter that we minimize is the dimension of
the Hilbert space to which a typical signal can be projected (i.e., ‘‘com-
pressed’’) with high ﬁdelity. In addition, it is expected that the interaction
1570-0755/02/0800-0257/0 # 2003 Plenum Publishing Corporation
Statistical Laboratory, DPMMS, Centre for Mathematical Sciences, University of Cambridge,
Cambridge CB3 0WB.
To whom correspondence should be addressed. E-mail: N.Datta@statslab.cam.ac.uk
Quantum Information Processing, Vol. 1, No. 4, August 2002 (# 2003)