2011

Quantum rate distortion, reverse Shannon theorems, and source-channel separation

Datta, Nilanjana, Hsieh, Min-Hsiu, Wilde, Mark M.

Understand

We derive quantum counterparts of two key theorems of classical information theory, namely, the rate distortion theorem and the source-channel separation theorem.

The rate-distortion theorem gives the ultimate limits on lossy data compression, and the source-channel separation theorem implies that a two-stage protocol consisting of compression and channel coding is optimal for transmitting a memoryless source over a memoryless channel.
In spite of their importance in the classical domain, there has been surprisingly little work in these areas for quantum information theory.
In the present paper, we prove that the quantum rate distortion function is given in terms of the regularized entanglement of purification.

Built on

A mathematical theory of communication
Claude E. Shannon · 1948
Earlier work this paper cites.
Coding theorems for a discrete source with a fidelity criterion
Claude E. Shannon · 1959
Earlier work this paper cites.
Rate Distortion Theory: A Mathematical Basis for Data Compression
Toby Berger · 1971
Earlier work this paper cites.
Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels
Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters · 1993
Earlier work this paper cites.
Quantum coding
Benjamin Schumacher · 1995
Earlier work this paper cites.
Scheme for reducing decoherence in quantum computer memory
Peter W. Shor · 1995
Earlier work this paper cites.
The source-channel separation theorem revisited
Sridhar Vembu, Sergio Verdu, and Yossef Steinberg · 1995
Earlier work this paper cites.
Good quantum error-correcting codes exist
A. Robert Calderbank and Peter W. Shor · 1996
Earlier work this paper cites.
Sending entanglement through noisy quantum channels
Benjamin Schumacher · 1996
Earlier work this paper cites.
Quantum data processing and error correction
Benjamin Schumacher and Michael A. Nielsen · 1996
Earlier work this paper cites.
von Neumann capacity of noisy quantum channels
Christoph Adami and Nicolas J. Cerf · 1997
Earlier work this paper cites.
Capacity of the noisy quantum channel
Seth Lloyd · 1997
Earlier work this paper cites.
Sending classical information via noisy quantum channels
Benjamin Schumacher and Michael D. Westmoreland · 1997
Earlier work this paper cites.
Information transmission through a noisy quantum channel
Howard Barnum, M. A. Nielsen, and Benjamin Schumacher · 1998
Earlier work this paper cites.
Quantum error correction via codes over GF(4)
A. Robert Calderbank, Eric M. Rains, Peter W. Shor, and N. J. A. Sloane · 1998
Earlier work this paper cites.
The capacity of the quantum channel with general signal states
Alexander S. Holevo · 1998
Earlier work this paper cites.
Quantum rate-distortion coding
Howard Barnum · 2000
Earlier work this paper cites.
On quantum fidelities and channel capacities
Howard Barnum, Emanuel Knill, and Michael A. Nielsen · 2000
Earlier work this paper cites.
A quantum analog of Huffman coding
Samuel L. Braunstein, Christopher A. Fuchs, Daniel Gottesman, and Hoi-Kwong Lo · 2000
Earlier work this paper cites.
Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem
Charles H. Bennett, Peter W. Shor, John A. Smolin, and Ashish V. Thapliyal · 2002
Earlier work this paper cites.
Lossless quantum data compression and variable-length coding
Kim Bostroem and Timo Felbinger · 2002
Earlier work this paper cites.

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Then

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