Coding theorems of quantum information theory

Coding theorems and (strong) converses for memoryless quantum communication channels and quantum sources are proved: for the quantum source the coding theorem is reviewed, and the strong converse proven. For classical information transmission via quantum channels we give a new proof of the coding theorem, and prove the strong converse, even under the extended model of nonstationary channels. As a by-product we obtain a new proof of the famous Holevo bound. Then multi-user systems are investigated, and the capacity region for the quantum multiple access channel is determined. The last chapter contains a preliminary discussion of some models of compression of correlated quantum sources, and a proposal for a program to obtain operational meaning for quantum conditional entropy. An appendix features the introduction of a notation and calculus of entropy in quantum systems.

[1]  C. Bendjaballah,et al.  Rate Distortion and Detection in Quantum Communication , 1998, IEEE Trans. Inf. Theory.

[2]  L. Ballentine,et al.  Quantum Theory: Concepts and Methods , 1994 .

[3]  M. Nielsen,et al.  Information transmission through a noisy quantum channel , 1997, quant-ph/9702049.

[4]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[5]  A. Wehrl.  General properties of entropy , 1978 .

[6]  Charles H. Bennett,et al.  Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. , 1992, Physical review letters.

[7]  H. Nagaoka,et al.  Algorithms of Arimoto-Blahut type for computing quantum channel capacity , 1998, Proceedings. 1998 IEEE International Symposium on Information Theory (Cat. No.98CH36252).

[8]  H. Umegaki.  Conditional expectation in an operator algebra. IV. Entropy and information , 1962 .

[9]  Asher Peres,et al.  Quantum Theory: Concepts and Methods , 1994 .

[10]  Rudolf Ahlswede,et al.  Multi-way communication channels , 1973 .

[11]  W. Wootters,et al.  A single quantum cannot be cloned , 1982, Nature.

[12]  Alexander Semenovich Holevo,et al.  Quantum coding theorems , 1998 .

[13]  Alexander S. Holevo,et al.  The Capacity of the Quantum Channel with General Signal States , 1996, IEEE Trans. Inf. Theory.

[14]  Samuel L. Braunstein,et al.  A quantum analog of huffman coding , 2000, IEEE Trans. Inf. Theory.

[15]  W. Arveson.  An Invitation To C*-Algebras , 1976 .

[16]  G. Ludwig.  Die Grundlagen der Quantenmechanik , 1954 .

[17]  A. Holevo.  Problems in the mathematical theory of quantum communication channels , 1977 .

[18]  M. Horodecki,et al.  Universal Quantum Information Compression , 1998, quant-ph/9805017.

[19]  E. B. Davies.  Quantum theory of open systems , 1976 .

[20]  Michael J. W. Hall,et al.  QUANTUM INFORMATION AND CORRELATION BOUNDS , 1997 .

[21]  Schumacher,et al.  Sending entanglement through noisy quantum channels. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[22]  G. Lindblad.  Completely positive maps and entropy inequalities , 1975 .

[23]  Andreas J. Winter,et al.  Coding theorem and strong converse for quantum channels , 1999, IEEE Trans. Inf. Theory.

[24]  J. Neumann.  Thermodynamik quantenmechanischer Gesamtheiten , 1927 .

[25]  W. Stinespring.  Positive functions on *-algebras , 1955 .

[26]  M. Horodecki.  Limits for compression of quantum information carried by ensembles of mixed states , 1997, quant-ph/9712035.

[27]  Schumacher,et al.  Classical information capacity of a quantum channel. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[28]  D. Petz,et al.  Quantum Entropy and Its Use , 1993 .

[29]  Albert Einstein,et al.  Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? , 1935 .

[30]  A. S. Holevo,et al.  Capacity of a quantum communication channel , 1979 .

[31]  Howard Barnum.  Quantum rate-distortion coding , 1998 .

[32]  A. Uhlmann.  Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory , 1977 .

[33]  L. Ballentine,et al.  Probabilistic and Statistical Aspects of Quantum Theory , 1982 .

[34]  A. Shimony,et al.  Bell’s theorem without inequalities , 1990 .

[35]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[36]  Michael D. Westmoreland,et al.  Sending classical information via noisy quantum channels , 1997 .

[37]  R. Ahlswede.  The Capacity Region of a Channel with Two Senders and Two Receivers , 1974 .

[38]  Jozsa,et al.  General fidelity limit for quantum channels. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[39]  Lev B. Levitin.  Conditional entropy and information in quantum systems , 1998, Proceedings. 1998 IEEE International Symposium on Information Theory (Cat. No.98CH36252).

[40]  A. El Gamal,et al.  Multiple user information theory , 1980, Proceedings of the IEEE.

[41]  Peter W. Shor,et al.  Algorithms for quantum computation: discrete logarithms and factoring , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[42]  Benjamin Schumacher,et al.  A new proof of the quantum noiseless coding theorem , 1994 .

[43]  Jacob Wolfowitz.  Coding Theorems of Information Theory , 1962 .

[44]  Claude E. Shannon,et al.  Two-way Communication Channels , 1961 .

[45]  A. E. Allahverdyan,et al.  Multi-access channels in quantum information theory , 1997 .

[46]  Rudolf Ahlswede,et al.  Coloring hypergraphs: A new approach to multi-user source coding, 1 , 1979 .

[47]  J. Bell.  On the Einstein-Podolsky-Rosen paradox , 1964 .

[48]  Tomohiro Ogawa,et al.  Strong converse to the quantum channel coding theorem , 1999, IEEE Trans. Inf. Theory.

[49]  Andreas J. Winter.  The capacity of the quantum multiple-access channel , 2001, IEEE Trans. Inf. Theory.

[50]  Schumacher,et al.  Quantum data processing and error correction. , 1996, Physical review. A, Atomic, molecular, and optical physics.