Entropy and Quantum Kolmogorov Complexity: A Quantum Brudno’s Theorem

In classical information theory, entropy rate and algorithmic complexity per symbol are related by a theorem of Brudno. In this paper, we prove a quantum version of this theorem, connecting the von Neumann entropy rate and two notions of quantum Kolmogorov complexity, both based on the shortest qubit descriptions of qubit strings that, run by a universal quantum Turing machine, reproduce them as outputs.

[1]  Paul M. B. Vitányi,et al.  Quantum Kolmogorov complexity based on classical descriptions , 2001, IEEE Trans. Inf. Theory.

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

[3]  I. Bjelakovic,et al.  Chained Typical Subspaces - a Quantum Version of Breiman's Theorem , 2003, quant-ph/0301177.

[4]  A. Holevo.  Statistical structure of quantum theory , 2001 .

[5]  Robert Alicki,et al.  Comparison of dynamical entropies for the noncommutative shifts , 1995 .

[6]  R. Feynman.  Simulating physics with computers , 1999 .

[7]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[8]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 2019, Texts in Computer Science.

[9]  D. Petz,et al.  Stationary quantum source coding , 1999, quant-ph/9912103.

[10]  Alexandros Eleftheriadis,et al.  Complexity distortion theory , 2003, IEEE Trans. Inf. Theory.

[11]  Harumichi Nishimura,et al.  Local transition functions of quantum Turing machines , 2000, RAIRO Theor. Informatics Appl..

[12]  V. Paulsen.  Completely Bounded Maps and Operator Algebras: Contents , 2003 .

[13]  Ray J. Solomonoff,et al.  A Formal Theory of Inductive Inference. Part I , 1964, Inf. Control..

[14]  Péter Gács,et al.  Quantum algorithmic entropy , 2000, Proceedings 16th Annual IEEE Conference on Computational Complexity.

[15]  D. Deutsch.  Quantum theory, the Church–Turing principle and the universal quantum computer , 1985, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[16]  F. Hiai,et al.  The proper formula for relative entropy and its asymptotics in quantum probability , 1991 .

[17]  Martin Mathieu.  COMPLETELY BOUNDED MAPS AND OPERATOR ALGEBRAS (Cambridge Studies in Advanced Mathematics 78) , 2004 .

[18]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[19]  V. Uspenskii,et al.  Can an individual sequence of zeros and ones be random? Russian Math , 1990 .

[20]  H. White.  Algorithmic complexity of points in dynamical systems , 1993, Ergodic Theory and Dynamical Systems.

[21]  Peter W. Shor,et al.  Algorithms for Quantum Computation: Discrete Log and Factoring (Extended Abstract) , 1994, FOCS 1994.

[22]  Igor Bjelakovic,et al.  The Data Compression Theorem for Ergodic Quantum Information Sources , 2005, Quantum Inf. Process..

[23]  David P. DiVincenzo,et al.  Quantum information and computation , 2000, Nature.

[24]  Jacob Ziv,et al.  Coding of sources with unknown statistics-I: Probability of encoding error , 1972, IEEE Trans. Inf. Theory.

[25]  En-Hui Yang,et al.  Universal compression of ergodic quantum sources , 2003, Quantum Inf. Comput..

[26]  Gregory J. Chaitin,et al.  On the Length of Programs for Computing Finite Binary Sequences , 1966, JACM.

[27]  Karl Svozil,et al.  Randomness and undecidability in physics , 1993 .

[28]  L. Levin,et al.  THE COMPLEXITY OF FINITE OBJECTS AND THE DEVELOPMENT OF THE CONCEPTS OF INFORMATION AND RANDOMNESS BY MEANS OF THE THEORY OF ALGORITHMS , 1970 .

[29]  P. Jorrand,et al.  Measurement-Based Quantum Turing Machines and their Universality , 2004, quant-ph/0404146.

[30]  C. Mora,et al.  ALGORITHMIC COMPLEXITY OF QUANTUM STATES , 2004 .

[31]  G. Segre.  Physical Complexity of Classical and Quantum Objects and Their Dynamical Evolution From an Information-Theoretic Viewpoint , 2004 .

[32]  A. Brudno.  Entropy and the complexity of the trajectories of a dynamical system , 1978 .

[33]  Ray J. Solomonoff,et al.  A Formal Theory of Inductive Inference. Part II , 1964, Inf. Control..

[34]  Umesh V. Vazirani,et al.  Quantum complexity theory , 1993, STOC.

[35]  Andrei N. Kolmogorov,et al.  Logical basis for information theory and probability theory , 1968, IEEE Trans. Inf. Theory.

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

[37]  Jozef Gruska,et al.  Quantum Computing , 2008, Wiley Encyclopedia of Computer Science and Engineering.

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

[39]  P. Rousseeuw,et al.  Wiley Series in Probability and Mathematical Statistics , 2005 .

[40]  John C. Kieffer,et al.  A unified approach to weak universal source coding , 1978, IEEE Trans. Inf. Theory.

[41]  V. Alekseev,et al.  Symbolic dynamics and hyperbolic dynamic systems , 1981 .

[42]  H. S. Allen.  The Quantum Theory , 1928, Nature.

[43]  Thomas M. Cover,et al.  Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing) , 2006 .

[44]  Leonard M. Adleman,et al.  Quantum Computability , 1997, SIAM J. Comput..

[45]  Sophie Laplante,et al.  Quantum Kolmogorov complexity , 2000, Proceedings 15th Annual IEEE Conference on Computational Complexity.

[46]  P. Billingsley,et al.  Ergodic theory and information , 1966 .

[47]  B. M. Fulk.  MATH , 1992 .

[48]  H. Briegel,et al.  Algorithmic complexity and entanglement of quantum states. , 2005, Physical review letters.

[49]  A. Kolmogorov.  Three approaches to the quantitative definition of information , 1968 .

[50]  Igor Bjelakovic,et al.  The Shannon-McMillan theorem for ergodic quantum lattice systems , 2002, ArXiv.