Quantum Information Theory

In this thesis I present a short review of ideas in quantum information theory. The first chapter contains introductory material, sketching the central ideas of probability and information theory. Quantum mechanics is presented at the level of advanced undergraduate knowledge, together with some useful tools for quantum mechanics of open systems. In the second chapter I outline how classical information is represented in quantum systems and what this means for agents trying to extract information from these systems. The final chapter presents a new resource: quantum information. This resource has some bewildering applications which have been discovered in the last ten years, and continually presents us with unexpected insights into quantum theory and the universe. The treatment is pedagogical and suitable for beginning graduates in the field.

[1]  Umesh V. Vazirani,et al.  Quantum Algorithms , 2001, LATIN.

[2]  R. Jozsa,et al.  On quantum coding for ensembles of mixed states , 2000, quant-ph/0008024.

[3]  Dominic Mayers,et al.  Unconditional security in quantum cryptography , 1998, JACM.

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

[5]  H. Inamori Security of EPR-based Quantum Key Distribution , 2000, quant-ph/0008064.

[6]  E. Lieb,et al.  A Fresh Look at Entropy and the Second Law of Thermodynamics , 2000, math-ph/0003028.

[7]  D. Deutsch,et al.  Machines, logic and quantum physics , 1999, Bull. Symb. Log..

[8]  R. Jozsa,et al.  Distinguishability of states and von Neumann entropy , 1999, quant-ph/9911009.

[9]  Samuel L. Braunstein,et al.  Criteria for continuous-variable quantum teleportation , 1999, quant-ph/9910030.

[10]  M. Horodecki Optimal compression for mixed signal states , 1999, quant-ph/9905058.

[11]  V. Vedral Landauer's erasure, error correction and entanglement , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[12]  John Preskill Plug-in quantum software , 1999, Nature.

[13]  M. Plenio The Holevo bound and Landauer's principle , 1999, quant-ph/9910086.

[14]  Michal Horodecki,et al.  Toward optimal compression for mixed signal states , 1999 .

[15]  C. Caves,et al.  Classical model for bulk-ensemble NMR quantum computation , 1999, quant-ph/9903101.

[16]  N. Gisin,et al.  Spin Flips and Quantum Information for Antiparallel Spins , 1999, quant-ph/9901072.

[17]  J. Eisert,et al.  Quantum Games and Quantum Strategies , 1998, quant-ph/9806088.

[18]  Charles H. Bennett,et al.  Quantum nonlocality without entanglement , 1998, quant-ph/9804053.

[19]  H. Chau,et al.  Unconditional security of quantum key distribution over arbitrarily long distances , 1998, Science.

[20]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[21]  R. D. Wolf Quantum Computation and Shor's Factoring Algorithm , 1999 .

[22]  E. Knill,et al.  Complete quantum teleportation using nuclear magnetic resonance , 1998, Nature.

[23]  Kimble,et al.  Unconditional quantum teleportation , 1998, Science.

[24]  Daniel R. Terno,et al.  Optimal distinction between non-orthogonal quantum states , 1998, quant-ph/9804031.

[25]  Ashish V. Thapliyal,et al.  Entanglement of Assistance , 1998, QCQC.

[26]  M. Mosca,et al.  The Hidden Subgroup Problem and Eigenvalue Estimation on a Quantum Computer , 1998, QCQC.

[27]  P. Knight,et al.  Multiparticle generalization of entanglement swapping , 1998 .

[28]  M. Horodecki,et al.  Mixed-State Entanglement and Distillation: Is there a “Bound” Entanglement in Nature? , 1998, quant-ph/9801069.

[29]  N. Gisin Quantum cloning without signaling , 1998, quant-ph/9801005.

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

[31]  Michael D. Westmoreland,et al.  Quantum Privacy and Quantum Coherence , 1997, quant-ph/9709058.

[32]  W. Wootters Entanglement of Formation of an Arbitrary State of Two Qubits , 1997, quant-ph/9709029.

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

[34]  M. Mosca Quantum Searching Counting and Amplitude Ampli cation by Eigenvector Analysis , 1998 .

[35]  H. Weinfurter,et al.  Experimental quantum teleportation , 1997, Nature.

[36]  N. Gisin,et al.  OPTIMAL EAVESDROPPING IN QUANTUM CRYPTOGRAPHY. I. INFORMATION BOUND AND OPTIMAL STRATEGY , 1997 .

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

[38]  C. Fuchs Nonorthogonal Quantum States Maximize Classical Information Capacity , 1997, quant-ph/9703043.

[39]  N. Mermin Quantum theory: Concepts and methods , 1997 .

[40]  M. Plenio,et al.  Quantifying Entanglement , 1997, quant-ph/9702027.

[41]  N. Gisin,et al.  Optimal Eavesdropping in Quantum Cryptography. I , 1997, quant-ph/9701039.

[42]  C. Adami,et al.  Accessible information in quantum measurement , 1996, quant-ph/9611032.

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

[44]  C. Fuchs Information Gain vs. State Disturbance in Quantum Theory , 1996, quant-ph/9605014.

[45]  Charles H. Bennett,et al.  Mixed-state entanglement and quantum error correction. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[46]  B. Schumacher Sending quantum entanglement through noisy channels , 1996, quant-ph/9604023.

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

[48]  C. Fuchs,et al.  Quantum information: How much information in a state vector? , 1996, quant-ph/9601025.

[49]  C. Fuchs Distinguishability and Accessible Information in Quantum Theory , 1996, quant-ph/9601020.

[50]  Shor,et al.  Good quantum error-correcting codes exist. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[51]  Pérès,et al.  Quantum-state disturbance versus information gain: Uncertainty relations for quantum information. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[52]  Charles H. Bennett,et al.  Concentrating partial entanglement by local operations. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[53]  Schumacher,et al.  Noncommuting mixed states cannot be broadcast. , 1995, Physical review letters.

[54]  C. Caves,et al.  Information-theoretic characterization of quantum chaos. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[55]  Barenco,et al.  Elementary gates for quantum computation. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[56]  DiVincenzo,et al.  Two-bit gates are universal for quantum computation. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[57]  Schumacher,et al.  Quantum coding. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[58]  Caves,et al.  Ensemble-dependent bounds for accessible information in quantum mechanics. , 1994, Physical review letters.

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

[60]  N. David Mermin,et al.  Quantum mysteries refined , 1994 .

[61]  Ueli Maurer,et al.  Generalized privacy amplification , 1994, Proceedings of 1994 IEEE International Symposium on Information Theory.

[62]  I. Pitowsky,et al.  George Boole's ‘Conditions of Possible Experience’ and the Quantum Puzzle , 1994, The British Journal for the Philosophy of Science.

[63]  Popescu,et al.  Bell's inequalities versus teleportation: What is nonlocality? , 1994, Physical review letters.

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

[65]  D. Deutsch,et al.  Rapid solution of problems by quantum computation , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[66]  Ekert,et al.  Quantum cryptography based on Bell's theorem. , 1991, Physical review letters.

[67]  N. Gisin Bell's inequality holds for all non-product states , 1991 .

[68]  W. Wootters,et al.  Optimal detection of quantum information. , 1991, Physical review letters.

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

[70]  W. Wootters Random quantum states , 1990 .

[71]  N. David Mermin,et al.  Boojums All The Way Through , 1990 .

[72]  R. T. Cox Probability, frequency and reasonable expectation , 1990 .

[73]  Werner,et al.  Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. , 1989, Physical review. A, General physics.

[74]  D. Deutsch Quantum computational networks , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[75]  N. David Mermin,et al.  What's Wrong with this Pillow? , 1989 .

[76]  C. Ray Smith,et al.  From Rationality and Consistency to Bayesian Probability , 1989 .

[77]  J. Bell,et al.  Speakable and Unspeakable in Quatum Mechanics , 1988 .

[78]  J. Rice Mathematical Statistics and Data Analysis , 1988 .

[79]  Wojciech Hubert Zurek,et al.  Maxwell’s Demon, Szilard’s Engine and Quantum Measurements , 2003, quant-ph/0301076.

[80]  Alain Aspect,et al.  Experimental Tests of Bell’s Inequalities with Pairs of Low Energy Correlated Photons , 1986 .

[81]  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.

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

[83]  Charles H. Bennett,et al.  The thermodynamics of computation—a review , 1982 .

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

[85]  J. Wheeler The computer and the universe , 1982 .

[86]  W. Wootters Statistical distance and Hilbert space , 1981 .

[87]  William K. Wootters The Acquisition of Information from Quantum Measurements. , 1980 .

[88]  D. Lane Fisher, Jeffreys, and the Nature of Probability , 1980 .

[89]  W. Unruh QUANTUM NONDEMOLITION AND GRAVITY WAVE DETECTION , 1979 .

[90]  E. B. Davies,et al.  Information and quantum measurement , 1978, IEEE Trans. Inf. Theory.

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

[92]  A. Uhlmann The "transition probability" in the state space of a ∗-algebra , 1976 .

[93]  A. Holevo Bounds for the quantity of information transmitted by a quantum communication channel , 1973 .

[94]  Lars-Ake Levin,et al.  Problems of Information Transmission , 1973 .

[95]  D. Haar,et al.  Statistical Physics , 1971, Nature.

[96]  A. Shimony,et al.  Proposed Experiment to Test Local Hidden Variable Theories. , 1969 .

[97]  R. A. Doney Introduction to Measure and Probability , 1967 .

[98]  L. Szilard On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings. , 1964, Behavioral science.

[99]  R. Landauer,et al.  Irreversibility and heat generation in the computing process , 1961, IBM J. Res. Dev..

[100]  L. Brillouin,et al.  Science and information theory , 1956 .

[101]  Yoichiro Takada ON THE MATHEMATICAL THEORY OF COMMUNICATION , 1954 .

[102]  Claude E. Shannon,et al.  The mathematical theory of communication , 1950 .

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

[104]  J. Preskill The future of quantum information science , 2022 .

[105]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .