Categorical quantum mechanics

This invited chapter in the Handbook of Quantum Logic and Quantum Structures consists of two parts: 1. A substantially updated version of quant-ph/0402130 by the same authors, which initiated the area of categorical quantum mechanics, but had not yet been published in full length; 2. An overview of the progress which has been made since then in this area.

[1]  C. Piron,et al.  On the Foundations of Quantum Physics , 1976 .

[2]  Peter Selinger,et al.  Towards a quantum programming language , 2004, Mathematical Structures in Computer Science.

[3]  Bill Edwards,et al.  Toy quantum categories , 2008, 0808.1037.

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

[5]  R Raussendorf,et al.  A one-way quantum computer. , 2001, Physical review letters.

[6]  D. A. Edwards.  The mathematical foundations of quantum mechanics , 1979, Synthese.

[7]  Edmund M. Clarke,et al.  Model Checking , 1999, Handbook of Automated Reasoning.

[8]  Isaac L. Chuang,et al.  Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations , 1999, Nature.

[9]  Rajagopal Nagarajan,et al.  Interaction categories and the foundations of typed concurrent programming , 1996, NATO ASI DPD.

[10]  R. A. G. Seely,et al.  Weakly distributive categories , 1997 .

[11]  Samson Abramsky,et al.  Abstract Physical Traces , 2009, ArXiv.

[12]  J. Baez.  Quantum Quandaries: a Category-Theoretic Perspective , 2004, quant-ph/0404040.

[13]  I. Chuang,et al.  Quantum Teleportation is a Universal Computational Primitive , 1999, quant-ph/9908010.

[14]  John Harding.  Orthomodularity in dagger biproduct categories , 2008 .

[15]  Bob Coecke,et al.  POVMs and Naimark's Theorem Without Sums , 2006, QPL.

[16]  A. Carboni,et al.  Cartesian bicategories I , 1987 .

[17]  M. Sipser,et al.  Quantum Computation by Adiabatic Evolution , 2000, quant-ph/0001106.

[18]  C. J. Isham,et al.  Topos Perspective on the Kochen-Specker Theorem: I. Quantum States as Generalized Valuations , 1998, quant-ph/9803055.

[19]  J. Neumann.  Mathematische grundlagen der Quantenmechanik , 1935 .

[20]  Robin Houston.  Finite products are biproducts in a compact closed category , 2008 .

[21]  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.

[22]  Samson Abramsky,et al.  Physical Traces: Quantum vs. Classical Information Processing , 2002, CTCS.

[23]  R. A. G. Seely,et al.  Linear Logic, -Autonomous Categories and Cofree Coalgebras , 1989 .

[24]  Ben Reichardt,et al.  Fault-Tolerant Quantum Computation , 2016, Encyclopedia of Algorithms.

[25]  Jamie Vicary,et al.  A Categorical Framework for the Quantum Harmonic Oscillator , 2007, 0706.0711.

[26]  L. Kauffman.  Knots And Physics , 1991 .

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

[28]  Louis H. Kauffman,et al.  Quantum entanglement and topological entanglement , 2002 .

[29]  Ross Duncan,et al.  Types for quantum computing , 2006 .

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

[31]  Edward Witten,et al.  Topological quantum field theory , 1988 .

[32]  John E. Roberts,et al.  A new duality theory for compact groups , 1989 .

[33]  H. Dishkant,et al.  Logic of Quantum Mechanics , 1976 .

[34]  J. Lambek,et al.  Introduction to higher order categorical logic , 1986 .

[36]  David N. Yetter,et al.  Braided Compact Closed Categories with Applications to Low Dimensional Topology , 1989 .

[37]  Robin Milner,et al.  Communication and concurrency , 1989, PHI Series in computer science.

[38]  Dominic R. Verity,et al.  Traced monoidal categories , 1996, Mathematical Proceedings of the Cambridge Philosophical Society.

[39]  Andr'e van Tonder,et al.  Quantum Computation, Categorical Semantics and Linear Logic , 2003, ArXiv.

[40]  R. Spekkens.  Evidence for the epistemic view of quantum states: A toy theory , 2004, quant-ph/0401052.

[41]  Bob Coecke,et al.  De-linearizing Linearity: Projective Quantum Axiomatics From Strong Compact Closure , 2005, QPL.

[42]  Simon Perdrix.  Bases in diagramatic quantum protocols , 2008 .

[43]  S. Braunstein,et al.  Impossibility of deleting an unknown quantum state , 2000, Nature.

[44]  C. J. Isham,et al.  A Topos Foundation for Theories of Physics: IV. Categories of Systems , 2008 .

[45]  Bob Coecke,et al.  Axiomatic Description of Mixed States From Selinger's CPM-construction , 2008, QPL.

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

[47]  Ekert,et al.  "Event-ready-detectors" Bell experiment via entanglement swapping. , 1993, Physical review letters.

[48]  M. Freedman,et al.  Topological Quantum Computation , 2001, quant-ph/0101025.

[49]  Louis Crane.  Categorical Geometry and the Mathematical Foundations of Quantum General Relativity , 2006 .

[50]  Samson Abramsky,et al.  A categorical semantics of quantum protocols , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

[51]  Dusko Pavlovic,et al.  Quantum measurements without sums , 2007 .

[52]  David N. Yetter,et al.  Functorial Knot Theory : Categories of Tangles, Coherence, Categorical Deformations and Topological Invariants , 2001 .

[53]  Louis H. Kauffman.  Teleportation topology , 2005 .

[54]  D. Dieks.  Communication by EPR devices , 1982 .

[55]  Ross Street.  Quantum groups: a path to current algebra , 2007 .

[56]  Lucas Dixon,et al.  Extending Graphical Representations for Compact Closed Categories with Applications to Symbolic Quantum Computation , 2008, AISC/MKM/Calculemus.

[57]  John C. Baez,et al.  Physics, Topology, Logic and Computation: A Rosetta Stone , 2009, 0903.0340.

[58]  D. Bouwmeester,et al.  The Physics of Quantum Information , 2000 .

[59]  Elham Kashefi,et al.  The measurement calculus , 2004, JACM.

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

[61]  H. Briegel,et al.  Measurement-based quantum computation on cluster states , 2003, quant-ph/0301052.

[62]  G. M. Kelly.  Many-variable functorial calculus. I. , 1972 .

[63]  B. Coecke,et al.  Categories for the practising physicist , 2009, 0905.3010.

[64]  P. Panangaden,et al.  Nuclear and trace ideals in tensored-categories , 1998, math/9805102.

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

[66]  Samson Abramsky,et al.  Abstract Scalars, Loops, and Free Traced and Strongly Compact Closed Categories , 2005, CALCO.

[67]  Samson Abramsky,et al.  A categorical quantum logic , 2006, Math. Struct. Comput. Sci..

[68]  Samson Abramsky,et al.  Retracing some paths in Process Algebra , 1996, CONCUR.

[69]  Temperley-Lieb.  Temperley-Lieb Algebra: From Knot Theory to Logic and Computation via Quantum Mechanics , 2007 .

[70]  Bob Coecke,et al.  Interacting Quantum Observables , 2008, ICALP.

[71]  G. M. Kelly,et al.  Coherence for compact closed categories , 1980 .

[72]  H. Prakash.  Quantum teleportation , 2009, 2009 International Conference on Emerging Trends in Electronic and Photonic Devices & Systems.

[73]  Roger Goodman,et al.  Oxford University , 1910, The Hospital.

[74]  R. Jozsa.  An introduction to measurement based quantum computation , 2005, quant-ph/0508124.

[75]  Joachim Kock,et al.  Frobenius Algebras and 2-D Topological Quantum Field Theories , 2004 .

[76]  J. Baez,et al.  Higher dimensional algebra and topological quantum field theory , 1995, q-alg/9503002.

[77]  Stephen G. Simpson,et al.  Subsystems of second order arithmetic , 1999, Perspectives in mathematical logic.

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

[79]  César Fernández,et al.  Non-sequential Processes , 1986, Advances in Petri Nets.

[80]  A. Joyal,et al.  The geometry of tensor calculus, I , 1991 .

[81]  C. J. Isham,et al.  A topos foundation for theories of physics: I. Formal languages for physics , 2007 .

[82]  Charles H. Bennett,et al.  Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. , 1993, Physical review letters.

[83]  Patrick Lincoln,et al.  Linear logic , 1992, SIGA.

[84]  N. Bohr.  II - Can Quantum-Mechanical Description of Physical Reality be Considered Complete? , 1935 .

[85]  Jean-Yves Girard,et al.  Linear Logic , 1987, Theor. Comput. Sci..

[86]  Samson Abramsky,et al.  High-level methods for quantum computation and information , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

[87]  César Fernández.  Non-sequential Processes , 1986, Advances in Petri Nets.