Categorical Quantum Mechanics I: Causal Quantum Processes

We derive the category-theoretic backbone of quantum theory from a process ontology. More specifically, we treat quantum theory as a theory of systems, processes and their interactions. In this first part of a three-part overview, we first present a general theory of diagrams, and in particular, of string diagrams. We discuss why diagrams are a very natural starting point for developing scientific theories. Then we define process theories, and define a very general notion of quantum type. We show how our process ontology enables us to assert causality, that is, compatibility of quantum theory and relativity theory, and we prove the no-signalling theorem. Other notable contributions include new, elegant derivations of the no-broadcasting theorem, unitarity of evolution, and Stinespring dilation, all for any 'quantum' type in a general class of process theories.

[1]  H. Barnum,et al.  Generalized no-broadcasting theorem. , 2007, Physical review letters.

[2]  Peter Selinger.  Finite Dimensional Hilbert Spaces are Complete for Dagger Compact Closed Categories (Extended Abstract) , 2011, Electron. Notes Theor. Comput. Sci..

[3]  P. Selinger.  A Survey of Graphical Languages for Monoidal Categories , 2009, 0908.3347.

[4]  Man-Duen Choi.  Completely positive linear maps on complex matrices , 1975 .

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

[6]  Aleks Kissinger.  Abstract Tensor Systems as Monoidal Categories , 2014, Categories and Types in Logic, Language, and Physics.

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

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

[9]  B. Coecke,et al.  Classical and quantum structuralism , 2009, 0904.1997.

[10]  Peter Selinger,et al.  Dagger Compact Closed Categories and Completely Positive Maps: (Extended Abstract) , 2007, QPL.

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

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

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

[14]  J. Kowski.  Linear transformations which preserve trace and positive semidefiniteness of operators , 1972 .

[15]  G. Ghirardi,et al.  A general argument against superluminal transmission through the quantum mechanical measurement process , 1980 .

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

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

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

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

[20]  Peter Selinger,et al.  Autonomous categories in which A ∼ = A ∗ , 2014 .

[21]  S. Lane.  Natural Associativity and Commutativity , 1979 .

[22]  Aleks Kissinger,et al.  Categorical Quantum Mechanics II: Classical-Quantum Interaction , 2016, 1605.08617.

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

[24]  Bob Coecke,et al.  Terminality implies non-signalling , 2014, QPL.

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

[26]  Samson Abramsky,et al.  Categorical quantum mechanics , 2008, 0808.1023.

[27]  Bob Coecke,et al.  The Logic of Entanglement , 2004, Horizons of the Mind.

[28]  G. D’Ariano,et al.  Probabilistic theories with purification , 2009, 0908.1583.

[29]  B. Coecke.  Kindergarten Quantum Mechanics , 2005, quant-ph/0510032.

[30]  Tobias Fritz,et al.  Beyond Bell’s Theorem II: Scenarios with Arbitrary Causal Structure , 2014, 1404.4812.

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

[32]  Aleks Kissinger,et al.  Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning , 2017 .

[33]  Bob Coecke,et al.  Deep Beauty: A Universe of Processes and Some of Its Guises , 2010, 1009.3786.

[34]  L. Hardy.  Foliable Operational Structures for General Probabilistic Theories , 2009, 0912.4740.

[35]  B. Coecke.  Quantum picturalism , 2009, 0908.1787.