Purity through Factorisation

We give a construction that identifies the collection of pure processes (i.e. those which are deterministic, or without randomness) within a theory containing both pure and mixed processes. Working in the framework of symmetric monoidal categories, we define a pure subcategory. This definition arises elegantly from the categorical notion of a weak factorisation system. Our construction gives the expected result in several examples, both quantum and classical.

[1]  Giulio Chiribella,et al.  Operational axioms for diagonalizing states , 2015, ArXiv.

[2]  G. M. Kelly,et al.  Categories of continuous functors, I , 1972 .

[3]  Simon Perdrix,et al.  Environment and Classical Channels in Categorical Quantum Mechanics , 2010, CSL.

[4]  Giulio Chiribella,et al.  Dilation of states and processes in operational-probabilistic theories , 2014, QPL.

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

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

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

[8]  Bob Coecke,et al.  Leaks: Quantum, Classical, Intermediate and More , 2017, Entropy.

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

[10]  Chris Heunen,et al.  Axiomatizing complete positivity , 2015, ArXiv.

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

[12]  Chris Heunen,et al.  Pictures of complete positivity in arbitrary dimension , 2011, Inf. Comput..

[13]  Bas Westerbaan,et al.  Paschke Dilations , 2016, QPL.

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

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

[16]  Giulio Chiribella,et al.  Distinguishability and Copiability of Programs in General Process Theories , 2014, Int. J. Softw. Informatics.

[17]  Jirí Adámek,et al.  Abstract and Concrete Categories - The Joy of Cats , 1990 .