Logic of Local Inference for Contextuality in Quantum Physics and Beyond

Contextuality in quantum physics provides a key resource for quantum information and computation. The topological approach in [Abramsky and Brandenburger, New J. Phys., 2011, Abramsky et al., CSL 2015, 2015] characterizes contextuality as "global inconsistency" coupled with "local consistency", revealing it to be a phenomenon also found in many other fields. This has yielded a logical method of detecting and proving the "global inconsistency" part of contextuality. Our goal is to capture the other, "local consistency" part, which requires a novel approach to logic that is sensitive to the topology of contexts. To achieve this, we formulate a logic of local inference by using context-sensitive theories and models in regular categories. This provides a uniform framework for local consistency, and lays a foundation for high-level methods of detecting, proving, and moreover using contextuality as computational resource.

[1]  P. Johnstone.  Sketches of an Elephant: A Topos Theory Compendium Volume 1 , 2002 .

[2]  L. Hardy,et al.  Nonlocality for two particles without inequalities for almost all entangled states. , 1993, Physical review letters.

[3]  Kohei Kishida,et al.  Contextuality, Cohomology and Paradox , 2015, CSL.

[4]  Kiel T. Williams,et al.  Extreme quantum entanglement in a superposition of macroscopically distinct states. , 1990, Physical review letters.

[5]  Jonathan A. Zvesper,et al.  From Lawvere to Brandenburger-Keisler: Interactive forms of diagonalization and self-reference , 2015, J. Comput. Syst. Sci..

[6]  Axel Poigné,et al.  Basic category theory , 1993, LICS 1993.

[7]  Tom Leinster,et al.  Basic Category Theory , 2014, 1612.09375.

[8]  I. Moerdijk,et al.  Sheaves in geometry and logic: a first introduction to topos theory , 1992 .

[9]  A. Zeilinger,et al.  Going Beyond Bell’s Theorem , 2007, 0712.0921.

[10]  Carsten Butz.  Regular Categories and Regular Logic , 1998 .

[11]  R. Raussendorf.  Quantum computation, discreteness, and contextuality , 2009 .

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

[13]  S. Popescu,et al.  Quantum nonlocality as an axiom , 1994 .

[14]  Jürg Kohlas,et al.  Generic local computation , 2012, J. Comput. Syst. Sci..

[15]  Isaac L. Chuang,et al.  Quantum Computation and Quantum Information (10th Anniversary edition) , 2011 .

[16]  C. J. Isham,et al.  Topos Perspective on the Kochen–Specker Theorem: IV. Interval Valuations , 2001 .

[17]  F. W. Lawvere.  QUANTIFIERS AND SHEAVES by , 2010 .

[18]  Samson Abramsky,et al.  The sheaf-theoretic structure of non-locality and contextuality , 2011, 1102.0264.

[19]  D. Gabbay,et al.  Interpolation and Definability: Modal and Intuitionistic Logic , 2005 .

[20]  Kohei Kishida,et al.  Stochastic Relational Presheaves and Dynamic Logic for Contextuality , 2014, QPL.

[21]  Samson Abramsky,et al.  Relational Databases and Bell's Theorem , 2012, In Search of Elegance in the Theory and Practice of Computation.

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

[23]  Dov M. Gabbay,et al.  Interpolation and definability , 2005 .

[24]  Victor Veitch,et al.  Contextuality supplies the ‘magic’ for quantum computation , 2014, Nature.

[25]  F. W. Lawvere,et al.  Diagonal arguments and cartesian closed categories , 1969 .