Quantum projective measurements and the CHSH inequality in Isabelle/HOL

We present a formalization in Isabelle/HOL of quantum projective measurements, a class of measurements involving orthogonal projectors that is frequently used in quantum computing. We also formalize the CHSH inequality, a result that holds on arbitrary probability spaces, which can used to disprove the existence of a local hidden-variable theory for quantum mechanics.

[1]  J. S. BELLt.  Einstein-Podolsky-Rosen Paradox , 2018 .

[2]  René Thiemann,et al.  Formalizing Jordan normal forms in Isabelle/HOL , 2016, CPP.

[3]  Andrei Popescu,et al.  From Types to Sets by Local Type Definition in Higher-Order Logic , 2018, Journal of Automated Reasoning.

[4]  Andrew Rosalsky,et al.  Laws of Large Numbers , 2011, International Encyclopedia of Statistical Science.

[5]  Florian Kammüller,et al.  Attack trees in Isabelle extended with probabilities for quantum cryptography , 2019, Comput. Secur..

[6]  Dominique Unruh,et al.  Quantum relational Hoare logic , 2018, Proc. ACM Program. Lang..

[7]  Johannes Hölzl,et al.  Construction and stochastic applications of measure spaces in higher-order logic , 2013 .

[8]  Clemens Ballarin.  Locales: A Module System for Mathematical Theories , 2013, Journal of Automated Reasoning.

[9]  Bohua Zhan,et al.  Formal Verification of Quantum Algorithms Using Quantum Hoare Logic , 2019, CAV.

[10]  Jennifer Paykin,et al.  QWIRE Practice: Formal Verification of Quantum Circuits in Coq , 2018, QPL.

[11]  R. Durrett.  Probability: Theory and Examples , 1993 .

[12]  Yijun He,et al.  Isabelle Marries Dirac: a Library for Quantum Computation and Quantum Information , 2020, Arch. Formal Proofs.

[13]  Tobias Nipkow,et al.  Concrete Semantics: With Isabelle/HOL , 2014 .

[14]  N. Mermin.  Hidden variables and the two theorems of John Bell , 1993, 1802.10119.

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

[16]  P. Fuhrmann.  Linear Systems and Operators in Hilbert Space , 1982 .

[17]  S. Wehner,et al.  Bell Nonlocality , 2013, 1303.2849.

[18]  B. Garraway,et al.  Tests for Einstein-Podolsky-Rosen steering in two-mode systems of identical massive bosons , 2020 .

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

[20]  Rajagopal Nagarajan,et al.  Formalization of Quantum Protocols using Coq , 2015, QPL.

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

[22]  Alain Aspect,et al.  Experimental Tests of Bell’s Inequalities in Atomic Physics , 1983 .