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Quantum logic as motivated by quantum computing

Published online by Cambridge University Press:  12 March 2014

J. Michael Dunn ,
Tobias J. Hagge ,
Lawrence S. Moss  and
Zhenghan Wang
J. Michael Dunn
Affiliation:
School of Informatics, Indiana University, Bloomington, Indiana 47408., USA, E-mail: [email protected]
Tobias J. Hagge
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405., USA, E-mail: [email protected]
Lawrence S. Moss
Affiliation:
Department of Computer Science, Indiana University, Bloomington, Indiana 47405., USA, E-mail: [email protected]
Zhenghan Wang
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405., USA, E-mail: [email protected]
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Extract

§1. Introduction. Our understanding of Nature comes in layers, so should the development of logic. Classic logic is an indispensable part of our knowledge, and its interactions with computer science have recently dramatically changed our life. A new layer of logic has been developing ever since the discovery of quantum mechanics. G. D. Birkhoff and von Neumann introduced quantum logic in a seminal paper in 1936 [1]. But the definition of quantum logic varies among authors (see [2]). How to capture the logic structure inherent in quantum mechanics is very interesting and challenging. Given the close connection between classical logic and theoretical computer science as exemplified by the coincidence of computable functions through Turing machines, recursive function theory, and λ-calculus, we are interested in how to gain some insights about quantum logic from quantum computing. In this note we make some observations about quantum logic as motivated by quantum computing (see [5]) and hope more people will explore this connection.

The quantum logic as envisioned by Birkhoff and von Neumann is based on the lattice of closed subspaces of a Hilbert space, usually an infinite dimensional one. The quantum logic of a fixed Hilbert space ℍ in this note is the variety of all the true equations with finitely many variables using the connectives meet, join and negation. Quantum computing is theoretically based on quantum systems with finite dimensional Hilbert spaces, especially the states space of a qubit ℂ2. (Actually the qubit is merely a convenience.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 2 , June 2005 , pp. 353 - 359
Copyright
Copyright © Association for Symbolic Logic 2005

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References

REFERENCES

[1] Birkhoff, G. and von Neumann, J., The logic of quantum mechanics, Annals of Mathematics, vol. 37 (1936), pp. 823–843.Google Scholar
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