Square root of ‘not’: a major difference between fuzzy and quantum logics

Abstract

Many authors have mentioned the similarity between quantum logic and fuzzy logic. In this paper, we show that, in spite of this similarity, these logics are not identical. Specifically, we emphasize that, while quantum logic has a special ‘square root of not’ operation, which is very useful in quantum computing, fuzzy logic lacks such an operation.

Keywords:

• fuzzy logic
• quantum logic
• quantum computing
• square root of ‘not’

Acknowledgements

This work was supported, in part, by NSF grants HRD-0734825, EAR-0225670 and EIA-0080940, by Texas Department of Transportation contract No. 0-5453, by the Japan Advanced Institute of Science and Technology (JAIST) International Joint Research Grant 2006-08 and by the Max Planck Institut für Mathematik.

The authors are thankful to all the participants of the 27th International Conference of the North American Fuzzy Information Processing Society NAFIPS’2008 (New York, New York, May 19–22, 2008) for valuable suggestions, especially to Rudolf Seising and Mark Wierman. The authors are also grateful to the anonymous referees for their useful advice.