Spectral theory in quantum logics

Donald E. Catlin

A. L. Ivanov, Dan Caragheorgheopol

Kenji Tokuo

We study the spectral sign implication for quantum logic, defined by the nonnegative spectral projection of the operator obtained by subtracting the antecedent projection from the consequent projection. The construction agrees with classical material implication on commuting projections and compares arbitrary pairs of projections through the spectral structure of their difference. It satisfies Hardegree's four minimal implicative conditions, his law of contraposition, and a falsity condition. It differs from the standard polynomial implications in that its value need not belong to the ortholattice generated by its arguments. Within a uniform class of Borel constructions for pairs of projections, the operations satisfying entailment, contraposition, and the falsity condition correspond exactly to measurable choices of spectral branch. In the continuous subclass, these three conditions determine the spectral sign implication uniquely. In finite dimensions, the operation is also the largest among the acceptance projections of optimal projective tests for Helstrom discrimination between subspace states with priors proportional to rank. These results connect quantum implication with the operator theory of two projections and binary quantum discrimination.

Gottfried T. Rüttimann

Anas N. Al‐Rabadi

This paper reviews recent developments in the use of reversible synthesis techniques in the reversible and quantum logic circuit design of binary and multiple valued switching functions. In particular, this paper reviews the specific developments and various uses of spectral transforms and functional expansions, group theory, and cellular automata in the reversible logic synthesis of digital functions.

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