Coherent spaces, Boolean rings and quantum gates

TL;DR

This paper introduces coherent spaces based on finite coherent states, explores their properties and algebraic structure, and discusses applications to classical and quantum CNOT gates using coherent states.

Contribution

It defines a new class of coherent spaces, analyzes their algebraic structure as a Boolean ring, and applies this framework to classical and quantum CNOT gate implementations.

Findings

Coherent spaces form a distributive lattice and a Boolean ring.

Projectors in these spaces resolve the identity and are invariant under transformations.

Applications include classical and quantum CNOT gates with coherent states.

Abstract

Coherent spaces spanned by a finite number of coherent states, are introduced. Their coherence properties are studied, using the Dirac contour representation. It is shown that the corresponding projectors resolve the identity, and that they transform into projectors of the same type, under displacement transformations, and also under time evolution. The set of these spaces, with the logical OR and AND operations is a distributive lattice, and with the logical XOR and AND operations is a Boolean ring (Stone\rq{}s formalism). Applications of this Boolean ring into classical CNOT gates with $n$-ary variables, and also quantum CNOT gates with coherent states, are discussed.