Some generalizations of fuzzy structures in quantum computational logic

Abstract

Quantum computational logics provide a fertile common ground for a unified treatment of vagueness and uncertainty. In this paper, we describe an approach to the logic of quantum computation that has been recently taken up and developed. After reporting on the state of the art, we explore some future research perspectives in the light of some recent limitative results whose general significance will be duly assessed.

Keywords:

• fuzzy logic
• quantum computation
• vagueness

Acknowledgements

We thank Hector Freytes and Marisa Dalla Chiara for the stimulating conversations on the topics covered in this paper.

Notes

^1. Meaning that there might be a sentence α, such that neither α nor its negation holds for the state at issue.

^2. For the sake of simplicity we henceforth denote by or .

^3. The set of all n-configurations is an orthonormal basis for . We call the computational basis of .

^4. A density operator is a positive, self-adjoint, trace class (linear) operator with trace 1.

^5. A projection operator is a self-adjoint operator s.t. .

^6. We remark that there exists unitary gates which may admit of entangled states as outputs, yet fail to be computationally entangled. A case in point is the XOR gate, which yields an entanglement only when applied to superposition states (cp. Nielsen and Chuang 2000).

^7. We remark that the meaning herein attached to the expression ‘logical consequence’ only partially overlaps with the standard Tarskian notion of logical consequence relation adopted in contemporary abstract algebraic logic. To cite only the most striking difference, the present relation may or may not hold between single formulae, whereas Tarski's may or may not hold between a set of formulae and a single formula.

^8. This ‘Stone–Weierstrass type’ result is actually much more general: every continuous function has a quantum analogue which can be approximated in such a way by means of polynomial quantum operations.