# Quantum Observables for Binary, Multi-Valued and Fuzzy Logic : Eigenlogic

Abstract : We propose a linear algebraic method, named Eigenlogic, for two-, many-valued and fuzzy logic using observables in Hilbert space. All logical connectives are represented by observables where the truth values correspond to eigenvalues and the atomic input propositional cases, i.e. the “models” of a propositional system, to the respective eigenvectors. In this way propositional logic can be formalized by using combinations of tensored elementary quantum observables. The outcome of a “measurement” of a logical observable will give the truth value of the associated logical proposition, and becomes “interpretable” when applied to vectors of its eigenspace, leading to an original insight into the quantum measurement postulate. Recently the concept of “quantum predicate” has been proposed in leading to similar concepts. We develop logical observables for binary logic and extend them to many-valued logic. For binary logic and truth values {0, 1} logical observables are commuting projector operators. For truth values {+1,−1} the logical observables are isometries formally equivalent to the ones of a composite quantum spin 1 2 system, these observables are reversible quantum logic gates. The analogy of many-valued logic with quantum angular momentum is then established using a general algebraic method, based on classical interpolation framework suggested by the finite-element method. Logical observables for three-valued logic are formulated using the orbital angular momentum observable Lz with ` = 1. The representative 3-valued 2-argument logical observables for the ternary threshold logical connectives Min and Max are then explicitly obtained. Also in this approach fuzzy logic arises naturally when considering vectors outside the eigensystem. The fuzzy membership function is obtained by the quantum mean value (Born rule) of the logical projector observable and turns out to be a probability measure. Fuzziness arises because of the quantum superposition of atomic propositional cases, the truth of a proposition being in this case a probabilistic value ranging from completely false to completely true. This method could be employed for developing algorithms in high-dimensional vector spaces for example in modern semantic theories, such as distributional semantics or in connectionist models of cognition. For practical implementation, due to the exponential growth of the vector space dimension, adapted logical reduction methods must then be used. LSA (Latent Semantic Analysis) algorithms are often used in quantum-like approaches, this was done using the HAL (Hyperspace Analogue to Language) algorithm. Our approach is also of interest for quantum computation because several of the observables in Eigenlogic are well-known quantum gates (for example CONTROL-Z) and other ones can be derived by unitary transformations. Ternary-logic quantum gates using qu-trits lead to less complex circuits, our formulation of multi-valued logical observables could help the development of new multi-valued quantum gate architectures.
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Submitted on : Saturday, November 19, 2016 - 11:52:31 AM
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Zeno Toffano, François Dubois. Quantum Observables for Binary, Multi-Valued and Fuzzy Logic : Eigenlogic. 7th Colloquium IQFA-2016, CNRS GDR-3322 on Q. Engineering, Foundations & Applications Ingénierie Quantique, des aspects Fondamentaux aux Applications – IQFA, Nov 2016, Paris, France. ⟨hal-01399528⟩

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