Abstract
We consider a semantics based on the peculiar holistic features of the quantum formalism. Any formula of the language gives rise to a quantum circuit that transforms the density operator associated to the formula into the density operator associated to the atomic subformulas in a reversible way. The procedure goes from the whole to the parts against the compositionality-principle and gives rise to a semantic characterization for a new form of quantum logic that has been called “Łukasiewicz quantum computational logic”. It is interesting to compare the logic based on qubit-semantics with that on qudit-semantics. Having in mind the relationships between classical logic and Łukasiewicz-many valued logics, one could expect that the former is stronger than the fragment of the latter. However, this is not the case. From an intuitive point of view, this can be explained by recalling that the former is a very weak form of logic. Many important logical arguments, which are valid either in Birkhoff and von Neumann’s quantum logic or in classical logic, are generally violated.
Highlights
Quantum computational logics are new forms of quantum logics inspired by the theory of quantum computation [1]
The procedure goes from the whole to the parts and not the other way around, against the compositionalityprinciple. This gives rise to a natural semantic characterization for a new form of quantum logic that has been called “Łukasiewicz quantum computational logic ” (ŁQCL)
Unlike standard compositional semantics, where the meaning of a whole is determined by the meanings of its parts, a holistic model determines the contextual meaning of any subformula
Summary
Quantum computational logics are new forms of quantum logics inspired by the theory of quantum computation [1]. The simplest piece of quantum information is a qubit (state): a unit-vector of the Hilbert space C2 that can be represented as a superposition. The procedure goes from the whole to the parts and not the other way around, against the compositionalityprinciple This gives rise to a natural semantic characterization for a new form of quantum logic that has been called “Łukasiewicz quantum computational logic ” (ŁQCL). In this semantics, the entropy of a formula is the average level of information or uncertainty inherent in the possible truth-values.
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