arXiv subjects
Publications and source records attributed to R. Giuntini.
A holistic extension of classical propositional logic is introduced in the framework of quantum computation with mixed states. The concepts of tautology and contradiction are investigated in this extensions. A special family of quantum states are investigated as particular cases of "holistic" contradiction.
In the qubit semantics the \emph{meaning} of any sentence $α$ is represented by a \emph{quregister}: a unit vector of the $n$--fold tensor product $\otimes^n \C^2$, where $n$ depends on the number of occurrences of atomic sentences in $α$. The logic characterized by this semantics, called {\it quantum computational logic} (QCL), is {\it unsharp}, because the non-contradiction principle is violated. We show that QCL does not admit any logical truth. In this framework, any sentence $α$ gives rise to a \emph{quantum tree}, consisting of a sequence of unitary operators. The quantum tree of $α$ can be regarded as a quantum circuit that transforms the quregister associated to the atomic subformulas of $α$ into the quregster associated to $α$.
Quantum computation has suggested new forms of quantum logic, called quantum computational logics. The basic semantic idea is the following: the meaning of a sentence is identified with a quregister, a system of qubits, representing a possible pure state of a compound quantum system. The generalization to mixed states, which might be useful to analyse entanglement-phenomena, is due to Gudder. Quantum computational logics represent non standard examples of unsharp quantum logic, where the non-contradiction principle is violated, while conjunctions and disjunctions are strongly non-idempotent. In this framework, any sentence of the language gives rise to a quantum tree: a kind of quantum circuit that transforms the quregister associated to the atomic subformulas of the sentence into the quregister associated to the sentence.
Logical gates studied in quantum computation suggest a natural logical abstraction that gives rise to a new form of unsharp quantum logic. We study the logical connectives corresponding to the following gates: the Toffoli gate, the NOT and the squareroot of NOT (which admit of natural physical models). This leads to a semantic characterization of a logic that we call computational quantum logic CQL.
We investigate some forms of quantum logic arising from the standard and the unsharp approach.