Electronic Proceedings in Theoretical Computer Science

A Basic Modal Logic for the Abstract Treatment of Quantum Mixed States

Tomoaki Kawano

Published Oct 25, 2026• 0 citations

Abstract

Quantum logic has developed in various directions since its inception.However, many existing logical approaches either focus only on pure states, or use Hilbert-space models in which the abstract relationship between pure states, mixed states, and probabilities is not made explicit.In this work, we introduce a model designed to handle quantum mixed states abstractly.Furthermore, we construct a proof system for the proposed logic and establish its completeness with respect to the intended semantics.

DOI: 10.4204/eptcs.455.4

Related papers

Quantum modal logic

Kenji Tokuo

Abstract A modal logic based on quantum logic is formalized in its simplest possible form. Specifically, a relational semantics and a sequent calculus are provided, and the soundness and the completeness theorems connecting both notions are demonstrated. This framework is intended to serve as a basis for formalizing various modal logics over quantum logic, such as quantum alethic logic, quantum temporal logic, quantum epistemic logic, and quantum dynamic logic.

arXiv:2511.10188Logic Journal of IGPLNov 12, 20243 citations

Systems of quantum logic

R I G Hughes

According to quantum mechanics, the pure states of a microsystem are represented by vectors in a Hilbert Space. Sentences of the form, "x є L" (where x is the state vector for a system, L a subspace of the appropriate Hilbert space), may be called Q-propositions: such sentences serve to summarise our information about the results of possible experiments on the system. Quantum logic investigates the relations which hold among the Q-propositions about a given physical sys tem. These logical relations correspond to algebraic relations among the subspaces of Hilbert space. The algebra of this set of subspaces is non-Boolean, and may be regarded either as an orthomodular lattice or as a partial Boolean algebra. With each type of structure we can associate a logic. A general approach to the semantics for such a logic is provided in terms of interpretations of a formal language within an algebraic structure; an interpretation maps sentences of the language homomorphically onto elements of the structure. When the structure in question is a Boolean algera, the resulting logic is classical; here we develop a semantics for the logic associated with partial Boolean algebras. Two systems of proof, based on the natural deduction systems of Gentzen, are shown for this logic. With respect to the given sematics, these calculi are sound and weakly complete. Strong completeness is conjectured. Quantum logic deals with the logical relations between sentences, and so is properly called a logic. However, it is the logic appropriate to a limited class of sentences: proposals that it should replace classical logic wherever the latter is used should be viewed with suspicion.

Jan 1, 20100 citations

A LOGIC-ALGEBRAIC FRAMEWORK FOR CONTEXTUALITY AND MODALITY IN QUANTUM SYSTEMS

Héctor Freytes

In this work we develop a modal structure for the simultaneous treatment of \nactual and possible properties of quantum systems. A logical system based on \northomodular lattices enriched with a modal operator is given, obtaining algebraic \ncompleteness and completeness with respect to a Kripke-style semantic. \nWe show that, in spite of the fact that, the language is enriched with the addition \nof a modal operator, contextuality remains a central feature of quantum \nsystems.

Mar 5, 20140 citations

The Logical Difference in Quantum Mathematics Separating Pure States from Mixed States

Steve Faulkner

Abstract I give a short explanation of how mathematics representing pure states is logically distinct from mixed states. This is intended as understandable to the undergraduate.

Mar 1, 20150 citations

A holistic quantum computational semantics

María Luisa Dalla Chiara, Roberto Giuntini, Roberto Leporini

Does quantum holism give rise to new forms of abstract holistic semantics? This question can be discussed in the framework of quantum computational logics, non-standard versions of quantum logic that have been suggested by the theory of logical gates in quantum computation. The basic idea of the quantum computational semantics is the following: meanings of sentences are identified with quantum information quantities, represented by pure or mixed states of special quantum systems. In particular cases, the meaning of a sentence may correspond to an entangled state. On this basis, one can formally describe some typical holistic semantic situations, where the meaning of the whole determines the meanings of its parts, but generally not vice versa.

Jan 1, 20062 citations

Quantum logic

Alireza Moazzen

This article employs a descriptive-analytical method with an algebraic and axiomatic approach to address quantum logic. Through several thought experiments, it is demonstrated that the distributive law for propositions does not hold in quantum mechanics. Therefore, the logic governing this domain of science is not classical logic. In fact, the algebra corresponding to the logic of quantum propositions is an orthomodular lattice, which is why the distributive law is not necessarily valid for such propositions. In quantum logic, various definitions are offered for the implication operator. Since none of these operators satisfy the deduction theorem, a suitable implication operator with this property must be defined. This objective is achieved by defining a unified quantum logic. By constructing a Lindenbaum-Tarski algebra for this unified quantum logic, it can be shown that it is indeed a quantum logic. Dishkant presented an embedding of quantum logic into the Br-modal logic system. This provides an appropriate semantics for quantum logic. A suitable model for this logic is the closed subspaces of a Hilbert space.

Nov 1, 20250 citations