Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2603.26739

Quantum Fuzzy Sets Revisited: Density Matrices, Decoherence, and the Q-Matrix Framework

Mirco A. Mannucci

quant-phcs.AISubmitted Mar 22, 2026• Announced Mar 24, 2026• 0 citations
arXivHTMLPDF

Abstract

In 2006 we proposed Quantum Fuzzy Sets, observing that states of a quantum register could serve as characteristic functions of fuzzy subsets, embedding Zadeh's unit interval into the Bloch sphere. That paper was deliberately preliminary. In the two decades since, the idea has been taken up by researchers working on quantum annealers, intuitionistic fuzzy connectives, and quantum machine learning, while parallel developments in categorical quantum mechanics have reshaped the theoretical landscape. The present paper revisits that programme and introduces two main extensions. First, we move from pure states to density matrices, so that truth values occupy the entire Bloch ball rather than its surface; this captures the phenomenon of semantic decoherence that pure-state semantics cannot express. Second, we introduce the Q-Matrix, a global density matrix from which individual quantum fuzzy sets emerge as local sections via partial trace. We define a category QFS of quantum fuzzy sets, establish basic structural properties (monoidal structure, fibration over Set), characterize the classical limit as simultaneous diagonalizability, and exhibit an obstruction to a fully internal Frobenius-algebra treatment.

Comments
11 pages

Related papers

Quantum Fuzzy Sets: Blending Fuzzy Set Theory and Quantum Computation

Mirco A. Mannucci

In this article we investigate a way in which quantum computing can be used to extend the class of fuzzy sets. The core idea is to see states of a quantum register as characteristic functions of quantum fuzzy subsets of a given set. As the real unit interval is embedded in the Bloch sphere, every fuzzy set is automatically a quantum fuzzy set. However, a generic quantum fuzzy set can be seen as a (possibly entangled) superposition of many fuzzy sets at once, offering new opportunities for modeling uncertainty. After introducing the main framework of quantum fuzzy set theory, we analyze the standard operations of fuzzification and defuzzification from our viewpoint. We conclude this preliminary paper with a list of possible applications of quantum fuzzy sets to pattern recognition, as well as future directions of pure research in quantum fuzzy set theory.

arXiv:cs/0604064•Apr 16, 2006•13 citations

Quantum Fuzzy Sets Revisited: Density Matrices, Decoherence, and the Q-Matrix Framework

Mirco A. Mannucci

•Jan 1, 2026•0 citations

Interpretations on Quantum Fuzzy Computing: Intuitionistic Fuzzy Operations × Quantum Operators

Renata Reiser, Alexandre Lemke, Anderson Ávila +3 more

Quantum processes provide a parallel model for fuzzy connectives. Calculations of quantum states may be simultaneously performed by the superposition of membership and non-membership degrees of each element regarding the intuitionistic fuzzy sets. This work aims to interpret Atanassov's intuitionistic fuzzy logic through quantum computing, where not only intuitionistic fuzzy sets, but also their basic operations and corresponding connectives (negation, conjuntion, disjuntion, difference, codifference, implication, and coimplication), are interpreted based on the traditional quantum circuit model.

Electronic Notes in Theoretical Computer Science•Sep 1, 2016•11 citations

Aggregation operations from quantum computing

Lidiane Visintin, Adriano Maron, Renata Reiser +1 more

Computer systems based on fuzzy logic are capable of generating a reliable output even when handling inaccurate input data by applying a rule based system. The main contribution of this paper is to show that quantum computing can be used to extend the class of fuzzy sets. The central idea associates the states of a quantum register to membership functions (mFs) of fuzzy subsets, and the rules for the processes of fuzzyfication are performed by unitary quantum transformations. Thus, this paper describes multi-dimensional quantum registers, associated to mFs on the unitary interval U, in order to introduce a novel interpretation of aggregations found in fuzzy set theory. In particular, t-norms and t-conorms based on quantum gates allows the modeling and interpretation of union, intersection, difference and implication among fuzzy sets, also including an expression for the class of fuzzy S-implications. Furthermore, an interpretation of the symmetric sum was achieved by considering the quantum register sum operator.

IEEE International Conference on Fuzzy Systems (FUZZ-IEEE)•Jul 1, 2013•13 citations

FBGT Meta-Theory: A Branch Growth Layer Framework Interpreting Classical Geometry from Outside ZFC

xun zhang

Abstract Within the Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), Gödel‘s incompleteness theorems and Tarski’s undefinability theorem are established via Gödel arithmetic encoding, embedding first-order formal languages into zero-dimensional natural number spaces. Geometric measure theory, in turn, defines Lebesgue measure and Hausdorff dimension as intrinsic invariant volume quantities on Euclidean manifolds, treating measure as an absolute property of space independent of observation. This paper, taking Amari‘s information geometry and its quantum generalization as an external meta-basis beyond ZFC, constructs a branch growth layer system with a sheaf-theoretic semantics embedding via a fully faithful functor, embedding the entire first-order formal syntax into the global section space of the structure sheaf on a compact simply connected 4D Poincaré topological manifold. We strictly distinguish two classes of mathematical structures: (i) global quantum joint pure states with no observational loss (corresponding to global sheaf sections, carrying complete intrinsic information), and (ii) marginal measures induced by local observational truncation (classical Lebesgue measure being the classical limit of the reduced density mapping from quantum partial trace). The central proof establishes that the formal-logical obstructions identified by Gödel and Tarski are equivalent to the strict Shannon / von Neumann information entropy loss induced by the local truncation functor of sheaves. We define a critical trajectory submanifold T, on whose neighborhood all sheaf restriction morphisms are surjective, thereby eliminating information loss due to projection and constructing a hierarchy-free global truth morphism. The Kakeya-Besicovitch paradox is employed as a case study to validate the framework: the Besicovitch set — of zero Lebesgue measure and full Hausdorff dimension — is categorically isomorphic to a two-body maximally entangled pure state, where local marginal observational information vanishes while global intrinsic correlation information reaches its maximum. This meta-framework takes information as the primary ontology, transcending the internal constraints of ZFC axioms, and coherently integrates sheaf cohomology, quantum information geometry, harmonic analysis, computability theory, and geometric measure theory. It demonstrates that the inherent limitations of classical mathematical logic and measure theory are merely categorical representations arising from dimensional projection reduction, and not essential constraints of purely mathematical objects themselves. License CC BY 4.0 Knowledge belongs to all humanity. Research work only indicates the directions explored and the places once reached by human mathematicians.

•Aug 7, 2026•0 citations

Classical and Quantum Structuralism

Bob Coecke, Éric Oliver Paquette, Duško Pavlović

In recent work, several researchers including the authors have developed a categorical formalization of quantum mechanics in terms of symmetric monoidal dagger categories. In this framework, classical data turned out to be represented by an algebraic structure, that of special commutative dagger Frobenius algebras. This structure captures the distinct capabilities that apply to classical data – that they can be copied and deleted. In the present paper, we provide categorical semantics and diagrammatic representations of deterministic, nondeterministic, and probabilistic operations over classical data represented in this way. Moreover, a combination of some fundamental categorical constructions (the Kleisli construction of the category of free algebras and the Grothendieck construction of the total category of an indexed category) with the specific categorical presentations of pure and mixed quantum states provides a resource-sensitive categorical account of classical control of quantum data and of classical data resulting from quantum measurements, as well as of the classical data processing that may happen in between measurements and controls. Along the way we also discover some apparently novel quantum typing structures. One of the salient features of categorical quantum mechanics is still its graphic calculus, which allows succinct presentations of diverse quantum protocols. The elements of an abstract stochastic calculus are beginning to emerge from it, pointing toward convenient refinements of resource-sensitive logics that are hoped to capture the probabilistic content and limited observability of quantum data.

arXiv:0904.1997•Cambridge University Press eBooks•Nov 30, 2009•72 citations
In 2006 we proposed Quantum Fuzzy Sets, observing that states of a quantum register could serve as characteristic functions of fuzzy subsets, embedding Zadeh's unit interval into the Bloch sphere. That paper was deliberately preliminary. In the two decades since, the idea has been taken up by researchers working on quantum annealers, intuitionistic fuzzy connectives, and quantum machine learning, while parallel developments in categorical quantum mechanics have reshaped the theoretical landscape. The present paper revisits that programme and introduces two main extensions. First, we move from pure states to density matrices, so that truth values occupy the entire Bloch ball rather than its surface; this captures the phenomenon of semantic decoherence that pure-state semantics cannot express. Second, we introduce the Q-Matrix, a global density matrix from which individual quantum fuzzy sets emerge as local sections via partial trace. We define a category QFS of quantum fuzzy sets, establish basic structural properties (monoidal structure, fibration over Set), characterize the classical limit as simultaneous diagonalizability, and exhibit an obstruction to a fully internal Frobenius-algebra treatment.
In this article we investigate a way in which quantum computing can be used to extend the class of fuzzy sets. The core idea is to see states of a quantum register as characteristic functions of quantum fuzzy subsets of a given set. As the real unit interval is embedded in the Bloch sphere, every fuzzy set is automatically a quantum fuzzy set. However, a generic quantum fuzzy set can be seen as a (possibly entangled) superposition of many fuzzy sets at once, offering new opportunities for modeling uncertainty. After introducing the main framework of quantum fuzzy set theory, we analyze the standard operations of fuzzification and defuzzification from our viewpoint. We conclude this preliminary paper with a list of possible applications of quantum fuzzy sets to pattern recognition, as well as future directions of pure research in quantum fuzzy set theory.
Quantum processes provide a parallel model for fuzzy connectives. Calculations of quantum states may be simultaneously performed by the superposition of membership and non-membership degrees of each element regarding the intuitionistic fuzzy sets. This work aims to interpret Atanassov's intuitionistic fuzzy logic through quantum computing, where not only intuitionistic fuzzy sets, but also their basic operations and corresponding connectives (negation, conjuntion, disjuntion, difference, codifference, implication, and coimplication), are interpreted based on the traditional quantum circuit model.
Computer systems based on fuzzy logic are capable of generating a reliable output even when handling inaccurate input data by applying a rule based system. The main contribution of this paper is to show that quantum computing can be used to extend the class of fuzzy sets. The central idea associates the states of a quantum register to membership functions (mFs) of fuzzy subsets, and the rules for the processes of fuzzyfication are performed by unitary quantum transformations. Thus, this paper describes multi-dimensional quantum registers, associated to mFs on the unitary interval U, in order to introduce a novel interpretation of aggregations found in fuzzy set theory. In particular, t-norms and t-conorms based on quantum gates allows the modeling and interpretation of union, intersection, difference and implication among fuzzy sets, also including an expression for the class of fuzzy S-implications. Furthermore, an interpretation of the symmetric sum was achieved by considering the quantum register sum operator.
Abstract Within the Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), Gödel‘s incompleteness theorems and Tarski’s undefinability theorem are established via Gödel arithmetic encoding, embedding first-order formal languages into zero-dimensional natural number spaces. Geometric measure theory, in turn, defines Lebesgue measure and Hausdorff dimension as intrinsic invariant volume quantities on Euclidean manifolds, treating measure as an absolute property of space independent of observation. This paper, taking Amari‘s information geometry and its quantum generalization as an external meta-basis beyond ZFC, constructs a branch growth layer system with a sheaf-theoretic semantics embedding via a fully faithful functor, embedding the entire first-order formal syntax into the global section space of the structure sheaf on a compact simply connected 4D Poincaré topological manifold. We strictly distinguish two classes of mathematical structures: (i) global quantum joint pure states with no observational loss (corresponding to global sheaf sections, carrying complete intrinsic information), and (ii) marginal measures induced by local observational truncation (classical Lebesgue measure being the classical limit of the reduced density mapping from quantum partial trace). The central proof establishes that the formal-logical obstructions identified by Gödel and Tarski are equivalent to the strict Shannon / von Neumann information entropy loss induced by the local truncation functor of sheaves. We define a critical trajectory submanifold T, on whose neighborhood all sheaf restriction morphisms are surjective, thereby eliminating information loss due to projection and constructing a hierarchy-free global truth morphism. The Kakeya-Besicovitch paradox is employed as a case study to validate the framework: the Besicovitch set — of zero Lebesgue measure and full Hausdorff dimension — is categorically isomorphic to a two-body maximally entangled pure state, where local marginal observational information vanishes while global intrinsic correlation information reaches its maximum. This meta-framework takes information as the primary ontology, transcending the internal constraints of ZFC axioms, and coherently integrates sheaf cohomology, quantum information geometry, harmonic analysis, computability theory, and geometric measure theory. It demonstrates that the inherent limitations of classical mathematical logic and measure theory are merely categorical representations arising from dimensional projection reduction, and not essential constraints of purely mathematical objects themselves. License CC BY 4.0 Knowledge belongs to all humanity. Research work only indicates the directions explored and the places once reached by human mathematicians.
In recent work, several researchers including the authors have developed a categorical formalization of quantum mechanics in terms of symmetric monoidal dagger categories. In this framework, classical data turned out to be represented by an algebraic structure, that of special commutative dagger Frobenius algebras. This structure captures the distinct capabilities that apply to classical data – that they can be copied and deleted. In the present paper, we provide categorical semantics and diagrammatic representations of deterministic, nondeterministic, and probabilistic operations over classical data represented in this way. Moreover, a combination of some fundamental categorical constructions (the Kleisli construction of the category of free algebras and the Grothendieck construction of the total category of an indexed category) with the specific categorical presentations of pure and mixed quantum states provides a resource-sensitive categorical account of classical control of quantum data and of classical data resulting from quantum measurements, as well as of the classical data processing that may happen in between measurements and controls. Along the way we also discover some apparently novel quantum typing structures. One of the salient features of categorical quantum mechanics is still its graphic calculus, which allows succinct presentations of diverse quantum protocols. The elements of an abstract stochastic calculus are beginning to emerge from it, pointing toward convenient refinements of resource-sensitive logics that are hoped to capture the probabilistic content and limited observability of quantum data.