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:1602.00567 · Electronic Proceedings in Theoretical Computer Science

Cohomology of Effect Algebras

Frank Roumen

Published Jan 1, 2017• 17 citations
arXivHTMLPDF

Abstract

We will define two ways to assign cohomology groups to effect algebras, which occur in the algebraic study of quantum logic. The first way is based on Connes' cyclic cohomology. The resulting cohomology groups are related to the state space of the effect algebra, and can be computed using variations on the Kunneth and Mayer-Vietoris sequences. The second way involves a chain complex of ordered abelian groups, and gives rise to a cohomological characterization of state extensions on effect algebras. This has applications to no-go theorems in quantum foundations, such as Bell's theorem.

DOI
10.4204/eptcs.236.12

Related papers

Algebraic and Mathematical Foundations of Quantum Computation: Hilbert Spaces, Group Structures, and Quantum Algorithms

Bhavesh B. Prajapati

Quantum computation is, at bottom, a branch of applied algebra and functional analysis: the state of a quantum register is a unit vector in a finite-dimensional complex Hilbert space, computation is the action of the unitary group on that space, and the structure of the most important quantum algorithms is dictated by the representation theory of finite and continuous groups. This paper develops the mathematical scaffolding of the field from first principles and uses it to explain why quantum computers can outperform classical ones. We review complex inner-product spaces, self-adjoint and unitary operators, the spectral theorem, and the tensor-product construction that gives rise to entanglement and the exponential dimension of the state space. We then describe the qubit and its geometry, the group-theoretic classification of quantum gates (the special unitary groups, the Pauli and Clifford groups, and their Lie-algebraic generators), and the universality results that justify finite gate sets together with the Solovay–Kitaev theorem. With this apparatus we analyse the quantum Fourier transform, phase estimation, and the hidden subgroup problem, showing how Shor’s factoring algorithm, Grover’s search, and the quantum linear-systems algorithm arise as instances of a small number of algebraic templates. Finally we treat the stabilizer formalism of quantum error correction—where finite abelian groups over GF(2) play the central role—and the variational, near-term (NISQ) algorithms that dominate current practice. Throughout, the emphasis is on the algebraic invariants that make quantum speed-ups possible.

Communications on Applied Nonlinear Analysis•Jan 24, 2025•0 citations

On realization of effect algebras

Josef Niederle, Jan Paseka

Abstract A well known fact is that there is a finite orthomodular lattice with an order determining set of states which is not order embeddable into the standard quantum logic, the lattice L(𝓗) of all closed subspaces of a separable complex Hilbert space. We show that a finite generalized effect algebra is order embeddable into the standard effect algebra E(𝓗) of effects of a separable complex Hilbert space iff it has an order determining set of generalized states iff it is order embeddable into the power of a finite MV-chain. As an application we obtain an algorithm, which is based on the simplex algorithm, deciding whether such an order embedding exists and, if the answer is positive, constructing it.

Mathematica Slovaca•Apr 1, 2016•1 citation

Topological Order from a Cohomological and Higher Gauge Theory perspective

Ricardo Costa de Almeida, J. P. Ibieta Jimenez, J. Lorca Espiro +1 more

In recent years, attempts to generalize lattice gauge theories to model topological order have been carried out through the so called $2$-gauge theories. These have opened the door to interesting new models and new topological phases which are not described by previous schemes of classification. In this paper we show that we can go beyond the $2$-gauge construction when considering chain complexes of abelian groups. Based on elements of homological algebra we are able to greatly simplify already known constructions for abelian theories under a single all encompassing framework. Furthermore, this formalism allows us to systematize the computation of the corresponding topological degeneracies of the ground states and establishes a connection between them and a known cohomology, which conveniently characterizes them with a suitable set of quantum numbers.

arXiv:1711.04186•Nov 11, 2017•13 citations

ON SPECTRAL THEORY IN EFFECT ALGEBRAS

Eissa D. Habil, Hossam F. Abu Lamdy

If one supposes a quantum logic Lto be a ˙-effect algebra, then the observables on Lare identified with the L-valued measures defined on the Borel subsets of the real line. In this structure (and without the aid of Hilbert spaceformalism) we show that (1) the spectrum of an observable Acan be completely characterized by studying theobservable (A )

•Jan 1, 2012•0 citations

Symmetries of Hilbert space effect algebras

Peter Šemrl

Let H be a Hilbert space and E(H) the effect algebra on H, that is, E(H) is the set of all self-adjoint operators A: H→H satisfying 0⩽A⩽I. The effect algebra can be equipped with several operations and relations that are important in mathematical foundations of quantum mechanics. Automorphisms with respect to these operations or relations are called symmetries. We present a new method that can be used to describe the general form of such maps. The main idea is to reduce this kind of problem to the study of adjacency-preserving maps. The efficiency of this approach is illustrated by reproving some known results as well as by obtaining some new theorems.

Journal of the London Mathematical Society•Jul 1, 2013•13 citations

Higher Abelian Quantum Double Models

Flores, Jorge Acuña, Giuseppe De Nittis, J. Lorca Espiro

This paper develops a rigorous $C^*$-algebraic framework for higher abelian quantum double models, generalizing Kitaev's construction to simplicial complexes of arbitrary finite dimension and local regularity. We fully characterize the frustration-free ground state space through an associated algebra of logical operators: its state space is shown to be homeomorphic to the space of frustration-free ground states, and to obey generalized canonical commutation relations. When the relevant homology and cohomology groups are finite, the logical algebra decomposes into a commutative factor and a full matrix algebra, thereby separating the classical and quantum parts of the frustration-free ground state structure. We further prove that the vanishing of these groups is necessary and sufficient for the core algebra of the model to form a Cartan pair with the full algebra of observables, a property expected to pave the way toward classifying the model's equilibrium (KMS) states, extending recent results for the planar case.

arXiv:2509.12864•Sep 16, 2025•0 citations
We will define two ways to assign cohomology groups to effect algebras, which occur in the algebraic study of quantum logic. The first way is based on Connes' cyclic cohomology. The resulting cohomology groups are related to the state space of the effect algebra, and can be computed using variations on the Kunneth and Mayer-Vietoris sequences. The second way involves a chain complex of ordered abelian groups, and gives rise to a cohomological characterization of state extensions on effect algebras. This has applications to no-go theorems in quantum foundations, such as Bell's theorem.
Quantum computation is, at bottom, a branch of applied algebra and functional analysis: the state of a quantum register is a unit vector in a finite-dimensional complex Hilbert space, computation is the action of the unitary group on that space, and the structure of the most important quantum algorithms is dictated by the representation theory of finite and continuous groups. This paper develops the mathematical scaffolding of the field from first principles and uses it to explain why quantum computers can outperform classical ones. We review complex inner-product spaces, self-adjoint and unitary operators, the spectral theorem, and the tensor-product construction that gives rise to entanglement and the exponential dimension of the state space. We then describe the qubit and its geometry, the group-theoretic classification of quantum gates (the special unitary groups, the Pauli and Clifford groups, and their Lie-algebraic generators), and the universality results that justify finite gate sets together with the Solovay–Kitaev theorem. With this apparatus we analyse the quantum Fourier transform, phase estimation, and the hidden subgroup problem, showing how Shor’s factoring algorithm, Grover’s search, and the quantum linear-systems algorithm arise as instances of a small number of algebraic templates. Finally we treat the stabilizer formalism of quantum error correction—where finite abelian groups over GF(2) play the central role—and the variational, near-term (NISQ) algorithms that dominate current practice. Throughout, the emphasis is on the algebraic invariants that make quantum speed-ups possible.
Abstract A well known fact is that there is a finite orthomodular lattice with an order determining set of states which is not order embeddable into the standard quantum logic, the lattice L(𝓗) of all closed subspaces of a separable complex Hilbert space. We show that a finite generalized effect algebra is order embeddable into the standard effect algebra E(𝓗) of effects of a separable complex Hilbert space iff it has an order determining set of generalized states iff it is order embeddable into the power of a finite MV-chain. As an application we obtain an algorithm, which is based on the simplex algorithm, deciding whether such an order embedding exists and, if the answer is positive, constructing it.
In recent years, attempts to generalize lattice gauge theories to model topological order have been carried out through the so called 222-gauge theories. These have opened the door to interesting new models and new topological phases which are not described by previous schemes of classification. In this paper we show that we can go beyond the 222-gauge construction when considering chain complexes of abelian groups. Based on elements of homological algebra we are able to greatly simplify already known constructions for abelian theories under a single all encompassing framework. Furthermore, this formalism allows us to systematize the computation of the corresponding topological degeneracies of the ground states and establishes a connection between them and a known cohomology, which conveniently characterizes them with a suitable set of quantum numbers.
Let H be a Hilbert space and E(H) the effect algebra on H, that is, E(H) is the set of all self-adjoint operators A: H→H satisfying 0⩽A⩽I. The effect algebra can be equipped with several operations and relations that are important in mathematical foundations of quantum mechanics. Automorphisms with respect to these operations or relations are called symmetries. We present a new method that can be used to describe the general form of such maps. The main idea is to reduce this kind of problem to the study of adjacency-preserving maps. The efficiency of this approach is illustrated by reproving some known results as well as by obtaining some new theorems.
This paper develops a rigorous C∗C^*C∗-algebraic framework for higher abelian quantum double models, generalizing Kitaev's construction to simplicial complexes of arbitrary finite dimension and local regularity. We fully characterize the frustration-free ground state space through an associated algebra of logical operators: its state space is shown to be homeomorphic to the space of frustration-free ground states, and to obey generalized canonical commutation relations. When the relevant homology and cohomology groups are finite, the logical algebra decomposes into a commutative factor and a full matrix algebra, thereby separating the classical and quantum parts of the frustration-free ground state structure. We further prove that the vanishing of these groups is necessary and sufficient for the core algebra of the model to form a Cartan pair with the full algebra of observables, a property expected to pave the way toward classifying the model's equilibrium (KMS) states, extending recent results for the planar case.