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

Linear algebra for quantum theory

Per‐Olov Löwdin

Published Jan 1, 1998• 113 citations
OpenAlex

Abstract

Elements of Set Theory. Linear Spaces. Binary Product Spaces. Axioms of Quantum Theory Formulated as a Trace Algebra. References. Appendices. Index.

Related papers

On the axioms of quantum theory formulated as a trace algebra

Per‐Olov Löwdin

International Journal of Quantum Chemistry•Jan 1, 1982•50 citations

Elements of quantum theory

Giuseppe Di Molfetta

In this chapter we will expose the fundamental principles of quantum mechanics alongside the mathematical formalism on which they are grounded, which is that of Hilbert spaces. In particular we will briefly present the necessary fundamentals of linear algebra, including the definitions of vector spaces, of scalar and tensorial products, and the definition of linear operators.

Institute of Physics eBooks•Sep 1, 2024•1 citation

Linear algebra

Steven Herbert

Abstract Quantum mechanics is mathematically formulated in terms of a Hilbert space, which is a vector space with a defined inner product. Linear algebra is the study of vector spaces and the (linear) operations thereon, and so it follows that having a solid grasp of the basics of linear algebra is essential for studying quantum mechanics and, therefore, quantum computation. All the linear algebra prerequisites necessary to understand the theoretical results presented in the remainder of the book are detailed in this chapter. The chapter also acts as a nomenclature reference for the book, by providing all the notations that are used thereafter.

•Dec 8, 2025•0 citations

The Mathematics of Quantum Mechanics

Jeffrey Alan Barrett

Abstract Quantum mechanics is written in the language of linear algebra. On the Schrodinger picture the theory represents quantum-mechanical states using the elements of a Hilbert space and represents observable physical properties and the standard dynamics using the linear operators on the state space. We consider the mathematical notions for understanding and working with the standard formulation of quantum mechanics. Each mathematical notion is characterized geometrically, algebraically, and physically. The mathematical representation of quantum-mechanical superpositions is discussed.

•Dec 26, 2019•1 citation

algebras in quantum theory bk chapter

Moffat, James

•Jan 1, 2018•0 citations

Group-Covariant Stochastic Products and Phase-Space Convolution Algebras

Paolo Aniello

Abstract A quantum stochastic product is defined as a binary operation on the convex set of quantum states that preserves the convex structure. We discuss a class of group-covariant, associative stochastic products, the twirled products, having remarkable connections with quantum measurement theory and with the theory of open quantum systems. By extending this binary operation from the density operators to the full Banach space of trace class operators, one obtains a Banach algebra. In the case where the covariance group is the group of phase-space translations, one has a quantum convolution algebra. The expression of the quantum convolution in terms of Wigner distributions and of the associated characteristic functions is analyzed.

International Journal of Theoretical Physics•Apr 22, 2023•3 citations
Abstract Quantum mechanics is mathematically formulated in terms of a Hilbert space, which is a vector space with a defined inner product. Linear algebra is the study of vector spaces and the (linear) operations thereon, and so it follows that having a solid grasp of the basics of linear algebra is essential for studying quantum mechanics and, therefore, quantum computation. All the linear algebra prerequisites necessary to understand the theoretical results presented in the remainder of the book are detailed in this chapter. The chapter also acts as a nomenclature reference for the book, by providing all the notations that are used thereafter.
Abstract Quantum mechanics is written in the language of linear algebra. On the Schrodinger picture the theory represents quantum-mechanical states using the elements of a Hilbert space and represents observable physical properties and the standard dynamics using the linear operators on the state space. We consider the mathematical notions for understanding and working with the standard formulation of quantum mechanics. Each mathematical notion is characterized geometrically, algebraically, and physically. The mathematical representation of quantum-mechanical superpositions is discussed.
Abstract A quantum stochastic product is defined as a binary operation on the convex set of quantum states that preserves the convex structure. We discuss a class of group-covariant, associative stochastic products, the twirled products, having remarkable connections with quantum measurement theory and with the theory of open quantum systems. By extending this binary operation from the density operators to the full Banach space of trace class operators, one obtains a Banach algebra. In the case where the covariance group is the group of phase-space translations, one has a quantum convolution algebra. The expression of the quantum convolution in terms of Wigner distributions and of the associated characteristic functions is analyzed.