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

Birkhäuser Boston eBooks

Analysis of One and Two Particle Quantum Systems using Geometric Algebra

Rachel Parker, Chris Doran

Published Jan 1, 2002• 11 citations
Publisher

Abstract

When two or more subsystems of a quantum system interact with each other they can become entangled. In this case the individual subsystems can no longer be described as pure quantum states. For systems with only two subsystems this entanglement can be described using the Schmidt decomposition. This selects a preferred orthonormal basis for expressing the wavefunction and gives a measure of the degree of entanglement present in the system. The extension of this to the more general case of n subsystems is not yet known. We present a review of this process using the standard representation and apply this method in the geometric algebra setting, which has the advantage of suggesting a generalisation to n subsystems. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-1-4612-0089-5_20

Related papers

Analysis of 1 and 2 Particle Quantum Systems using Geometric Algebra

Rachel Parker, Chris Doran

When two or more subsystems of a quantum system interact with each other they can become entangled. In this case the individual subsystems can no longer be described as pure quantum states. For systems with only 2 subsystems this entanglement can be described using the Schmidt decomposition. This selects a preferred orthonormal basis for expressing the wavefunction and gives a measure of the degree of entanglement present in the system. The extension of this to the more general case of n subsystems is not yet known. We present a review of this process using the standard representation and apply this method to the geometric algebra representation. This latter form has the advantage of suggesting a generalisation to n subsystems.

arXiv:quant-ph/0106055•Jun 11, 2001•1 citation

Entangled quantum systems and the Schmidt decomposition

ARTUR K. EKERT, P. L. Knight

Quantum systems comprised of interacting subsystems become highly correlated and their individual identities become entangled. This entanglement can be described using the Schmidt decomposition, in which a pair of preferred orthonormal bases can be constructed to emphasize the tight correlations between two quantum subsystems. Examples are given of how the Schmidt basis can be exploited to shed new light on entangled subsystems in quantum optics, paying special attention to two-mode squeezed states and to atom–field interaction.

American Journal of Physics•May 1, 1995•452 citations

Formulas for Schmidt decompositions of mutually orthogonal quantum states in two-qubit systems

Yonghae Lee, Youngho Min, Sunghyun Bae +1 more

Abstract We present Schmidt decomposition formulas for mutually orthogonal two-qubit pure states and classify orthonormal sets according to their entanglement structure. We begin by presenting explicit formulas for the Schmidt decomposition of arbitrary two-qubit pure states. Building on this, we construct and analyze all possible configurations of two mutually orthogonal pure states, providing corresponding Schmidt decomposition expressions for each case. For orthonormal sets of three and four states, we derive explicit formulas for selected representative types and discuss the difficulties in obtaining closed-form expressions for more general cases. In particular, we focus on special classes of orthonormal bases that include one or two maximally entangled states. Finally, we prove that no orthonormal basis can contain three product states and one entangled state, revealing a structural constraint that restricts the composition of entangled orthonormal sets.

arXiv:2503.06988•Journal of Physics A Mathematical and Theoretical•Oct 20, 2025•1 citation

Entanglement distribution in pure non-Gaussian tripartite states: a Schmidt decomposition approach

Abdeldjalil Merdaci, Ahmed Jellal

We study entanglement in a system of three coupled quantum harmonic oscillators. Specifically, we use the Schmidt decomposition to analyze how the entanglement is distributed among the three subsystems. The Schmidt decomposition is a powerful mathematical tool for characterizing bipartite entanglement in composite quantum systems. It allows to write a multipartite quantum state as a sum of product states between the subsystems, with coefficients known as Schmidt coefficients. We apply this decomposition to the general quantum state of three coupled oscillators and study how the Schmidt coefficients evolve as the interaction strengths between the oscillators are varied. This provides insight into how entanglement is shared between the different bipartitions of the overall three-particle system. Our results advance the fundamental understanding of multipartite entanglement in networked quantum systems. They also have implications for quantum information processing using multiple entangled nodes.

arXiv:2409.18923•Sep 27, 2024•1 citation

Quantum system Schmidt decomposition and its geometric analysis

Shuang Cong, Feng Xian-yong

Quantum entangled state and some common methods to measure entanglement are introduced, and Schmidt decomposition of 2-qubit quantum state and the method to identify the entangled state are induced. A method to calculate the degree of entanglement is also given, which is obtained very easily by using the Schmidt decomposition. An example about NMR quantum computation is discussed to illustrate the application of the Schmidt decomposition. The evolution of the state of sub-system is described by using the Bloch vector, which demonstrates the geometric prospect of quantum system.

Chinese Control Conference (CCC)•Jul 1, 2008•1 citation

Geometry of the Schmidt decomposition and Hardy’s theorem

P. K. Aravind

The Schmidt decomposition theorem allows us to express an entangled wave function of two quantum systems in a particularly simple form. The purpose of this paper is to describe a geometrical method of performing the Schmidt decomposition that complements the usual algebraic method. This geometrical method is worked out in detail for a pair of spin-1/2 particles and its utility is illustrated by giving a simple discussion of several aspects of Hardy’s version of Bell’s theorem.

American Journal of Physics•Sep 1, 1996•16 citations
When two or more subsystems of a quantum system interact with each other they can become entangled. In this case the individual subsystems can no longer be described as pure quantum states. For systems with only two subsystems this entanglement can be described using the Schmidt decomposition. This selects a preferred orthonormal basis for expressing the wavefunction and gives a measure of the degree of entanglement present in the system. The extension of this to the more general case of n subsystems is not yet known. We present a review of this process using the standard representation and apply this method in the geometric algebra setting, which has the advantage of suggesting a generalisation to n subsystems. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
When two or more subsystems of a quantum system interact with each other they can become entangled. In this case the individual subsystems can no longer be described as pure quantum states. For systems with only 2 subsystems this entanglement can be described using the Schmidt decomposition. This selects a preferred orthonormal basis for expressing the wavefunction and gives a measure of the degree of entanglement present in the system. The extension of this to the more general case of n subsystems is not yet known. We present a review of this process using the standard representation and apply this method to the geometric algebra representation. This latter form has the advantage of suggesting a generalisation to n subsystems.
Quantum systems comprised of interacting subsystems become highly correlated and their individual identities become entangled. This entanglement can be described using the Schmidt decomposition, in which a pair of preferred orthonormal bases can be constructed to emphasize the tight correlations between two quantum subsystems. Examples are given of how the Schmidt basis can be exploited to shed new light on entangled subsystems in quantum optics, paying special attention to two-mode squeezed states and to atom–field interaction.
Abstract We present Schmidt decomposition formulas for mutually orthogonal two-qubit pure states and classify orthonormal sets according to their entanglement structure. We begin by presenting explicit formulas for the Schmidt decomposition of arbitrary two-qubit pure states. Building on this, we construct and analyze all possible configurations of two mutually orthogonal pure states, providing corresponding Schmidt decomposition expressions for each case. For orthonormal sets of three and four states, we derive explicit formulas for selected representative types and discuss the difficulties in obtaining closed-form expressions for more general cases. In particular, we focus on special classes of orthonormal bases that include one or two maximally entangled states. Finally, we prove that no orthonormal basis can contain three product states and one entangled state, revealing a structural constraint that restricts the composition of entangled orthonormal sets.
We study entanglement in a system of three coupled quantum harmonic oscillators. Specifically, we use the Schmidt decomposition to analyze how the entanglement is distributed among the three subsystems. The Schmidt decomposition is a powerful mathematical tool for characterizing bipartite entanglement in composite quantum systems. It allows to write a multipartite quantum state as a sum of product states between the subsystems, with coefficients known as Schmidt coefficients. We apply this decomposition to the general quantum state of three coupled oscillators and study how the Schmidt coefficients evolve as the interaction strengths between the oscillators are varied. This provides insight into how entanglement is shared between the different bipartitions of the overall three-particle system. Our results advance the fundamental understanding of multipartite entanglement in networked quantum systems. They also have implications for quantum information processing using multiple entangled nodes.
Quantum entangled state and some common methods to measure entanglement are introduced, and Schmidt decomposition of 2-qubit quantum state and the method to identify the entangled state are induced. A method to calculate the degree of entanglement is also given, which is obtained very easily by using the Schmidt decomposition. An example about NMR quantum computation is discussed to illustrate the application of the Schmidt decomposition. The evolution of the state of sub-system is described by using the Bloch vector, which demonstrates the geometric prospect of quantum system.