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:1301.1300 · Physical Review A

Algebraic approach to entanglement and entropy

A. P. Balachandran, T. R. Govindarajan, Amilcar R. de Queiroz, Andrés Fernando Reyes-Lega

Published Aug 1, 2013• 79 citations
arXivHTMLPDF

Abstract

We present a general approach to quantum entanglement and entropy that is based on algebras of observables and states thereon. In contrast to more standard treatments, Hilbert space is an emergent concept, appearing as a representation space of the observable algebra, once a state is chosen. In this approach, which is based on the Gel'fand-Naimark-Segal construction, the study of subsystems becomes particularly clear. We explicitly show how the problems associated with the partial trace for the study of entanglement of identical particles are readily overcome. In particular, a suitable entanglement measure is proposed that can be applied to systems of particles obeying Fermi, Bose, para-, and even braid-group statistics. The generality of the method is also illustrated by study of the time evolution of subsystems emerging from restriction to subalgebras. Problems related to anomalies and quantum epistemology are also discussed.

DOI
10.1103/physreva.88.022301

Related papers

Algebraic formulation of quantum theory, particle identity and entanglement

T. R. Govindarajan

Quantum theory as formulated in conventional framework using statevectors in Hilbert spaces misses the statistical nature of the underlying quantum physics. Formulation using operators [Formula: see text] algebra and density matrices appropriately captures this feature in addition leading to the correct formulation of particle identity. In this framework, Hilbert space is an emergent concept. Problems related to anomalies and quantum epistemology are discussed.

International Journal of Quantum Information•Aug 3, 2016•0 citations

An algebraic formulation of nonassociative quantum mechanics

Peter Schupp, Richard J. Szabo

Abstract We develop a version of quantum mechanics that can handle nonassociative algebras of observables and which reduces to standard quantum theory in the traditional associative setting. Our algebraic approach is naturally probabilistic and is based on using the universal enveloping algebra of a general nonassociative algebra to introduce a generalized notion of associative composition product. We formulate properties of states together with notions of trace, and use them to develop Gel’fand–Naimark–Segal constructions. We describe Heisenberg and Schrödinger pictures of completely positive dynamics, and we illustrate our formalism on the explicit examples of finite-dimensional matrix Jordan algebras as well as the octonion algebra.

arXiv:2311.03647•Journal of Physics A Mathematical and Theoretical•May 9, 2024•3 citations

Entanglement and Particle Identity: A Unifying Approach

A. P. Balachandran, T. R. Govindarajan, Amilcar R. de Queiroz +1 more

It has been known for some years that entanglement entropy obtained from partial trace does not provide the correct entanglement measure when applied to systems of identical particles. Several criteria have been proposed that have the drawback of being different according to whether one is dealing with fermions, bosons, or distinguishable particles. In this Letter, we give a precise and mathematically natural answer to this problem. Our approach is based on the use of the more general idea of the restriction of states to subalgebras. It leads to a novel approach to entanglement, which is suitable to be used in general quantum systems and especially in systems of identical particles. This settles some recent controversy regarding entanglement for identical particles. The prospects for applications of our criteria are wide ranging, from spin chains in condensed matter to entropy of black holes.

arXiv:1303.0688•Physical Review Letters•Feb 22, 2013•115 citations

Target space entanglement entropy

Edward A. Mazenc, Daniel Ranard

Abstract We define a notion of target space entanglement entropy. Rather than partitioning the base space on which the theory is defined, we consider partitions of the target space. This is the physical case of interest for first-quantized theories, such as worldsheet string theory. We associate to each subregion of the target space a suitably chosen subalgebra of observables $$ \mathcal{A} $$ A . The entanglement entropy is calculated as the entropy of the density matrix restricted to $$ \mathcal{A} $$ A . As an example, we illustrate our framework by computing spatial entanglement in first-quantized many-body quantum mechanics. The algebra $$ \mathcal{A} $$ A is chosen to reproduce the entanglement entropy obtained by embedding the state in the fixed particle sub-sector of the second-quantized Hilbert space. We then generalize our construction to the quantum field-theoretical setting.

arXiv:1910.07449•Journal of High Energy Physics•Mar 16, 2023•28 citations

Emergent gauge symmetries and quantum operations

A P Balachandran, Iván Mauricio Burbano, A F Reyes-Lega +1 more

Abstract The algebraic approach to quantum physics emphasizes the role played by the structure of the algebra of observables and its relation to the space of states. An important feature of this point of view is that subsystems can be described by subalgebras, with partial trace being replaced by the more general notion of restriction to a subalgebra. This, in turn, has recently led to applications to the study of entanglement in systems of identical particles. In the course of those investigations on entanglement and particle identity, an emergent gauge symmetry has been found by Balachandran, de Queiroz and Vaidya. In this letter we establish a novel connection between that gauge symmetry, entropy production and quantum operations. Thus, let A be a system described by a finite dimensional observable algebra and a mixed faithful state. Using the Gelfand–Naimark–Segal (GNS) representation we construct a canonical purification of , allowing us to embed A into a larger system C . Using Tomita–Takesaki theory, we obtain a subsystem decomposition of C into subsystems A and B , without making use of any tensor product structure. We identify a group of transformations that acts as a gauge group on A while at the same time giving rise to entropy increasing quantum operations on C . We provide physical means to simulate this gauge symmetry/quantum operation duality.

arXiv:1906.05464•Journal of Physics A Mathematical and Theoretical•Dec 12, 2019•0 citations

Entanglement Entropy in Quantum Mechanics: An Algebraic Approach

Andrés Fernando Reyes-Lega

An algebraic approach to the study of entanglement entropy of quantum systems is reviewed. Starting with a state on a $C^*$-algebra, one can construct a density operator describing the state in the GNS representation state. Applications of this approach to the study of entanglement measures for systems of identical particles are outlined. The ambiguities in the definition of entropy within this approach are then related to the action of unitaries in the commutant of the representation and their relation to modular theory explained.

arXiv:2212.04601•WORLD SCIENTIFIC eBooks•Apr 1, 2023•2 citations
We present a general approach to quantum entanglement and entropy that is based on algebras of observables and states thereon. In contrast to more standard treatments, Hilbert space is an emergent concept, appearing as a representation space of the observable algebra, once a state is chosen. In this approach, which is based on the Gel'fand-Naimark-Segal construction, the study of subsystems becomes particularly clear. We explicitly show how the problems associated with the partial trace for the study of entanglement of identical particles are readily overcome. In particular, a suitable entanglement measure is proposed that can be applied to systems of particles obeying Fermi, Bose, para-, and even braid-group statistics. The generality of the method is also illustrated by study of the time evolution of subsystems emerging from restriction to subalgebras. Problems related to anomalies and quantum epistemology are also discussed.
Abstract We develop a version of quantum mechanics that can handle nonassociative algebras of observables and which reduces to standard quantum theory in the traditional associative setting. Our algebraic approach is naturally probabilistic and is based on using the universal enveloping algebra of a general nonassociative algebra to introduce a generalized notion of associative composition product. We formulate properties of states together with notions of trace, and use them to develop Gel’fand–Naimark–Segal constructions. We describe Heisenberg and Schrödinger pictures of completely positive dynamics, and we illustrate our formalism on the explicit examples of finite-dimensional matrix Jordan algebras as well as the octonion algebra.
It has been known for some years that entanglement entropy obtained from partial trace does not provide the correct entanglement measure when applied to systems of identical particles. Several criteria have been proposed that have the drawback of being different according to whether one is dealing with fermions, bosons, or distinguishable particles. In this Letter, we give a precise and mathematically natural answer to this problem. Our approach is based on the use of the more general idea of the restriction of states to subalgebras. It leads to a novel approach to entanglement, which is suitable to be used in general quantum systems and especially in systems of identical particles. This settles some recent controversy regarding entanglement for identical particles. The prospects for applications of our criteria are wide ranging, from spin chains in condensed matter to entropy of black holes.
Abstract We define a notion of target space entanglement entropy. Rather than partitioning the base space on which the theory is defined, we consider partitions of the target space. This is the physical case of interest for first-quantized theories, such as worldsheet string theory. We associate to each subregion of the target space a suitably chosen subalgebra of observables A \mathcal{A} A A . The entanglement entropy is calculated as the entropy of the density matrix restricted to A \mathcal{A} A A . As an example, we illustrate our framework by computing spatial entanglement in first-quantized many-body quantum mechanics. The algebra A \mathcal{A} A A is chosen to reproduce the entanglement entropy obtained by embedding the state in the fixed particle sub-sector of the second-quantized Hilbert space. We then generalize our construction to the quantum field-theoretical setting.
Abstract The algebraic approach to quantum physics emphasizes the role played by the structure of the algebra of observables and its relation to the space of states. An important feature of this point of view is that subsystems can be described by subalgebras, with partial trace being replaced by the more general notion of restriction to a subalgebra. This, in turn, has recently led to applications to the study of entanglement in systems of identical particles. In the course of those investigations on entanglement and particle identity, an emergent gauge symmetry has been found by Balachandran, de Queiroz and Vaidya. In this letter we establish a novel connection between that gauge symmetry, entropy production and quantum operations. Thus, let A be a system described by a finite dimensional observable algebra and a mixed faithful state. Using the Gelfand–Naimark–Segal (GNS) representation we construct a canonical purification of , allowing us to embed A into a larger system C . Using Tomita–Takesaki theory, we obtain a subsystem decomposition of C into subsystems A and B , without making use of any tensor product structure. We identify a group of transformations that acts as a gauge group on A while at the same time giving rise to entropy increasing quantum operations on C . We provide physical means to simulate this gauge symmetry/quantum operation duality.
An algebraic approach to the study of entanglement entropy of quantum systems is reviewed. Starting with a state on a C∗C^*C∗-algebra, one can construct a density operator describing the state in the GNS representation state. Applications of this approach to the study of entanglement measures for systems of identical particles are outlined. The ambiguities in the definition of entropy within this approach are then related to the action of unitaries in the commutant of the representation and their relation to modular theory explained.