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:1906.07309

Computational Many-Body Physics via $\mathcal M_{2^q}$ Algebra

Emil V. Prodan

Published Jun 17, 2019• 1 citation
arXivHTMLPDF

Abstract

The many-body Hamiltonians and other fermionic physical observables are expressed in terms of fermionic creation and annihilation operators, which form the algebra of canonical anti-commutation relations (CAR). In this work we use a canonical isomorphism between CAR and $\mathcal M_{2^\infty}$ algebras to derive analytic matrix representations of many-fermion operators. Code-lines implementing these matrix representations are supplied and Hubbard-type Hamiltonians are worked out explicitly.

DOI
10.48550/arxiv.1906.07309

Related papers

Density Matrix of Bipartite Fermi Systems in the Framework of Local CAR-Algebras

Аirat Sitdikov, A. S. Nikitin, D. V. Bushtets

An algebraic model for studying quantum Fermi systems has been formulated. An embedding of the algebra of canonical anticommutation relations ( $$CAR$$ ) into the Cuntz algebra $$\mathcal{O}_{2}$$ was used. Representing this embedding on an infinite-dimensional Hilbert space forms a Fock space, on which creation and annihilation operators act, expressed in terms of the generators of the Cuntz algebra. This approach, in some cases, simplifies the calculation of various physical quantities within the second quantization method. As an example, the calculation of a two-mode partial density matrix of a bipartite fermionic system within the proposed model is presented.

Lobachevskii Journal of Mathematics•Apr 1, 2026•0 citations

Algebra, Logic and Qubits: Quantum Abacus

Alexander Yu. Vlasov

The canonical anticommutation relations (CAR) for fermion systems can be represented by finite-dimensional matrix algebra, but it is impossible for canonical commutation relations (CCR) for bosons. After description of more simple case with representation of CAR and (bounded) quantum computational networks via Clifford algebras in the paper are discussed CCR. For representation of the algebra it is not enough to use quantum networks with fixed number of qubits and it is more convenient to consider Turing machine with essential operation of appending new cells for description of infinite tape in finite terms --- it has straightforward generalization for quantum case, but for CCR it is necessary to work with symmetrized version of the quantum Turing machine. The system is called here quantum abacus due to understanding analogy with the ancient counting devices (abacus).

arXiv:quant-ph/0001100•Jan 27, 2000•1 citation

The structure of gauge invariant Gaussian quantum operations on finite Fermion systems

Eric A. Carlen

Let ${\mathcal H}_1$ be a finite dimensional complex Hilbert space. Let $ψ\mapsto Z(ψ)$ be a canonical anti-commutation relations (CAR) field over ${\mathcal H}_1$ acting irreducibly on a Hilbert space ${\mathord{\mathscr K}}$. The $*$-algebra ${\mathscr A}_{{\mathcal H}_1}$ generated by the $Z(ψ)$, $ψ\in {\mathcal H}_1$, is simply all operators on ${\mathscr K}$. However, the CAR field endows ${\mathscr A}_{{\mathcal H}_1}$ with additional structure, and we are concerned with quantum operations acting in harmony with this structure. In particular, there is a "gauge" automorphism group generated by "second quantizing'' $ψ\mapsto e^{it}ψ$. The fixed point algebra of the gauge group, ${\mathscr G}_{{\mathcal H}_1}$, is a sub-algebra of ${\mathscr A}_{{\mathcal H}_1}$ studied by Araki and Wyss. It contains the density matrices of an important class of states, the {\em gauge invariant Gaussian states}, ${\mathfrak S}_{GIG}$. Our focus is on semigroups $\{e^{t{\mathscr L}}\}_{t\geq 0}$ of quantum operations on ${\mathscr A}_{{\mathcal H}_1}$ that map ${\mathfrak S}_{GIG}$ into itself. Each $e^{t{\mathscr L}}$ is one-to-one, and our first main result is a structure theorem for such quantum operations on ${\mathscr G}_{{\mathcal H}_1}$ that map ${\mathfrak S}_{GIG}$ into itself. We apply this to study semigroups of quantum operations on ${\mathscr G}_{{\mathcal H}_1}$ that map ${\mathfrak S}_{GIG}$ into itself. Our second main result is a structure theorem showing that they are parameterized by pairs $(G,A)$ where $G$ is a contraction semigroup generator on ${\mathcal H}_1$, and $0 \leq A \leq -G -G^*$. We then show that each of these semigroups has a natural extension to the full CAR algebra ${\mathscr A}_{{\mathcal H}_1}$. Further results are obtained under further assumptions on the pair $(G,A)$.

math.FACross-listedarXiv:2605.00784•Advanced Nonlinear Studies•May 4, 2026•0 citations

CAR on Fock spaces

Jan Dereziński, Christian Gérard

CAR on Fock spacesThis chapter is devoted to the study of Fock representations of the canonical anticommutation relations, a basic tool of quantum many-body theory and quantum field theory.It is parallel to Chap. 9, where Fock CCR representations were studied.The basic framework is almost the same as in Chap.9. Throughout this chapter Z is a Hilbert space, called the one-particle space.We will consider the Fock CAR representation acting on the fermionic Fock space Γ a (Z).As in Sect.1.3, we introduce the space Y = Re(Z ⊕ Z) := {(z, z) : z ∈ Z}, which will serve as the dual phase space of our system.Recall that in the bosonic case we equipped Y with the structure of the Kähler space consisting of the anti-involution j, the Euclidean scalar product • and the symplectic form ω: j(z, z) := (iz, iz), (13.1) (z, z) • (w, w) := 2Re(z|w), (13.2) (z, z)•ω(w, w) := 2Im(z|w) = -(z, z) • j(w, w).(13.3)In our presentation of the fermionic case, we will need the Kähler anti-involution j.The symplectic form ω will not be used.Instead of the scalar product (13.2) we will use another scalar product,Again we will avoid identifing Z with Y. CY is identified with Z ⊕ Z by the mapY # , the space dual to Y, is canonically identified with Re(Z ⊕ Z) by using the scalar product (9.2), and CY # is identified with Z ⊕ Z. Fock CAR representationConsider the fermionic Fock space Γ a (Z).Recall that, for z ∈ Z, a * (z), resp.a(z) denote the corresponding creation, resp.annihilation operators defined in Sect.3.4.

Cambridge University Press eBooks•Nov 4, 2022•0 citations

The algebra of Grassmann canonical anticommutation relations and its applications to fermionic systems

Michael Keyl, Dirk-M. Schlingemann

We present an approach to a noncommutativelike phase space which allows to analyze quasifree states on the algebra of canonical anti-commutation relations (CAR) in analogy to quasifree states on the algebra of canonical commutation relations (CCR). The used mathematical tools are based on a new algebraic structure the “Grassmann algebra of canonical anticommutation relations” (GAR algebra) which is given by the twisted tensor product of a Grassmann and a CAR algebra. As a new application, the corresponding theory provides an elegant tool for calculating the fidelity of two quasifree fermionic states which is needed for the study of entanglement distillation within fermionic systems.

arXiv:0906.2929•Journal of Mathematical Physics•Feb 1, 2010•9 citations

A mapping between the spin and fermion algebra

Felix Meier, Daniel Waltner, Petr A. Braun +1 more

Abstract We derive a formalism to express the spin algebra <?CDATA $\mathfrak{s}\mathfrak{u}(2)$?> s u ( 2 ) in a spin s representation in terms of the algebra of L fermionic operators that obey the canonical anti-commutation relations. We also give the reverse direction of expressing the fermionic operators as polynomials in the spin operators of a single spin. We extend here to further spin values the previous investigations by Dobrov (2003 J. Phys. A: Math. Gen. 36 L503) who in turn clarified on an inconsistency within a similar formalism in the works of Batista and Ortiz (2001 Phys. Rev. Lett. 86 1082). We then consider a system of L fermion flavors and apply our mapping in order to express it in terms of the spin algebra. Furthermore we investigate a possibility to simplify certain Hamiltonian operators by means of the mapping.

arXiv:2101.10119•Journal of Physics A Mathematical and Theoretical•Jul 13, 2021•0 citations
Computational Many-Body Physics via M2q\mathcal M_{2^q}M2q​ Algebra
The many-body Hamiltonians and other fermionic physical observables are expressed in terms of fermionic creation and annihilation operators, which form the algebra of canonical anti-commutation relations (CAR). In this work we use a canonical isomorphism between CAR and M2∞\mathcal M_{2^\infty}M2∞​ algebras to derive analytic matrix representations of many-fermion operators. Code-lines implementing these matrix representations are supplied and Hubbard-type Hamiltonians are worked out explicitly.
An algebraic model for studying quantum Fermi systems has been formulated. An embedding of the algebra of canonical anticommutation relations ( CARCARCAR ) into the Cuntz algebra O2\mathcal{O}_{2}O2​ was used. Representing this embedding on an infinite-dimensional Hilbert space forms a Fock space, on which creation and annihilation operators act, expressed in terms of the generators of the Cuntz algebra. This approach, in some cases, simplifies the calculation of various physical quantities within the second quantization method. As an example, the calculation of a two-mode partial density matrix of a bipartite fermionic system within the proposed model is presented.
The canonical anticommutation relations (CAR) for fermion systems can be represented by finite-dimensional matrix algebra, but it is impossible for canonical commutation relations (CCR) for bosons. After description of more simple case with representation of CAR and (bounded) quantum computational networks via Clifford algebras in the paper are discussed CCR. For representation of the algebra it is not enough to use quantum networks with fixed number of qubits and it is more convenient to consider Turing machine with essential operation of appending new cells for description of infinite tape in finite terms --- it has straightforward generalization for quantum case, but for CCR it is necessary to work with symmetrized version of the quantum Turing machine. The system is called here quantum abacus due to understanding analogy with the ancient counting devices (abacus).
Let H1{\mathcal H}_1H1​ be a finite dimensional complex Hilbert space. Let ψ↦Z(ψ)ψ\mapsto Z(ψ)ψ↦Z(ψ) be a canonical anti-commutation relations (CAR) field over H1{\mathcal H}_1H1​ acting irreducibly on a Hilbert space K{\mathord{\mathscr K}}K. The ∗*∗-algebra AH1{\mathscr A}_{{\mathcal H}_1}AH1​​ generated by the Z(ψ)Z(ψ)Z(ψ), ψ∈H1ψ\in {\mathcal H}_1ψ∈H1​, is simply all operators on K{\mathscr K}K. However, the CAR field endows AH1{\mathscr A}_{{\mathcal H}_1}AH1​​ with additional structure, and we are concerned with quantum operations acting in harmony with this structure. In particular, there is a "gauge" automorphism group generated by "second quantizing'' ψ↦eitψψ\mapsto e^{it}ψψ↦eitψ. The fixed point algebra of the gauge group, GH1{\mathscr G}_{{\mathcal H}_1}GH1​​, is a sub-algebra of AH1{\mathscr A}_{{\mathcal H}_1}AH1​​ studied by Araki and Wyss. It contains the density matrices of an important class of states, the {\em gauge invariant Gaussian states}, SGIG{\mathfrak S}_{GIG}SGIG​. Our focus is on semigroups {etL}t≥0\{e^{t{\mathscr L}}\}_{t\geq 0}{etL}t≥0​ of quantum operations on AH1{\mathscr A}_{{\mathcal H}_1}AH1​​ that map SGIG{\mathfrak S}_{GIG}SGIG​ into itself. Each etLe^{t{\mathscr L}}etL is one-to-one, and our first main result is a structure theorem for such quantum operations on GH1{\mathscr G}_{{\mathcal H}_1}GH1​​ that map SGIG{\mathfrak S}_{GIG}SGIG​ into itself. We apply this to study semigroups of quantum operations on GH1{\mathscr G}_{{\mathcal H}_1}GH1​​ that map SGIG{\mathfrak S}_{GIG}SGIG​ into itself. Our second main result is a structure theorem showing that they are parameterized by pairs (G,A)(G,A)(G,A) where GGG is a contraction semigroup generator on H1{\mathcal H}_1H1​, and 0≤A≤−G−G∗0 \leq A \leq -G -G^*0≤A≤−G−G∗. We then show that each of these semigroups has a natural extension to the full CAR algebra AH1{\mathscr A}_{{\mathcal H}_1}AH1​​. Further results are obtained under further assumptions on the pair (G,A)(G,A)(G,A).
CAR on Fock spacesThis chapter is devoted to the study of Fock representations of the canonical anticommutation relations, a basic tool of quantum many-body theory and quantum field theory.It is parallel to Chap. 9, where Fock CCR representations were studied.The basic framework is almost the same as in Chap.9. Throughout this chapter Z is a Hilbert space, called the one-particle space.We will consider the Fock CAR representation acting on the fermionic Fock space Γ a (Z).As in Sect.1.3, we introduce the space Y = Re(Z ⊕ Z) := {(z, z) : z ∈ Z}, which will serve as the dual phase space of our system.Recall that in the bosonic case we equipped Y with the structure of the Kähler space consisting of the anti-involution j, the Euclidean scalar product • and the symplectic form ω: j(z, z) := (iz, iz), (13.1) (z, z) • (w, w) := 2Re(z|w), (13.2) (z, z)•ω(w, w) := 2Im(z|w) = -(z, z) • j(w, w).(13.3)In our presentation of the fermionic case, we will need the Kähler anti-involution j.The symplectic form ω will not be used.Instead of the scalar product (13.2) we will use another scalar product,Again we will avoid identifing Z with Y. CY is identified with Z ⊕ Z by the mapY # , the space dual to Y, is canonically identified with Re(Z ⊕ Z) by using the scalar product (9.2), and CY # is identified with Z ⊕ Z. Fock CAR representationConsider the fermionic Fock space Γ a (Z).Recall that, for z ∈ Z, a * (z), resp.a(z) denote the corresponding creation, resp.annihilation operators defined in Sect.3.4.
We present an approach to a noncommutativelike phase space which allows to analyze quasifree states on the algebra of canonical anti-commutation relations (CAR) in analogy to quasifree states on the algebra of canonical commutation relations (CCR). The used mathematical tools are based on a new algebraic structure the “Grassmann algebra of canonical anticommutation relations” (GAR algebra) which is given by the twisted tensor product of a Grassmann and a CAR algebra. As a new application, the corresponding theory provides an elegant tool for calculating the fidelity of two quasifree fermionic states which is needed for the study of entanglement distillation within fermionic systems.
Abstract We derive a formalism to express the spin algebra <?CDATA su(2)\mathfrak{s}\mathfrak{u}(2)su(2)?> s u ( 2 ) in a spin s representation in terms of the algebra of L fermionic operators that obey the canonical anti-commutation relations. We also give the reverse direction of expressing the fermionic operators as polynomials in the spin operators of a single spin. We extend here to further spin values the previous investigations by Dobrov (2003 J. Phys. A: Math. Gen. 36 L503) who in turn clarified on an inconsistency within a similar formalism in the works of Batista and Ortiz (2001 Phys. Rev. Lett. 86 1082). We then consider a system of L fermion flavors and apply our mapping in order to express it in terms of the spin algebra. Furthermore we investigate a possibility to simplify certain Hamiltonian operators by means of the mapping.