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arXiv:2305.02788 · Mathematical Physics Analysis and Geometry

Relative Entropy of Fermion Excitation States on the CAR Algebra

Stefano Galanda, Albert Much, Rainer Verch

Published Aug 30, 2023• 12 citations
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Abstract

Abstract The relative entropy of certain states on the algebra of canonical anticommutation relations (CAR) is studied in the present work. The CAR algebra is used to describe fermionic degrees of freedom in quantum mechanics and quantum field theory. The states for which the relative entropy is investigated are multi-excitation states (similar to multi-particle states) with respect to KMS states defined with respect to a time-evolution induced by a unitary dynamical group on the one-particle Hilbert space of the CAR algebra. If the KMS state is quasifree, the relative entropy of multi-excitation states can be explicitly calculated in terms of 2-point functions, which are defined entirely by the one-particle Hilbert space defining the CAR algebra and the Hamilton operator of the dynamical group on the one-particle Hilbert space. This applies also in the case that the one-particle Hilbert space Hamilton operator has a continuous spectrum so that the relative entropy of multi-excitation states cannot be defined in terms of von Neumann entropies. The results obtained here for the relative entropy of multi-excitation states on the CAR algebra can be viewed as counterparts of results for the relative entropy of coherent states on the algebra of canonical commutation relations which have appeared recently. It turns out to be useful to employ the setting of a self-dual CAR algebra introduced by Araki.

DOI
10.1007/s11040-023-09464-7

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Abstract The relative entropy of certain states on the algebra of canonical anticommutation relations (CAR) is studied in the present work. The CAR algebra is used to describe fermionic degrees of freedom in quantum mechanics and quantum field theory. The states for which the relative entropy is investigated are multi-excitation states (similar to multi-particle states) with respect to KMS states defined with respect to a time-evolution induced by a unitary dynamical group on the one-particle Hilbert space of the CAR algebra. If the KMS state is quasifree, the relative entropy of multi-excitation states can be explicitly calculated in terms of 2-point functions, which are defined entirely by the one-particle Hilbert space defining the CAR algebra and the Hamilton operator of the dynamical group on the one-particle Hilbert space. This applies also in the case that the one-particle Hilbert space Hamilton operator has a continuous spectrum so that the relative entropy of multi-excitation states cannot be defined in terms of von Neumann entropies. The results obtained here for the relative entropy of multi-excitation states on the CAR algebra can be viewed as counterparts of results for the relative entropy of coherent states on the algebra of canonical commutation relations which have appeared recently. It turns out to be useful to employ the setting of a self-dual CAR algebra introduced by Araki.
We study the relative entropy, in the sense of Araki, for the representation of a self-dual CAR algebra ASDC(H,Γ)\mathfrak{A}_{SDC}(\mathcal{H},Γ)ASDC​(H,Γ). We notice, for a specific choice of f∈Hf \in \mathcal{H}f∈H, that the associated element in ASDC(H,Γ)\mathfrak{A}_{SDC}(\mathcal{H},Γ)ASDC​(H,Γ) is unitary. As a consequence, we explicitly compute the relative entropy between a quasifree state over ASDC(H,Γ)\mathfrak{A}_{SDC}(\mathcal{H},Γ)ASDC​(H,Γ) and an excitation of it with respect to the abovely mentioned unitary element. The generality of the approach, allows us to consider H\mathcal{H}H as the Hilbert space of solutions of the classical Dirac equation over globally hyperbolic spacetimes, making our result, a computation of relative entropy for a Fermionic Quantum Field Theory. Our result extends those of Longo and Casini et al. for the relative entropy between a quasifree state and a coherent excitation for a free Scalar Quantum Field Theory, to the case of fermions. As a first application, we computed such a relative entropy for a Majorana field on an ultrastatic spacetime.
We show how one can use the convexity of non-commutative LpL^pLp norms to bound the relative entropy between a faithful state on a von Neumann algebra and an arbitrary excitation thereof. Our results hold for general von Neumann algebras, including the local algebras of type III that are ubiquitous in quantum field theory, and do not require knowledge of the relative modular operator. As an application of our results, we prove that for the chiral current on a light ray, the relative entropy between the vacuum and a dense set of single-particle states is uniformly bounded.
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.
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.
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).