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arXiv:2310.06691 · Communications in Mathematical Physics

Freedman’s Theorem for Unitarily Invariant States on the CCR Algebra

Vitonofrio Crismale, Simone Del Vecchio, Tommaso Monni, Stefano Rossi

Published Feb 1, 2024• 3 citations
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Abstract

Abstract The set of states on $$\textrm{CCR}({{\mathcal {H}}})$$ CCR ( H ) , the CCR algebra of a separable Hilbert space $${{\mathcal {H}}}$$ H , is here looked at as a natural object to obtain a non-commutative version of Freedman’s theorem for unitarily invariant stochastic processes. In this regard, we provide a complete description of the compact convex set of states of $$\textrm{CCR}({{\mathcal {H}}})$$ CCR ( H ) that are invariant under the action of all automorphisms induced in second quantization by unitaries of $${{\mathcal {H}}}$$ H . We prove that this set is a Bauer simplex, whose extreme states are either the canonical trace of the CCR algebra or Gaussian states with variance at least 1.

DOI
10.1007/s00220-024-04932-9

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Abstract The set of states on CCR(H)\textrm{CCR}({{\mathcal {H}}})CCR(H) CCR ( H ) , the CCR algebra of a separable Hilbert space H{{\mathcal {H}}}H H , is here looked at as a natural object to obtain a non-commutative version of Freedman’s theorem for unitarily invariant stochastic processes. In this regard, we provide a complete description of the compact convex set of states of CCR(H)\textrm{CCR}({{\mathcal {H}}})CCR(H) CCR ( H ) that are invariant under the action of all automorphisms induced in second quantization by unitaries of H{{\mathcal {H}}}H H . We prove that this set is a Bauer simplex, whose extreme states are either the canonical trace of the CCR algebra or Gaussian states with variance at least 1.
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).
The features of the paper are these: 1) we discuss {\bf especially} quadratic (alias bilinear) Bose-Hamiltonians, the related Bogoliubov transformations and {\bf especially} quasi-free-like (alias coherent or Fock-like) states 2) we discuss {\bf any} quadratic Bose-Hamiltonians and Bogoliubov transformations, whether diagonalizable or not, whether proper or improper, and {\bf arbitrary} quasi-free-like states, whether regular or non-regular they are 3) we associate notions and terms of the CCRs theory {CCRs = Canonical Commutation Relations} with notions and terms of the indefinite inner product spaces theory. Then, we apply the corresponding `bilingual dictionary' so as to construct invariant states of some of the quadratic Hamiltonians.
Entanglement in C∗^*∗-algebras: tensor products of state spaces
We analyze the Namioka-Phelps minimal and maximal tensor products of compact convex sets which arise as the state spaces of unital C∗^*∗-algebras. Relatedly, we study entanglement in (infinite dimensional) C∗^*∗-algebras. While the minimal Namioka-Phelps tensor product of the state spaces of two C∗^*∗-algebras is the well-known set of separable (= un-entangled) states on the (minimal) tensor product of the C∗^*∗-algebras, we also describe the more elusive maximal Namioka-Phelps tensor product of state spaces of C∗^*∗-algebras. We show that the minimal and maximal tensor products of state spaces of C∗^*∗-algebras agree precisely when one of the two C∗^*∗-algebras is commutative, which confirms Barker's conjecture in the case where the compact convex sets are state paces of C∗^*∗-algebras. Further, the Namioka-Phelps tensor product of the trace simplexes of two or more unital C∗^*∗-algebras is shown to be the trace simplex of the (minimal or maximal) tensor product of the C∗^*∗-algebras. This enables a systematic way of determining the trace simplex of a tensor product of C∗^*∗-algebras.
We consider Hilbert space representations of a generalization of canonical commutation relations , where 's are the elements of an algebra with identity , is the imaginary unit, and is a real number with antisymmetry . Some basic aspects on Hilbert space representations of the generalized CCR (GCCR) are discussed. We define a Schrödinger-type representation of the GCCR by an analogy with the usual Schrödinger representation of the CCR with degrees of freedom. Also, we introduce a Weyl-type representation of the GCCR. The main result of the present paper is a uniqueness theorem on Weyl representations of the GCCR.
Let [Formula: see text] be a unital C*-subalgebra of the C*-algebra ℬ(ℋ) of all bounded operators on a complex separable Hilbert space ℋ. Let [Formula: see text] denote the convex set of all unital, linear, completely positive and normal maps of [Formula: see text] into itself. Using Stinespring's theorem, we present a criterion for an element [Formula: see text] to be extremal. When [Formula: see text], this criterion leads to an explicit description of the set of all extreme points of [Formula: see text]. We also obtain a quantum probabilistic analogue of the classical Birkhoff's theorem2 that every bistochastic matrix can be expressed as a convex combination of permutation matrices.