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

Abstract

Let ${\mathcal{H}}_1$ be a finite dimensional complex Hilbert space. Let $\psi \mapsto Z(\psi )$ be a canonical anti-commutation relations (CAR) field over ${\mathcal{H}}_1$ acting irreducibly on a Hilbert space $\mathcal{K}$. The $\ast$-algebra ${\mathcal{A}}_{{\mathcal{H}}_1}$ generated by the $Z(\psi )$, $\psi \in {\mathcal{H}}_1$, is simply all operators on $\mathcal{K}$. However, the CAR field endows ${\mathcal{A}}_{{\mathcal{H}}_1}$ 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'' $\psi \mapsto e^{it}\psi$. The fixed point algebra of the gauge group, ${\mathcal{G}}_{{\mathcal{H}}_1}$, is a sub-algebra of ${\mathcal{A}}_{{\mathcal{H}}_1}$ 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 $\{e^{t\mathcal{L}}{\}}_{t\ge 0}$ of quantum operations on ${\mathcal{A}}_{{\mathcal{H}}_1}$ that map ${\mathfrak{S}}_{GIG}$ into itself. Each $e^{t\mathcal{L}}$ is one-to-one, and our first main result is a structure theorem for such quantum operations on ${\mathcal{G}}_{{\mathcal{H}}_1}$ that map ${\mathfrak{S}}_{GIG}$ into itself. We apply this to study semigroups of quantum operations on ${\mathcal{G}}_{{\mathcal{H}}_1}$ 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 $0\le A\le -G-G^{\ast }$. We then show that each of these semigroups has a natural extension to the full CAR algebra ${\mathcal{A}}_{{\mathcal{H}}_1}$. Further results are obtained under further assumptions on the pair $(G,A)$.

This revision is the version accepted for publication in Advanced Nonlinear Studies