On the state extension and quantum correlation for CAR systems

Hajime Moriya

Huzihiro Araki, Hajime Moriya

Prasanta Kumar Panigrahi, Chiranjib Mitra

H. Hajihoseinlou, Bahram Ahansaz, F. Eghbalifam +1 more

Hajime Moriya

For given two regions of the lattice, assume that our state has a pure-state restriction for each of them. We discuss whether its restrictions to the intersection and the union regions are pure states or not. It is immediate to see that the answer is affirmative for any tensor-product systems. (We, however, include its proof for the completeness sake.) In this note, we consider mostly CAR systems. We show that the assertion above holds for any such a state of finite-dimensional CAR systems. For infinite-dimensional CAR systems, assuming additionally the product properties between each of the regions and its complement region on the states satisfying the standing assumption, we can show the purity on the intersection region for them. However, the latter part of the assertion will not always hold for the infinite-dimensional CAR systems unlike for the finite-dimensional CAR or any dimensional tensor-product systems. We establish the criterions for the purity and the non-purity on the union region for those states of the infinite-dimensional CAR 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 $(G,A)$.