# Quantum Operations on Conformal Nets

April 29, 2022

On a conformal net $\mathcal{A}$, one can consider collections of unital
completely positive maps on each local algebra $\mathcal{A}(I)$, subject to
natural compatibility, vacuum preserving and conformal covariance conditions.
We call \emph{quantum operations} on $\mathcal{A}$ the subset of extreme such
maps. The usual automorphisms of $\mathcal{A}$ (the vacuum preserving
invertible unital *-algebra morphisms) are examples of quantum operations, and
we show that the fixed point subnet of $\mathcal{A}$ under all quantum
operations is the Virasoro net generated by the stress-energy tensor of
$\mathcal{A}$. Furthermore, we show that every irreducible conformal subnet
$\mathcal{B}\subset\mathcal{A}$ is the fixed points under a subset of quantum
operations.
When $\mathcal{B}\subset\mathcal{A}$ is discrete (or with finite Jones
index), we show that the set of quantum operations on $\mathcal{A}$ that leave
$\mathcal{B}$ elementwise fixed has naturally the structure of a compact (or
finite) hypergroup, thus extending some results of [Bis17]. Under the same
assumptions, we provide a Galois correspondence between intermediate conformal
nets and closed subhypergroups. In particular, we show that intermediate
conformal nets are in one-to-one correspondence with intermediate subfactors,
extending a result of Longo in the finite index/completely rational conformal
net setting [Lon03].

Keywords:

*none*