On the existence of derivations as square roots of generators of state-symmetric quantum Markov semigroups

TL;DR

This paper investigates the existence of derivations as square roots of generators in quantum Markov semigroups, highlighting differences between tracial and non-tracial cases and providing explicit solutions in finite dimensions.

Contribution

It demonstrates that the construction of derivations cannot be universally extended to non-tracial symmetric quantum Markov semigroups and characterizes their form in finite dimensions.

Findings

Derivations can be assumed to have a concrete form.

Existence of derivations is equivalent to positive solutions of linear systems in finite dimensions.

Explicit solutions are obtained using Mathematica for specific examples.

Abstract

Cipriani and Sauvageot have shown that for any $L^2$-generator $L^{(2)}$ of a tracially symmetric quantum Markov semigroup on a C*-algebra $\mathcal{A}$ there exists a densely defined derivation $\delta$ from $\mathcal{A}$ to a Hilbert bimodule $H$ such that $L^{(2)}=\delta^*\circ \overline{\delta}$. Here we show that this construction of a derivation can in general not be generalised to quantum Markov semigroups that are symmetric with respect to a non-tracial state. In particular we show that all derivations to Hilbert bimodules can be assumed to have a concrete form, and then we use this form to show that in the finite-dimensional case the existence of such a derivation is equivalent to the existence of a positive matrix solution of a system of linear equations. We solve this system of linear equations for concrete examples using Mathematica to complete the proof.