General fixed points of quasi-local frustration-free quantum semigroups: from invariance to stabilization

TL;DR

This paper provides a linear-algebraic criterion and explicit methods for stabilizing certain quantum states as unique fixed points of frustration-free semigroup dynamics under quasi-local constraints, with applications to quantum information and statistical mechanics.

Contribution

It introduces a necessary and sufficient condition for stabilizing full-rank target states and an explicit construction of dynamics, extending stabilization results to non-commuting and entangled states.

Findings

Full-rank states are stabilizable under the criterion.

Thermal graph states and Gibbs states of commuting Hamiltonians are stabilizable.

Examples of non-commuting Gibbs states stabilizable despite non-commutativity.

Abstract

We investigate under which conditions a mixed state on a finite-dimensional multipartite quantum system may be the unique, globally stable fixed point of frustration-free semigroup dynamics subject to specified quasi-locality constraints. Our central result is a linear-algebraic necessary and sufficient condition for a generic (full-rank) target state to be frustration-free quasi-locally stabilizable, along with an explicit procedure for constructing Markovian dynamics that achieve stabilization. If the target state is not full-rank, we establish sufficiency under an additional condition, which is naturally motivated by consistency with pure-state stabilization results yet provably not necessary in general. Several applications are discussed, of relevance to both dissipative quantum engineering and information processing, and non-equilibrium quantum statistical mechanics. In particular, we show that a large class of graph product states (including arbitrary thermal graph states) as well as Gibbs states of commuting Hamiltonians are frustration-free stabilizable relative to natural quasi-locality constraints. Likewise, we provide explicit examples of non-commuting Gibbs states and non-trivially entangled mixed states that are stabilizable despite the lack of an underlying commuting structure, albeit scalability to arbitrary system size remains in this case an open question.