Generic pure quantum states as steady states of quasi-local dissipative dynamics

TL;DR

This paper explores the conditions under which generic pure quantum states can be uniquely stabilized as steady states of quasi-local dissipative dynamics, revealing measure-based properties and providing analytical characterizations.

Contribution

It offers a comprehensive analytical framework for understanding when pure states can be stabilized via local dissipative processes, including measure-theoretic and dimension-based criteria.

Findings

Set of stabilizable pure states has measure zero or one depending on subsystem dimensions.

Complete characterization for tripartite systems with a qubit central subsystem.

Random pure states from t-designs share stabilizability properties with Haar-random states.

Abstract

We investigate whether a generic multipartite pure state can be the unique asymptotic steady state of locality-constrained purely dissipative Markovian dynamics. In the simplest tripartite setting, we show that the problem is equivalent to characterizing the solution space of a set of linear equations and establish that the set of pure states obeying the above property has either measure zero or measure one, solely depending on the subsystems' dimension. A complete analytical characterization is given when the central subsystem is a qubit. In the N-partite case, we provide conditions on the subsystems' size and the nature of the locality constraint, under which random pure states cannot be quasi-locally stabilized generically. Beside allowing for the possibility to approximately stabilize entangled pure states that cannot be exact steady states in settings where stabilizability is generic, our results offer insights into the extent to which random pure states may arise as unique ground states of frustration free parent Hamiltonians. We further argue that, to high probability, pure quantum states sampled from a t-design enjoy the same stabilizability properties of Haar-random ones as long as suitable dimension constraints are obeyed and t is sufficiently large. Lastly, we demonstrate a connection between the tasks of quasi-local state stabilization and unique state reconstruction from local tomographic information, and provide a constructive procedure for determining a generic N-partite pure state based only on knowledge of the support of any two of the reduced density matrices of about half the parties, improving over existing results.