Classifying One-Dimensional Quantum States Prepared by a Single Round of Measurements

TL;DR

This paper develops a framework for classifying and understanding the entanglement patterns of one-dimensional quantum states created by a single round of measurements, revealing constraints and phase connections.

Contribution

It introduces necessary and sufficient tensor conditions for state preparability, enabling classification and analysis of physical constraints of measurement-induced quantum states.

Findings

Identifies tensor conditions for state preparability.

Establishes a trade-off between entanglement richness and correlations.

Connects preparation protocols to quantum phases of matter.

Abstract

Measurements and feedback have emerged as powerful resources for creating many-body quantum states. However, a detailed understanding has been restricted to fixed-point representatives of phases of matter. Here, we go beyond this and characterize the patterns of many-body entanglement that can be deterministically created from measurement. Focusing on one spatial dimension, a framework is developed for the case where a single round of measurements is the only entangling operation. We show this creates matrix product states and identify necessary and sufficient tensor conditions for preparability, which uniquely determine the preparation protocol. We use these conditions to both classify preparable quantum states and characterize their physical constraints. In particular, we find a trade-off between the richness of the preparable entanglement spectrum and correlation functions, which leads to a no-go theorem for preparing certain quantum states. More broadly, we connect properties of the preparation protocol to the resulting phase of matter, including trivial, symmetry-breaking, and symmetry-protected topological phases -- for both uniform and modulated symmetries. This work offers a resource-theoretic perspective on preparable quantum entanglement and shows how to systematically create states of matter, away from their fixed points, in quantum devices.