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

Measurements and feedback have emerged as a powerful resource for creating quantum states. However, a detailed understanding is restricted to fixed-point representatives of phases of matter. Here, we go beyond this and ask which types of many-body entanglement can be created from measurement. Focusing on one spatial dimension, a framework is developed for the case where a single round of measurements are the only entangling operations. We show this creates matrix product states and identify necessary and sufficient tensor conditions for preparability, which uniquely determine the preparation protocol. These conditions are then used to characterize the physical constraints on preparable quantum states. First, we find a trade-off between the richness of the preparable entanglement spectrum and correlation functions, which moreover leads to a powerful no-go theorem. Second, in a subset of cases, where undesired measurement outcomes can be independently paired up and corrected, we are able to provide a complete classification for preparable quantum states. Finally, 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.