Phases of matter in quantum information and error correction

Abstract

This thesis investigates phases of matter and phase transitions through the lens of quantum information, with an emphasis on phenomena not fully captured by conventional local observables or equilibrium order parameters. While the traditional framework of phase transitions relies on correlation functions and order parameters, entanglement and other information-theoretic quantities provide a broader language for characterizing both equilibrium and non-equilibrium many-body systems. A central perspective developed here is that such quantities furnish sharp diagnostics of phases and criticality, particularly in topological phases subjected to noise and measurement.

First, we study how measurements on topological quantum states reshape entanglement structure and induce phase transitions. Focusing on the toric code, we show that measuring part of the system generates distinct entanglement phases in the remaining degrees of freedom, and that tuning the measurement protocol drives transitions between them. To analyze these phenomena, we develop analytical tools that track the entanglement structure of the post-measurement state and reveal a rich phase diagram.

Next, we turn to topological codes in the presence of noise, where information-theoretic probes reveal forms of non-equilibrium criticality invisible to conventional observables. In this setting, we identify extended critical behavior in mixed states and show that conditional mutual information diagnoses transitions between distinct regimes of information retention and loss. Interpreted through quantum error correction, these transitions distinguish phases in which logical information is robustly preserved, only partially accessible, or completely lost. Building on this connection, we extend the mixed-state perspective from static codes to fault-tolerant dynamics by relating faulty syndrome-extraction circuits to the mixed-state structure of an associated higher-dimensional resource state. This leads to a decoder-independent diagnostic of fault tolerance based on the conditional mutual information of syndrome data across spacetime. The resulting spacetime Markov length diverges at the fault-tolerance threshold, providing an intrinsic information-theoretic characterization of the preservation and breakdown of logical information in noisy quantum circuits.

Finally, we develop structural results for thermal and symmetry-constrained mixed states. We show that symmetry can obstruct the sudden death of entanglement in thermal states: for canonical ensembles and for Gibbs states subject to superselection rules, entanglement persists, and in broad settings remains nonzero at arbitrarily high temperatures. In fermionic systems, this identifies parity superselection as a generic mechanism protecting mixed-state entanglement and fermionic negativity. Complementing this perspective, we study extendibility as a tractable probe of entanglement structure in fermionic Gaussian states, showing that it admits an efficient characterization and provides practical criteria for mixed-state entanglement, including an extendibility transition in the disordered Kitaev chain. Taken together, these results support a unified picture in which information-theoretic quantities serve as fundamental diagnostics of phase transitions and criticality in both equilibrium and non-equilibrium quantum systems.