Quantum Matter Seminar

Defining mixed-state phases of matter: What replaces the idea of "gap"?

Topological order in closed systems is classified by equivalence classes of ground states of gapped local Hamiltonians. Extending this to open systems has suffered due to lack of the notion, parent Hamiltonian. To resolve this issue, we propose three axioms—(i) local recoverability, (ii) no long-range correlations, and (iii) spatial uniformity. States obeying them are fixed points; imposing the axioms only after coarse-graining promotes each fixed point to an equivalence class (a phase), initiating an axiomatic "mixed-state bootstrap."

From these axioms, robust topological data emerge: each topological mixed state hosts locally indistinguishable classical and/or quantum memories with distinct responses to topological operations, providing phase labels. We also find a hierarchy of secret-sharing constraints: in non-Abelian phases, reliable recovery requires coordinated subregions, reflecting fusion rules that persist under decoherence. Large-scale numerics show weakly decohered fixed points satisfy the axioms after coarse-graining, supporting stability and a systematic classification of open-system topological phases.