Topological Mixed States: Phases of Matter from Axiomatic Approaches

TL;DR

The paper frames topological mixed states within an axiomatic bootstrap, introducing three principles—local recoverability ($P0$), absence of long-range order ($M0$), and uniformity ($M1$)—to define fixed points and classify phases via coarse-graining. It constructs the information-convex set as the central diagnostic, ties memory capacity to a generalized topological entropy $\gamma_{LW}$, and demonstrates concrete fixed points for Abelian and non-Abelian quantum doubles, including their memory and boundary structures. Phases are refined through coarse-graining and the notion of transparent domain walls, leading to the concept of topological channel connectivity; numerical results support stability of invariants under approximate axioms. The work reveals a rich landscape of mixed-state phases, including a hierarchy of secret-sharing capacities in non-Abelian models and a higher-dimensional extension, establishing a foundational framework for robust classification of topological mixed states with potential relevance to decoherence-robust memories and quantum information tasks.

Abstract

For closed quantum systems, topological orders are understood through the equivalence classes of ground states of gapped local Hamiltonians. The generalization of this conceptual paradigm to open quantum systems, however, remains elusive, often relying on operational definitions without fundamental principles. Here, we fill this gap by proposing an approach based on three axioms: ($i$) local recoverability, ($ii$) absence of long-range correlations, and ($iii$) spatial uniformity. States that satisfy these axioms are fixed points; requiring the axioms only after coarse-graining promotes each fixed point to an equivalence class, i.e., a phase, presenting the first step towards the axiomatic classification of mixed-state phases of matter: mixed-state bootstrap program. From these axioms, a rich set of topological data naturally emerges; importantly, these data are robust under relaxation of axioms. For example, each topological mixed state supports locally indistinguishable classical and/or quantum logical memories with distinct responses to topological operations. These data label distinct mixed-state phases, allowing one to distinguish them. We further uncover a hierarchy of secret-sharing constraints: in non-Abelian phases, reliable recovery-even of information that looks purely classical-demands a specific coordination among spatial subregions, a requirement different across non-Abelian classes. This originates from non-Abelian fusion rules that can stay robust under decoherence. Finally, we performed large-scale numerical simulations to corroborate stability: weakly decohered fixed points respect the axioms once coarse-grained. These results lay the foundation for a systematic classification of topological states in open quantum systems.

Figures (27)

• Figure 1: Phases of matter defined through (a) equivalence relations (adiabatic connection without closing gaps of a local Hamiltonian), which specify boundaries as well, and (b) fixed points and allowed deformations, which are finite-depth (quasi)-local operations.
• Figure 2: Bootstrap axioms. Given a reference state $\sigma$, we impose axioms (a) P0, (b) M0, and (c) M1 on it. The axiom P0 requires the entire system, while keeping $BC$ local. The axioms P0 and P1 are stated on local patches indicated by green regions.
• Figure 3: Information convex set. For topological mixed states encoding (a) a single qubit (quantum memory) and (b) a single bit (classical memory). While quantum memories reside on the surface of the Bloch sphere, classical memories reside in the linear combination of extreme points. Mathematically, the convex sets in (a) is the convex hull ${\rm conv} (\{|\psi\rangle \langle \psi|\, |\, |\psi\rangle \in \mathcal{H}\})$, and the convex set in (b) is ${\rm conv}(\{|0\rangle \langle 0|, |1\rangle \langle 1|\})$.
• Figure 4: Finite depth local channel (FDLC). For each site, we have two physical indices corresponding to bra and ket. The red block represents a local quantum channel acting on two sites. The entire channel has a finite depth of three.
• Figure 5: Cases to check for (a) M0 and P0, and (b) M1. In both cases, a local region $C$ is separated from the rest by an annulus ($B$ or $BD$).

Theorems & Definitions (40)

• Definition 1: Information convex set, closed manifold
• Definition 2: Finite depth local channel
• Definition 3: Classically preparable
• Proposition 4: torus capacity
• Example 1: Product state
• Example 2: Ground states of topological order
• Example 3: Dephased toric code
• Example 4: Fermionic decohered toric code
• Example 5: Stacked layers of fixed points
• Definition 5: Canonically decohered Abelian quantum double