Fractional quantum Hall states under density decoherence

TL;DR

The paper analyzes how density-dephasing decoherence impacts quantum information stored in the Laughlin and Moore-Read fractional quantum Hall states. By mapping the problem to a bilayer plasma and to 1+1D CFTs, it identifies a critical filling factor $ν_c$ that separates a topological memory phase from a decohered critical phase, with Laughlin memory degrading in the latter while Moore-Read memory in the fusion space remains fully recoverable for any finite decoherence. The authors derive bounds on $ν_c$ for von Neumann quantities ($1/8≤ν_c(1)≤1/4$; Laughlin and MR both show similar bounds) and demonstrate that the Moore-Read fusion space is intrinsically protected against decoherence even beyond the BKT transition, contrasting with the Abelian sector that can undergo a decoherence-driven transition. These results reinforce the promise of non-Abelian FQH states for topological quantum computation and provide a framework for understanding and potentially correcting errors in such systems.

Abstract

Fractional quantum Hall states are promising platforms for topological quantum computation due to their capacity to encode quantum information in topologically degenerate ground states and in the fusion space of non-abelian anyons. We investigate how the information encoded in two paradigmatic states, the Laughlin and Moore-Read states, is affected by density decoherence -- coupling of local charge density to non-thermal noise. We identify a critical filling factor $ν_c$, above which the quantum information remains fully recoverable for arbitrarily strong decoherence. The $ν=1/3$ Laughlin state and $ν= 1/2$ Moore-Read state both lie within this range. Below $ν_c$ both classes of states undergo a decoherence induced Berezinskii-Kosterlitz-Thousless (BKT) transition into a critical decohered phase. For Laughlin states, information encoded in the topological ground state manifold degrades continuously with decoherence strength inside this critical phase, vanishing only in the limit of infinite decoherence strength. On the other hand, quantum information encoded in the fusion space of non-abelian anyons of the Moore-Read states remains fully recoverable for arbitrary strong decoherence even beyond the BKT transition. These results lend further support to the promise of non-Abelian FQH states as platforms for topological quantum computation and raises the question of how errors in such states can be corrected.

Figures (3)

• Figure 1: (a) Schematics for the purity of the decohered state. Two layers are the two copies of states, orange and blue dots are the electrons, black circles represent their tight binding at the strong decoherence limit, dashed circles is the unbinding event when deviating from the limit. (b) Renormalization-group flow for decohered Laughlin and Moore-Read states. Below a critical filling the system enters a critical phase through a BKT transition. Above it the system remains topological. (c) Encoding a single qubit in the two-dimensional fusion space of four well-separated fundamental quasiholes of the Moore-Read state.
• Figure 2: Rényi-2 quantum coherent information as a function of the decoherence strength. (a) For Laughlin states, the information is encoded in the degenerate ground states. For systems below the critical filling it starts to decrease after a finite time and becomes zero only at the strong decoherence limit. (b) For Moore-Read states, the information is encoded in the fusion space and is infinitely robust against the decoherence.
• Figure 3: Inverse dielectric constant of the Ising plasma at $|\mathbf{k}|=2\pi/L$. The error bar is invisible in the plot.