Small quantum networks in the qudit stabilizer formalism

TL;DR

This paper investigates the noise tolerance of qudit stabilizer states, especially graph states, under global and local white noise, deriving thresholds and relating them to graph theory, with new results for GHZ states.

Contribution

It introduces a generalized framework for noise thresholds in qudit stabilizer states, including novel solutions for specific graph state families and GHZ states under local noise.

Findings

High noise thresholds from PPT and reduction criteria.

Coarse thresholds from sector length criteria based on stabilizer count.

First-time noise threshold for GHZ states ruling out semiseparability.

Abstract

How much noise can a given quantum state tolerate without losing its entanglement? For qudits of arbitrary dimension, I investigate this question for two noise models: Global white noise, where a depolarizing channel is applied to all qudits simultaneously, and local white noise, where a single qudit depolarizing channel is applied to every qudit individually. Using a unitary generalization of the Pauli group, I derive noise thresholds for stabilizer states, with an emphasis on graph states, and compare different entanglement criteria. The PPT and reduction criteria generally provide high noise thresholds, however, it is difficult to apply them in the case of local white noise. Entanglement criteria based on so-called sector lengths, on the other hand, provide coarse noise thresholds for both noise models. The only thing one has to know about a state to compute this threshold is the number of its full-weight stabilizers. In the special case of qubit graph states, I relate this question to a graph-theoretical puzzle and solve it for four important families of states. For Greenberger-Horne-Zeilinger states under local white noise, I obtain for the first time a noise threshold below which so-called semiseparability is ruled out.