Small quantum networks in the qudit stabilizer formalism

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.