Entanglement in the stabilizer formalism

Abstract

We define a multi-partite entanglement measure for stabilizer states, which can be computed efficiently from a set of generators of the stabilizer group. Our measure applies to qubits, qudits and continuous variables.

Figures (3)

• Figure 1: Canonical set of generators for a stabilizer group $S(\psi)$ with respect to a given partition $\{A,B\}$ of the qubits. $S_A$ and $S_B$ contain the purely local information of $\left| {\psi} \right\rangle$. $S_{AB}$ is generated by $p$ pairs $(g_k, \bar{g}_k)$ whose projections on $A$ (or $B$) anticommute, but commute with all other generators of $S$ including elements of other pairs.
• Figure 2: Application of the measure $e_{\mathcal{A}}$ to the classification of graph (stabilizer) states. The measure is shown only for the relevant partitions. A complete study would have to consider all partitions and relabelling of the qubits. From state (a) (GHZ) to state (d) (cluster), entanglement becomes more "localized" and robust against measurement of local operators.
• Figure 3: Structure of a subgroup G of the Pauli group. $C(G)$ is a maximum abelian subgroup of S. It always includes the center $Z(G)$ of G.