Multi-invariants in stabilizer states

TL;DR

This work develops a practical framework to compute multi-invariants for stabilizer states, enabling quantitative analysis of multipartite entanglement in a class of states central to quantum information and condensed matter. By reformulating multi-invariants for graph states as inner products on a constructed big graph and providing an O($n^3$) projector-based algorithm, the authors achieve efficient computation in the stabilizer setting. They prove a tripartite analytic decomposition, showing ${\cal Z}$ factors into GHZ- and Bell-type contributions with a Rényi multi-entropy ${\cal E}_n^{\text{ME}}$ that admits closed forms and simple $n=2$ counting, and they connect these invariants to Coxeter groups and topology, including a genus-dependent piece. They further simplify the framework for X-stabilizer states (e.g., toric code and X-cube ground states) and conjecture a general closed form for Coxeter invariants, with implications for entanglement in stabilizer-based quantum codes and topological models.

Abstract

Multipartite entanglement is a natural generalization of bipartite entanglement, but is relatively poorly understood. In this paper, we develop tools to calculate a class of multipartite entanglement measures - known as multi-invariants - for stabilizer states. We give an efficient numerical algorithm that computes multi-invariants for stabilizer states. For tripartite stabilizer states, we also obtain an explicit formula for any multi-invariant using the GHZ-extraction theorem. We then present a counting argument that calculates any Coxeter multi-invariant of a q-partite stabilizer state. We conjecture a closed form expression for the same. We uncover hints of an interesting connection between multi-invariants, stabilizer states and topology. We show how our formulas are further simplified for a restricted class of stabilizer states that appear as ground states of interesting models like the toric code and the X-cube model.

Figures (18)

• Figure 1: Graphical notation for the coefficients of $\psi$ and the conjugate $\bar{\psi}$.
• Figure 2: The multi-invariant corresponding to $\sigma_1 = (1)(2)(3), \sigma_2 = (123), \sigma_3 = (12)(3)$.
• Figure 3: Multi-invariant from figure \ref{['fig:perms']} after tracing out party $1$. Each vertex stands for a reduced density matrix on the remaining $(q-1)$ parties.
• Figure 4: The multi-invariant graph that computes the multi-entropy for $n = 3, q = 3$, and $n = 3, q = 2$ respectively.
• Figure 5: A simple graph $G$ with three vertices $V = \{A, B, C\}$ and two edges $E = \{(A,B), (B,C)\}$.

Theorems & Definitions (1)

• Theorem 1