Tripartite entanglement in the HaPPY code is not holographic

TL;DR

The paper investigates whether holographic entropic constraints extend to genuine multipartite entanglement by formulating the GHZ-forbidding inequality using the reflected entropy $S^{(R)}$ and the genuine tripartite multi-entropy $S^{(3)}$. It shows that stabilizer states cannot strictly satisfy this inequality, implying that stabilizer toy models like the HaPPY code do not capture holographic tripartite entanglement. The authors derive explicit expressions for $S^{(R)}$ and $S^{(3)}$ for stabilizer states via the GHZ extraction theorem and apply them to HaPPY's tensor networks, demonstrating the boundary states remain stabilizer and thus non-holographic in this sense. The discussion points to non-stabilizer AME tensors or random tensor networks as potential holographic candidates and frames future work on multipartite holographic inequalities and their connections to quantum magic and complexity.

Abstract

Holographic states satisfy several entropic inequalities owing to the Ryu-Takayangi formula. A drawback of these inequalities is that they only use bipartite entanglement in their formulation. We investigate a recently proposed "GHZ-forbidding" inequality, built out of the reflected entropy and the tripartite multi-entropy, that holds for holographic states. We show that the inequality is either violated or saturated, but never strictly satisfied, by stabilizer states, thereby showing that stabilizer states are not holographic. As a consequence, we show that tripartite entanglement in the HaPPY code is not holographic.

Figures (11)

• Figure 1: The RT surface $\gamma$ is the minimal area surface in the bulk that is homologous to $A$.
• Figure 2: The operator $\sqrt{\rho_{AB}}$ takes $\mathcal{H}_A \otimes \mathcal{H}_B$ as input, and gives $\mathcal{H}_A \otimes \mathcal{H}_B$ as output. This is depicted with incoming $A$ and $B$ arrows and outgoing $A$ and $B$ arrows. The state $\ket{\sqrt{\rho_{AB}}}$, on the other hand, is depicted with four incoming arrows.
• Figure 3: (a) Graphical representation of a $q$-partite state $\ket{\psi} \in \mathcal{H}_1 \otimes \dots \otimes \mathcal{H}_q$ and its conjugate $\bra{\psi}$. (b) An example of the lattice with $n = 3$ sites in each of the $q = 2$ directions used to define the multi-entropy.
• Figure 4: (a) The Ryu-Takayangi surfaces for a tripartite holographic state. The pink surface is the entanglement wedge cross-section surface. (b) The minimal area Mercedes-Benz surface whose area (divided by $4G_N$) is the tripartite multi-entropy for a holographic state.
• Figure 5: The $\{6, 4\}$ tiling of the hyperbolic disk with hexagons. We place a six-legged perfect tensor at the center of each hexagon.

Theorems & Definitions (1)

• Theorem 1: GHZ extraction