Tripartite Correlation Signal from Multipartite Entanglement of Purification

TL;DR

This work introduces semi-definite signals for multipartite entanglement in quantum systems, constructing a tripartite correlation signal Δ_p^(3) from multipartite entanglement of purification and a holographic counterpart Δ_w^(3) from multipartite entanglement wedge cross sections. It proves Δ_p^(3) ≥ 0 and that it vanishes for pure states and fully bipartite-product states, while being nonzero for certain tripartite-entangled configurations beyond GHZ-type, with concrete analysis for GHZ_N and W_N states. The paper then explores the holographic side under the conjecture E_p^(n) = E_w^(n), deriving Δ_w^(3) properties, analyzing pure AdS_3 in symmetric configurations, and identifying phase transitions where Δ_w^(3) changes behavior, including a discontinuity at the bipartite-to-tripartite transition. A forward path to n-partite signals Δ_p^(n) is sketched, noting sign-indefiniteness for Δ_p^(4) in holographic contexts and outlining broader implications for multipartite entanglement distillation, holographic codes, and bulk reconstruction. Overall, the work provides a framework to diagnose and compare multipartite entanglement structures in both finite-dimensional and holographic settings, linking information-theoretic measures to geometric spacetime features.

Abstract

We propose a signal $Δ^{(3)}_p$ for genuine tripartite entanglement in finite-dimensional quantum systems and $Δ^{(3)}_w$ for holographic systems. We prove that $Δ^{(3)}_p$ is non-negative for any tripartite entangled mixed states. Based on the conjecture, the equality between an entanglement wedge cross section $E_w$ and entanglement of purification $E_p$, i.e., $E_w = E_P$ in the semiclassical limit, we apply the tripartite entanglement measure to study the structures of tripartite entanglement in AdS$_3$/CFT$_2$, especially for pure AdS$_3$. We comment on a generalization to $n$-partite entanglement signals $Δ^{(n)}_p(A_1:\cdots:A_n)$.

Figures (7)

• Figure 1: (a) The upper bound and the candidate lower bounds in \ref{['eq:uppr-lower-bounds-3']} are plotted as a function of total number $N$ of qubits for $\ket{W_N}$\ref{['eq:Wn']}. The blue dots are the upper bound. The orange and green dots are the candidate lower bounds. $E^{(3)}_p(A:B:C)$ as a function of $N$ takes a value between the blue dots and the orange dots because the orange dots maximize the lower bound. (b) Similarly, the candidate upper bounds and the lower bound of \ref{['eq:uppr-lower-bounds-2']} is plotted as a function of $N$. $E^{(2)}_p(A:B:C)$ as a function of $N$ takes a value between the orange dots and the green dots because the orange dots minimize the upper bound. (c) We plotted \ref{['eq:upper-lower-bounds-delta']} as a function of $n$.
• Figure 2: (a) The red curves are $l_a$ and $l_b$. The blue curves are $L_a$ and $L_b$. The figure is exaggerated so that one can see the explicit contributions of $L_a,L_b,l_a,l_b$ to the corresponding MEoP. See \ref{['eq:bipartite_connected_phase']} and \ref{['eq:bipartite_connected_phase_EWCS']}. (b) A graphical proof using the Beltrami-Klein model. Black lines are the minimal surfaces of the entanglement wedge $\Gamma^{(3)}_{min}(A:B:C)$ in a fully connected phase. Green lines are tangent to the boundary of the boundary subregions $A,B,C$. Red points represent the intersections of the green lines. $L_a,L_b,L_c$ form a triangle, whereas $l_a,l_b,l_c$ cannot form a triangle within the disk unless $ABC$ covers the whole boundary of the disk. (c) The exaggerated figure representing the phase just right before ABC covers the whole boundary. Black dotted curves are the RT surfaces $\Gamma_{min}(A),\Gamma_{min}(B),\Gamma_{min}(C)$ of boundary subregion $A,B,C$. When $ABC$ becomes the whole boundary, we have \ref{['eq:pure_state_ewcs']}.
• Figure 3: The connected entanglement wedge of $ABC$ in a static time-slice of $AdS_3/CFT_2$. The red curves denote $E_w^{(2)}(A:BC) = 2E_w(A:BC)=2l_a$ and permutations, $E_w^{(2)}(B:AC)=2E_w(B:AC)=2l_b$ and $E_w^{(2)}(C:AB)=2E_w(C:AB)=2l_c$. The blue curve denotes $E_w^{(3)}(A:B:C)=L_a+L_b+L_c$, which form a hyperbolic triangle. $\theta_1,\theta_2,\theta_3$ are the angles of the hyperbolic triangle. $\tilde{A}\supset A$ and $\hat{A}\supset A$ are the optimal boundary subregions to compute $l_a$ and $L_a$. $\tilde{B},\hat{B}$ and $\tilde{C},\hat{C}$ are defined similarly, although they are suppressed from the figure.
• Figure 4: We have set $L_{AdS}/4_{G_N}=1$. At $\alpha=\frac{\pi}{6}$, there is a discontinuous jump in $\Delta_w^{(3)}(A:B:C)$, due to the disconnected–connected phase transition of the tripartite entanglement wedge $M_{ABC}$, after which $\Delta_w^{(3)}(A:B:C)$ monotonically decreases. (a) A comparison of $E_w^{(3)}(A:B:C)$, $E_w^{(2)}(A:BC)$ and $\Delta_w^{(3)}(A:B:C)$. The $x$-axis is cut-off at $\alpha=\pi/3-10^{-1}$. (b) $\Delta_w^{(3)}(A:B:C)$ as a function of $\alpha$. The $x$-axis is cut-off at $\alpha=\pi/3-10^{-2}$.
• Figure 5: We have set $L_{AdS}/4G_N=1$. The transition points of the blue curves happen at $\alpha^{(3)}=\pi/6$, and are the same as in figure \ref{['fig:Delta3vsalpha_comparisons']}. The transition points of the orange curves happen at $\alpha^{(2)}=\pi/3\sqrt{2}$. The $x$-axis is cut-off at $\alpha=\pi/3-10^{-4}$. (a) The phase $\alpha^{(2)}\leq \alpha \leq \alpha^{(3)}$ exhibits the genuine tripartite entanglement. (b) The polygamy of $E^{(2)}(A:BC)$.

Theorems & Definitions (15)

• Definition 2.1: Multipartite entanglement of purification(MEoP)Umemoto-2018-MultipartiteEoP
• Definition 2.2: Tripartite correlation signal
• Proposition 2.1: Properties of $\Delta^{(3)}_p(A:B:C)$
• proof
• Definition 3.1: Multipartite entanglement wedge cross sectionUmemoto-2018-MultipartiteEoP
• Definition 3.2: Tripartite correlation signal for a holographic state
• Conjecture 3.1: Umemoto-2018-MultipartiteEoPBao-2019-conditionalEoP
• Proposition 3.1: Properties of $\Delta^{(3)}_w(A:B:C)$
• proof
• Definition 4.1: A candidate four-partite correlation signal for a density matrix