Emergent geometry from entanglement structure

TL;DR

The paper introduces an entanglement adjacency matrix (EAM) $J_{ij}$ that encodes the entanglement structure of an $N$-partite pure state, enabling the bipartition von Neumann entropies to be approximated by $S_A \ approx \sum_{i\in A, j\in A^c} J_{ij} + s_0$ and thereby revealing an emergent geometry from entanglement. It provides exact results for simple states (dimer, rainbow, GHZ) and a numerical least-squares framework to obtain $J_{ij}$ in general, accompanied by a natural entanglement contour $s_A(i)=\sum_{j\in A^c} J_{ij}$ that extends to interacting systems. The correspondence between $J_{ij}$ and geometry is further enriched by a contour comparison with prior formalisms and by a field-theoretic interpretation in conformal field theory, where $J(x,y)$ acts as the two-point correlator of an entanglement current, yielding a flow-like picture of entanglement. Overall, the work demonstrates that entanglement distributions can define geometry distinct from the Hamiltonian and provides tools to quantify and interpret entanglement through contours and current-like correlators, with potential implications for understanding quantum many-body structure and conformal dynamics.

Abstract

We attempt to reveal the geometry, emerged from the entanglement structure of any general $N$-party pure quantum many-body state by representing entanglement entropies corresponding to all $2^N $ bipartitions of the state by means of a generalized adjacency matrix. We show this representation is often exact and may lead to a geometry very different than suggested by the Hamiltonian. Moreover, in all the cases, it yields a natural entanglement contour, similar to previous proposals. The formalism is extended for conformal invariant systems, and a more insightful interpretation of entanglement is presented as a flow among different parts of the system.

Figures (3)

• Figure 1: Schematic of the entanglement entropy obtained for an arbitrary bipartition ($A, A^c$) by removing the links connecting the sites. Here the links represent the constants $J_{ij}$.
• Figure 2: Similar to the illustration of the different entropies and the mutual information between two variables $X$ and $Y$ using Venn diagrams, as shown in the upper panel, a graphical representation of the quantum version of different entropic relations is presented in the lower panel (a)-(c).
• Figure 3: In panel (a), we compare the contour functions for the entanglement entropy $s_A(i)$, those obtained using the Eqs. (\ref{['eq:contour_J']}) and (\ref{['eq:vidal_contour']}), for the ground state of dimerized Hamiltonian ($t_{ij} = (1+\delta (-1)^i), |i-j| = 1,\delta=0.5$, $N=14$). Whereas, in panel (b), we plot the contour functions for the entanglement entropy obtained using Eq. (\ref{['eq:contour_J']}) for ground state of $XXZ$ Hamiltonian for different values of the parameter $\Delta$, and for $N=12$. Additionally, in the inset of both the figures, the scaling of the error ($\mathcal{E}$) with the system size ($N$) have been shown for all the parameter values considered.