Bond Additivity and Persistent Geometric Imprints of Entanglement in Quantum Thermalization

TL;DR

The paper introduces multi-bipartition entanglement tomography to reveal the geometric structure of entanglement in quantum many-body dynamics. It defines a bond-additive law where the entanglement entropy decomposes into a bulk term $S_0$ plus a geometric sum $\sum_j \omega_j n_j$, with $n_j$ counting crossed $j$-order bonds and $\{\omega_j\}$ the entanglement bond tensions. For Hamiltonian thermalization, a persistent hierarchy $\omega_1 \gg \omega_{j>1}$ emerges, signaling a geometric imprint of locality enforced by energy conservation, while random quantum circuits and Floquet dynamics erase this imprint. The framework distinguishes thermalization mechanisms, links entanglement geometry to interaction locality, and offers experimentally accessible diagnostics via randomized measurements or classical shadows to reconstruct Hamiltonian features from $\{\omega_j\}$.

Abstract

Characterizing the intricate structure of entanglement in quantum many-body systems remains a central challenge, as standard measures often obscure underlying geometric details. In this Letter, we introduce a powerful framework, termed multi-bipartition entanglement tomography, which probes the fine structure of entanglement across an exhaustive ensemble of distinct bipartitions. Our cornerstone is the discovery of a ``bond-additive law'', which reveals that the entanglement entropy can be precisely decomposed into a bulk volume-law baseline plus a geometric correction formed by a sum of local contributions from crossed bonds of varying ranges. This law distills complex entanglement landscapes into a concise set of entanglement bond tensions $\{ω_j\}$, serving as a quantitative fingerprint of interaction locality. By applying this tomography to Hamiltonian dynamics, random quantum circuits, and Floquet dynamics, we resolve a fundamental distinction between thermalization mechanisms: Hamiltonian thermalized states retain a persistent geometric imprint characterized by a significantly non-zero $ω_1$, while this structure is completely erased in random quantum circuit and Floquet dynamics. Our work establishes multi-bipartition entanglement tomography as a versatile toolbox for the geometric structure of quantum information in many-body systems.

Figures (26)

• Figure 1: (a) Schematic illustration of a bipartition for a chain with periodic boundary condition. The spins represented by the green and blue circles constitute subsystem $\text{A}$ and its complement, respectively. Examples of crossed bonds with different orders connecting to the top spin are shown. (b) Saturated EEs of equal-sized bipartitions as a function of the crossed $1$-order bonds number $n_1$ for chain length $L=16$. The NN thermal dynamics governed by $\hat{H}_{\text{NN}}(W=0.5)$ (evolution time $t=1000.0$) shows a structured distribution that remains below the Haar measure average value. In contrast, RQC (evolution depth $2000$) and Floquet (evolution period number $100$) dynamics make saturated EEs perfectly reproduce the Haar measure average. All these results are averaged over $1000$ random samples.
• Figure 2: Evolution of the entanglement structure and mutual information in NN thermal dynamics for $L=16$. The panels compare the entanglement bond tensions $\left\lbrace \omega_j\right\rbrace$ and the mutual informations $\left\lbrace I_j\right\rbrace$ at three distinct stages: (a) early time ($t=0.1$), (b) intermediate time ($t=2.0$), and (c) late time ($t=1000.0$) at which the entanglement has saturated. The entanglement bond tensions $\left\lbrace \omega_j\right\rbrace$ are extracted from linear regression on the EEs of equal-sized bipartitions ($n_0=L/2=8$), whose raw distributions against $n_1$ are shown in the corresponding insets. The excellent quality of the fits is confirmed by the high coefficients of determination, $R^2 \approx 0.9993$ (a), $R^2 \approx 0.9959$ (b), and $R^2 \approx 0.9995$ (c). The persistence of the hierarchy $\omega_1 \gg \omega_j$ for $j>1$ in panel (c) demonstrates a lasting imprint of locality in the thermalized state. All the data of mutual informations and EEs used for fitting are averaged over $1000$ random samples.
• Figure 3: Entanglement bond tensions $\left\lbrace \omega_j\right\rbrace$ with $n_0=L/2$ and mutual informations $\left\lbrace I_j\right\rbrace$ at saturation of other Hamiltonian dynamics. (a) NNN thermal dynamics ($L=16$, $t=1000.0$, $R^2\approx0.9986$). (b) MBL dynamics ($L=16$, $t=10^{12}$, $R^2\approx0.9984$). (c) mixed-field dynamics ($L=12$, $t=1000.0$, $R^2\approx0.9990$). All these results are averaged over $1000$ random samples.
• Figure S1: Schematic illustrations of Hamiltonian dynamics (where $\hat{H}_{\text{NN}}(W=0.5)$, $\hat{H}_{\text{NN}}(W=5.0)$$\hat{H}_{\text{NNN}}$ and $\hat{H}_{\text{MF}}$ are all represented by $\hat{H}$) (a), RQC dynamics (b), and Floquet dynamics (c). Blue circles represent spins.
• Figure S2: Evolution of the HCEE for the various dynamics studied. The top row (a-c) displays the HCEE evolution for the NN thermal dynamics, showing the (a) initial linear growth, (b) subsequent slowing, and (c) final saturation. Similarly, the second row (d-f), the fourth row (j-l) and the fifth row (m-o) respectively show the evolution for the NNN thermal dynamics, the mixed-field dynamics and the dynamics governed by $\hat{H}_{\mathrm{NN}}(W=0.5)$ starting from random product states across the same three regimes. The third row presents the complete HCEE evolution for (g) MBL dynamics, (h) RQC dynamics and (i) Floquet dynamics. For the Hamiltonian dynamics (a-g) and (j-o), the horizontal axis is evolution time $t$; for the RQC (h) and Floquet (i) dynamics, it represents the circuit depth and the number of evolution periods, respectively. The simulations for mixed-field dynamics and the dynamics governed by $\hat{H}_{\mathrm{NN}}(W=0.5)$ starting from random product state are performed on a chain of length $L=12$, while all other dynamics are performed for $L=16$. The results for MBL dynamics are averaged over $1000$ disorder and initial state realizations, while all other dynamics are averaged over $200$ realizations.