Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures

TL;DR

This work proves that conditional mutual information (CMI) in quantum Gibbs states decays exponentially with the separation of regions A and C, at arbitrary temperatures and dimensions, establishing an approximate quantum Markov property and advancing a quantum version of the Hammersley–Clifford framework. The authors develop a three-pronged proof strategy combining quantum belief propagation (BP) in 1D, a partial-trace projection (PTP) approach in higher dimensions, and a connected-exponential/entanglement-Hamiltonian formalism to show quasi-locality of effective interactions. They further translate CMI decay into clustering statements for genuine multipartite entanglement, deriving bounding expressions for entanglement of formation (E_F) beyond PPT, with stronger 1D results that hold in the thermodynamic limit. A quasi-local entanglement Hamiltonian in 1D is proven with bounds on the approximation error that grow only sub-exponentially with inverse temperature, providing insights into Hamiltonian learning and quantum Gibbs sampling. The framework sets the stage for deeper understanding of long-range entanglement at finite temperature and suggests avenues to sharpen subsystem-size dependence and extend locality results to broader classes of systems.

Abstract

Recent investigations have unveiled exotic quantum phases that elude characterization by simple bipartite correlation functions. In these phases, long-range entanglement arising from tripartite correlations plays a central role. Consequently, the study of multipartite correlations has become a focal point in modern physics. In these, Conditional Mutual Information (CMI) is one of the most well-established information-theoretic measures, adept at encapsulating the essence of various exotic phases, including topologically ordered ones. Within the realm of quantum many-body physics, it has been a long-sought goal to establish a quantum analog to the Hammersley--Clifford theorem that bridges the two concepts of the Gibbs state and the Markov network. This theorem posits that the correlation length of CMI remains short-range across all thermal equilibrium quantum phases. In this work, we demonstrate that CMI exhibits exponential decay with respect to distance, with its correlation length increasing polynomially with respect to the inverse temperature. While this clustering theorem has previously been believed to hold for high temperatures devoid of thermal phase transitions, it has remained elusive at low temperatures, where genuine long-range entanglement can exist due to the quantum topological order. Our findings unveil that, even at low temperatures, a broad class of tripartite entanglement cannot manifest in the long-range regime. To achieve the proof, we establish a comprehensive formalism for analyzing the locality of effective Hamiltonians on subsystems, commonly known as the entanglement Hamiltonian or Hamiltonian of mean force. As an outcome of our analyses, we improve the prior clustering theorem for bipartite entanglement. In essence, this means that we investigate genuine bipartite entanglement that extends beyond the limitations of the Positive Partial Transpose (PPT) class.

Figures (19)

• Figure 1: Illustration of the main problem. We consider a finite-dimensional lattice system, depicted in 2D. The system is partitioned into three subsystems: $A$, $B$, and $C$, and we examine the conditional mutual information (CMI) between $A$ and $C$, conditioned on $B$. When the system Hamiltonian is classical or commuting, the Hammersley--Clifford theorem indicates that subsystems $A$ and $C$ are conditionally independent. This independence implies that the CMI is zero when the distance between regions $A$ and $C$ exceeds the interaction length. The quantum analog suggests a rapid decay of the CMI with respect to the distance, as conjectured in Eq. \ref{['quantum_Markov_Conj_ineq']}. A positive resolution of this conjecture would imply that generic quantum Gibbs states form approximate Markov networks, which, in turn, implies the existence of a local recovery map capable of reconstructing the global Gibbs state from the reduced density matrices Fawzi2015. Our main result offers a partial solution to this conjecture, encapsulated in the primary statements Eqs. \ref{['main_high_dim_CMI']} and \ref{['main_1D_dim_CMI']}.
• Figure 2: Illustration of 1D entanglement Hamiltonian. For a contiguous region $L\subset \Lambda$, we define the entanglement Hamiltonian $H^\ast_L$ by $\beta^{-1} \log(\rho_{\beta,L})$ as in Eq. \ref{['def_rho_L_H^ast_L_main']}. The effective interaction terms in $H^\ast_L$ (i.e., $H^\ast_L-H_L$) are expected to be quasi-local around the boundary of $L$, with the length scale of the quasi-locality given by $\xi_\beta$. The quasi-locality of the entanglement Hamiltonian imposes a much stronger structural constraint than the CMI decay \ref{['quantum_Markov_Conj_ineq']}. In one-dimensional systems with finite-range interactions, we rigorously prove the error bound \ref{['Hamiltonian_eff_approx_main']}, which yields $\xi_\beta = e^{e^{\mathcal{O}(\beta)}}$.
• Figure 3: Belief propagation formalism for the entanglement Hamiltonian. By tracing out the subsystem $L$, we aim to derive the connected exponential form given in Eq. \ref{['approx_exponential_reduced']} to approximate the reduced density matrix $\rho_{L^{\rm c}}$. For this purpose, we apply the quantum belief propagation method to segregate the boundary interaction term $\partial h_Y$ from the quantum Gibbs state, which is represented as $e^{\beta H} = e^{\beta (H_{LX} + H_Y + \partial h_Y)}$. In this setup, the quantum-belief-propagation operator $\Phi_{\partial h_Y}$ serves as a bridge linking $e^{\beta (H_{LX} + H_Y)}$ to $e^{\beta H}$. By approximating this operator within the region $XY$ with $\tilde{\Phi}_{XY}$, we can manage the partial trace ${\rm tr}_{L}(\cdots)$ independently from the quantum belief propagation operation. This approximation leads to the desired form of Eq. \ref{['BP_approx_reduce_den']}.
• Figure 4: Quasi-locality of the effective interaction terms. We are examining the operator logarithm given by $\log\left( e^{\tau V_{i_0}} e^{\beta H} e^{\tau V_{i_0}} \right)$ and begin by establishing the quasi-locality of the unitary operator $U_\tau$ in Eq. \ref{['eff_Ham_U_B_mC']}, which is instrumental in defining the effective Hamiltonian presented in Eq. \ref{['effecetve_mA_tau']}. The quasi-locality of $U_\tau$ is evaluated based on the norm $\left \| \left[ U_\tau, u_i \right] \right \|$, where $u_i$ represents any arbitrary unitary operator located on a site $i$ ($d_{i,i_0} = r$). However, confirming the quasi-locality of $U_\tau$ alone does not fulfill all our requirements. We further stipulate that $U_\tau$ should primarily be influenced by the neighboring region around the site $i_0$. This requirement leads us to approximate $U_\tau$ by $U_{\tau, i_0[r]}$, as expressed in Eq. \ref{['U_tau_ball_region_approx']}. Here, $U_{\tau, i_0[r]}$ is specifically determined by the ball region centered at the site $i_0$ with radius $r$ (inside the dashed circle).
• Figure 5: Roadmap (from Sec. S I to Sec. S VII)

Theorems & Definitions (63)

• Lemma 1
• Conjecture 1
• Lemma 2
• Lemma 3
• Lemma 4: Lieb--Robinson bound ref:Nachtergaele2006-LR
• Lemma 5
• Corollary 6
• Lemma 7
• Lemma 8
• Lemma 9