Conditional Independence of 1D Gibbs States with Applications to Efficient Learning

TL;DR

This work investigates how 1D quantum Gibbs states exhibit a strong form of conditional independence, quantified through Belavkin-Staszewski-based conditional mutual information (BS-CMI). The authors prove that BS-CMI decays superexponentially with the shielding region size for finite-range, translation-invariant 1D Hamiltonians at any $\beta>0$, and they provide an explicit upper bound on the BS-entropy data-processing inequality via a BS-recovery framework. They further establish an approximate factorization for the global Gibbs state and its Rényi-2 purity, enabling efficient learning of classical tensor-network (MPO) representations from local data, and show how the global purity can be estimated with polynomial sample complexity. The results extend to exponentially-decaying interactions above a threshold temperature, and they include technical bounds on recovery maps and Lipschitz control for concatenated recovery channels, underpinning practical learning algorithms for 1D quantum thermal states with potential impact on scalable classical representations and tomography-based learning of quantum many-body systems.

Abstract

We show that spin chains in thermal equilibrium have a correlation structure in which individual regions are strongly correlated at most with their near vicinity. We quantify this with alternative notions of the conditional mutual information, defined through the so-called Belavkin-Staszewski relative entropy. We prove that these measures decay superexponentially at every positive temperature, under the assumption that the spin chain Hamiltonian is translation-invariant. Using a recovery map associated with these measures, we sequentially construct tensor network approximations in terms of marginals of small (sublogarithmic) size. As a main application, we show that classical representations of the states can be learned efficiently from local measurements with a polynomial sample complexity. We also prove an approximate factorization condition for the purity of the entire Gibbs state, which implies that it can be efficiently estimated to a small multiplicative error from a small number of local measurements. The results extend from strictly local to exponentially-decaying interactions above a threshold temperature, albeit only with exponential decay rates. As a technical step of independent interest, we show an upper bound to the decay of the Belavkin-Staszewski relative entropy upon the application of a conditional expectation.

Figures (5)

• Figure 1: Regions $A$ and $C$ shielded by a region $B$.
• Figure 2: Representation of an interval $\Lambda$ split into three subintervals $\Lambda=ABC$, where $B$ shields $A$ from $C$.
• Figure 3: Representation of an interval $\Lambda$ split into five subintervals $\Lambda=A'ABCC'$.
• Figure 4: Representation of an interval $\Lambda$ split into three subintervals $\Lambda=ABC$, with $B$ further split into $B_1$, $B_2$ and $B_3$ such that $|B_1|, |B_2|, |B_3| \geq \ell$.
• Figure 5: Representation of an interval $\Lambda$ split into multiple subintervals $\Lambda=A_1 A_2 \ldots A_n$.

Theorems & Definitions (59)

• Proposition 2.1: Bluhm.2022
• Proposition 2.2: Bluhm.2022
• Theorem 2.3: Bluhm.2022
• Lemma 2.4
• Lemma 2.5
• Remark 2.6
• Corollary 2.7
• Proposition 2.8: Decay of correlations Bluhm.2022
• Theorem 2.9: Local indistinguishability Bluhm.2022
• Remark 2.10