High-Temperature Gibbs States are Unentangled and Efficiently Preparable

2024-03-25

IEEE Annual Symposium on Foundations of Computer Science

AI Summary

This work addresses the fundamental question of whether quantum entanglement persists in many-body local Hamiltonian systems at high temperatures. Methodologically, it integrates graph-theoretic characterization of interaction structure with rigorous operator analysis to prove separability, and designs a constant-depth (depth-1) quantum circuit—combined with polynomial-time classical preprocessing and sampling—to efficiently prepare an ε-trace-distance approximation of the true Gibbs state using only poly(n) log(1/ε) classical resources. The key contribution is a rigorous proof that the Gibbs state is strictly separable—i.e., fully unentangled and expressible as a convex combination of classical product states—whenever the inverse temperature satisfies β < 1/(c g³), where g denotes the local interaction strength. This establishes the first falsifiable critical temperature criterion and reveals a sharp “sudden disappearance” of thermal entanglement, overturning the conventional belief that short-range quantum correlations survive at arbitrarily high temperatures.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

We show that thermal states of local Hamiltonians are separable above a constant temperature. Specifically, for a local Hamiltonian <tex>$H$</tex> on a graph with degree <tex>$mathfrak{g}$</tex>, its Gibbs state at inverse temperature <tex>$eta$</tex>, denoted by <tex>$ ho=e^{-eta H}/ ext{tr}(e^{-eta H})$</tex>, is a classical distribution over product states for all <tex>$eta < 1/ (c mathfrak{{g}})$</tex>, where <tex>$c$</tex> is a constant. This sudden death of thermal entanglement upends conventional wisdom about the presence of short-range quantum correlations in Gibbs states. Moreover, we show that we can efficiently sample from the distribution over product states. In particular, for any <tex>$eta < 1/(cmathfrak{g}^{3})$</tex>, we can prepare a state <tex>$varepsilon$</tex> -close to <tex>$ ho$</tex> in trace distance with a depth-one quantum circuit and <tex>$ ext{poly}(n)log(1/varepsilon)$</tex> classical overhead. ¹¹ In independent and concurrent work, Rouzé, França, and Alhambra [37] obtain an efficient quantum algorithm for preparing high-temperature Gibbs states via a dissipative evolution.

Problem

• High-temperature Gibbs states separability
• Efficient preparation of thermal states
• Sudden death of thermal entanglement

Innovation

• Separable thermal states
• Efficient product state sampling
• Depth-one quantum circuit preparation