High-Temperature Gibbs States are Unentangled and Efficiently Preparable (2403.16850v2)

Abstract: We show that thermal states of local Hamiltonians are separable above a constant temperature. Specifically, for a local Hamiltonian $H$ on a graph with degree $\mathfrak{d}$, its Gibbs state at inverse temperature $\beta$, denoted by $\rho = e^{-\beta H}/ \operatorname{tr}(e^{-\beta H})$, is a classical distribution over product states for all $\beta < 1/(c\mathfrak{d})$, where $c$ is a constant. This proof of sudden death of thermal entanglement resolves the fundamental question of whether many-body systems can exhibit entanglement at high temperature. Moreover, we show that we can efficiently sample from the distribution over product states. In particular, for any $\beta < 1/( c \mathfrak{d}^2)$, we can prepare a state $\varepsilon$-close to $\rho$ in trace distance with a depth-one quantum circuit and $\operatorname{poly}(n, 1/\varepsilon)$ classical overhead.

• The paper demonstrates that above a temperature threshold, Gibbs states lose entanglement and become classical product states for β < 1/(cΔ).
• The methodology introduces a depth-one quantum circuit algorithm that prepares an ε-close Gibbs state with polynomial computational overhead.
• The research employs cluster expansion and tree-based sampling techniques to bridge classical simulations with quantum many-body thermodynamics.

Analysis of High-Temperature Gibbs States: Unentanglement and Efficient Preparation

The paper "High-Temperature Gibbs States are Unentangled and Efficiently Preparable" examines the thermal states of local Hamiltonians, showing that above a certain temperature, these states demonstrate remarkable properties of separability and efficient preparability. Through rigorous mathematical proofs, the authors challenge established notions about the entanglement characteristics of quantum systems at thermal equilibrium.

Unentanglement at High Temperatures

The key assertion of the paper is the absence of entanglement in Gibbs states at high temperatures. For a local Hamiltonian $H$ with degree $\Delta$, the Gibbs state $\rho = e^{-\beta H} / \text{tr}(e^{-\beta H})$ becomes a classical distribution over product states for inverse temperatures $\beta < 1/(c\Delta)$, where $c$ is a constant. This finding contradicts traditional views that allow for short-range quantum correlations in such thermal states. The authors provide a formal proof, demonstrating that these correlations vanish above a fixed temperature threshold, thus enabling the Gibbs states to be expressed as distributions over tensor products of stabilizer states.

Efficient State Preparation

The paper further explores the computational impact of this structural simplicity by addressing the efficient preparation of Gibbs states. For temperatures above a certain constant threshold $\beta < 1/(c \Delta^3)$, the authors develop an algorithm capable of preparing a state $\epsilon$-close to the true Gibbs state in trace distance. This is achieved with a depth-one quantum circuit and a polynomial in $n$ computational overhead. The algorithm's approach refutes the potential for super-polynomial quantum speedups in Gibbs state preparation at high temperatures, aligning quantum and classical computational capabilities in this domain.

Technical Contributions

Key technical contributions include a polynomial approximation of a restricted Gibbs state, as well as the design of a tree-based sampling method that approximates the Gibbs state preparation problem. These results leverage cluster expansion techniques and establish connections to classical abstract polymer models.

Implications and Future Directions

The paper provides a significant theoretical advance in understanding the thermal properties of quantum many-body systems, suggesting that in certain thermal regimes, quantum systems can be fully described by classical physics. Practically, this implies that efficient algorithms for high-temperature regimes might be adapted for large-scale quantum simulations.

The work invites future exploration into questions such as whether the entanglement and computational thresholds can be further tightened or extended. It also opens avenues for leveraging classical frameworks in other quantum thermodynamic contexts and using these results as a benchmark for quantum advantage.

In summary, this paper presents valuable insights into the nature of quantum Gibbs states at high temperatures, potentially influencing both theoretical research and practical applications in quantum simulation and computation.