One of the most fundamental problems in quantum many-body physics is the characterization of correlations among thermal states. Of particular relevance is the thermal area law, which justifies the tensor network approximations to thermal states with a bond dimension growing polynomially with the system size. In the regime of sufficiently low temperatures, which is crucially important for practical applications, the existing techniques do not yield optimal bounds. Here, we propose a new thermal area law that holds for generic many-body systems on lattices. We improve the temperature dependence from the original $\mathcal{O}(\beta )$ to $\mathcal{O}({\beta }^{2/3})$ up to a logarithmic factor, thereby suggesting subballistic propagation of entanglement by imaginary-time evolution. This qualitatively differs from the real-time evolution, which usually induces linear growth of entanglement. We also prove analogous bounds for the Rényi entanglement of purification and the entanglement of formation. Our analysis is based on a polynomial approximation to the exponential function which provides a relationship between the imaginary-time evolution and random walks. Moreover, for one-dimensional (1D) systems with $n$ spins, we prove that the Gibbs state is well approximated by a matrix product operator with a sublinear bond dimension for $\beta =o[\log (n)]$. This proof allows us to rigorously establish, for the first time, a quasilinear time classical algorithm for constructing a matrix product state representation of 1D quantum Gibbs states at arbitrary temperatures of $\beta =o[\log (n)]$. Our new technical ingredient is a block decomposition of the Gibbs state that bears a resemblance to the decomposition of real-time evolution given by Haah et al. [Proceedings of the 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, New York, 2018), pp. 350–360].

中文翻译：

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量子吉布斯态的改进的热面积定律和拟线性时间算法

量子多体物理学中最基本的问题之一是热态之间相关性的表征。特别相关的是热面积定律，该热定律证明张量网络近似于热态，其结合尺寸随系统尺寸成倍增长。在足够低的温度范围内，这对于实际应用至关重要，现有技术无法产生最佳界限。在这里，我们提出了一个新的热面积定律，适用于晶格上的一般多体系统。我们从原始的角度改善了温度依赖性$\mathcal{Ø}（\beta ）$ 到 $\mathcal{Ø}（{\beta }^{\mathrm{2个}/3}）$高达对数因子，从而通过假想时间演化来暗示纠缠的子弹道传播。从本质上讲，这与实时演化不同，后者通常会引起纠缠的线性增长。我们还证明了Rényi纯化的纠缠与形成的纠缠的相似界。我们的分析基于指数函数的多项式逼近，该函数提供了虚时演化和随机游动之间的关系。此外，对于一维（1D）系统，$ñ$ 自旋，我们证明吉布斯状态被矩阵乘积算子很好地近似，其中子线性键维为 $\beta =Ø[\mathrm{日志}（ñ）]$。该证明使我们能够首次严格建立准线性时间经典算法，以在任意温度下构建一维量子吉布斯态的矩阵乘积态表示。$\beta =Ø[\mathrm{日志}（ñ）]$。我们的新技术成分是吉布斯状态的块分解，这类似于Haah等人给出的实时演化分解。[ 《 2018年IEEE第59届计算机科学基础年度研讨会（FOCS）的议事录》（IEEE，纽约，2018年，第350–360页）。