Abstract
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 particularly 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 $\tilde{\mathcal{O}}(\beta^{2/3})$, thereby suggesting diffusive 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 R\'enyi 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 of $e^{\sqrt{\tilde{\mathcal{O}}(\beta \log(n))}}$. This proof allows us to rigorously establish, for the first time, a quasi-linear time classical algorithm for constructing an MPS 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 resemblance to the decomposition of real-time evolution given by Haah et al., FOCS'18.
Highlights
Throughout the paper, we focus on the Gibbs state ρβ with an inverse temperature β: ρβ e−βH trðe−βHÞ : To extend the concept of the entanglement area law from the ground states to finite temperatures, we often utilize the mutual information IðL∶RÞρ as in Ref. [28]
For a more detailed characterization of the structure of the quantum Gibbs state, we focus on the matrix product operators (MPOs) representation
The first one is the improved thermal area law that gives a scaling of Oðβ2=3Þ over all lattices (Theorem 1)
Summary
It is widely accepted that the area law plays a crucial role [18,19] in the characterization of low-temperature physics of many-body systems This law states that the entanglement entropy between two subsystems is at most as large as the size of their boundaries. The widely known relation between area laws and tensor networks suggests that the identification of the minimum γc would lead to optimal representations of Gibbs states This outcome would result in faster algorithms for computing local expectation values and evaluating the partition functions. Below a threshold temperature where the cluster expansion technique does not work [86], little is known about the universal properties of Gibbs states that may hold independent of the system’s details This lack provides a strong motivation to identify the optimal thermal area laws
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