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arXiv:2207.07760 · Quantum

Thermal Area Law for Lattice Bosons

Marius Lemm, Oliver Siebert

Published Aug 16, 2023• 4 citations
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Abstract

A physical system is said to satisfy a thermal area law if the mutual information between two adjacent regions in the Gibbs state is controlled by the area of their boundary. Lattice bosons have recently gained significant interest because they can be precisely tuned in experiments and bosonic codes can be employed in quantum error correction to circumvent classical no-go theorems. However, the proofs of many basic information-theoretic inequalities such as the thermal area law break down for bosons because their interactions are unbounded. Here, we rigorously derive a thermal area law for a class of bosonic Hamiltonians in any dimension which includes the paradigmatic Bose-Hubbard model. The main idea to go beyond bounded interactions is to introduce a quasi-free reference state with artificially decreased chemical potential by means of a double Peierls-Bogoliubov estimate.

DOI
10.22331/q-2023-08-16-1083

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A physical system is said to satisfy a thermal area law if the mutual information between two adjacent regions in the Gibbs state is controlled by the area of their boundary. Lattice bosons have recently gained significant interest because they can be precisely tuned in experiments and bosonic codes can be employed in quantum error correction to circumvent classical no-go theorems. However, the proofs of many basic information-theoretic inequalities such as the thermal area law break down for bosons because their interactions are unbounded. Here, we rigorously derive a thermal area law for a class of bosonic Hamiltonians in any dimension which includes the paradigmatic Bose-Hubbard model. The main idea to go beyond bounded interactions is to introduce a quasi-free reference state with artificially decreased chemical potential by means of a double Peierls-Bogoliubov estimate.
The entanglement area law is a universal principle that characterizes quantum many-body phases and underpins tensor network algorithms. Traditionally, its validity has been limited to systems with short-range interactions and bounded local energy. Achieving a complete generalization that removes both of these constraints has been a longstanding goal in quantum many-body theory, especially for interacting boson systems where unbounded energy presents intrinsic difficulties. In this work, we rigorously prove the area law for one-dimensional interacting boson systems with long-range interactions, covering broad models including the Bose-Hubbard and ϕ4 classes. Furthermore, we establish an efficiency guarantee for Matrix-Product-State approximations of the ground states, offering a practical route to numerical simulation. One of our main technical contributions is a general method for Hilbert space dimension reduction, whose applicability extends to arbitrary spatial dimensions. These results address two major challenges simultaneously and provide important foundations for simulating long-range cold atomic systems. Entanglement area law has been established in systems with short range interactions and bounded local energy. Here the authors extend the proof to 1D bosonic systems with long-range interactions and unbounded local energy and outline implications for matrix product state approximations of the ground states.
We study the scaling of entanglement in low-energy states of quantum many-body models on lattices of arbitrary dimensions. We allow for unbounded Hamiltonians such that systems with bosonic degrees of freedom are included. We show that, if at low enough temperatures the specific heat capacity of the model decays exponentially with inverse temperature, the entanglement in every low-energy state satisfies an area law (with a logarithmic correction). This behavior of the heat capacity is typically observed in gapped systems. Assuming merely that the low-temperature specific heat decays polynomially with temperature, we find a subvolume scaling of entanglement. Our results give experimentally verifiable conditions for area laws, show that they are a generic property of low-energy states of matter, and constitute proof of an area law for unbounded Hamiltonians beyond those that are integrable.
We provide a rigorous and asymptotically exact expression of the mutual information of translationally invariant free fermionic lattice systems in a Gibbs state. In order to arrive at this result, we introduce a novel framework for computing determinants of Töplitz operators with smooth symbols, and for treating Töplitz matrices with system size dependent entries. The asymptotically exact mutual information for a partition of the 1D lattice satisfies an area law, with a prefactor which we compute explicitly. As examples, we discuss the fermionic XX model in one dimension and free fermionic models on the torus in higher dimensions in detail. Special emphasis is put on the discussion of the temperature dependence of the mutual information, scaling like the logarithm of the inverse temperature, hence confirming an expression suggested by conformal field theory. We also comment on the applicability of the formalism to treat open systems driven by quantum noise. In the appendix, we derive useful bounds to the mutual information in terms of purities. Finally, we provide a detailed error analysis for finite system sizes. This analysis is valuable in its own right for the abstract theory of Töplitz determinants.
We prove a β2β^{2}β2 area law for the quantum mutual information of thermal states of local lattice Hamiltonians: for any bipartitioning A∣BA|BA∣B and all inverse temperatures βββ below a critical value β∗β^{*}β∗, we show Iβ(A,B)⩽f~(β) β2 ∣∂AB∣\mathcal{I}_β(A,B) \leqslant \tilde{f}(β)\, β^{2}\,|\partial_{AB}|Iβ​(A,B)⩽f~​(β)β2∣∂AB​∣, where ∣∂AB∣|\partial_{AB}|∣∂AB​∣ denotes the size of the boundary region and f~(β)\tilde{f}(β)f~​(β) remains finite as β→0β\to 0β→0. This quadratic scaling improves the relationship between high-temperature correlations and macroscopic thermodynamics. While a linear area law implicitly permits a loose, temperature-independent upper bound on extractable work and binding energy, the quadratic scaling ensures that the nonequilibrium free energy available after decoupling regions and the total energy of the interface both are proportional to the area of the interface of A∣BA|BA∣B and vanish as βββ at high temperatures. These results show that the high-temperature mutual information and correlation are suppressed, respectively, quadratically and linearly in inverse temperature, which are consistent with the expected thermal decoupling in macroscopic thermodynamic quantities.
Abstract We establish the exponential clustering of correlation functions for the high-temperature Gibbs states of Bose–Hubbard type models. To overcome the technical difficulties arising from the unboundedness of bosonic operators, we develop the interaction-picture cluster-expansion technique. This method also allows us to systematically bound the moments of the local particle number. This result provides an analytical justification for the low-boson-density condition frequently assumed in the study of bosonic many-body systems. As direct mathematical consequences of the clustering property, we derive a uniform upper bound on the specific heat density and establish a bosonic thermal area law with improved temperature dependence.
While recent advances have established efficient quantum algorithms for preparing Gibbs states of finite-dimensional systems, comparable complexity results for bosonic and other infinite-dimensional models remain unexplored. We introduce the first general rigorous Gibbs sampling framework for bosonic many-body systems, showing that physically relevant bosonic models admit gapped dissipative generators, enabling efficient preparation of thermal states, provided that the spectral gap scales favourably with the number of modes. Although our results hold for broad classes of models, we illustrate them using Bose-Hubbard Hamiltonians, both within and beyond the mean-field regime. In both cases, we show that the associated dissipative generators maintain a positive spectral gap, thereby implying exponential convergence to the thermal state. For the Gibbs sampler corresponding to the full Bose-Hubbard model, we obtain a spectral-gap lower bound Ce−cnCe^{-cn}Ce−cn, where nnn is the number of modes. We apply our results to provide a Gibbs-state preparation algorithm on qubit hardware, with runtime polynomial in the number of modes and the inverse spectral gap, and thereby obtain a quantum algorithm to compute thermal properties of the model. This provides the first mathematically controlled route to Gibbs sampling in infinite-dimensional systems, with implications for quantum simulation, thermalization, and many-body complexity, where quantum advantages may arise.