Thermal Area Law for Lattice Bosons

Marius Lemm and Oliver Siebert

Department of Mathematics, University of Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany

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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.

In the realm of quantum many-body and quantum information theory a central problem lies in the determination of entanglement and correlation for systems of many interacting quantum particles. A system at zero temperature that is weakly entangled satisfies a so-called area law. This means that for any partition into two subsystems, the entanglement between the two subsystems grows at most proportional to the size of their mutual boundary region, in contrast to a trivial bound with respect to the subsystems' volume. At positive temperature, which is a more realistic assumption for applications, one speaks of a thermal area law if the mutual information of the equilibrium (Gibbs) state satisfies a similar property.

Thermal area laws control the amount of quantum entanglement and correlation. Consequently, they are expected to be potentially very useful in practice for the efficient approximation of Gibbs states by tensor network states (matrix product operators) with fixed bond dimension. The complexity of such states grows only polynomially and not exponentially in the system size, which is crucial for any numerical algorithm.

While earlier works established thermal area laws for quantum spin systems, our paper presents the first thermal area law for bosonic lattice systems. Our proof is technically more involved than in the spin case due to unbounded interactions and infinite-dimensional Hilbert spaces, which require special trace inequalities. Notably, our findings encompass the Bose-Hubbard model, a versatile description of optical lattices hosting cold neutral atoms trapped in laser-induced interference patterns. This model's fine tunability holds promising applications in quantum simulation and computation.

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