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.
Given a compact hyperbolic surface X=Γ\ℍ^2 and a linear non-unitary representation ϱ :Γ→GL(V) , we investigate the spectrum of the twisted Laplacian Δ _ϱ acting on sections of the associated flat vector bundle E_ϱ→ X . We show that this non self-adjoint operator has its spectrum inside a parabola related to a critical exponent δ of the representation. In the case where ϱ is of Teichmüller type we then exhibit explicit parabolic regions determined by another constant δ _0 with an asymptotic spectral density which improves Weyl’s law by a power factor. Both δ and δ _0 are uniquely determined by the so-called “Manhattan curve” related to the representation ϱ .
Information stored in quantum states naturally decays under noise. Many earlier works have quantified and demonstrated lower bounds on decay rates of quantum relative entropy, showing exponential decay in a wide variety of contexts. Here we study the converse question: are there uniform upper bounds on the ratio of post-noise to initial information when noise is sufficiently weak? In several scenarios, including when a quantum channel induces decay of relative entropy to its own fixed point, we find such multiplicative converse bounds. However, there are other cases in which the ratio can fall by an arbitrary extent. Even for simple noise such as qubit dephasing or depolarizing, mutual information may fall by an unbounded factor under arbitrarily weak noise. As an application, we find families of channels with non-zero private capacity despite arbitrarily high probability of transmitting an arbitrarily good copy of the input to the environment.
