By extending the concept of energy-constrained diamond norms, we obtain continuity bounds on the dynamics of both closed and open quantum systems in infinite-dimensions, which are stronger than previously known bounds. We extensively discuss applications of our theory to quantum speed limits, attenuator and amplifier channels, the quantum Boltzmann equation, and quantum Brownian motion. Next, we obtain explicit log-Lipschitz continuity bounds for entropies of infinitedimensional quantum systems, and classical capacities of infinite-dimensional quantum channels under energy-constraints. These bounds are determined by the high energy spectrum of the underlying Hamiltonian and can be evaluated using Weyl's law.1 T * t is the adjoint of T t with respect to the Hilbert-Schmidt inner product.1 arXiv:1810.00863v3 [quant-ph]
In this article we study the Bistritzer-MacDonald (BM) model with external magnetic field. We study the spectral properties of the Hamiltonian in an external magnetic field with a particular emphasis on the flat band of the chiral model at magic angles. Our analysis includes different types of interlayer tunneling potentials, the so-called chiral and anti-chiral limits. One novelty of our article is that we show that using a magnetic field one can discriminate between flat bands of different multiplicities, as they lead to different Chern numbers in the presence of magnetic fields, while for zero magnetic field their Chern numbers always coincide.
Along the ideas of Curtain and Glover [CG86], we extend the balanced truncation method for infinite-dimensional linear systems to arbitrary-dimensional bilinear and stochastic systems. In particular, we apply Hilbert space techniques used in many-body quantum mechanics to establish new fully explicit error bounds for the truncated system and prove convergence results. The functional analytic setting allows us to obtain mixed Hardy space error bounds for both finite-and infinitedimensional systems, and it is then applied to the model reduction of stochastic evolution equations driven by Wiener noise.
We consider a quantum graph as a model of graphene in magnetic fields and give a complete analysis of the spectrum, for all constant fluxes. In particular, we show that if the reduced magnetic flux Φ/2π through a honeycomb is irrational, the continuous spectrum is an unbounded Cantor set of Lebesgue measure zero.
We prove a variety of new and refined uniform continuity bounds for entropies of both classical random variables on an infinite state space and of quantum states of infinite-dimensional systems. We obtain the first tight continuity estimate on the Shannon entropy of random variables with a countably infinite alphabet. The proof relies on a new mean-constrained Fano-type inequality and the notion of maximal coupling of random variables. We then employ this classical result to derive the first tight energy-constrained continuity bound for the von Neumann entropy of states of infinite-dimensional quantum systems, when the Hamiltonian is the number operator, which is arguably the most relevant Hamiltonian in the study of infinitedimensional quantum systems in the context of quantum information theory. The above scheme works only for Shannon-and von Neumann entropies. Hence, to deal with more general entropies, e.g. α-Rényi and α-Tsallis entropies, with α ∈ (0, 1), for which continuity bounds are known only for finite-dimensional systems, we develop a novel approximation scheme which relies on recent results on operator Hölder continuous functions. This approach is, as we show, motivated by continuity bounds for α-Rényi and α-Tsallis entropies of random variables that follow from the Hölder continuity of the entropy functionals. We also provide bounds for α > 1. Finally, we settle an open problem on related approximation questions posed in the recent works by Shirokov [1, 2] on the so-called Finite-dimensional Approximation (FA) property.
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