Dynamical error bounds for continuum discretisation via Gauss quadrature rules—A Lieb-Robinson bound approach

Abstract

Instances of discrete quantum systems coupled to a continuum of oscillators are ubiquitous in physics. Often the continua are approximated by a discreate set of modes. We derive error bounds on expectation values of system observables that have been time evolved under such discretised Hamiltonians. These bounds take on the form of a function of time and the number of discrete modes, where the discrete modes are chosen according to Gauss quadrature rules. The derivation makes use of tools from the field of Lieb-Robinson bounds and the theory of orthonormal polynominals. 1Instances of discrete quantum systems coupled to continua are ubiquitous in physics as they describe open quantum systems, i.e. well-characterised systems under the control of the experimenter that are in contact with a much larger and typically uncontrolled environment 1 . Examples can be found in quantum optics 2 , solid state and condensed matter physics 3 and recently quantum biology 4,5 to name just a few. In numerical studies environments with continuous spectra are often modeled by a discrete spectrum while in analytical work the reverse, i.e. replacing a discrete environmental spectrum by a continuous one in a "continuum limit" 3 , is often convenient. There have been many suggestions about how to best approximate continuous spectra by discrete spectra for the evaluation of dynamical quantities and numerical studies into their efficiency. These studies were, to the best of our knowledge, initiated by Rice in 1929 [6][7][8] in which the continuum was discretised to form a point spectrum with support at equally spaced points. Later, it was suggested by Burkey and Cantrell 9 that a different choice of discretisation would lead to a more accurate description of the dynamics. This later idea was based on the fact that approximating integrals by discrete sums using Gauss quadrature rules is often more efficient than the trapezoidal rule. A bibliographical review of the subject can be found in 10 . While sometimes estimates to some of the committed errors are given 9,11 , exact bounds for the quantitywhereÔ is an observable,̺ 0 is any initial state of potential interest,Ĥ con is a Hamiltonian with absolutely continuous spectra andĤ dis is a Hamiltonian with pure point spectra, do not appear to exist in the literature. Bounds of this form are of particular relevance because they concern precisely the quantities of physical interest -the expectation values of local observables. In this article, we will derive bounds on the quantity Eq. (1) for physically relevant unbounded Hamiltonians, for discretisation schemes based on Gauss quadrature rules.In a different vein of research, in the context of lattice quantum systems, a bound introduced More precisely, the Lieb-Robinson bound states that there is a velocity v > 0 and constants µ > 0 and K > 0, which depend on the details of the lattice and the Hamiltonian, such that the operator norm · of the commutator of two local observables andB, separated by a distance |x|, is bounded byOften, the developme...

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