Strong convergence rates of probabilistic integrators for ordinary differential equations

Probabilistic integration of a continuous dynamical system is a way of systematically introducing model error, at scales no larger than errors introduced by standard numerical discretisation, in order to enable thorough exploration of possible responses of the system to inputs. It is thus a potentially useful approach in a number of applications such as forward uncertainty quantification, inverse problems, and data assimilation. We extend the convergence analysis of probabilistic integrators for deterministic ordinary differential equations, as proposed by Conrad et al.\ (\textit{Stat.\ Comput.}, 2017), to establish mean-square convergence in the uniform norm on discrete- or continuous-time solutions under relaxed regularity assumptions on the driving vector fields and their induced flows. Specifically, we show that randomised high-order integrators for globally Lipschitz flows and randomised Euler integrators for dissipative vector fields with polynomially-bounded local Lipschitz constants all have the same mean-square convergence rate as their deterministic counterparts, provided that the variance of the integration noise is not of higher order than the corresponding deterministic integrator. These and similar results are proven for probabilistic integrators where the random perturbations may be state-dependent, non-Gaussian, or non-centred random variables.Comment: 25 page

Intersubband transitions in pseudomorphic InGaAs/GaAs/AlGaAs multiple step quantum wells

Intersubband transitions from the ground state to the first and second excited states in pseudomorphic AlGaAs/InGaAs/GaAs/AlGaAs multiple step quantum wells have been observed. The step well structure has a configuration of two AlGaAs barriers confining an InGaAs/GaAs step. Multiple step wells were grown on GaAs substrate with each InGaAs layer compressively strained. During the growth, a uniform growth condition was adopted so that inconvenient long growth interruptions and fast temperature ramps when switching the materials were eliminated. The sample was examined by cross‐sectional transmission electron microscopy, an x‐ray rocking curve technique, and the results show good crystal quality using this simple growth method. Theoretical calculations were performed to fit the intersubband absorption spectrum. The calculated energies are in good agreement with the observed peak positions for both the 1→2 and 1→3 transitions

Invariants of differential equations defined by vector fields

We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the second order. A result on the characterization of classes of these equations by the invariant functions is also given.Comment: 13 page

Simultanagnosia: When a Rose Is Not Red

Information regarding object identity (‘‘what’’) and spatial location (‘‘where/how to’’) is largely segregated in visual processing. Under most circumstances, however, object identity and location are linked. We report data from a simultanagnosic patient (K.E.) with bilateral posterior parietal infarcts who was unable to ‘‘see’’ more than one object in an array despite relatively preserved object processing and normal preattentive processing. K.E. also demonstrated a finding that has not, to our knowledge, been reported: He was unable to report more than one attribute of a single object. For example, he was unable to name the color of the ink in which words were written despite naming the word correctly. Several experiments demonstrated, however, that perceptual attributes that he was unable to report influenced his performance. We suggest that binding of object identity and location is a limited-capacity operation that is essential for conscious awareness for which the posterior parietal lobe is crucial

Ordinary differential equations which linearize on differentiation

In this short note we discuss ordinary differential equations which linearize upon one (or more) differentiations. Although the subject is fairly elementary, equations of this type arise naturally in the context of integrable systems.Comment: 9 page

Catalysis always degrades external quantum correlations

Catalysts used in quantum resource theories need not be in isolation and therefore are possibly correlated with external systems, which the agent does not have access to. Do such correlations help or hinder catalysis, and does the classicality or quantumness of such correlations matter? To answer this question, we first focus on the existence of a non-invasively measurable observable that yields the same outcomes for repeated measurements, since this signifies macro-realism, a key property distinguishing classical systems from quantum systems. We show that a system quantumly correlated with an external system so that the joint state is necessarily perturbed by any repeatable quantum measurement, also has the same property against general quantum channels. Our full characterization of such systems called totally quantum systems, solves the open problem of characterizing tomographically sensitive systems raised in [Lie and Jeong, Phys. Rev. Lett. 130, 020802 (2023)]. An immediate consequence is that a totally quantum system cannot catalyze any quantum process, even when a measure of correlation with its environment is arbitrarily low. It generalizes to a stronger result, that the mutual information of totally quantum systems cannot be used as a catalyst either. These results culminate in the conclusion that, out of the correlations that a generic quantum catalyst has with its environment, only classical correlations allow for catalysis, and therefore using a correlated catalyst is equivalent to using an ensemble of uncorrelated catalysts.Comment: 5+7 pages, 1 figure, Comments are welcom

Uniqueness of quantum state over time function

A fundamental asymmetry exists within the conventional framework of quantum theory between space and time, in terms of representing causal relations via quantum channels and acausal relations via multipartite quantum states. Such a distinction does not exist in classical probability theory. In effort to introduce this symmetry to quantum theory, a new framework has recently been proposed, such that dynamical description of a quantum system can be encapsulated by a static quantum state over time. In particular, Fullwood and Parzygnat recently proposed the state over time function based on the Jordan product as a promising candidate for such a quantum state over time function, by showing that it satisfies all the axioms required in the no-go result by Horsman et al. However, it was unclear if the axioms induce a unique state over time function. In this work, we demonstrate that the previously proposed axioms cannot yield a unique state over time function. In response, we therefore propose an alternative set of axioms that is operationally motivated, and better suited to describe quantum states over any spacetime regions beyond two points. By doing so, we establish the Fullwood-Parzygnat state over time function as the essentially unique function satisfying all these operational axioms.Comment: 5+4 pages, comments welcom