The presence of correlations in physical systems can be a valuable resource for many quantum information tasks. They are also relevant in thermodynamic transformations, and their creation is usually associated to some energetic cost. In this work, we study the role of correlations in the thermodynamic process of state formation in the single-shot regime, and find that correlations can also be viewed as a resource. First, we show that the energetic cost of creating multiple copies of a given state can be reduced by allowing correlations in the final state. We obtain the minimum cost for every finite number of subsystems, and then we show that this feature is not restricted to the case of copies. More generally, we demonstrate that in the asymptotic limit, by allowing a logarithmic amount of correlations, we can recover standard results where the free energy quantifies this minimum cost.
We study the selection of adjustment sets for estimating the interventional mean under a point exposure dynamic treatment regime, that is, a treatment rule that depends on the subject’s covariates. We assume a non-parametric causal graphical model with, possibly, hidden variables and at least one adjustment set comprised of observable variables. We provide the definition of a valid adjustment set for a point exposure dynamic treatment regime, which generalizes the existing definition for a static intervention. We show that there exists an adjustment set, referred to as optimal minimal, that yields the non-parametric estimator of the interventional mean with the smallest asymptotic variance among those that are based on observable minimal adjustment sets. An observable minimal adjustment set is a valid adjustment set such that all its variables are observable and the removal of any variable from it destroys validity. We provide similar optimality results for the class of observable minimum adjustment sets, that is, valid observable adjustment sets of minimum cardinality among the observable adjustment sets. Moreover, we show that if either no variables are hidden or if all the observable variables are ancestors of either treatment, outcome, or the variables that are used to decide treatment, a globally optimal adjustment set exists. We provide polynomial time algorithms to compute the globally optimal, optimal minimal, and optimal minimum adjustment sets. Because static interventions can be viewed as a special case of dynamic regimes, all our results also apply for static interventions.
Consider an i.i.d. sample from an unknown density function supported on an unknown manifold embedded in a high dimensional Euclidean space. We tackle the problem of learning a distance between points, able to capture both the geometry of the manifold and the underlying density. We prove the convergence of this microscopic distance, as the sample size goes to infinity, to a macroscopic one that we call Fermat distance as it minimizes a path functional, resembling Fermat principle in optics. The proof boils down to the study of geodesics in Euclidean first-passage percolation for nonhomogeneous Poisson point processes.
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