Concentration of Measure Phenomena for Quantum States on a Higher Dimensional Equator

Abstract

We revisit Lévy's lemma, a widely used analytical tool in quantum information theory. Concentration inequalities quantify the phenomenon in which Lipschitz observables concentrate around a median or mean, and serve as fundamental analytical tools across information theory, statistical physics, and learning theory. In particular, Lévy's lemma provides a crucial framework for describing functionals on pure quantum states, with applications in quantum entanglement and quantum statistical query learning. In this work, we isolate the hyper-equatorial part of the standard spherical concentration argument. The resulting estimate is a Lévy-type bound for Lipschitz functions on a fixed hyperequator, with the natural dimension parameter $d-1$. We also formulate the accompanying geometric localization in terms of neighborhoods of the boundary, hyperequator, and a codimension-two antipodal great subsphere. This viewpoint clarifies the structure of the usual proof and points to the measure-theoretic formulation needed for sharper constant-level statements.