Localization of quantum states within subspaces

TL;DR

This paper defines a rigorous localization probability for quantum states within subspaces, offering a more restrictive measure than traditional overlap and exploring implications in quantum information and cryptography.

Contribution

It introduces a novel operator decomposition to quantify quantum state localization, with properties useful for quantum information and cryptography.

Findings

Defined a new localization probability for quantum states

Derived properties like concavity and super-additivity of the probability

Presented a cryptographic masking scheme based on the decomposition

Abstract

This work introduces a rigorous notion of localization probability of a quantum state within a given subspace of its Hilbert space. A non-negative operator A is uniquely decomposed as A=B+C, where B is the maximal positive operator supported inside the chosen subspace and C has support disjoint from it. The localized component B can be expressed via the Schur complement and characterized through an A-dependent inner product and suitable trace inequalities. For quantum states, this yields a probability lambda that a state rho be completely contained in a subspace, which is strictly more restrictive than the usual overlap probability Tr(P rho) and enjoys concavity and super-additivity properties. The resulting framework admits natural interpretations in quantum information, including entropic aspects and a simple cryptographic masking scheme based on the uniqueness of the decomposition.