Symmetric orthogonalization and probabilistic weights in resource quantification

Abstract

Transforming non-orthogonal bases into orthogonal ones often compromises essential properties or physical meaning in quantum systems. Here, we demonstrate that L\"owdin symmetric orthogonalization (LSO) outperforms the widely used Gram-Schmidt orthogonalization (GSO) in characterizing and quantifying quantum resources, with particular emphasis on coherence and superposition. We employ LSO both to construct an orthogonal basis from a non-orthogonal one and to obtain a non-orthogonal basis from an orthogonal set, thereby mitigating ambiguity related to the basis choice in defining quantum coherence. Unlike GSO, which depends on the ordering of input states, LSO applies a symmetric transformation that treats all vectors equally and minimizes deviation from the original basis. This procedure yields basis sets with enhanced stability, preserving the closest possible correspondence to the original physical states while satisfying orthogonality. Building on LSO, we also introduce L\"owdin weights -- probabilistic weights for non-orthogonal representations that provide a consistent measure of resource content. We explicitly contrast these with Chirgwin-Coulson weights, demonstrating that L\"owdin weights ensure non-negativity, a prerequisite for information-theoretic measures. These weights further enable the quantification of coherence and the characterization of superposition, providing a degree of superposition as a distinct measure, as well as facilitating the assessment of state delocalization through entropy and participation ratios. Our theoretical and numerical analyses confirm LSO's superior preservation of quantum state symmetry and resource characteristics, underscoring the critical role of orthogonalization methods and L\"owdin weights in resource theory frameworks involving non-orthogonal bases.