Distances between pure quantum states induced by a distance matrix

Abstract:Given an $n \times n$ distance matrix, we construct an infinite family of distances $d_p$ (where $p \geq 2$) on the complex projective space $\mathbb{P}(\mathbb{C}^n)$ modelling the space of pure states of an $n$-level quantum system. The construction can be seen as providing a natural way to isometrically embed any given finite metric space into the space of pure quantum states 'spanned' upon it. In order to show that the maps $d_p$ are indeed distance functions -- in particular, that they satisfy the triangle inequality -- we employ methods of analysis, multilinear algebra and convex geometry, obtaining a nontrivial auxiliary convexity result in the process. In addition, a way of extending the distances $d_p$ to mixed states is proposed for a broad class of distance matrices.

arXiv:2509.14727 [math-ph]

(or arXiv:2509.14727v4 [math-ph] for this version)

https://doi.org/10.48550/arXiv.2509.14727 Focus to learn more arXiv-issued DOI via DataCite

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