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arXiv · 2509.14727

Distances between pure quantum states induced by a distance matrix

Tomasz Miller·Rafał Bistroń
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Abstract

With the help of a given distance matrix of size $n$, 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 distances $d_p$ onto mixed states is proposed for a broad class of distance matrices.

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Tomasz Miller, Rafał Bistroń. 2025-09-18. Distances between pure quantum states induced by a distance matrix. https://arxiv.org/abs/2509.14727

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