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arXiv:2509.14727

Distances between pure quantum states induced by a distance matrix

Tomasz Miller, Rafał Bistroń

Published Sep 18, 2025• 0 citations
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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.

DOI
10.48550/arxiv.2509.14727

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Given an n×nn \times nn×n distance matrix, we construct an infinite family of distances dpd_pdp​ (where p≥2p \geq 2p≥2) on the complex projective space P(Cn)\mathbb{P}(\mathbb{C}^n)P(Cn) modelling the space of pure states of an nnn-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 dpd_pdp​ 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 dpd_pdp​ to mixed states is proposed for a broad class of distance matrices.
We introduce a new family of quantum distances based on symmetric Csiszár divergences, a class of distinguishability measures that encompass the main dissimilarity measures between probability distributions. We prove that these quantum distances can be obtained by optimizing over a set of quantum measurements followed by a purification process. Specifically, we address in the first place the case of distinguishing pure quantum states, solving an optimization of the symmetric Csiszár divergences over von Neumann measurements. In the second place, by making use of the concept of purification of quantum states, we arrive at a new set of distinguishability measures, which we call extended quantum Csiszár distances. In addition, as it has been demonstrated that a purification process can be physically implemented, the proposed distinguishability measures for quantum states could be endowed with an operational interpretation. Finally, by taking advantage of a well-known result for classical Csiszár divergences, we show how to build quantum Csiszár true distances. Thus, our main contribution is the development and analysis of a method for obtaining quantum distances satisfying the triangle inequality in the space of quantum states for Hilbert spaces of arbitrary dimension.
We prove that the square root of the separable quantum optimal transport cost associated with the orthogonal projection onto the antisymmetric subspace defines a genuine distance between density matrices. Equivalently, this establishes the triangle inequality for the order-two Beatty-França quantum optimal transport construction induced by the Hilbert-Schmidt distance between pure states. The result also proves metricity of the corresponding distance derived from separable SWAP fidelity. The proof replaces the unavailable gluing argument by convex-roof duality and a dimension-independent interpolation result for Hermitian operators.
Abstract Our purpose here is to draw attention to an interesting relationship that can be shown to hold between (a) information geometry, and (b) quantum geometry. By ‘information geometry’, we mean the natural geometry associated with families of probability distributions. This is a subject that goes back more than half a century to the pioneering work of Rao (1945), who showed that the Fisher information matrix associated with a parametrised family of probability dis tributions gives rise to a natural Riemannian metric on the parameter space. This then allows one to speak in a precise way of the ‘distance’ between two probability distributions, an idea that turns out to be useful in problems of statistical inference. An extensive literature has thus developed, following this line of enquiry, on applications of differential geometry to statistics (see, e.g., Amari (1985), Barndorff-Nielsen et al. (1986), Murray and Rice (1993)). By ‘quantum geometry’, on the other hand, we mean the geometry of the manifold of states associated with a given quantum mechanical system. In particular, we consider the manifold of ‘pure’ states associated with a given complex Hilbert space. This is the space of ‘rays’ through the origin in the Hilbert space, which has the structure of a complex projective space. It is known that the usual rules of quantum mechanics allow one to construct a natural metric on this space, namely the Fubini-Study metric, the specification of which is equivalent to all the familiar structure associated with ordinary quantum mechanics.
This paper presents an introduction to geometric representations of quantum states in which each distinct quantum state, pure and mixed, corresponds to a unique point in a Euclidean space. Beginning with a review of some underappreciated properties of the most commonly used geometric representation, the Bloch sphere visualization of qubit states, we show how concepts, algorithms, and spatial relations viewable on this geometric representation can be extended to representations of qudit states of any finite quantum dimension ddd and on to the infinite-dimensional limit. A primary goal of the work is helping the reader develop a visual intuition of these spaces, which can complement the understanding of the algebraic formalism of quantum mechanics for learners, teachers, and researchers at any level. Particular emphasis is given both to understanding states in a basis-independent way and to understanding how probability amplitudes and density matrix elements used to algebraically represent states in a particular basis correspond to line segments and angles in the geometric representations. In addition to providing visualizations for such concepts as superpositions, mixtures, decoherence, and measurement, we demonstrate how the representations can be used to substitute simple geometrical calculations for more cumbersome linear algebra ones, which may be of particular use in introducing mixed states and density matrices to beginning quantum students at an early stage. The work concludes with the geometrical interpretation of some commonly used metrics such as the purity of states and their relation to real, Euclidean vectors in the infinite-dimensional limit of the space, which contains all lower-dimensional qudit spaces as subspaces.
Topological and geometrical properties of the set of mixed quantum states in the N -dimensional Hilbert space are analysed. Assuming that the corresponding classical dynamics takes place on the sphere we use the vector SU (2) coherent states and the generalized Husimi distributions to define the Monge distance between two arbitrary density matrices. The Monge metric has a simple semiclassical interpretation and induces a non-trivial geometry. Among all pure states the distance from the maximally mixed state ρ * , proportional to the identity matrix, admits the largest value for the coherent states, while the delocalized `chaotic' states are close to ρ * . This contrasts the geometry induced by the standard (trace, Hilbert-Schmidt or Bures) metrics, for which the distance from ρ * is the same for all pure states. We discuss possible physical consequences including unitary time evolution and the process of decoherence. We introduce also a simplified Monge metric, defined in the space of pure quantum states and more suitable for numerical computation.
The totality of normalised density matrices of order N forms a convex set Q_N in R^(N^2-1). Working with the flat geometry induced by the Hilbert-Schmidt distance we consider images of orthogonal projections of Q_N onto a two-plane and show that they are similar to the numerical ranges of matrices of order N. For a matrix A of a order N one defines its numerical shadow as a probability distribution supported on its numerical range W(A), induced by the unitarily invariant Fubini-Study measure on the complex projective manifold CP^(N-1). We define generalized, mixed-states shadows of A and demonstrate their usefulness to analyse the structure of the set of quantum states and unitary dynamics therein.