Triangle inequality for the distance d₂ on the complex projective space induced by a distance matrix

Prove that for any integer n ≥ 2 and any n × n distance matrix (E_{ij}), the function d₂: P × P → ℝ defined on the complex projective space P by d₂(x, y) = (∑_{i<j} E_{ij}^2 |x_i y_j − x_j y_i|^2)^{1/2} satisfies the triangle inequality; specifically, show that for all pure states x, y, z ∈ P, d₂(x, y) ≤ d₂(x, z) + d₂(z, y).

Background

In earlier work [llaproc], the authors introduced a general class of distances on the complex projective space P, representing pure states of an n-dimensional quantum system, constructed from any given n × n distance matrix (E_{ij}). The specific case d₂(x, y) = (∑{i<j} E{ij}2 |x_i y_j − x_j y_i|2){1/2} arises naturally in quantum optimal transport and generalizes the Hilbert–Schmidt metric.

The central challenge is to establish that d₂ is a bona fide metric by showing it satisfies the triangle inequality for all choices of (E_{ij}). This was formulated as Conjecture 2.1 in [llaproc]; the present paper proves a stronger result for d_p with p ≥ 2, thereby resolving the conjecture.

References

In , we formulated this central challenge as a conjecture. Let $n \geq 2$. For any $n \times n$ distance matrix $(E_{ij})$ the function $d_{2} : P P \to $ defined by eq:old_dist satisfies the triangle inequality. That is, for any pure states $, , \in P$, d_{2}(,) \leq d_{2}(,) + d_{2}(,).

eq:old_dist:

d2(,)≔∥(∧)∥=(∑i<jEij2∣xiyj−xjyi∣2)1/2,d_{2}(,) \coloneqq \|( \wedge )\| = \Big(\sum\limits_{i<j} E^2_{ij}|x_iy_j - x_jy_i|^2 \Big)^{1/2},

— Distances between pure quantum states induced by a distance matrix  (2509.14727 - Miller et al., 18 Sep 2025) in Conjecture (Conjecture 2.1 from [llaproc]), Section 1: Introduction and main result

The above reasoning seems to be specific to the quadratic cost in an essential way: the proof of Lemma~\ref{lem:interpolation} uses a Hermitian block matrix, Schur complements, and resolvents of positive operators. It does not by itself settle the corresponding metricity question for other orders $p$ or for general metrics on projective space. Nevertheless, Lemma~\ref{lem:interpolation} suggests a broader strategy for such problems: replace primal gluing by an operator-valued interpolation of Hermitian operators. Whether an analogous result holds beyond the quadratic setting or beyond the underlying Hilbert--Schmidt pure-state distance remains open.

— Metricity of separable quantum optimal transport with the Hilbert-Schmidt cost  (2609.03769 - Miller, 3 Sep 2026) in Section 7, “Equivalent formulations and outlook”