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.