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Triangle inequality conjecture for distance-matrix-induced pure-state distances

Let n≥2n\geq 2. Let (Eij)(E_{ij}) be an n×nn\times n distance matrix, and let P(Cn)\mathbb{P}(\mathbb{C}^n) denote the pure-state projective space. For pure states x,y∈P(Cn)\textup{\textbf{x}},\textup{\textbf{y}}\in\mathbb{P}(\mathbb{C}^n), define

d2(x,y)=(∑i<jEij2∣xiyj−xjyi∣2)1/2.d_{2}(\textup{\textbf{x}},\textup{\textbf{y}})=\left(\sum_{i<j}E_{ij}^2|x_i y_j-x_j y_i|^2\right)^{1/2}.

Triangle inequality conjecture. For any pure states x,y,z∈P(Cn)\textup{\textbf{x}},\textup{\textbf{y}},\textup{\textbf{z}}\in\mathbb{P}(\mathbb{C}^n),

d2(x,z)≤d2(x,y)+d2(y,z).d_{2}(\textup{\textbf{x}},\textup{\textbf{z}})\leq d_{2}(\textup{\textbf{x}},\textup{\textbf{y}})+d_{2}(\textup{\textbf{y}},\textup{\textbf{z}}).

The function is already nondegenerate and symmetric by construction, so this conjecture addresses the remaining condition for d2d_2 to be a distance. Its validity is presented as a central challenge for the general class of distances induced by distance matrices.

Sources & referencesView supporting material

Primary source

Tomasz Miller and Rafał Bistroń, “Distances between pure quantum states induced by a distance matrix”, arXiv:2509.14727 (2026).

distance-matricesmetric-geometryquantum-information

Progress summary

Refreshed
Claimed solved

A 2025 preprint claims to prove the conjecture in full, but no independent verification has been found.

The problem was formulated in 2023 as a conjecture about distances on complex projective space. The general case was left open after partial results for special distance matrices.

Known results

  • Rank-one distance matrices: the triangle inequality was previously known.
  • Euclidean distance matrices: the 2023 paper proves the triangle inequality.
  • It suffices to establish the inequality for triples of orthonormal vectors.

September 2025 claimed proof

A preprint claims a stronger theorem: for every distance matrix and every p≥2p\geq 2, the associated dpd_p satisfies the triangle inequality on P(Cn)\mathbb{P}(\mathbb{C}^n). Its p=2p=2 case would resolve this conjecture. The claim is not independently verified in the retrieved sources; no counterexample, withdrawal, or reported gap was found.

Current status (as of August 2026): the conjecture is claimed solved by the 2025 preprint, but independent verification is absent, so its general validity remains unconfirmed.

Sources

Solutions 0

No solutions have been posted yet.