From minimal informationally complete measurements to orthocentric simplices and back again

Abstract

The reconstruction of unknown quantum states via minimal informationally complete measurements (MICs) is a cornerstone of quantum tomography. Although the statistical properties of these measurements are well-understood, their geometric structure has remained elusive. In this work, we establish a correspondence between the class of minimal $s$-tight informationally complete measurements, encompassing, among others, tight IC and morphophoric measurements, and the classical geometry of orthocentric simplices. In particular, we prove a three-way equivalence: a MIC is $s$-tight if and only if its measurement vectors, upon suitable rescaling, form the vertices of an acute orthocentric simplex with the orthocentre at the origin, and such simplices are precisely the homothetically self-dual ones. This geometric manifestation of operational ''tightness'' provides a bridge between the physical world and Euclidean geometry. Furthermore, the $s$-tight class is fully characterised by its measurement directions: the angles between them must be obtuse and satisfy a cross-ratio condition. We determine the space of admissible direction configurations: modulo rotations, every such configuration is encoded by a single probability vector, the ''skeleton'' of the measurement, together with an orientation class, so that the moduli space of $s$-tight MIC directions is $Δ^{\circ}_{d+1}\times\{\pm 1\}$. Conversely, every acute orthocentric simplex with the orthocentre at the origin can be anchored in the state space, generating a class of minimal $s$-tight IC measurements that contains exactly one tight IC measurement up to overall rescaling.

The paper asks how frame-theoretic tightness of minimal informationally complete measurements is reflected in Euclidean geometry. It works in finite-dimensional geometric generalised probabilistic theories, representing measurements by vectors $\psi_j\in S^\star$ and positive probabilities $c_j$ satisfying $\sum_jc_j\psi_j=0$. A MIC has $d+1$ outcomes, where $d$ is the state-space dimension; quantum systems of Hilbert-space dimension $D$ have $d=D^2-1$. The main result is a three-way equivalence: a MIC is $s$-tight—its vectors can be positively rescaled into a tight frame—if and only if, for a unique probability vector $p$, the vectors $(c_j/p_j)\psi_j$ form an acute orthocentric simplex with orthocentre at zero, equivalently a simplex homothetic to its negative polar. Tight IC measurements require no rescaling, while morphophoric MICs yield regular simplices. The proof combines tight-frame identities, barycentric coordinates and polar geometry, providing explicit Gram matrices and rescaling formulas. A direction-only criterion requires strictly negative pairwise inner products and the cross-ratio identities $\langle\psi_i,\psi_j\rangle\langle\psi_k,\psi_l\rangle=\langle\psi_i,\psi_l\rangle\langle\psi_k,\psi_j\rangle$ for distinct indices. The paper sketches a classification of labelled directions modulo rotations by $\Delta_{d+1}^{\circ}\times\{\pm1\}$, with full details deferred. Every such simplex generates physical measurements after sufficiently small scaling, with exactly one tight IC family up to common scaling. The results concern minimal measurements and a fixed inner product. Nonminimal geometric analogues, stronger quantum realisability questions related to SICs, and compatible state-update instruments remain open. No SIC-existence result or tomography-performance bound is established.