Combined-probability space and certainty or uncertainty relations for a finite-level quantum system

Abstract

The Born rule provides a probability vector (distribution) with a quantum state for a measurement setting. For two settings, we have a pair of vectors from the same quantum state. Each pair forms a combined-probability vector that obeys certain quantum constraints, which are triangle inequalities in our case. Such a restricted set of combined vectors, titled combined-probability space, is presented here for a $d$-level quantum system (qudit). The combined space turns out a compact convex subset of a Euclidean space, and all its extreme points come from a family of parametric curves. Considering a suitable concave function on the combined space to estimate the uncertainty, we deliver an uncertainty relation by finding its global minimum at the curves for a qudit. If one chooses an appropriate concave (or convex) function, then there is no need to search for the absolute minimum (maximum) on the whole space, it will be at the parametric curves. So these curves are quite useful for establishing an uncertainty (or a certainty) relation for a general pair of settings. In the paper, we also demonstrate that many known tight (un)certainty relations for a qubit can be obtained with the triangle inequalities.