# A combined-probability space and (un)certainty relations for a finite-level quantum system

## TL;DR

This paper introduces a new geometric framework called combined-probability space for finite-level quantum systems, deriving uncertainty relations using convex analysis and parametric curves, unifying and extending known results.

## Contribution

It develops a novel combined-probability space for qudits, characterizes its extreme points, and derives uncertainty relations without exhaustive search by focusing on parametric curves.

## Key findings

- The combined-probability space is a compact convex set with extreme points on parametric curves.
- Uncertainty relations are obtained by minimizing concave functions on these curves.
- Many known tight (un)certainty relations for qubits are recovered through triangle inequalities.

## 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.