# Conditional states and entropy in qudit-qubit systems

## TL;DR

This paper investigates the geometry of conditional qubit states resulting from local measurements on qudit-qubit systems, revealing complex shapes and providing analytic results for measurement-dependent entropy.

## Contribution

It introduces a detailed geometric analysis of post-measurement states in qudit-qubit systems and derives explicit formulas for the minimal conditional entropy.

## Key findings

- Conditional qubit states can form complex solid regions beyond ellipsoids.
- The set of post-measurement states can be convex hulls of ellipsoids, leading to cone-like and triangle-like shapes.
- Analytic expressions for the minimum conditional entropy and optimal measurements are provided.

## Abstract

We examine, in correlated mixed states of qudit-qubit systems, the set of all conditional qubit states that can be reached after local measurements at the qudit based on rank-1 projectors. While for a similar measurement at the qubit, the conditional post-measurement qudit states lie on the surface of an ellipsoid, for a measurement at the qudit we show that the set of post-measurement qubit states can form more complex solid regions. In particular, we show the emergence, for some classes of mixed states, of sets which are the convex hull of solid ellipsoids and which may lead to cone-like and triangle-like shapes in limit cases. We also analyze the associated measurement dependent conditional entropy, providing a full analytic determination of its minimum and of the minimizing local measurement at the qudit for the previous states. Separable rank-2 mixtures are also discussed.