# Coherence transformations in single qubit systems

## TL;DR

This paper explores how various coherence-free operations transform single qubits, establishing hierarchical relationships, providing explicit transformation conditions, and illustrating these processes geometrically on the Bloch sphere.

## Contribution

It offers a comprehensive analysis of coherence-free operations in single qubits, including hierarchical relations, explicit transformation criteria, and geometric visualization methods.

## Key findings

- Hierarchical relationship: CPO ⊂ PIO ⊂ SIO=IO
- Explicit conditions for single qubit transformations via IO and SIO
- Geometric Bloch sphere depiction of transformation regions

## Abstract

We investigate the single qubit transformations under several typical coherence-free operations, such as, incoherent operation (IO), strictly incoherent operation (SIO), physically incoherent operation (PIO), and coherence preserving operation (CPO). Quantitative connection has been built between IO and SIO in single qubit systems. Moreover, these coherence-free operations have a clear hierarchical relationship in single qubit systems: CPO $\subset$ PIO $\subset$ SIO=IO. A new and explicit proof for the necessary and sufficient condition of single qubit transformation via IO or SIO has been provided, which indicates that SIO with only two Kraus operators are enough to realize this transformation. The transformation regions of single qubits via CPO and PIO are also given. Our method provides a geometric illustration to analyze single qubit coherence transformations by introducing the Bloch sphere depiction of the transformation regions, and tells us how to construct the corresponding coherence-free operations.