# General symmetry in the reduced dynamics of two-level system

## TL;DR

This paper investigates general transformations on two-level quantum systems that preserve observable expectations, introducing a set of generators that produce non-completely positive, trace-preserving maps capable of rotating, dilating, or translating the Bloch vector, with applications to amplitude and phase damping.

## Contribution

It introduces a comprehensive set of generators for invariant-preserving transformations on two-level systems, extending to higher levels and analyzing their properties and effects.

## Key findings

- Transformations can rotate, dilate, or translate the Bloch vector.
- Transformations are generally not factorized or completely positive.
- Application to symmetry analysis of amplitude and phase damping.

## Abstract

We study general transformation on the density matrix of two-level system that keeps the expectation value of observable invariant. We introduce a set of generators that yields hermiticity and trace preserving general transformation which casts the transformation into simple form. The general transformation is in general not factorized and not completely positive. Consequently, either the parameter of transformation or the density matrix it acts on needs to be restricted. It can transform the system in the forward and backward direction with regard to its parameter, not as a semigroup in the time translation symmetry of dynamical maps. The general transformation can rotate the Bloch vector circularly or hyperbolically, dilate it or translate it. We apply the general transformation to study the general symmetry of amplitude damping and phase damping in two-level system. We generalize the generators to higher level systems.