# Unified quantum no-go theorems and transforming of quantum states in a restricted set

## TL;DR

This paper unifies various quantum no-go theorems through a general framework based on the superposition principle, establishing forbidden transformations, and providing a scheme for perfect and imperfect quantum tasks with success bounds.

## Contribution

It introduces a unified approach to quantum no-go theorems, including a no-encoding theorem, and presents a scheme for quantum tasks with success probability bounds.

## Key findings

- Proves a no-encoding theorem forbidding superposition of unknown and fixed states.
- Provides a unified scheme for quantum cloning and deleting tasks.
- Derives upper bounds for success probabilities in quantum state transformations.

## Abstract

The linear superposition principle in quantum mechanics is essential for several no-go theorems such as the no-cloning theorem, the no-deleting theorem and the no-superposing theorem. It remains an open problem of finding general forbidden principles to unify these results. In this paper, we investigate general quantum transformations forbidden or permitted by the superposition principle for various goals. First, we prove a no-encoding theorem that forbids linearly superposing of an unknown pure state and a fixed state in Hilbert space of finite dimension. Two general forms include the no-cloning theorem, the no-deleting theorem, and the no-superposing theorem as special cases. Second, we provide a unified scheme for presenting perfect and imperfect quantum tasks (cloning and deleting) in a one-shot manner. This scheme may yield to fruitful results that are completely characterized with the linear independence of the input pure states. The generalized upper bounds for the success probability will be proved. Third, we generalize a recent superposing of unknown states with fixed overlaps when multiple copies of the input states are available.