# A quantum version of Wielandt's inequality

## TL;DR

This paper extends Wielandt's inequality to quantum channels, providing bounds on their behavior and implications for quantum information capacity and Hamiltonian ground states.

## Contribution

It introduces a quantum version of Wielandt's inequality, establishing bounds on the application of quantum channels and deriving new results in quantum information and Hamiltonian ground states.

## Key findings

- Bound on the number of channel applications to reach full rank
- Dichotomy theorems for quantum zero-error capacity
- Bounds on interaction-range of Hamiltonians with MPS ground states

## Abstract

In this paper, Wielandt's inequality for classical channels is extended to quantum channels. That is, an upper bound to the number of times a channel must be applied, so that it maps any density operator to one with full rank, is found. Using this bound, dichotomy theorems for the zero--error capacity of quantum channels and for the Matrix Product State (MPS) dimension of ground states of frustration-free Hamiltonians are derived. The obtained inequalities also imply new bounds on the required interaction-range of Hamiltonians with unique MPS ground state.