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.

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.