# Detecting non-Markovianity via uncertainty relations

## TL;DR

This paper introduces a formalism using uncertainty relations to detect non-Markovian dynamics in quantum systems, linking negative eigenvalues of Choi-states to the breakdown of uncertainty relations.

## Contribution

It proposes a novel method to identify non-Markovianity via uncertainty relation violations and introduces a non-linear witness based on the variance of hermitian operators.

## Key findings

- Uncertainty relations break down when non-Markovianity is present.
- Negative eigenvalues in Choi-states indicate information back-flow.
- Variance of certain operators can serve as a non-Markovianity witness.

## Abstract

We present a formalism for detection of non-Markovianity through uncertainty relations. We show that when there is an information back-flow to the system from its environment through CP-divisibility breaking, the Choi-states corresponding to the reduced system evolution contain at least one negative eigenvalue. The consequent break down of uncertainty relations for such states can be used to witness non-Markovian dynamics. We present some relevant examples of the phenomenon for qubit channels. We further prove that square of the variance of a suitable hermitian operator can act as a non-linear witness of non-Markovianity. We finally show that non-Markovianity is necessary in order to decrease the uncertainty of the states undergoing unital dynamics for qubits. This provides another method of certifying non-Markovianity.