# The Measurement-Disturbance Relation and the Disturbance Trade-off Relation in Terms of Relative Entropy

## TL;DR

This paper uses quantum relative entropy to formulate measurement-disturbance and trade-off relations for incompatible observables, revealing how quantum memory influences these relations and extending them to multiple measurements.

## Contribution

It introduces new measurement-disturbance and trade-off relations based on quantum relative entropy, accounting for quantum memory and multiple measurements, advancing understanding of quantum measurement effects.

## Key findings

- Without quantum memory, disturbance exceeds measurement uncertainty.
- With quantum memory, relations depend on conditional entropy.
- Relations are demonstrated through two illustrative examples.

## Abstract

We employ quantum relative entropy to establish the relation between the measurement uncertainty and its disturbance on a state in the presence (and absence) of quantum memory. For two incompatible observables, we present the measurement-disturbance relation and the disturbance trade-off relation. We find that without quantum memory the disturbance induced by the measurement is never less than the measurement uncertainty and with quantum memory they depend on the conditional entropy of the measured state. We also generalize these relations to the case with multiple measurements. These relations are demonstrated by two examples.