# Uncertainty under Quantum Measures and Quantum Memory

## TL;DR

This paper explores how quantum memory affects entropic uncertainty relations, deriving new bounds and generalizations that improve understanding of quantum correlations and their applications in entanglement detection.

## Contribution

It introduces generalized entropic uncertainty bounds incorporating quantum memory, extending previous bounds and providing stronger relations for multiple measurements.

## Key findings

- Derived new bounds for entropic uncertainty with quantum memory.
- Generalized existing bounds to include quantum side information.
- Discussed applications in entanglement witnessing.

## Abstract

The uncertainty principle restricts potential information one gains about physical properties of the measured particle. However, if the particle is prepared in entanglement with a quantum memory, the corresponding entropic uncertainty relation will vary. Based on the knowledge of correlations between the measured particle and quantum memory, we have investigated the entropic uncertainty relations for two and multiple measurements, and generalized the lower bounds on the sum of Shannon entropies without quantum side information to those that allow quantum memory. In particular, we have obtained generalization of Kaniewski-Tomamichel-Wehner's bound for effective measures and majorization bounds for noneffective measures to allow quantum side information. Furthermore, we have derived several strong bounds for the entropic uncertainty relations in the presence of quantum memory for two and multiple measurements. Finally, potential applications of our results to entanglement witnesses are discussed via the entropic uncertainty relation in the absence of quantum memory.