Improved Uncertainty Relation in the Presence of Quantum Memory

TL;DR

This paper generalizes the entropic uncertainty relation with quantum memory, providing tighter bounds based on all overlaps between measurements, which enhances applications in quantum cryptography.

Contribution

It introduces a new entropic uncertainty relation that accounts for all overlaps, improving bounds in the presence of quantum memory over previous results.

Findings

Derived exact dependence on all overlaps between measurements.

Established tighter entropic bounds than previous work.

Applied results to quantum cryptography scenarios.

Abstract

In Coles-Piani's recent remarkable version of the entropic uncertainty principle, the entropic sum is controlled by the first and second maximum overlaps between the two projective measurements. We generalize the entropic uncertainty relation in the presence of quantum memory and find the exact dependence on all $d$ largest overlaps between two measurements on any $d$-dimensional Hilbert space. The corresponding entropic uncertainty principle in the absence of quantum memory is also derived. Our bounds are strictly tighter than previous entropic bounds in the presence of quantum memory, which have potential applications to quantum cryptography with entanglement witnesses and quantum key distributions.