Extended quantum conditional entropy and quantum uncertainty inequalities

TL;DR

This paper introduces a simplified derivation of quantum uncertainty inequalities based on extended quantum conditional entropy, generalizing previous results and providing a more direct proof technique that bypasses the need for strong subadditivity.

Contribution

It presents a new, streamlined derivation method for quantum uncertainty inequalities that extends previous results to tensor product spaces without relying on SSA.

Findings

Simplified derivation of quantum uncertainty inequalities

Generalization to tensor product Hilbert spaces

Avoidance of strong subadditivity in proofs

Abstract

Quantum states can be subjected to classical measurements, whose incompatibility, or uncertainty, can be quantified by a comparison of certain entropies. There is a long history of such entropy inequalities between position and momentum. Recently these inequalities have been generalized to the tensor product of several Hilbert spaces and we show here how their derivations can be shortened to a few lines and how they can be generalized. All the recently derived uncertainty relations utilize the strong subadditivity (SSA) theorem; our contribution relies on directly utilizing the proof technique of the original derivation of SSA.