The Conditional Uncertainty Principle

TL;DR

This paper introduces a measure-independent framework for conditional uncertainty using conditional majorization, providing new uncertainty relations that are independent of traditional entropic measures.

Contribution

It formalizes conditional majorization, characterizes it with monotones, and derives new memory-assisted uncertainty relations that are measure-independent.

Findings

Derived two types of memory-assisted uncertainty relations.

Showed the relations are at least independent of entropic counterparts.

Provided a thorough characterization of conditional majorization for classical memory.

Abstract

We develop a general operational framework that formalizes the concept of conditional uncertainty in a measure-independent fashion. Our formalism is built upon a mathematical relation which we call conditional majorization. We define conditional majorization and, for the case of classical memory, we provide its thorough characterization in terms of monotones, i.e., functions that preserve the partial order under conditional majorization. We demonstrate the application of this framework by deriving two types of memory-assisted uncertainty relations: (1) a monotone-based conditional uncertainty relation, (2) a universal measure-independent conditional uncertainty relation, both of which set a lower bound on the minimal uncertainty that Bob has about Alice's pair of incompatible measurements, conditioned on arbitrary measurement that Bob makes on his own system. We next compare the obtained relations with their existing entropic counterparts and find that they are at least independent.