Uncertainty Relation for Mutual Information

TL;DR

This paper introduces a universal uncertainty relation linking quantum and classical mutual information, called the CQC relation, which is validated for various states and enhances quantum cryptography security analysis.

Contribution

It proposes and proves the CQC relation, a new uncertainty relation between quantum and classical mutual information, applicable to multiple quantum states and measurement scenarios.

Findings

CQC relation holds for pure states and certain mixed states.

Monte Carlo simulations suggest the CQC relation is generally valid.

The CQC relation improves entropic uncertainty principles with quantum memory.

Abstract

We postulate the existence of a universal uncertainty relation between the quantum and classical mutual informations between pairs of quantum systems. Specifically, we propose that the sum of the classical mutual information, determined by two mutually unbiased pairs of observables, never exceeds the quantum mutual information. We call this the complementary-quantum correlation (CQC) relation and prove its validity for pure states, for states with one maximally mixed subsystem, and for all states when one measurement is minimally disturbing. We provide results of a Monte Carlo simulation suggesting the CQC relation is generally valid. Importantly, we also show that the CQC relation represents an improvement to an entropic uncertainty principle in the presence of a quantum memory, and that it can be used to verify an achievable secret key rate in the quantum one-time pad cryptographic protocol.