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Conjectured Strong Complementary Information Tradeoff
Joseph M. Renes, Jean-Christian Boileau
TL;DR
The paper asks whether entropic uncertainty relations extend to complementary observables with quantum side information. It conjectures the generalized tradeoff, proves it for Fourier-conjugate observables using a quantum-channel argument, and applies it to decoupling and quantum-cryptographic security.
Problem
The paper investigates whether complementary-observable uncertainty can be bounded when the side information is a quantum state rather than classical measurement data.
Method
The authors provide numerical evidence for arbitrary observables and prove the tradeoff for Fourier-conjugate observables by adapting Christandl and Winter’s quantum-channel proof.
Results
The strong tradeoff implies the weak tradeoff and the original entropic uncertainty principle, while a derived criterion links small conditional entropies to decoupling of quantum systems.
Takeaways & Limitations
The result supports applications in quantum cryptography and establishes that near-maximal AB correlations can correspond to near-decoupling of AE.
Takeaways & Limitations
Whether an analogous bound holds for smoothed conditional min- and max-entropies remains open, and saturation without extremal values is also unresolved.
Abstract
from arXiv · showhide
We conjecture a new entropic uncertainty principle governing the entropy of complementary observations made on a system given side information in the form of quantum states, generalizing the entropic uncertainty relation of Maassen and Uffink [Phys. Rev. Lett. 60, 1103 (1988)]. We prove a special case for certain conjugate observables by adapting a similar result found by Christandl and Winter pertaining to quantum channels [IEEE Trans. Inf. Theory 51, 3159 (2005)], and discuss possible applications of this result to the decoupling of quantum systems and for security analysis in quantum cryptography.