The Conditional Uncertainty Principle

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.

5 pages main + 10 pages appendix. Changes since v1: new results demonstrating independence of our results from existing conditional uncertainty relations; significant revision in notation and presentation; expanded bibliography

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• Entropic Uncertainty Relations and their Applications
• The uncertainty principle determines the non-locality of quantum mechanics
• Conjectured Strong Complementary Information Tradeoff
• Strong Majorization Entropic Uncertainty Relations
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• Entropic Uncertainty Relations and their Applications
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• The resource theory under conditioned thermal operations
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• Symmetric distinguishability as a quantum resource
• Continuity of entropies via integral representations
• What is Entropy? A new perspective from games of chance
• Inevitability of knowing less than nothing
• Enhanced quantum channel uncertainty relations by skew information
• Inevitable Negativity: Additivity Commands Negative Quantum Channel Entropy
• Uncertainty relation and the constrained quadratic programming
• Quantum conditional entropy from information-theoretic principles
• Uniform continuity bound for sandwiched Rényi conditional entropy
• A Universal Quantum Certainty Relation for Arbitrary Number of Observables
• The uncertainty of quantum states with respect to the projective measurement
• Probabilistic Coherence Transformation Under Strictly Incoherent Operation