The Conditional Uncertainty Principle

The uncertainty principle, which states that certain sets of quantum-mechanical measurements have a minimal joint uncertainty, has many applications in quantum cryptography. But in such applications, it is important to consider the effect of a (sometimes adversarially controlled) memory that can be correlated with the system being measured: The information retained by such a memory can in fact diminish the uncertainty of measurements. Uncertainty conditioned on a memory was considered in the past by Berta et al. (Ref. 1), who found a specific uncertainty relation in terms of the von Neumann conditional entropy. But this entropy is not the only measure that can be used to quantify conditional uncertainty. In the spirit of recent work by several groups (Refs. 2--6), here we develop a general operational framework that formalizes the concept of conditional uncertainty in a measure-independent form. Our formalism is built around a mathematical relation that we call conditional majorization. We define and characterize conditional majorization, and use it to develop tools for the construction of measures of the conditional uncertainty of individual measurements, and also of the joint conditional uncertainty of sets of measurements. We demonstrate the use of this framework by deriving measure-independent conditional uncertainty relations of two types: (1) A lower bound on the minimal joint uncertainty that two remote parties (Bob and Eve) have about the outcome of a given pair of measurements performed by a third remote party (Alice), conditioned on arbitrary measurements that Bob and Eve make on their own systems. This lower bound is independent of the initial state shared by the three parties; (2) An initial state--dependent lower bound on the minimal joint uncertainty that Bob has about Alice's pair of measurements in a bipartite setting, conditioned on Bob's quantum system.

Comments: 17 pages main + 17 pages appendix; 5 figures

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