The paper examines relationships between the conditional Shannon entropy and the expectation of `α-norm for joint probability distributions. More precisely, we investigate the tight bounds of the expectation of `α-norm with a fixed conditional Shannon entropy, and vice versa. As applications of the results, we derive the tight bounds between the conditional Shannon entropy and several informati...

Conditional expectation is one of the most useful tools of probability. The Radon-Nikodym theorem enables us to construct conditional expectations. Definition 1. A measure µ on (Ω, F) is σ-finite iff there is A n ∈ F satisfying A n ↑ Ω where µ(A n) < ∞ for all n. Definition 2. Let µ, ν be measures on (Ω, F). Then ν is absolutely continuous with respect to µ (ν << µ) iff for all A ∈ F, µ(A) = 0 ...

The objective of this work is to develop a recursive, discrete time quantum filtering equation for a system that interacts with a probe, on which measurements are performed according to the Positive Operator Valued Measures (POVMs) framework. POVMs are the most general measurements one can make on a quantum system and although in principle they can be reformulated as projective measurements on ...