Mutual Uncertainty, Conditional Uncertainty and Strong Sub-Additivity

We introduce a new concept called as the mutual uncertainty between two observables in a given quantum state which enjoys similar features like the mutual information for two random variables. Further, we define the conditional uncertainty as well as conditional variance and show that conditioning on more observable reduces the uncertainty. Given three observables, we prove a strong sub-additivity relation for the conditional uncertainty under certain condition. As an application, we show that using the conditional variance one can detect bipartite higher dimensional entangled states. The efficacy of our detection method lies in the fact that it gives better detection criteria than most of the existing criteria based on geometry of the states. Interestingly, we find that for $N$-qubit product states, the mutual uncertainty is exactly equal to $N-sqrt{N}$, and if it is other than this value, the state is entangled. We also show that using the mutual uncertainty between two observables, one can detect non-Gaussian steering where Reids criteria fails to detect. Our results may open up a new direction of exploration in quantum theory and quantum information using the mutual uncertainty, conditional uncertainty and the strong sub-additivity for multiple observables.