  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 Reid's 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.
