  In this work, we derive Robertson-Heisenberg like uncertainty relation for
two incompatible observables in a pre- and post-selected (PPS) system. The
newly defined standard deviation and the uncertainty relation in the PPS system
have physical meanings which we present here. We demonstrate two unusual
properties in the PPS system using our uncertainty relation. First, for
commuting observables, the lower bound of the uncertainty relation in the PPS
system does not become zero even if the initially prepared state i.e.,
pre-selection is the eigenstate of both the observables when specific
post-selections are considered. This implies that for such case, two commuting
observables can disturb each other's measurement results which is in fully
contrast with the Robertson-Heisenberg uncertainty relation. Secondly, unlike
the standard quantum system, the PPS system makes it feasible to prepare
sharply a quantum state (pre-selection) for non-commuting observables {(to be
detailed in the main text)}. Some applications of uncertainty and uncertainty
relation in the PPS system are provided: $(i)$ detection of mixedness of an
unknown state, $(ii)$ stronger uncertainty relation in the standard quantum
system, ($iii$) ``purely quantum uncertainty relation" that is, the uncertainty
relation which is not affected (i.e., neither increasing nor decreasing) under
the classical mixing of quantum states, $(iv)$ state dependent tighter
uncertainty relation in the standard quantum system, and $(v)$ tighter upper
bound for the out-of-time-order correlation function.

Quantum Physics