  For a pair of incompatible quantum measurements, the total uncertainty can be
bounded by a state-independent constant. However, such a bound can be violated
if the quantum system is entangled with another quantum system (called memory);
the quantum correlation between the systems can reduce the measurement
uncertainty. On the other hand, in a curved spacetime, the presence of the
Hawking radiation can increase the uncertainty in quantum measurement. The
interplay of quantum correlation in the curved spacetime has become an
interesting arena for studying quantum uncertainty relations. Here we
demonstrate that the bounds of the entropic uncertainty relations, in the
presence of memory, can be formulated in terms of the Holevo quantity, which
limits how much information can be encoded in a quantum system. Specifically,
we considered two examples with Dirac fields, near the event horizon of a
Schwarzschild black hole, the Holevo bound provides a better bound than the
previous bound based on the mutual information. Furthermore, if the memory
moves away from the black hole, the difference between the total uncertainty
and the Holevo bound remains a constant, not depending on any property of the
black hole.
