  We investigate an entangled system, which is analogous to a composite system
of a black hole and Hawking radiation. If Hawking radiation is well
approximated by an outgoing particle generated from pair creation around the
black hole, such a pair creation increases the total number of states. There
should be a unitary mechanism to reduce the number of states inside the horizon
for black hole evaporation. Because the infalling antiparticle has negative
energy, as long as the infalling antiparticle finds its partner such that the
two particles form a separable state, one can trace out such a zero energy
system by maintaining unitarity. In this paper, based on some toy model
calculations, we show that such a unitary tracing-out process is only possible
before the Page time while it is impossible after the Page time. Hence, after
the Page time, if we assume that the process is unitary and the Hawking pair
forms a separable state, the internal number of states will monotonically
increase, which is supported by the Almheiri-Marolf-Polchinski-Sully (AMPS)
argument. In addition, the Hawking particles cannot generate randomness of the
entire system; hence, the entanglement entropy cannot reach its maximum. Based
on these results, we modify the correct form of the Page curve for the remnant
picture. The most important conclusion is this: if we assume unitarity,
semi-classical quantum field theory, and general relativity, then the black
hole should violate the Bekenstein-Hawking entropy bound around the Page time
at the latest; hence, the infinite production arguments for remnants might be
applied for semi-classical black holes, which seems very problematic.
