  The unitary evaporation of a black hole (BH) in an initially pure state must
lead to the eventual purification of the emitted radiation. It follows that the
late radiation has to be entangled with the early radiation and, as a
consequence, the entanglement among the Hawking pair partners has to decrease
continuously from maximal to vanishing during the BH's life span. Starting from
the basic premise that both the horizon radius and the center of mass of a
finite-mass BH are fluctuating quantum mechanically, we show how this process
is realized. First, it is shown that the horizon fluctuations induce a small
amount of variance in the total linear momentum of each created pair. This is
in contrast to the case of an infinitely massive BH, for which the total
momentum of the produced pair vanishes exactly on account of momentum
conservation. This variance leads to a random recoil of the BH during each
emission and, as a result, the center of mass of the BH undergoes a quantum
random walk. Consequently, the uncertainty in its momentum grows as the square
root of the number of emissions. We then show that this uncertainty controls
the amount of deviation from maximal entanglement of the produced pairs and
that this deviation is determined by the ratio of the cumulative number of
emitted particles to the initial BH entropy. Thus, the interplay between the
horizon and center-of-mass fluctuations provides a mechanism for teleporting
entanglement from the pair partners to the BH and the emitted radiation.
