  We present a feedback protocol that is able to confine a system to a single
micro-state without heat dissipation. The protocol adjusts the Hamiltonian of
the system in such a way that the Bayesian posterior distribution after
measurement is in equilibrium. As a result, the whole process satisfies
feedback reversibility -- the process is indistinguishable from its time
reversal -- and assures the lowest possible dissipation for confinement. In
spite of the whole process being reversible it can surprisingly be implemented
in finite time. We illustrate the idea with a Brownian particle in a harmonic
trap with increasing stiffness and present a general theory of reversible
feedback confinement for systems with discrete states.
