  We introduce and analyze a task in which a tripartite quantum state is
transformed to an approximately recoverable state by a randomizing operation on
one of the three subsystems. We consider cases where the initial state is a
tensor product of $n$ copies of a tripartite state $\rho^{ABC}$, and is
transformed by a random unitary operation on $A^n$ to another state which is
approximately recoverable from its reduced state on $A^nB^n$ (Case 1) or
$B^nC^n$ (Case 2). We analyze the minimum cost of randomness per copy required
for the task in an asymptotic limit of infinite copies and vanishingly small
error of recovery, mainly focusing on the case of pure states. We prove that
the minimum cost in Case 1 is equal to the Markovianizing cost of the state,
for which a single-letter formula is known. With an additional requirement on
the convergence speed of the recovery error, we prove that the minimum cost in
Case 2 is also equal to the Markovianizing cost. Our results have an
application for distributed quantum computation.
