  In a recent work [D. K. Burgarth et al., Nat. Commun. 5, 5173 (2014)] it was
shown that a series of frequent measurements can project the dynamics of a
quantum system onto a subspace in which the dynamics can be more complex. In
this subspace even full controllability can be achieved, although the
controllability over the system before the projection is very poor since the
control Hamiltonians commute with each other. We can also think of the
opposite: any Hamiltonians of a quantum system, which are in general
noncommutative with each other, can be made commutative by embedding them in an
extended Hilbert space, and thus the dynamics in the extended space becomes
trivial and simple. This idea of making noncommutative Hamiltonians commutative
is called "Hamiltonian purification." The original noncommutative Hamiltonians
are recovered by projecting the system back onto the original Hilbert space
through frequent measurements. Here we generalize this idea to open-system
dynamics by presenting a simple construction to make Lindbladians, as well as
Hamiltonians, commutative on a larger space with an auxiliary system. We show
that the original dynamics can be recovered through frequently measuring the
auxiliary system in a non-selective way. Moreover, we provide a universal pair
of Lindbladians which describes an "accessible" open quantum system for generic
system sizes. This allows us to conclude that through a series of frequent
non-selective measurements a nonaccessible open quantum system generally
becomes accessible. This sheds further light on the role of measurement
backaction on the control of quantum systems.
