  Entanglement in angular momentum degrees of freedom is a precious resource
for quantum metrology and control. Here we study the conversions of this
resource, focusing on Bell pairs of spin-J particles, where one particle is
used to probe unknown rotations and the other particle is used as reference.
When a large number of pairs are given, we show that every rotated spin-J Bell
state can be reversibly converted into an equivalent number of rotated spin
one-half Bell states, at a rate determined by the quantum Fisher information.
This result provides the foundation for the definition of an elementary unit of
information about rotations in space, which we call the Cartesian refbit. In
the finite copy scenario, we design machines that approximately break down Bell
states of higher spins into Cartesian refbits, as well as machines that
approximately implement the inverse process. In addition, we establish a
quantitative link between the conversion of Bell states and the simulation of
unitary gates, showing that the fidelity of probabilistic state conversion
provides upper and lower bounds on the fidelity of deterministic gate
simulation. The result holds not only for rotation gates, but also to all sets
of gates that form finite-dimensional representations of compact groups. For
rotation gates, we show how rotations on a system of given spin can simulate
rotations on a system of different spin.
