  Orbital angular momentum (OAM) of light is a promising degree of freedom for
quantum communication and quantum computation. However, efficient methods to
transform and measure OAM modes are still not available readily. Here, we
propose a method for full quantum state tomography for arbitrary OAM states of
light. The key component in our scheme is the helicity sorter, a device that
can sort OAM modes of a light beam based on their Helicity. We present two
setups to implement the Helicity sorted, one requires $\mathcal{O}(N)$ number
of optical elements for $2^N$ OAM modes, and the other uses spatial light
modulators and is independent of the number of OAM modes. Ours is the first
method to provide the efficient and scalable full quantum state tomography of
the OAM modes and can be extremely important for quantum communication and
quantum computation.
