As we mentioned before, a density matrix with weight w for an ensemble is in a from:
For any Operator Q, the expectation value is:
for a random polarization state:
where I is identity matrix. any density matrix can be expressed by the polarization vector π.
where λ is a normalization factor and σ is the Pauli’s matrix.
from this post, we knew that the out-spin and in-spin is related by the “scattering amplitude matrix” F.
the in-density matrix for an ensemble is:
thus, the out-density matrix is:
which is just the usual transformation rule for density matrix or a general tensor.
For a single particle, we have :
For an ensemble, we have: