  We calculate the statistical distribution P_2(I_2) of the speckle pattern
produced by a photon pair current I_2 transmitted through a random medium, and
compare with the single-photon speckle distribution P_1(I_1). We show that the
purity Tr rho^2 of a two-photon density matrix rho can be directly extracted
from the first two moments of P_1 and P_2. A one-to-one relationship is derived
between P_1 and P_2 if the photon pair is in an M-dimensional entangled pure
state. For M>>1 the single-photon speckle disappears, while the two-photon
speckle acquires an exponential distribution. The exponential distribution
transforms into a Gaussian if the quantum entanglement is degraded to a
classical correlation of M>>1 two-photon states. Two-photon speckle can
therefore discriminate between multi-dimensional quantum and classical
correlations.
