  We describe in detail the theory underpinning the measurement of density
matrices of a pair of quantum two-level systems (``qubits''). Our particular
emphasis is on qubits realized by the two polarization degrees of freedom of a
pair of entangled photons generated in a down-conversion experiment; however
the discussion applies in general, regardless of the actual physical
realization. Two techniques are discussed, namely a tomographic reconstruction
(in which the density matrix is linearly related to a set of measured
quantities) and a maximum likelihood technique which requires numerical
optimization (but has the advantage of producing density matrices which are
always non-negative definite). In addition a detailed error analysis is
presented, allowing errors in quantities derived from the density matrix, such
as the entropy or entanglement of formation, to be estimated. Examples based on
down-conversion experiments are used to illustrate our results.
