We describe in detail the theory underpinning the measurement of density ma
trices of a pair of quantum two-level systems ("qubits"). Our particular em
phasis is on qubits realized by the two polarization degrees of freedom of
a pair of entangled photons generated in a down-conversion experiment; howe
ver, the discussion applies in general, regardless of the actual physical r
ealization. Two techniques are discussed, namely, a tomographic reconstruct
ion (in which the density matrix is linearly related to a set of measured q
uantities) and a maximum likelihood technique which requires numerical opti
mization (but has the advantage of producing density matrices that are alwa
ys non-negative definite). In addition, a detailed error analysis is presen
ted, 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.