Measurement of qubits

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