Measurement of qubits

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 that 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.