Measurement of qubits - art. no. 052312

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.