The paper by James, Kwiat, Munro, and White provides a detailed theoretical framework for measuring the density matrices of pairs of quantum two-level systems, known as qubits. The focus is on qubits realized through the polarization degrees of freedom of entangled photons generated in a down-conversion experiment. Two main techniques are discussed: tomographic reconstruction and maximum likelihood estimation. Tomographic reconstruction involves relating the density matrix to a set of measured quantities, while maximum likelihood estimation uses numerical optimization to produce non-negative definite density matrices. The paper also includes a detailed error analysis, allowing for the estimation of errors in quantities derived from the density matrix, such as entropy and entanglement measures. Examples based on down-conversion experiments illustrate the methods and results.The paper by James, Kwiat, Munro, and White provides a detailed theoretical framework for measuring the density matrices of pairs of quantum two-level systems, known as qubits. The focus is on qubits realized through the polarization degrees of freedom of entangled photons generated in a down-conversion experiment. Two main techniques are discussed: tomographic reconstruction and maximum likelihood estimation. Tomographic reconstruction involves relating the density matrix to a set of measured quantities, while maximum likelihood estimation uses numerical optimization to produce non-negative definite density matrices. The paper also includes a detailed error analysis, allowing for the estimation of errors in quantities derived from the density matrix, such as entropy and entanglement measures. Examples based on down-conversion experiments illustrate the methods and results.