This paper presents a detailed theory for measuring the density matrices of pairs of quantum two-level systems, or qubits. The focus is on qubits realized by the polarization degrees of freedom of entangled photons generated in a down-conversion experiment, though the discussion applies generally. Two techniques are discussed: tomographic reconstruction, where the density matrix is linearly related to measured quantities, and a maximum likelihood technique that uses numerical optimization to ensure the density matrix is always non-negative definite. An error analysis is also provided, allowing estimation of errors in quantities derived from the density matrix, such as entropy and entanglement. Examples based on down-conversion experiments illustrate the results. The paper begins by discussing the analogy between the measurement of the polarization state of light and the measurement of the density matrix of an ensemble of two-level systems. The Stokes parameters are introduced, which allow the determination of the density matrix of a light beam. The paper then generalizes this to multiple qubits, showing how the state of an n-qubit system can be determined by 4^n measurements. The paper also discusses the experimental setup used to measure the polarization state of two photons, including the use of optical elements to project the light beams onto desired polarization states. The paper then presents a tomographically complete set of measurements, which allows the reconstruction of the density matrix. The density matrix is expressed as a linear combination of 16 matrices, and the parameters of the density matrix are determined by solving a system of equations. The paper also discusses the maximum likelihood estimation of the density matrix, which involves finding the set of parameters that maximizes the likelihood of the measured data. This method ensures that the density matrix is non-negative definite and provides a more accurate estimate of the quantum state. The paper concludes with an error analysis, showing how errors in the measured coincidence counts and in the angular settings of the waveplates affect the accuracy of the density matrix reconstruction. The errors in quantities derived from the density matrix, such as entropy and entanglement, are also discussed. The paper demonstrates that the maximum likelihood method provides a more accurate estimate of the quantum state than the linear tomographic method, especially for low entropy, highly entangled states. The results are illustrated with examples based on down-conversion experiments.This paper presents a detailed theory for measuring the density matrices of pairs of quantum two-level systems, or qubits. The focus is on qubits realized by the polarization degrees of freedom of entangled photons generated in a down-conversion experiment, though the discussion applies generally. Two techniques are discussed: tomographic reconstruction, where the density matrix is linearly related to measured quantities, and a maximum likelihood technique that uses numerical optimization to ensure the density matrix is always non-negative definite. An error analysis is also provided, allowing estimation of errors in quantities derived from the density matrix, such as entropy and entanglement. Examples based on down-conversion experiments illustrate the results. The paper begins by discussing the analogy between the measurement of the polarization state of light and the measurement of the density matrix of an ensemble of two-level systems. The Stokes parameters are introduced, which allow the determination of the density matrix of a light beam. The paper then generalizes this to multiple qubits, showing how the state of an n-qubit system can be determined by 4^n measurements. The paper also discusses the experimental setup used to measure the polarization state of two photons, including the use of optical elements to project the light beams onto desired polarization states. The paper then presents a tomographically complete set of measurements, which allows the reconstruction of the density matrix. The density matrix is expressed as a linear combination of 16 matrices, and the parameters of the density matrix are determined by solving a system of equations. The paper also discusses the maximum likelihood estimation of the density matrix, which involves finding the set of parameters that maximizes the likelihood of the measured data. This method ensures that the density matrix is non-negative definite and provides a more accurate estimate of the quantum state. The paper concludes with an error analysis, showing how errors in the measured coincidence counts and in the angular settings of the waveplates affect the accuracy of the density matrix reconstruction. The errors in quantities derived from the density matrix, such as entropy and entanglement, are also discussed. The paper demonstrates that the maximum likelihood method provides a more accurate estimate of the quantum state than the linear tomographic method, especially for low entropy, highly entangled states. The results are illustrated with examples based on down-conversion experiments.