Once a quantum sample space has been chosen the question arises of how probabilities are to be assigned to the different projectors of which it is composed. As in other applications of probability theory such assignments may represent pure guesswork, or contain parameters which are to be adjusted according to experimental data, or be based upon some other information about the system, such as prior preparation or a later measurement. Information about the preparation of a quantum system is often expressed using a density operator \(\rho\), a positive semidefinite operator (Hermitian with nonnegative eigenvalues) with trace (sum of the eigenvalues) equal to 1, used as a pre-probability in the terminology of Sec. 9.4 of Griffiths (2002a), a mathematical object that can be used to generate a probability distribution. In particular if \(\{P_j\}\) is a projective decomposition of the identity, a quantum sample space, one assigns a probability