A statistical ensemble in quantum mechanics (also known as a mixed state) is most often represented by a density matrix, denoted by ρ ρ ^ ^ . The density matrix provides a fully general tool that can incorporate both quantum uncertainties (present even if the state of the system were completely known) and classical uncertainties (due to a lack of knowledge) in a unified manner. Any physical observable X in quantum mechanics can be written as an operator, X ^ ^ . The expectation value of this operator on the statistical ensemble ρ ρ is given by the following trace :