as follows. Consider a "black box" apparatus that spits electrons towards an observer. The electrons' Hilbert spaces are identical. The apparatus might produce electrons that are all in the same state; in this case, the electrons received by the observer are then a pure ensemble. However, the apparatus could produce electrons in different states. For example, it could produce two populations of electrons: one with state with spins aligned in the positive direction, and the other with state with spins aligned in the negative direction. Generally, this is a mixed ensemble, as there can be any number of populations, each corresponding to a different state. Following the definition above, for a bipartite composite system, mixed states are just density matrices on . That is, it has the general form where the wi are positively valued probabilities, , and the vectors are unit vectors. This is self-adjoint and positive and has trace 1. Extending the definition of separability from the pure case, we say that a mixed state is separable if it can be written as where the are positively valued probabilities and the 's and 's are themselves mixed states (density operators) on the subsystems and respectively. In other words, a state is separable if it is a probability distribution over uncorrelated states, or product states. By writing the density matrices as sums of pure ensembles and expanding, we may assume without loss of generality that and are themselves pure ensembles. A state is then said to be entangled if it is not separable. In general, finding out whether or not a mixed state is entangled is considered difficult. The general bipartite case has been shown to be NP-hard. For the and cases, a necessary and sufficient criterion for separability is given by the famous Positive Partial Transpose (PPT) condition. Reduced density matrices The idea of a reduced density matrix was introduced by Paul Dirac in 1930. Consider as above systems and each with a Hilbert space . Let the state of the composite system be As indicated above, in general there is no way to associate a pure state to the component system . However, it still is possible to associate a density matrix. Let . which is the projection operator onto this state. The state of is the partial trace of over the basis of system : The sum occurs over and the identity operator in . is