systems product states correlated density matrices
where the wi are positively valued probabilities and the 




ρ

i


A




{\displaystyle \rho _{i}^{A}}

's and 




ρ

i


B




{\displaystyle \rho _{i}^{B}}

's are themselves mixed states (density operators) on the subsystems A and B 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 




ρ

i


A




{\displaystyle \rho _{i}^{A}}

 and 




ρ

i


B




{\displaystyle \rho _{i}^{B}}

 are themselves pure ensembles. A state is then said to be entangled if it is not separable.
