As in the classical case, section 5.2, the quantum BBGKY equations are the expression of the reduction of the full system to a susbsystem of the full system. Consider an orthonormal basis denoted by the set of vectors 
, and define the subsystem by the condition 
. Analogous to the classical case, the Hamiltonian may be written as a part specific to the subsystem and the remaining part, each term of which includes a coupling to the rest of the system,
For example, the Hamiltonian
with 
and 
, and where each term in 
contains a basis vector with index in 
, is a possible expression for a Hamiltonian with pairwise state interactions.
Take the quantum coordinate independent form of the density evolution equation, equation 7.12, insert the Hamiltonian in the form of equation 7.13, and trace over all of the states outside the subsystem to find the quantum BBGKY equations for the reduced density matrices
where 
. Compare this equation to equation 5.9. As in the classical case, the entropy of the reduced system is a constant unless there is information flow from the rest of the system into the subsystem.