the fact that the conditional quantum entropy may be negative, while the classical conditional entropy may never be. Consider a maximally entangled state such as a Bell state. If is a Bell state, say, then the total system is a pure state, with entropy 0, while each individual subsystem is a maximally mixed state, with maximum von Neumann entropy . Thus the joint entropy of the combined system is less than that of subsystems. This is because for entangled states, definite states cannot be assigned to subsystems, resulting in positive entropy. Notice that the above phenomenon cannot occur if a state is a separable pure state. In that case, the reduced states of the subsystems are also pure. Therefore all entropies are zero. Relations to other entropy measures The joint quantum entropy can be used to define of the conditional quantum entropy: and the quantum mutual information: These definitions parallel the use of the classical joint entropy to define the conditional entropy and mutual information. See also Quantum relative entropy Quantum mutual information References Nielsen, Michael A. and Isaac L. Chuang, Quantum Computation and Quantum Information. Cambridge University Press, 2000. Quantum mechanical entropy Quantum information theory