Properties
The classical joint entropy is always at least equal to the entropy of each individual system. This is not the case for the joint quantum entropy. If the quantum state exhibits quantum entanglement, then the entropy of each subsystem may be larger than the joint entropy. This is equivalent to 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.
Read more about this topic: Joint Quantum Entropy
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