# Von Neumann entropy of a quantum state vs Von Neumann entropy of a system

Using the Chuang and Nielsen Quantum Computation and Quantum Information book, for an introductory course in Quantum Information, I stumbled on an ambiguous (at least for me) use of the Von Neumann entropy, being used sometimes for quantum state (for example $$S(\rho)$$) and sometimes for quantum systems:

Suppose distinct quantum systems A and B have a joint state ρAB. Then the joint entropy for the two systems satisfies the inequalities $$S(A,B) \le S(A)+SA(B)$$

Sometimes it is said that $$S(A)$$ is the entropy of a state in the system $$A$$, but the example cited above seems not the case.

My question is, what is the meaning of the Von Neumann entropy of a system? Does it always is related to the entropy of a state in the aforementioned system or has it a different meaning?

• what do you mean by "system" as opposed to "state"? – glS Sep 28 '18 at 9:13

Maximize over all states $\rho$ on A. For the classical example, one bit can be in two states but we have not stated the probabilities. The maximum entropy is when $p=1/2$. So by an abuse of notation we can call it 1 bit of information even if it might really store less information.