$$I(A:B)=S(A)+S(B)-S(AB),$$
where \(S(A)\), \(S(B)\), and \(S(AB)\) are the von Neumann entropies of Alice's system, Bob's system, and the joint system AB, respectively.
If Eve has no access to the quantum system, then the quantum mutual information between Alice and Bob is preserved. However, if Eve performs eavesdropping operations, such as intercepting and measuring some of the qubits, then the quantum mutual information between Alice and Bob will decrease. The amount of decrease in quantum mutual information quantifies how much quantum information has been eavesdropped by Eve.
To gain a better understanding, let's consider a simple example. Suppose Alice and Bob share a two-qubit entangled state, such as the singlet state:
$$|\psi^{-}\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle).$$
Initially, the quantum mutual information between Alice and Bob is \(I(A:B)=1\), which represents the maximum amount of quantum correlation. If Eve intercepts and measures one of the qubits, say Alice's qubit, she gains some information about the state. Consequently, the quantum mutual information between Alice and Bob decreases to \(I(A:B)=\frac{1}{2}\) after Eve's eavesdropping.
In general, the amount of quantum information that can be eavesdropped depends on the specific eavesdropping strategy employed by Eve. However, there are fundamental limits on eavesdropping due to the no-cloning theorem and the uncertainty principle. These limits ensure that Eve cannot obtain perfect information about the quantum system without disturbing it, and thus, the quantum mutual information between Alice and Bob can never be completely compromised.
