One way to quantify the quantum entropy is to use the von Neumann entropy. Given a density matrix \(\rho\) representing the quantum state of the system, the von Neumann entropy is defined as:
$$S(\rho) = -Tr(\rho \log_2 \rho)$$
where \(Tr\) is the trace operator and \(\log_2\) is the logarithm base 2.
The von Neumann entropy ranges from 0 to \(log_2 d\), where \(d\) is the dimension of the quantum system. A higher value of quantum entropy indicates a more mixed or uncertain state, while a lower value indicates a more pure or certain state.
In the context of eavesdropping, the quantum entropy of the eavesdropper's accessible information can be used to quantify the amount of quantum information that they can obtain. If the quantum entropy of the eavesdropper's accessible information is high, it means that they have a significant amount of information about the secret communication, and the security of the communication is compromised. On the other hand, if the quantum entropy of the eavesdropper's accessible information is low, it means that they have a limited amount of information, and the security of the communication is better preserved.
Therefore, quantifying the quantum entropy of the eavesdropper's accessible information provides a measure of how much quantum information can be eavesdropped, allowing us to assess the security of quantum communication protocols and identify potential vulnerabilities.
