I am now studying about E91 protocol. In the process, I saw that E91 protocol utilizes the CHSH inequality instead of QBER. Other protocol like BB84 protocol uses QBER for the rate of the existence of eve. Then, why does only E91 protocol use CHSH inequality instead of QBER? Which means, what is the advantage of using CHSH inequality?
Quantum key distribution (QKD) algorithms can be roughly divided in two categories: prepare and measure (P&M) and entanglement-based (EB). The aim is the same: distribute information-theoretically secure keys among two distant nodes. However, they are based on different physical arguments to guarantee the security of the algorithm.
Ekert's protocol is an entanglement-based protocol. The idea is to make the security rely on the so-called monogamy of entanglement: if two particles (i.e., qubits) are entangled, they will not be entangled to any external system. In other terms, no "information" about the particles (the secret key) leaks to the environment. If an eavesdropper tries to manipulate one of the two halves, then entanglement will be lost. Therefore, to check the security of the QKD system, we need to check if the two particles are entangled. How can this be done? The most simple way is to use the CHSH inequality to check if the two particles share some kind of non-classical correlations.
Conversely, P&M protocols (e.g., BB84) are based on the uncertainty principle which characterize two non-commuting observable. In this case we don't need to send entangled particles, but the security-checking process is completely different from the one of E91. Indeed, in this case, we need to check if some one has eavesdropped the qubits by measuring some the wrong observable (i.e., the QBER).
In other terms, the "security checks" are different because the physical properties which are used to "hide the key" are different.
P.S. Interestingly, it turns out that P&M protocols can be transformed into EB protocols and viceversa. This is particularly useful for the proofs of security.