Ekert protocol revisited

Recently, I have been working on quantum algorithms that apply various quantum gates. There, the direction in which the quantum is measured is fixed, often by projecting it onto the Z axis. On the other hand, quantum algorithms that change the measurement method, that is, change the orthonormal basis and use it, are also important. For example, there is the Ekert protocol, which is the basis of quantum key distribution. I've already written about Ekert in this article, so I won't go into any further explanation, but I've revised the app that demos it this time. The situation is shown below.

The left side of the figure shows the case where there is no eavesdropping, and the right side shows the case where eavesdropping was detected. An important aspect of the Ekert protocol is the use of Bell's test, or quantum entanglement.

Animation added!

If there is no eavesdropping and Alice and Bob measure in the same orthonormal basis, the result will be 00 or 11. That is, the result is the same. Quantum entanglement is at work. On the other hand, if Eve eavesdrops first, the same thing happens between Eve and Alice. However, since quantum entanglement has already disappeared, there is no such relationship between Alice and Bob. This fact is tied to eavesdropping detection.

Continuously measure quantum states with different orthonormal bases

Quantum entanglement plays an important role in Bell's theorem and the quantum key distribution protocol Ekert. But there is one more important thing. It is to successively measure the quantum state with different orthonormal bases. Both Fig.1 and Fig.2 give information about it, but isn't the graphical Fig.1 clearer than the mathematical representation of Fig.2?

Suppose we apply the following replacement rules:

|0> ⇒ 0,  |1> ⇒ 1

The probability that the measurement result by the current Basis becomes |0> can be get by the function below. This function is equivalent to Fig.1.