What is the pros and cons of E91 protocol and BB84 protocol? Which one is faster than the other, and which one is more safe?


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As I explained in your previous question, both protocol are safe, as every other QKD protocol in the literature (e.g., SARG04, Decoy states, ...). The difference in the protocol is only in the practical implementation. Furthermore, most of the recent protocols aim to correct the non idealities of the sources and measurement devices (e.g., MDI-QKD). Roughly speaking, entanglement-based protocols are more difficult to implement than prepare and measure just because the quantum channel destroys entanglement over long distances. However, both type of protocols have been tested among a 1000 kilometer link using a LEO satellite. Take a look at this and this.

Now, you may ask what does it mean that a protocol 'is safe'? There is a precise definition about the secrecy of a protocol which involves information theory (I don't know if you're familiar with these concepts). Roughly speaking, a protocol is (information theoretically) 'secure' if the information an eavesdropper can extract about the key, without being detected, is arbitrary low. That is, if $A$ is the random variable associated to the key and $E$ is the r.v. associated to the information extracted by the eavesdropper, we have $I(A:E)<\varepsilon$ for every $\varepsilon>0$, where $I(A:E)$ is the mutual information. Of course, this is a limit result, and it could be obtained only for infinite length keys. The precise defition of secrecy is a little bit more complicated to take into account the effect of a quantum memory.

It turns out that this security definition 'affects' the secret key rate $K$ of the protocol, i.e., the ratio between the number of secret key bits and the number of sifted bits. All QKD protocols differ by their secret key rate. That is, for the BB84, given by the Shor-Preskill formula: $K=1-2H_2(e)$ where $e$ is the QBER. At the moment, I don't remember the secret key rate formula for the E91 protocol but you can find it on the internet.


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