Draft:Ekert 91 Protocol

Ekert-91(E91) protocol, is a QKD(Quantum Key Distribution Protocol) developed by Artur Ekert in 1991.

It's a entanglement-based protocol with foundations in the CHSH Inequality and the monogamy of entanglement which states that if Alice and Bob share a maximally entangled state it's impossible for their measured state to share correlations with a third party.

This protocol is designed to be used in complement with the classical networks as it provides an interface for sharing a P2P secret key without public keys for data encryption, which is not vulnerable to quantum attacks in asymmetric keys(PPK).

CHSH Game

In simple terms, the CHSH game a referee sends the bits $x$ and $y$ to the non-communicating players $A$ and $B$ respectively, and they answer with the bits $a$ and $b$ also respectively. The condition for $A$ and $B$ to win is that:$$x*y=a\oplus b$$

CHSH Game Illustration

Optimal classical strategy

As $A$ and $B$ can't communicate, the best classical strategy by checking the truth table for these variables is $(a=b=1)$ or $(a=b=0)$ which has a success rate of $75\mathrm{\%}$.

But this strategy does not account the values of $x$ and $y$

Optimal quantum strategy

If $A$ and $B$ share an entangled bell state $|{\mathrm{\Phi }}^{+}⟩$, then according to the CHSH Inequality it's possible to have a ${\cos }^2\left(\frac{\pi }{8}\right)\approx 85.35\mathrm{\%}$ success rate.

Details and Assumptions

Alice and Bob should use for measuring the bell state a basis according with the values of $x$ and $y$.

CHSH Game with entangled bell state

So it's the proof of concept that players $A$ and $B$ can acquire information about each other through the quantum channel.

Basis choosing

For choosing the basis, Alice and Bob will choose randomly one of 3 basis for measuring the value of each entangled qubit.

Basis for measuring qubits

Although bits can only be transferred if the basis Alice and Bob coincide, the results of the measurement now can be used for validating the quantum connection.

After the transmission of the qubits, similarly to the BB84 protocol, they share the information in a public classical channel and the measurements which were not made in the same basis can be shared in this channel for calculating how close is the connection to a maximally entangled state, if it's above a certain threshold defined by the users, they can proceed to use the generated key for cryptography or start over.

Characteristics

• On average, 1 in 3 qubits are going to carry information about the key
• No qubits are discarded (except for the ones which decay)
• The entangled state is verifiable and measurable.
• The secret key bits are never shared through a classical channel