The E91 protocol, also known as the Ekert91 protocol, is a quantum key distribution (QKD) protocol proposed by Artur Ekert in 1991. Unlike the BB84 protocol, which depend on on the Heisenberg uncertainty principle, the security of the E91 protocol is fixed in the fundamental properties of quantum entanglement and the violation of Bell’s inequalities. This distinct approach provides a different perspective on achieving secure communication based on the completeness of quantum mechanics.
Principles of E91 Protocol
Entanglement: When two or more quantum particles get entangled, their states are directly related to each other’s states regardless of the distance between them. These interesting and basic phenomena is known as quantum entanglement, and they occur in quantum mechanics.
For two qubits, the system can be entangled into a state like:
∣ψ⟩=1/√2 (|00⟩+ |11⟩)
- ∣00⟩ Both particles are in state 0.
- ∣11⟩ Both particles are in state 1.
The idea of quantum coupling is at the heart of the E91 protocol. There are entangled quantum states when there are two or more particles whose quantum traits are linked in a way that makes them connected, even if they are far apart physically. When you measure one particle’s property, it instantly changes the matching property of the other particle that is linked with it. The Bell state
|Φ+⟩= 1/√ 2 (|00⟩ + |11⟩)
is a key example used in E91. In this state, if Alice checks her qubit and finds it to be |0⟩, Bob’s qubit will also be |0⟩ right away, and the same goes for |1⟩. This is called perfect correlation.
In order to keep the E91 protocol safe, anyone listening in (Eve) will have to break up the interaction of the particles if they try to get information about them. Bell’s inequalities can be broken, which lets Alice and Bob notice this change.
There are some mathematical rules that every physical theory based on local realism must follow. Local realism is the idea that physical systems have fixed traits that don’t change when they are measured, and that forces can’t move faster than light. Quantum physics says that these inequalities will be broken in some linked states. Alice and Bob can test for these violations in the real world by taking specific measurements on their linked particles and looking at how the results of those measurements are related. A strong violation proves that there is a real connection and suggests that there was probably only a small attempt to listen in. On the other hand, if Eve intercepts and counts the entangled particles, she will create correlations that will weaken or get rid of the violation of Bell’s inequalities. This will let Alice and Bob know that Eve is there.
How does the E91 Protocol Work?
To set up a secret key between Alice and Bob using the E91 protocol, these steps are usually taken:
- Entangled Pair Generation: The first step is to make a source of entangled photon pairs. This source is usually in one of the Bell states, like | +⟩. It then makes a series of these pairs. This source could be trusted or, more importantly, not trusted, because the security depends on the connections that have been seen rather than the source’s honesty.
- Distribution: Two photons are sent from each entangled pair. One is sent to Alice through a quantum channel, and the other is sent to Bob through a different quantum channel.
- Choices for Measurement: Both Alice and Bob pick at random one of three possible measurement bases for each photon they receive. Some of these bases are picked because they are fair to both sides. These are the rectilinear basis (for example, horizontal/vertical polarization, shown as {|0⟩, |1⟩} or Bz) and the diagonal basis (for example, +45°/-45° polarization, shown as {|+⟩, |-⟩} = 1/√ 2 (|0⟩ + |1⟩), 1/√ 2 (|0⟩ – |1⟩) or Bx). It was important to pick the third measurement base that would respond to a different association in order to test Bell’s inequality. Most of the time, these three bases are set up at angles of 0°, 45°, and 22.5°, or something close.
- Recording Outcomes and Bases: Alice and Bob write down the result of each measurement (like 0 or 1) and the base they used for that measurement for each photon they watch.
- Public Discussion of Bases: Once Alice and Bob have measured enough photon pairs, they tell everyone (through a traditional, authenticated channel that can be quietly eavesdropped) the order of measurement bases they used for each photon they got. At this point, they don’t say what the results of their measurements were.
- Sifting: Alice and Bob throw away the measurement results for the times they used different bases of measurement. The only findings they keep are the ones where they used the same base. The raw secret key is made up of these linked test results. Because they are entangled in the | +⟩ state, their measurement results on the same base should be perfectly matched (they will get the same bit value).
- Eavesdropping Check (Bell Test): To see if anyone is listening in on them, Alice and Bob use a subset of the measurement data that includes only certain combinations of their three measurement bases. For example, Alice measures in one basis and Bob measures in another, in order to cover all possible pairs combinations. After getting these results, they figure out the connection values. To test Bell’s inequality, these association values are used. If the measured connections go against Bell’s inequality by a large amount, it means that the entanglement has been kept and the level of eavesdropping is probably low enough to go ahead. If there is a weak violation or no violation at all, it means that someone is listening in and causing a lot of trouble. Alice and Bob would probably end the protocol and start over.
- Key Extraction: If the Bell Test Works, Alice and Bob use the sorted measurement results (where they used the same base) as their original secret key.
- Error Correction and Privacy Amplification: Just like with the BB84 protocol, Alice and Bob may have gotten a raw key that has some mistakes in it because of flaws in the quantum channel or the measuring tools. They can publicly compare a sample of their key bits to get an idea of the error rate. To get bit strings that are exactly the same, they can then use error correction methods. Lastly, they use privacy amplification methods (by using the right hash function) to turn the error-corrected key into a shorter, more secure final key. This makes it less likely that Eve could have gotten any information during the transfer.
Advantages of the E91 Protocol
The E91 procedure has a number of important benefits, including:
- Entanglement-Based Security: The security is based on the basic rules of quantum entanglement and Bell’s theory, which has been around for a long time. The BB84 protocol is based on the uncertainty principle. This is a different basis that could provide a different level of security guarantee.
- Possible for Device Independent Quantum Key Distribution (DIQKD): Some Device Independent QKD (DIQKD) methods are based on the E91 algorithm. In DIQKD, security can be reached even if Alice and Bob don’t fully believe or understand how the quantum devices they use work on the inside. Security comes from the fact that Bell’s inequalities are broken, which makes it resistant to some types of hardware threats.
- Bell Test for Eavesdropping: The Bell test is a direct way to find eavesdropping by looking at the level of confusion caused by breaking Bell’s inequalities. In theory, this method can find any attempt to get information because it would break the web. When you look at BB84, you can tell that someone is listening in because the error rate goes up after the base correction. But E91 tests the quantum correlations directly.
- Source Not Trusted: It is possible to show that E91 is safe even if the source of the linked pairs is not trusted. No matter who made the linked pairs, Alice and Bob can be sure that their generated key is safe as long as they notice the Bell inequality violation.
Disadvantages of E91 Protocol
The E91 system has some problems, even though it is based on good ideas:
• Less efficient: The basic E91 protocol is thought to be less efficient than the BB84 protocol. Here in E91, the raw key is only made up of measurement results where Alice and Bob choose the same base. The chance of this happening is smaller with three measurement bases than with BB84, where the chance of using the same basis from two choices is higher. Before being fixed and made more private, the sorted key is about L = 2/9 N long, where N is the total number of linked pairs.
• How Hard It Is to Use the Bell Test: To use the Bell test correctly, you need to carefully choose your measurement bases and gather enough statistics to calculate correlation coefficients and look for violations. This makes the procedure more complicated than the error rate estimate that is used in standard BB84 implementations.
• Need for High-Quality Entangled Source and Channels: The protocol can possibly work with an unknown source, but the quality of the entangled states and the coherence of the quantum channels are still very important for getting a big Bell violation and a key rate that can be used. Noise and lack of consistency in the channels can weaken the coupling, making it harder to find people listening in and make a long, safe key.
What is the difference between E91 and BB84?
There are two different ways to send quantum keys: the BB84 protocol and the E91 protocol. BB84 uses conjugate quantum states and random basis choices for both encoding and measuring to store classical bits on a single qubit. In BB84, security is set by comparing a selection of the chosen bases after the broadcast to find mistakes made by people who might be listening in.
E91 is a different kind of protocol that uses entangled pairs of qubits, which are usually Bell states. Not following Bell’s rules is essential to the safety of E91. Attempts to listen in that mess up the connection would lessen or get rid of this violation, letting Alice and Bob know about a possible security breach. For safe key generation, E91 uses the non-local correlations of entangled particles instead of preparing and recording single qubit states like BB84 does.
What is the difference between E91 and B92?
The E91 protocol uses entangled pairs of qubits. Alice and Bob each receive one qubit from an entangled pair and perform measurements. They check for eavesdropping by verifying the violation of Bell’s inequalities. If an eavesdropper disturbs the entanglement, the Bell inequality violation will be reduced or disappear.
The B92 protocol, on the other hand, uses single photons prepared in two non-orthogonal states. Alice sends photons in one of two polarizations to represent the key bits. Bob measures these photons using detectors that are not perfectly aligned with Alice’s preparation bases. They detect eavesdropping by analyzing the error rate in the transmission. B92 is considered simpler than BB84 as it uses only two quantum states. While E91 relies on entanglement and Bell’s theorem, B92 depends on the properties of non-orthogonal quantum states to detect disturbances.
What is Bell’s theorem in simple terms?
Bell’s theorem is a fundamental result in quantum physics that demonstrates the incompatibility between quantum mechanics and the theory of local realism. Local realism posits that physical systems have definite, pre-existing properties independent of measurement (realism) and that influences cannot travel faster than light (locality).
Bell derivative Bell’s inequalities, mathematical limitations that any theory based on local realism must satisfy. Quantum mechanics predicts that for certain entangled quantum states, these inequalities can be violated.
Experiments, starting with Aspect’s work, have consistently shown violations of Bell’s inequalities, providing strong evidence against local realism and in favor of quantum mechanics. This implies that the correlations observed between entangled particles cannot be explained by predetermined local properties, suggesting a non-local connection.
The E91 quantum key distribution protocol, as discussed previously, leverages the violation of Bell’s inequalities to detect eavesdropping. Any attempt to measure or intercept the entangled particles would reduce or eliminate this violation, alerting the communicating parties. Bell’s theorem and its experimental verification are considered among the deepest results of 20th-century physics, challenging our classical intuition about reality.