Entanglement-Based Quantum Key Distribution 

Prepare-and-measure quantum key distribution and entanglement-based quantum key distribution each use principles of quantum mechanics to generate and distribute cryptographic keys between two parties. The goal of these key distribution processes is to enable the secure exchange of encryption keys in such a way that any attempt by an eavesdropper to intercept or manipulate the keys would be detected,  thus ensuring the confidentiality and integrity of the communication.

In prepare-and-measure QKD, secure keys are encoded into the quantum properties of photons. One common approach involves encoding key bits into the polarization states of photons which are sent from one party to another. The two communicating parties are able to identify and discard any compromised keys, ensuring that only secure keys are used for encrypting their communication.

Unlike prepare-and-measure QKD, entanglement-based QKD uses entangled photon sources to send entangled photon pairs to the communicating parties. When a pair of entangled photons is generated, the quantum state of each photon is interconnected such that the state of one photon instantly influences the state of the other, regardless of the distance between them. Each party uses their entangled photon from the pair to form a shared secret key. The benefit of utilizing entanglement for key distribution is that entanglement-based quantum networks which enable this type of key distribution are capable of a variety of use cases in addition to creating and sharing secret keys, such as networking quantum processing units for greater computational power. 

The Security of Entanglement-Based Quantum Key Distribution 

Classical Correlation
Quantum entanglement is a difficult concept to fully grasp. In particular, we should be careful not to confuse it with normal classical correlation.

Let's consider the classical source that is shown in the diagram below.

When it emits a photon pair, there is a 50 percent probability that the photon going left is horizontally polarized and the photon going right is vertically polarized. There is also a 50 percent probability that it is the other way around: the photon going left is vertically polarized and the photon going right is horizontally polarized.

The crucial distinction here is that as soon as the photon pair leaves the source, it is already one or the other. Of course, Alice and Bob don't know which it is until they actually measure the photon after it arrives. That doesn't change the fact that it was already one or the other as soon as the photon pair left the source before Alice and Bob did the measurement. We say that the state is hidden until Alice and Bob do the measurement.

This is not quantum entanglement. This is just classical correlation.

Entanglement
What, then, is entanglement?
In the diagram below we have a source that does produce real entangled photon pairs.

In the classical correlation diagram, each photon was either horizontally polarized or vertically polarized as soon as it left the source. In this diagram, each photon is in some sense both horizontally polarized and at the same time also vertically polarized.

It only becomes one or the other when someone, either Alice or Bob, performs the first measurement. A typical way of describing this process is that entangled photons' states are represented by a wave function, which encompasses all possible states of the photons. When an observation or measurement is made, this wave function collapses into one specific state, determining the polarization of the photons.

When Alice measures her entangled photon, it causes her photon to randomly collapse into one or the other polarization, but it also causes the photon traveling to Bob to immediately collapse into the orthogonal polarization.

This is bizarre because it happens instantaneously, which seems to imply that information travels faster than the speed of light. This is a conundrum, and raises questions as to whether entanglement could be empirically tested and differentiated from classical correlation. 

The Bell Test
Shockingly, in 1964, physicist John Steward Bell devised an experiment to distinguish classical correlation from quantum entanglement. This groundbreaking experiment, known as a Bell test, provided a method to determine whether the correlations observed in entangled particles could be explained by classical physics or if they truly reflected quantum entanglement.

In a Bell test, the polarization states of entangled photon pairs are measured under various settings. In a typical setup, a source generates pairs of entangled photons, which are then sent to two distant observers.

According to classical physics, measurement outcomes should be correlated based on local hidden variables, which predetermine the states of the photons before measurement. However, quantum mechanics predicts stronger correlations that can't be explained by these hidden variables. Bell introduced an inequality, known as Bell's inequality, which sets a limit on the correlations possible in classical theories. If experimental results violate this inequality, it confirms the presence of quantum entanglement, showing that the observed correlations are too strong to be explained by classical physics. 

A simpler version of this experiment was later discovered, called the CHSH inequality after its inventors.

The CHSH Inequality
The experimental setup is very simple and shown below.

There is a photon pair source located between Alice and Bob, and we want to determine whether the photons that it produces are really entangled or not. To find out, Alice and Bob each perform a series of Bell state measurements using a polarizing beam splitter and two single photon detectors.

Alice and Bob each have a polarization rotator which allows them to independently choose their measurement basis. For each series of measurements, we keep track of the coincident click counts. For example, N₁₀ is the number of times that the Single Photon Detector 1 at Alice’s location clicked at the same time as the Single Photon Detector 0 at Bob’s location. After a series of these coincident click counts,  the value E is computed using the formula shown at the bottom of the diagram above.

The polarization rotators allow Alice and Bob to independently choose their measurement basis. In the example above, Alice randomly chooses a measurement basis for each arriving photon: either 22.5 degrees or minus 22.5 degrees. Bob also randomly chooses a measurement basis for each arriving photon, but he picks either 0 degrees or 45 degrees.

Given that Alice and Bob each randomly choose between two measurement bases, we end up with four possible combinations as shown below.

We can compute the E value separately for each possible combination of bases. These four values of E can then be used in the formula for S shown on the right in the diagram above. If the absolute value of S is two or less, it means that the photons are not entangled, they are only classically correlated. If the absolute value of S is greater than two, it proves that the photons are quantum entangled.

The higher the value of S above 2, the more entangled the photons are. When S is two times the square root of 2, the photons are maximally entangled.

In 2022, Alain Aspect, John Clauser, and Anton Zeilinger were awarded the Nobel prize in physics for doing the CHSH experiment proving that entanglement is real.

Monogamy of Entanglement
There is one more concept that helps to explain the security of entanglement-based QKD.

Previously we said that two photons are maximally entangled if the S value in the CHSH experiment is equal to 2 times the square root of 2. This is also referred to as maximal bi-partite entanglement. The four Bell states, including for example psi-plus state, which is shown in the image below, are examples of such states.

The concept of monogamy of entanglement says that if two photons are maximally entangled with each other, then it is impossible for a third photon to be entangled as well. This exclusivity is a critical element of the security of entanglement-based QKD: it ensures that the entangled photon pairs used for key generation and distribution remain shared between the legitimate parties only. Any attempt by an eavesdropper to gain information by becoming entangled with the legitimate communicating parties would be detected. Monogamy of entanglement provides a rigorous basis for demonstrating that the security of the distributed key is not compromised by any third-party attempts to gain information about the entangled states.

The principles underlying entanglement-based QKD offer a promising path forward for achieving secure communications in an increasingly digital world. By harnessing the fundamental properties of quantum mechanics with entanglement-based quantum networks, we can build systems that are inherently resistant to eavesdropping and other security threats. Entanglement-based QKD provides a provable security framework. As these technologies mature, the potential for physics-enhanced security will grow, offering more possibilities for protecting our most sensitive information from sophisticated attacks.