What is entanglement swapping?

The goal of quantum networking is to distribute high-quality entanglement—typically in the form of Bell pairs. By consuming this entdvancedvancedanglement, quantum networking applications can achieve considerable advantages over any of their current or future classical counterparts. However, achieving high-quality distributed entanglement turns out to be no easy task.

Quantum information—such as the entanglement we are trying to distribute—is generally transmitted in a quantum network using particles of light—known as photons.Photons experience exponential loss of information with respect to the length of the optical fiber. Classically we get around issues of loss of information in long-distance transmission by using error correction techniques that involve cloning the classical data. For example, on long classical links, a repeater copies and retransmits a signal.. However, we are unable to clone quantum data. We thus need a different approach to get around the loss of quantum information.

Entanglement swapping works as follows: given two pairs of entangled qubits \(A_1-A_2\) and \(B_1-B_2\) we can transfer the entanglement to \(A_2-B_2\) by doing a joint measurement, known as a Bell state measurement, on \(A_1-B_1\). It actually turns out that the Bell state in which we measure \(A_1-B_1\) will be the entangled state in which \(A_2-B_2\) collapses.

For networking purposes, we can create a long-distance entangled pair by doing multiple entanglement swaps on a chain of shorter-distance entangled pairs. In this post, we will explore Bell state measurements, the mechanics and mathematics of the entanglement swapping protocol, and how we can use entanglement swapping to distribute long-distance entanglement in greater detail.

Bell state measurements

Bell state measurements are necessary in several quantum protocols in addition to entanglement swapping. For example, they are vital in the quantum communication protocols: quantum teleportation and superdense coding. A Bell state measurement is actually a specific example of the extraordinarily important quantum tool of being able to measure in different bases.

Measurement bases

To acquire any information from a quantum-mechanical system, we need to make measurements on it. Each time we make such a measurement, we need to pick which basis we are measuring in. Typically, we measure our quantum states in the computational basis—that is, in terms of 0’s and 1’s—but we could choose any other orthonormal basis. It turns out that there are infinitely many different measurement bases. This will work to our advantage.

Several quantum protocols require use of more than one measurement basis. For example, the BB84 quantum key distribution protocol requires use of exactly two different measurement bases. Other quantum protocols require measuring quantum states in bases other than the computational basis. For example, in entanglement swapping, quantum teleportation, and superdense coding, we need to make a (Bell state) measurement in the Bell basis.

The Bell basis

The Bell basis consists of the four orthogonal Bell states:

A Bell state measurement is a measurement of a 2-qubit system in the Bell basis. We can express any two-qubit state \(\ket{\Psi}\) as a superposition of Bell states. Whenever we measure a 2-qubit system in the Bell basis, we will get exactly one of the four Bell states back as the result.

Performing Bell state measurements in the computational basis

As long as we have access to the appropriate quantum gates and can measure in the computational basis, we can simulate measurements in any other basis. For example, we can actually perform Bell state measurements by using a Controlled-NOT (CNOT) gate (control on qubit 1 target on qubit 2) and then a Hadamard (H) gate (on qubit 1) in succession and then measuring in the computational basis. We will see in the next section how we can actually implement and apply these two gates to perform a Bell state measurement on photonic qubits.

Applying the Controlled-Not (CNOT) gate (control on qubit 1 target on qubit 2) and then the Hadamard (H) gate (on qubit 1) will send \(\ket{\Phi^+}\) to \(\ket{00}\), \(\ket{\Phi^-}\) to \(\ket{10}\), \(\ket{\Psi^+}\) to \(\ket{01}\), and \(\ket{\Psi^-}\) to \(\ket{11}\). We see that use of these two gates will map each Bell basis state to a computational basis state. By applying these two gates to any two-qubit system and measuring in the computational basis, we actually get the measurement of the system in the Bell basis. For example, measuring \(\ket{11}\) in the computational basis right after applying these two gates gives us a measurement of \(\ket{\Psi^-}\) in the Bell basis.

Now that we better understand Bell state measurements, we will explore the entanglement swapping protocol in more detail and see exactly how it is enabled by them.

The entanglement swapping protocol

We explore how the protocol works mechanically and then get into the math for why it works. In this section, we will consider two entangled pairs \(A_1-A_2\) and \(B_1-B_2\) that are in the \(\ket{\Phi^+} = \frac{1}{\sqrt{2}} \ket{00} +\ket{11}\) Bell state. We will then perform a Bell state measurement on \(A_1-B_1\) to transfer the entanglement to \(A_2-B_2\).

The mechanics of entanglement swapping

Let’s look at one method of performing entanglement swapping, using two entangled pairs and a Bell state measurement station. First, we send one qubit from each of the entangled pairs to a Bell state measurement station. The design of a Bell state measurement set-up is dependent upon which type of qubits we are using. Let’s consider the scenario in which we are trying to perform entanglement swapping on photonic qubits. In this case, the devices holding our entangled pairs will be connected to the Bell state measurement station via optical fibers. We can then easily transmit the photonic qubits to the Bell state measurement station via these optical fibers. The Bell state measurement station will consist of the gates and measurement devices needed to perform a Bell state measurement on photonic qubits. Complete Bell state measurement set-ups (those that are able to distinguish between the four Bell states) often include beam splitters, photonic CNOT gates, and photon detectors. However, it is much more common to use partial set-ups (in this case, those only being able to reveal two of the four Bell states) that include only beam splitters and photon detectors. This is because partial set-ups of this type are generally less expensive and easier to implement.

Schematic of entanglement swapping using a Bell state measurement station implemented with a beam splitter and single photon detectors.

It turns out that in addition to what we have already mentioned we will also need to use classical communication—known as heralding—to implement entanglement swapping. In the example given above, we need to use heralding to communicate the measurement result from the Bell state measurement station to the end nodes.

Heralding is an awesome tool that allows us to implement entanglement swapping and the other quantum networking protocols. However, our reliance on heralding in first-generation quantum networks will turn out to be the big limiting factor of their performance and potential. In later generations of quantum networks, we will use different protocols, such as moving away from entanglement swapping, so as to do away with heralding as much as possible and eventually completely.

The math of entanglement swapping

Our initial state for the two Bell pairs, \(A_1 A_2 B_1 B_2\), can be expressed by:

$$(1) \Psi_{A_1 A_2 B_1 B_2}=\frac{1}{2}\ket{0000}+\ket{0011}+\ket{1100}+\ket{1111}$$.

Let’s rewrite equation (1), so that we can place the qubits that we will perform a Bell state measurement on next to one another – on the left – and place the qubits we want to entangle with one another – on the right. The state itself has not changed, but rearranging the order of the qubits in our equation as  \(A_1 B_1 A_2 B_2 \), our state can now be expressed by the equation:

$$(2) \Psi_{A_1 B_1 A_2 B_2}=\frac{1}{2}(\ket{0000}+\ket{0101}+\ket{1010}+\ket{1111})$$.

We will now do a little arithmetic magic. For clarity, we will break-up this arithmetic into two steps. We first add each state from equation (2) to the system. This multiplies each state in equation (2) by two. However, we do not want to change the value of our system, so we will cancel out this effect by dividing the whole system by two—that is, changing the scalar from in front of equation (2) from \(\frac{1}{2}\) to \(\frac{1}{4}\). Here we have effectively multiplied our system by 1. We now have our state in the form:

$$(3a) \Psi_{A_1 B_1 A_2 B_2}=\frac{1}{4}(\ket{0000}+\ket{0101}+\ket{1010}+\ket{1111}+\ket{0000}+\ket{0101}+\ket{1010}+\ket{1111})$$.

We now add \(\ket{0011}-\ket{0011},\ket{1100}-\ket{1100}, \ket{0110}-\ket{0110}\), and \(\ket{1001}-\ket{1001}\) to this resulting system. We have not changed the state of the system since we immediately subtract each state that we added. Here, we have effectively added zero to the system. Doing these additions and subtractions, and re-ordering the terms, our system can now be written as:

$$(3) \Psi_{A_1 B_1 A_2 B_2}=\frac{1}{4}(\ket{0000}+\ket{0011}+\ket{1100}+\ket{1111)} +(\ket{0101}+\ket{0110}+\ket{1001}+\ket{1010}) \\ +(\ket{0101}-\ket{0110}-\ket{1001}+\ket{1010})+(\ket{0000}-\ket{0011}-\ket{1100}+\ket{1111})$$

So why did we go through this trouble of rewriting our equation and why did we group the states in our superposition in fours?

In equation (3), we have actually written \(\Psi_{A_1 B_1 A_2 B_2}\) as:

 $$ \Psi_{A_1 B_1 A_2 B_2}=(\ket{\Phi^+}\ket{\Phi^+})(\ket{\Psi^+}\ket{\Psi^+})(\ket{\Psi^-}\ket{\Psi^-})(\ket{\Phi^-}\ket{\Phi^-})$$

This tells us that if we measure \(A_1-B_1\) in one of the four Bell states, then \(A_2-B_2\) must collapse into that exact same Bell state. Hence, performing a Bell state measurement on \(A_1-B_1\) will leave \(A_2-B_2\) in an entangled state.

Entanglement swapping in quantum networks

The diagram below demonstrates how we can create a long-distance entangled pair by doing multiple entanglement swaps on a chain of shorter-distance entangled pairs.

In this diagram, we are given four entangled pairs of qubits \(A_1-A_2, B_1-B_2, C_1-C_2, D_1-D_2\). Each of these pairs can be on a single quantum device—in which case we perform elementary entanglement generation in our first step—or on neighboring quantum devices.

We first perform two entanglement swaps by doing Bell state measurements on \(A_2-B_1\) and \(C_2-D_1\) to obtain entangled pairs \(A_1-B_2\) and \(C_1-D_2\) respectively. Next we perform a Bell state measurement on \(B_2-C_1\) to obtain the entangled pair \(A_1-D_2\), which were the two furthest qubits away from each other in our initial system. This process can be abstracted to much larger chains of entangled pairs.

Quantum repeaters are the part of the quantum network that distribute entanglement. First-generation quantum-repeaters will contain devices to enable entanglement swapping. Quantum repeaters need to ensure that we have high-quality distributed entanglement. Because of this, first-generation quantum repeaters will also contain devices to enable entanglement distillation, a way to increase the quality of entangled systems by consuming more lower-quality entangled systems. Entanglement distillation is incredibly important for first-generation quantum networks and we will explore it in much greater detail in its own post.