# Elementary entanglement generation with beam splitters

The first step in generating long-distance entanglement is to entangle two directly connected quantum nodes, known as elementary entanglement generation. Then, in quantum repeater networks, these point-to-point links can be “glued” together using a process known as entanglement swapping. Finally, these long-range pairs can be consumed by a wide range of applications, such as quantum secure communication and distributed quantum computing.

But, how do we generate entanglement in the first place? In this post, we’ll answer this question by taking a closer look at one specific way of generating point-to-point entanglement using an optical tool called the beam splitter

## What is a photonic qubit?

A photon is an individual particle of light. We can manipulate properties of photons to physically realize qubits. For example, we can encode qubits in the polarization of photons. To do this, we must first explore how the linear polarization of a photon behaves like a qubit.

Just like qubits can be in the $$\ket{0}$$ state, the $$\ket{1}$$ state, or a superposition of these two states, photons can have vertical, horizontal, or a superposition of these two states of linear polarizations. These polarization-encoded qubits—in the absence of noise—show all the properties of qubits:

1. By measuring the linear polarization of a photon—in terms of vertical or horizontal—the photon will end up with either exactly vertical or exactly horizontal polarization.
2. Measurement collapsing the linear polarization from a superposition of vertical and horizontal into exactly one of them is probabilistic.
3. The closer the original linear polarization is to vertical, then the higher the probability that the photon will be measured with vertical polarization. Similarly, the closer the original polarization is to horizontal, then the higher the probability that the photon will be measured with horizontal polarization.
4. Whenever a photon is in a superposition of horizontal and vertical polarization, these two states will interfere with one another. The way the two states interact is parameterized by the measurement probabilities for the linear polarization of the photon collapsing into vertical and horizontal polarization and by a property known as phase, which occurs in all quantum objects.
5. A system of at least two polarized photonic qubits can be entangled. This means that measuring the linear polarization of one of the entangled photonic qubits will have an effect on the measurement probabilities of the other entangled photonic qubits.

It turns out that photonic qubits work well in quantum devices and even better in quantum networks.

## Photonic qubits in quantum networks

The goal of quantum networks is to distribute entanglement between multiple quantum devices, which can be far distances apart. Because of how effectively we can transmit photonic qubits via optical fibers, we almost always use them as the communication qubits in the network—that is, the qubits that are used to communicate information between quantum devices. In general, photons have many valuable attributes as qubits, which also makes them a popular choice as device qubits—the qubits that a quantum device uses locally.

In this post, we will discuss one method to entangle a pair of photonic qubits. We will then discuss how we can make use of this method that acts on and can entangle photonic qubits to entangle a pair of non-photonic qubits that never directly interact. Both of these protocols are implemented using a device known as a beam splitter.

## What is a beam splitter?

A beam splitter is an optical tool that splits an incoming beam of light into two (or more) beams of light. The resulting beams of light do not need to be split evenly. A beam splitter partitions the individual photons from the initial beam of light based on these photon’s properties. Different beam splitters partition photons based on different properties, such as polarization, wave-length, or intensity. Beam splitters can also affect the phase of the photons. The resulting state of a photon that entered from one input port of the beam splitter will always differ by a constant phase shift from the resulting state had the photon entered from another input port of the beam splitter. In fact, the specifics of this phase shift will play a key part in both of the entanglement protocols we are going to discuss.

In this post, we will focus on polarizing beam splitters, which partition photons based on their linear polarization. Photons measured with horizontal polarization will be sent in one direction and photons measured with vertical polarization will be sent in another direction. If we are using polarization-encoded photonic qubits in an optical set-up, then we will use polarizing beam splitters. For other photonic qubits, we will use a different type of beam splitter that partitions photons based on the two states that defines that specific type of photonic qubit.

## Entangling photonic qubits using beam splitters

Consider a beam splitter with two sides that has an input port and output port on each of its two sides that sends vertically-polarized photons through the first output port and sends horizontally-polarized photons through the second output port. Also consider two physically indistinguishable (that is, identical in their observable physical properties) input photons, photon A and photon B, in the two input ports of this beam splitter. This set-up would look something like this:

Let’s say photon A and photon B have linear polarization exactly in the middle of vertical and horizontal. This means that there is a 50% chance photon A will exit the beam splitter through Output Port 1 and a 50% chance that photon A will exit the beam splitter through Output Port 2. Photon B has these same odds. We will examine the probabilities of which output ports the photons will exit through given that both photons entered the beam splitter at the same time.

If we flipped two coins at the same time, we would have a 25% chance of each outcome: HH, TT, TH, and HT, where H and T stand for heads and tails, respectively.  One might then expect that if the two photons enter the beam splitter at the same time, we would have a 25% chance of each outcome:

It turns out, however, that this is not what happens in reality.

## Hong-Ou-Mandel interference

Due to a quantum-mechanical property known as “Hong-Ou-Mandel interference,” there actually turns out to be a 50% chance of outcome 1 (where photon A and photon B both exit the beam splitter through Output Port 1) and a 50% chance of outcome 4 (where both photon A and photon B exit the beam splitter through Output Port 2). We will still find this same result whenever we have this same set-up but with a different type of photonic qubit and corresponding beam splitter. So, why does this happen?

When the two photons go through the beam splitter, they are in a superposition of the four possible outcomes states. Because the photons A and B are indistinguishable and both photons enter and exit the beam splitter at the exact same time, the outcome states 2 and 3 (where the two photons exit the beam splitter in different output ports) will also be indistinguishable. We can only determine that one photon exited from each of the two output ports, but we cannot determine which photon exited from which output port. We treat outcomes 2 and 3 as the same outcome and, amazingly enough, this is exactly what nature does too. These two outcome states, however, have inverse phases due to the two photons entering ports on opposite sides of the beam splitter. We can represent the resulting superposition of our system by the following diagram:

Just like when we subtract a number by itself we obtain the value 0, when two identical states with mutually inverse phases interfere with one another, they will cancel each other out completely. This eliminates any chance of ending up with either of these two outcome states. This means that we can more simply represent the state of our system by:

This state is actually known as the $$\ket{\Phi^-}$$ state and it is one of the four Bell states.

## Why is the resulting state entangled?

The resulting superposition state of the two photonic qubits after going through the beam splitter has a 50% chance of being measured in outcome 1 (with both photons having vertical polarization) and a 50% chance of being measured in outcome 4 (with both photons having horizontal polarization). We will discuss why this is an entangled state.

If we measure the linear polarization of one of the two qubits, then we will know the linear polarization of the other qubit. For example, if we measure one of the qubits as having vertical polarization, then we know that the other qubit will also have vertical polarization upon measurement. We know this because in either of the two possible outcome states the two qubits have the same linear polarization. Similarly, if we measure one of the qubits as having horizontal polarization, then we know that the other qubit must also have horizontal polarization when measured. This means that the unmeasured qubit’s probability distributions for linear polarization change from 50% and 50% to 100% and 0% when we measure the other qubit. Thus, by definition, these two qubits are entangled.

## Non-photonic qubits

There is not currently—and may never be—a universal choice of qubit for quantum technology. In fact, there is little overlap between industry leaders’ choices for qubits in their quantum devices. Communication qubits in quantum networks are typically photonic, but another type of qubit might be used on the quantum devices in the network. We thus need photonic qubits to facilitate communication between the other types of qubits. To understand this better, we will explore a method in which photonic qubits are used to entangle non-photonic qubits. In this case, we will use the intrinsic spin of an electron that is trapped inside a nitrogen-vacancy (NV) defect in diamond as our choice of non-photonic qubit. However, analogous set-ups have also been used for several other types of non-photonic qubits.

## Entangling electron spin states using photonic qubits

We start this protocol with two electrons, each isolated in an NV defect in diamond. Using lasers—in a method like this one—we excite the electrons in such a way that they emit a photon whose linear polarization is entangled with the spin of its generating electron. This entangled superposition state has a 50% chance of being measured with the electron in the spin up state and the photon having vertical polarization and a 50% chance of being measured with the electron in the spin down state and the photon having horizontal polarization. Measuring the spin of the electron would tell us what the linear polarization of the photon will be when measured. Similarly, measuring the linear polarization of the photon would tell us what spin state the electron will be in when measured. For example, if the spin of the electron is measured to be in the spin up state, then the photon will have vertical polarization when measured.

These two photons will be indistinguishable upon generation. We transmit these two photons via optical fibers to the polarizing beam splitter described in the previous section. We make sure that these two photons arrive at the two distinct input ports of the beam splitter at the same time—to ensure that Hong-Ou-Mandel interference occurs. How to ensure their precise timing is a technical challenge in its own right, which can be addressed using specialized controller software. Depending on the measurement results (which output ports of the beam splitter the photons exit through) the spin states of the electrons can become entangled.

If only the left photon detector clicks, then the spin states of the electrons will be in the $$\Psi^+$$ Bell-state. If only the right photon detector clicks, then the spin states of the electrons will be in the $$\Psi^-$$ Bell-state. If neither or both detectors click, then the spin states of the electrons will not be entangled and we try again.

Here is a schematic for the set-up we just described:

We now explore these different outcome scenarios and why some of them produce an entangled state and others do not.

## Entanglement swapping

Though we were considering this protocol in the context of elementary entanglement generation, it turns out that this specific protocol can also be used to extend entanglement across a network. 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. Let’s examine the linear polarization states of the two photons in each of the outcome scenarios.

If only the left photon detector clicks, then the linear polarization states of the photons will be measured in the $$\Psi^+$$ Bell-state. If only the right photon detector clicks, then the linear polarization states of the photons will be measured in the $$\Psi^-$$ Bell-state. If neither or both detectors click, then the linear polarization states of the photons will not be measured in any of the Bell-states. We can see that the outcome results for the spin states of the electrons are due entirely to the Bell-state measurements on the linear polarization states of the photons and entanglement swapping.

Entanglement swapping can be used for elementary entanglement generation, but it is most famous and vital for its ability to help distribute high-quality long-distance quantum entanglement.