**Qumodes vs Qubits explained, part II: information processing with qumodes**

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*by Jelmer Renema*

It’s time to continue our discussion about qumodes. In the previous discussion, we talked about qumodes vs qubits from the point of view of state spaces, i.e.: how much information can you store in how many quantum particles? This time around, I will discuss how entanglement forms in such a processor. For that, I first have to explain what a linear quantum photonic processor is.

**Linear quantum photonic processors**

A linear quantum processor (which is QuiX’s product) sits at the heart of an optical quantum information processing device. Mathematically, a linear quantum photonic processor is a *set of controlled, coupled qumodes*. As we discussed in the previous section, a qumode is a single quantized light path. We also discussed how it turns out these qumodes have the same physics as harmonic oscillators.

Let’s start by talking about how to make these qumodes happen in the first place. In principle, a qumode is anything that can carry a beam of light (for example, some arrangement of mirrors), but if you want to manipulate the light, you want to build up your qumodes in such a way that you can control them.

At QuiX, we implement our qumodes in something called integrated optics. The idea of integrated optics is that rather than have the light go through the air (‘free space’) the light is guided through channels called waveguides. These channels consist of material which traps the light and which allow it to be guided, even around corners.

We work with two-dimensional structures, where all the light is guided on a single plane of a chip. Therefore, you can almost think of these channels as railway tracks, and the photons as trains following the railway tracks. If you’d look at one of these chips from the top, it’d be just like looking at a Tube map, with paths, intersections, and so on.

As we said previously, the key to doing something useful with these qumodes is to couple them. Coupling means that the light can ‘jump’ from one qumode to another qumode. In practice, this is achieved by placing two waveguides in very close proximity to each other. The light then undergoes quantum tunneling, jumping from one waveguide to the next. In this way, you can physically realize intersections: the light can take a left or a right at an intersection, and jump from one track to another.

On top of just being able to couple the light, it is also vital that this coupling is done in a controlled way. How this is done is beyond the scope of this post (there’ll be another one talking in detail about integrated optics soon!) and it’s one of the things that the engineers at QuiX spend a lot of time and effort on, but let’s for the moment step over the question of how to tune the coupling between one waveguide and another, and assume that we’ve also solved that problem. What you can then build is a tunable switch (point) between two waveguides, that allows you to shunt the light from one path to another.

So what you end up with is something very much like a marshalling yard for light: a series of waveguides going in at one end, then a whole series of switches connecting all the inputs to all of the outputs, and finally a series of output waveguides on the other end. The whole thing can be controlled by tuning the probability that the light jumps from one waveguide to the next at each waveguide intersection. This complete device is what we call a linear photonic processor.

**Superposition**

Now it’s time to bring quantum mechanics back into things. Something that doesn’t neatly fit into the story of the train yard is that the light can divide itself over two paths, forming a quantum superposition. In fact, it’s possible to flip a switched to a halfway position, so that part of the light does jump from one waveguide to another, and part of the light stays in its original path. If you do this with single photons, you end up with a superposition.

In fact, that light can do this is something you experience in everyday life if you look out the window at night. The window lets through some light but also reflects some, meaning that you see both the reflection of the room behind you and what’s going on outside.

**Entanglement**

Having explained how a photonic processor works, we can finally talk about entanglement. Entanglement is one of the key features that makes quantum mechanics different from our classical experience of the world. Entanglement between two objects means that you can no longer assign a state to each object individually, but only to both objects jointly.

In a linear optical processor, entanglement is formed through something called the Hong-Ou-Mandel effect (HOM effect) and its generalizations. The HOM effect works as follows: if you have two photons and they encounter each other on one of the switches in the photonic processor, both photons must take the same path, going either into one waveguide, or into the other. Since there is now no longer a definite state for either of the two photons, but there is one for both of them together, the whole state is entangled.

The HOM effect is at the heart of the quantum nature of a photonic processor, and it is a quantum effect that has no classical analogue. There also exist equivalent effects for more photons, but for our purposes it is sufficient only to consider the two-photon case for the moment.

The interference effect which HOM relies on occurs because there is an interference of histories between the process of the first photon being transmitted and the second being reflected and the first being reflected and the second being transmitted. These two processes produce an identical outcome (namely one photon in each qumode) and since in quantum mechanics, two processes whose outcomes cannot be distinguished add up as probability amplitudes rather than as probabilities, the two processes interfere. In this case, it turns out they precisely interfere destructively, so the event is never observed, leaving the two outcomes for which both photons end up in the same qumode.

What is important to note about HOM is that it relies crucially on the quantum states which are present at the input of the system, namely single photons. It turns out that if you would have used for example laser light, the effect would not occur and there would not be any entanglement. It also relies crucially on the two photons being exactly identical. This places severe demands on the source that is producing the photons, which we’ll go into in a future post.

In a photonic processor, this process repeats at each node. As the light makes its way from one layer of switches to the next, entanglement grows, until eventually all the photons have become entangled. And that’s how entanglement builds up in a photonic processor.