Quantum Teleportation from scratch to magic. Part 4 — How it actually works: Quantum Pictorialism
At the end of part 3 of this series, after getting severely tangled in the weeds of trying to understand entanglement (let alone teleportation) I ended with a joke saying I would be back for part 4 to explain how it all REALLY works.
Well, guess what! With even more surprise than you (though probably less scepticism), I am back with a very different presentation of quantum computing that does away with matrices and kets and replaces them with lovely little diagrams. What’s more, teleportation actually makes sense in these diagrams!
Even better news is that you won’t have to hear me rant about time either. (To be fair, I wrote the last one in the middle of COVID lockdown and was beginning to forget the days of the week). Instead, I will be giving a very brief overview of the wonderful textbook “Picturing Quantum Processes” by Aleks Kissinger and Bob Coecke. These guys are part of a group at Oxford doing some very exciting, unconventional work who I call “Bob and the Bad Boys of Quantum” (aka Bob and the BBQs).
(NB I’ve stolen all of the diagrams in this article from these slides because trust me, you don’t want to see my handwritten drawings)
Why Processes?
The key motivation for this “Process Theory” is rather than asking “what is this thing?”, it asks “what can I do with this thing?”. This naturally leads to a higher-level (and more intuitive) way of presenting quantum circuits which basically amounts to plugging different boxes together with wires.
With that said, let’s check these diagrams out!
Wires and States
When asking what we can do with things, we first need a way of doing nothing. We represent the identity wire as follows:
Then, to do some random operation f on the wire, we simply put f in a box:
In this case, f has 1 input and 1 output since there is one line going in and one line going out. However, this is not always the case. We can take operations with multiple inputs or outputs and combine them together, for example like this:
By the way, time goes upwards in these diagrams, so you should read the above diagram like this:
Boxes and wires and boxes and wires, so far so good. What else do we need?
States and Effects
There are some special operations which don’t have any inputs, and so always output the same thing. These special operations are called states (for obvious reasons) and are even given their own special shape, a triangle! This means our old friend ψ the arbitrary state is now:
Remember that states used to be represented using kets (|ψ>). This is a why a triangle was chosen:
For the purposes of this article, you can think of states as qubits, however, these process diagrams can actually be used to represent much more than quantum circuits so in general, a state could be 0, 1, cheese, the word “cheese” and much more! (A little more about nouns at the end)
“But what about operations with no outputs?” How very precogniscent of you! In the previous part, I pointed out in a very roundabout way that measuring something is the opposite of preparing it. This observation was not as radical as I had thought and in fact, the Process Theory uses the same observation to define a measurement (aka “effect” or “test”) as an upside-down triangle:
This means that a specific effect, π, is observed. This can also be thought of as testing for π, since you need to ask the system in some way about what’s going on. Of course, depending on what is connected to that wire underneath, we will not necessarily always measure π. Instead, combining a state with an effect gives us a number which corresponds to the probability of a certain state giving us a certain effect:
The astute among you will recognise this as our other old friend, the Born Rule.
“Very nice! Can we finally do teleportation then?” Very nearly, but we need one final ingredient.
Bell States and Effect
Remember that the magic ingredient to whole teleportation stuff was that weird thing called a Bell pair. You might think that because it is made of two qubits, it could then just be two triangles next to each other. Nice try, but that would be wrong.
This is because a very crucial implication of entanglement is that entangled qubits are inseparable. Informally, this means that they are firm friends, but we already knew that. Formally, this means that there is no way they can be described independently. There is a very short, fairly marvellous proof that a Bell pair cannot be written as two separate kets, e.g.|a>|b>, however, unfortunately, Medium does not give me a margin in which to write it. In any case, the same holds true for these diagrams, meaning that a Bell pair cannot simply be two adjacent triangles.
Instead, *drum roll please*, it is a single triangle with two outputs:
Perhaps not as amazing as you were hoping, but don’t worry there’s more to come. First of all, the notation is simplified by removing the triangle:
This Bell state (known as a cup for obvious reasons) obeys the following relation:
Crossing the wires like that means swapping them, so if swapping the wires is the same as not swapping them, it is clear that the two outputs (whatever they may be) are identical.
Secondly, there is, of course, a corresponding Bell effect aka “cap”:
When cups and caps are combined together, they give us what’s known as the “yanking equations”:
Here’s why they’re called the yanking equations:
This means that you can transpose an operation f. To make it clearer when an operation has been transposed, we cut off the bottom-right corner of the square and so the transpose of an operation is literally just the “yanked” version :
Therefore, it is perfectly legitimate to slide operations around the bendy wires anywhere you like. One weird consequence of this, which I hinted at in part 3, is that Bell states and effects do some weird stuff to the flow of time (or at least causality). For example, the following equation shows that the order of operations in the diagram on the left is time-reversed since f is applied “before” g:
Is that weird enough for you?
In which case, we are ready for teleportation!
Teleportation Diagram
Recall the set-up is defined as follows. Alice (now called Aleks, after the textbook’s co-author) has a qubit which he would like to send to Bob. They also have a Bell pair between them. Alright, so let’s draw this set-up:
(NB a corner is also chopped off the state triangle so we can tell whether its transposed)
All we need is to somehow fill in the question marks so that the qubit can slide for Aleks to Bob. If you have been paying any attention or have the drawing ability of a 4-year-old, you will immediately realise all we need is a little cap:
This is just the yanking equations back in action
And then we’re done! Right??
Not quite, though very nearly. Remember that we can’t guarantee what measurement outcome we will obtain. Instead, there are four different possible outcomes, three of which add some error to the qubit. We can model each possible outcome as some operation (U_1, U_2, U_3 or U_4) on the middle qubit. This error will also slide to Bob:
So all we need is for Aleks to tell Bob (classically) which outcome he observed, and then Bob can apply the appropriate correction, as we saw the first time I described teleportation.
It might be a little frustrating that the new teleportation diagram still does not seem to actually explain what it is about entanglement that means this can happen, but I think what it does is show that a Bell pair’s property of “slidability” is as essential as its inseparability. In other words, continuing to persist with the question “how can entanglement do that?” would be like asking “why is an electron negatively charged?” — it kind of just is.
The sticklers among you might also still want to know how to represent the classical communication. Some may even be as curious as Brad Pitt in the film se7en and want to know the internal wiring of the correction operation. In which case I’ll make you a deal: if enough people like and comment on this article, maybe I’ll follow up with part 4.5 of this series and tell you what’s in the box (and perhaps how teleportation REALLY REALLY works). This requires a bit of a deeper looker into the Process Theory at what’s known as the “ZX-calculus” and so would lengthen this article without offering any huge insights on the teleportation side of things.
93 ’til infinity
And there you have it! I was inspired to write this article because of how intuitive the Process Theory makes teleportation. Firstly, the insertion of the Bell measurement used to puzzle me, but now is almost trivial. Keep in mind that while Quantum Mechanics was first formulated in the 1920s (and Dirac notation in the 1930s), teleportation is so counter-intuitive that it took until 1993 for it to be discovered. If we’d had these diagrams since the 1920s, imagine all the other discoveries that could have been made by now! Admittedly, the diagrammatic notation was made with teleportation in mind, however other mind-blowing features of entanglement such as entanglement swapping also drop right out of cups and caps.
Furthermore, one incredible discovery that Bob and the BBQ’s have already made is that grammatical meaning has the exact same structure as the quantum processes we have seen here. Nouns are states, verbs are operations, adjectives are states plugged into nouns with cups, and so on. This has given birth to the field of QNLP (Quantum Natural Language Processing) which promises to be both more efficient and more powerful than classical NLP (which is rapidly heading towards the limits of Moore’s Law). You can check out the recent “QNLP manifesto” here or a medium blog post about real experimental results here.
In any case, thank you very much for reading. Of course, I owe this entire article to Bob and the BBQ’s, whose textbook I once again highly recommend. The website for the “Categorical Quantum Mechanics” group at Oxford can also be found here: http://www.cs.ox.ac.uk/activities/quantum/ (unfortunately, their nomenclature isn't as catchy as their notation…)
