For those of you following the Network Theory series, we’ve been trying to unify concepts across an apriori seemingly distinct range of topics. For this reason, I jokingly and seriously at the same time call this the grand unified network theory project. This post is more of a news item related to some potentially interesting work.
Before getting suckered into working on the network project by John Baez, I was considering topics related to quantum networks. Today I want to mention some recent work I took part in, related to quantum networks and initiated by Chris Wood from the Institute for Quantum Computing (IQC) in Canada. A future direction of the network theory project will be to consider open quantum systems. We might build on and use some of the results appearing in the following preprint.
There is a story behind how this project all got started, and if you have a moment, you can read it right now. Mike Mosca invited me to IQC to teach my course on tensor networks. Chris Wood must have been bored, but regardless of the reason, he showed up. He was not even enrolled in the course initially, but he liked it enough that he signed up. Chris was already an expert in open quantum systems, he wrote what I consider a very solid honours thesis on the topic
In his thesis, Chris explains a lot of the background in open quantum systems before going into several reserach results. You might be thinking, “that’s one heck of a masters thesis”, but in fact, this is his undergraduate thesis! He got a 1st class degree and a university medal for this, ended up doing a masters at Perimeter Institute and is now working towards his PhD at IQC. In his thesis, he made use of the so called quantum circuits model, and as is typical in the field of quantum information, he drew pictures such as
Here the wedge shaped diagram with edges labelled and depicts the so called Bell-state. He could have used a curved line like we did in our paper, but it’s just syntax. He drew other diagrams too, for increasingly complicated scenarios including
Where did all these diagrams come from and what do they mean? Well, we’re not going to have time to explain that here, but for those that are curious about quantum network theory, I can shamelessly recommend my own lecture series on the topic.
If you’re not happy with my course, I suggest you make a better one. I even placed all of the LaTeX source for my lectures notes online to download if you wanted to base parts of your new course on what I did. In my course, we talked a lot about using Penrose graphical notation for tensor network states. For instance,
Here Oxford Professor, Roger Penrose is expressing a so called density operator using the graphical tensor notation he pioneered. One of the key citations to his work includes
To get an idea of what sorts of things you can find in this 1971 paper, consider
Here Penrose is explaining what we call “Penrose wire bending duality”. As he explains, the input and output of diagram can be changed at will, by simply bending inputs to outputs and vise versa.
To get an idea of what this means exactly, consider the following figure from the paper.
What this is showing is known as the Kraus picture of open systems evolution. To explain this diagram, we have a quantum state acted on by operators .
Of course, expressing the known pictures of evolution into string diagrams would not get published in a journal. It is well known that one can express quantum equations in terms of string diagrams, and follows from work done as early as the 1960’s and 1970’s by Penrose and others. What we did was something different.
We can use Penrose duality and bend one of the wires the other way around. We can then slide a box around the bent wire and manipulate the diagram a bit to arrive at the following
The form we arrive at already has a name. It is called the superoperator picture of open systems evolution. We translated from one picture to another, using pictures. This was the point of the paper. There are several so called pictures of open systems evolution, and we considered how the Penrose graphical notation can be used to transform between them.
This is perhaps the simplest case, but it illustrates the key idea. If you are very interested, we encourage you to read the paper and take a look at figure 1.
The boxes are the different pictures we consider and for each arrow, we give a transformation between them. This is even explained a bit in the abstract.
We develop a graphical calculus for completely positive maps and in doing so cast the theory of open quantum systems into the language of tensor networks. We tailor the theory of tensor networks to pictographically represent the Liouville-superoperator, Choi-matrix, process-matrix, Kraus, and system-environment representations for the evolution of open-system states, to expose how these representations interrelate, and to concisely transform between them. Several of these transformations have succinct depictions as wire bending dualities in our graphical calculus --- reshuffling, vectorization, and the Choi-Jamiolkowski isomorphism. The reshuffling duality between the Choi-matrix and superoperator is bi-directional, while the vectorization and Choi-Jamiolkowski dualities, from the Kraus and system-environment representations to the superoperator and Choi-matrix respectively, are single directional due to the non-uniqueness of the Kraus and system-environment representations. The remaining transformations are not wire bending duality transformations due to the nonlinearity of the associated operator decompositions. Having new tools to investigate old problems can often lead to surprising new results, and the graphical calculus presented in this paper should lead to a better understanding of the interrelation between CP-maps and quantum theory.'
If you have ideas, we’d like to hear them: please feel free to email us. If you have a few quick questions about the paper, Chris Wood will be around today to respond to them. He lives in Waterloo Canada right now which is on the East Coast. I’m in Singapore, so if it’s the middle of the night in Waterloo, I’ll answer them.
Now before we go, I should mention what some of you might have noticed. We are using what David Cory suggested as the color convention. In the diagrams, like colored pictures are summed over. This could of course be replaced by attaching colored diagrams with a connecting wire in the Penrose graphical notation. However, the color convention proved helpful for our work considering open systems. To get an idea of how nice it looks, here is a proof of Penrose’s snake equation.