The Harvard trio of Arthur Jaffe, the Landon T. Clay Professor of Mathematics and Theoretical Science, postdoctoral fellow Zhengwei Liu, and researcher Alex Wozniakowski has developed a 3-D picture-language for mathematics with potential as a tool across a range of topics, from pure math to physics.
We present a 3D, topological picture-language for quantum information. Our approach combines charged excitations carried by strings, with topological properties that arise from embedding the strings in the interior of a three-dimensional manifold with boundary. A quon is a composite that acts as a particle. Specifically a quon is a hemisphere containing a neutral pair of open strings with opposite charge.
We interpret multi-quons and their transformations in a natural way. We obtain a new type of relation, a string-genus “joint relation,” involving both a string and the 3D manifold. We use the joint relation to obtain a topological interpretation of the C∗ Hopf algebra relations, that are widely used in tensor networks. We obtain a 3D representation of the Controlled NOT or CNOT gate (that is considerably simpler than earlier work) and a 3D topological protocol for teleportation.
Though not the first pictorial language of mathematics, the new one, called quon, holds promise for being able to transmit not only complex concepts, but also vast amounts of detail in relatively simple images. The language is described in a February 2017 paper published in the Proceedings of the National Academy of Sciences.
“We learn to use algebra, and we use letters to represent variables or other values that might be altered,” Liu said. “Now, when we look at research work, we see fewer numbers and more letters and formulas. One of our aims is to replace ‘symbol proof’ by ‘picture proof.’”
The new language relies on images to convey the same information that is found in traditional algebraic equations — and in some cases, even more.
“An image can contain information that is very hard to describe algebraically,” Liu said. “It is very easy to transmit meaning through an image, and easy for people to understand what they see in an image, so we visualize these concepts and instead of words or letters can communicate via pictures.”
“So this pictorial language for mathematics can give you insights and a way of thinking that you don’t see in the usual, algebraic way of approaching mathematics,” Jaffe said. “For centuries there has been a great deal of interaction between mathematics and physics because people were thinking about the same things, but from different points of view. When we put the two subjects together, we found many new insights, and this new language can take that into another dimension.”
In their most recent work, the researchers moved their language into a more literal realm, creating 3-D images that, when manipulated, can trigger mathematical insights.
Among their pictorial feats, Jaffe said, are the complex equations used to describe quantum teleportation. The researchers have pictures for the Pauli matrices, which are fundamental components of quantum information protocols. This shows that the standard protocols are topological, and also leads to discovery of new protocols.
“It turns out one picture is worth 1,000 symbols,” Jaffe said.
“We could describe this algebraically, and it might require an entire page of equations,” Liu added. “But we can do that in one picture, so it can capture a lot of information.”