Study Notes

Once in a while, I try to type up notes on subjects that I’ve been studying during my master’s and post them in this page. Bear in mind that most of them are works still in progress. If you spot any mistakes, feel free to contact me.

Tensor Product of Vector Spaces

Despite their ubiquitousness, during my undergraduate years, tensors were usually presented as mysterious objects $T_{j_1…j_n}^{i_1…i_m}$ that have tons of components and are characterized by the way they transform under changes of coordinate systems. To me, such presentation is a bit unsatisfying for two reasons: first, it does not define what kind of mathematical object a tensor is; second, it prevents students from getting a (fairly) simple exposition to the mathematical notion of ‘‘taking a quotient’’. In these notes, I try to carefully construct the tensor product of vector spaces and characterize it by its universal property.

Generalized Probabilistic Theories (Slides)

I presented a seminar on the basics of GPTs to my research group, here are the slides.

Notes on Category Theory (incomplete)

Category theory is a fascinating subject in and of itself. Remarkably, quantum theory can be formulated in a category-theoretical language and, since it is a particular type of generalized probabilistic theory (GPT), this gives me the impression that GPTs can be formulated within such framework, too. With that in mind, I am studying a bit of category theory in the hopes that I will get a better picture of the Choi-Jamiołkowski isomorphism and how to extend it to GPTs without resorting to purification.