Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

February 28, 2019

Sporadic SICs and Exceptional Lie Algebras II

Posted by John Baez

guest post by Blake C. Stacey

Today, we carry forward with the project we began last week: exploring Symmetric Informationally Complete quantum measurements, otherwise known as SICs. They’re really just maximal sets of equiangular lines in a complex vector space!

In our first post, we laid the groundwork and studied one example: a set of four equiangular lines in 2\mathbb{C}^2. Now, we move up a dimension and investigate a set of 9 equiangular lines in 3\mathbb{C}^3. This will bring the exceptional Lie algebras into the narrative, and we’ll also get a chance to greet a biodiversity measure and a polytope known as the 24-cell.

Posted at 10:00 AM UTC | Permalink | Followups (13)

February 20, 2019

Tychonoffication

Posted by John Baez

Joshua Meyers is a grad student in my real analysis class. We had an interesting conversation about topology and came up with some conjectures. Maybe someone has already proved them. I just want to write them down somewhere.

Posted at 5:17 PM UTC | Permalink | Followups (26)

Sporadic SICs and Exceptional Lie Algebras I

Posted by John Baez

guest post by Blake C. Stacey

Sometimes, mathematical oddities crowd in upon one another, and the exceptions to one classification scheme reveal themselves as fellow-travelers with the exceptions to a quite different taxonomy. I am grateful to John for giving me the opportunity to discuss one such confluence, where quantum information theory comes together with geometry, root systems and even the octonions.

In what follows, I will be divvying up these notes into blog posts. The first step is to introduce the geometrical statement of the SIC problem. Then, we’ll establish some basics about quantum theory, which may be fairly standard if you learned out of Mike and Ike while being a little un-standard outside of quantum information.

Posted at 5:21 AM UTC | Permalink | Followups (4)

February 7, 2019

Applied Category Theory 2019

Posted by John Baez

I hope to see you at this conference!

Here’s some information about it, such as how to submit papers.

Posted at 7:42 AM UTC | Permalink | Post a Comment

February 4, 2019

Jacobi Manifolds

Posted by John Baez

Here at the conference Foundations of Geometric Structures of Information 2019, Aïssa Wade of Penn State gave a talk about Jacobi manifolds. She got my attention with these words: “Poisson geometry is a good framework for classical mechanics, while contact geometry is the right framework for classical thermodynamics. Jacobi manifolds are a natural bridge between these.”

So what’s a Jacobi manifold?

Posted at 7:40 PM UTC | Permalink | Followups (8)