Quantum Information in Math Departments
Sounds like Applied Math.  In terms of a specific PI doing quantum information*, I don't know which school.  But when you mention your course interests and wanting to collaborate with people on real problems that is applied math all the way.

Here is a list of top programs:  https://www.usnews.com/best-graduate-schools/top-science-schools/applied-mathematics-rankings  Not passing judgment, but list gets you started.  I would look at some of their webpages and reach out via email to one or tow and just start educating yourself on what they're like versus what you want.  Maybe do a visit if one near.

You could also consider statistics grad programs.  That is a wonderfully applied field and one where there is some connection to industry, the sciences, etc.

*Music on CDs as opposed to vinyl?