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Now that quantum information theory (QIT) reaches the point that group theory have deeply combined with its applications and theoretical understandings, such as random benchmarking, quantum scrambling, and measurement induced phase transition (MITP). The celebrated Schur-Wely duality and tools like Pauli twirling have found ubiquitous use in the area. So, I am wondering are there some references recommended for systematically learn the group theory related to the topics mentioned above? Especially the Schur-Wely duality because I am interested in the MITP, which closely related to this topics. p.s. It is feverable that the math is not too heavy, and QIT-oriented.

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    $\begingroup$ Related. $\endgroup$
    – narip
    Commented Aug 17, 2022 at 4:56
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    $\begingroup$ Two courses with notes available: 1. Michael Walter's course on Symmetry and Quantum Information: qi.rub.de/qit18 2. Felix Leditzky's course on Representation-theoretic methods in quantum information theory: felixleditzky.info/teaching/FT22/math595_reptheory.html There's also a book by Hayashi titled "A Group Theoretic Approach to Quantum Information". $\endgroup$
    – HerrWarum
    Commented May 7 at 20:47

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Short answer: there are no shortcuts. Things like Schur–Weyl duality are from representation theory. This generally assumes you understand group theory to begin with. Most physics-oriented texts I've seen which try to introduce group theory go through it so fast it's unclear if the reader can actually get anything out of it before moving on. That makes it very difficult by the time they get to the "higher" subjects like representation theory. I would absolutely recommend actually taking the time to read the first several chapters of Fraleigh (easiest), Herstein's Topics in Algebra (more theoretical), or Dummit & Foote (upper undergraduate/beginning graduate level textbook). I'd claim the last one is required just to read the wikipedia article on Schur–Weyl duality, although there are plenty of better explanations of it once you get a firm grasp on groups or modules. From there you can try representation theory.

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  • $\begingroup$ Hi, thanks for the reply. I'm a bit familiar with group theory through algebraic graph theory. I was wondering about Fraleigh's book. Do you think it is enough to work through chapters related to groups and stop at rings and fields? I'm familiar with rings and fields, though I'm very rusty. $\endgroup$
    – MonteNero
    Commented Aug 16, 2022 at 21:38
  • $\begingroup$ @MonteNero you aren't the original poster, so I have to ask, worth it for what? My understanding of algebraic graph theory is limited to Godsil & Royle. There they concentrate on a single section of a group theory text, although a very important one. I don't remember them using it often during the text. Rings are groups plus a second operation though so to say you know rings but not groups is odd. But what is your goal? Same as above? $\endgroup$
    – esabo
    Commented Aug 17, 2022 at 23:06
  • $\begingroup$ Yes, let's suppose that the goal is to understand (not necessarily do meaningful work) the topics in the original post. $\endgroup$
    – MonteNero
    Commented Aug 17, 2022 at 23:30
  • $\begingroup$ To get an idea of what's going on, probably. To understand the equations, maybe not. Things like Pauli twirling are just the average of a group action. So if you understand group actions, you can understand that they are using it to randomize with respect to that action. But I imagine the author would have to be decently explicit about their math for that to come across sometimes since things like Fraleigh aren't that applied. $\endgroup$
    – esabo
    Commented Aug 18, 2022 at 13:11
  • $\begingroup$ Thank you very much $\endgroup$
    – MonteNero
    Commented Aug 18, 2022 at 16:04

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