# Linear algebra underlying quantum entanglement?

Hope this question is appropriate. I think I saw certain claims that quantum entanglement is a certain phenomena that can be explained (or modelled) in terms of tensor products in linear algebra. I wonder if this is the case, and if yes, is there some nice mathematical source? If you have your own insight in the question, I would be very happy to learn about it.

I am asking the question because want to mention it in an undergraduate course on representation theory to cheer up students.

PS. Since the proposed suggestions are mainly books (or physics literature), I start to suspect that what I was looking for doesn't exist. I guess, I wanted some short piece of text (say 1-20 pages long), that would be additionally purely mathematical. Basically, some compression of information is needed. How to make a 5 minutes talk out of 10 books?

• This is true and is covered is mostly any book on quantum information theory. Nielsen and Chuang is possibly the most standard. If you want a linear algebra book instead, Woerdeman's "Advanced Linear Algebra" has a digression at some point about the difference between separable and entangled states via the tensor product. – Nathaniel Johnston Mar 5 at 13:29
• Thank you! As far as I can see, this is Section 7.8 in Woerdeman? – aglearner Mar 5 at 13:38
• That's correct. I'll provide an expanded answer now. – Nathaniel Johnston Mar 5 at 13:43
• This expository piece on Quanta (yesterday!) may be of interest quantamagazine.org/… – J.J. Green Mar 5 at 14:53
• You can read the short chapter on quantum probability in William's book "Weighing the odds: a course in probability and statistics". – Stéphane Laurent Mar 6 at 8:39

Many introductory books on quantum information theory go over the linear algebraic tools necessary to study the topic, including the tensor product (since it indeed models quantum entanglement). Taking the tensor product of two or more quantum states (pieces of quantum information) is analogous to forming a bitstring out of two or more bits (pieces of classical information).

I'll summarize some references that go into this topic below.

Quantum information books:

Linear algebra books:

Survey papers:

• Thank you Nathanel. I had a look in Section 2.1.7, it only speaks about tensor products, there is no mentioning of entanglement. Searching for the word entanglement in this book, I see that it is mentioned multiple times, but I am afraid that I will not be able to extract mathematical "gist" from it. Do you think you can point me to a more precise place where "maths of entanglement" is explained (whatever this means). Maybe my initial idea is not realistic - wanted to say some non-trivial maths statement about tensor products to students next week and add two words about entanglement. – aglearner Mar 5 at 16:41

Maybe these books be interesting:

Linear Algebra for Quantum Theory

Per-Olov Löwdin

Quantum Algorithms via Linear Algebra: A Primer

Richard J. Lipton Kenneth W. Regan

Quantum Computing: From Linear Algebra to Physical Realizations

Mikio Nakahara

Quantum Mechanics: The Theoretical Minimum

Leonard Susskind

• Thanks. Among these books, which one do you think is the most appropriate, if I have to choose one? I am a mathematician who wants to tell to maths students (3-4th year of university) a 5-10 minutes story about entanglement while introducing tensor product in a lecture. First, of course, I need to understand what is this myself. (maybe what I am asking is not feasible...) – aglearner Mar 6 at 8:07
• @aglearner I think for a mathematician, Keyl's review is particularly suited. – lcv Mar 6 at 9:20
• When I want to talk about entanglement, I try to give to students some visual prespective. I begin with hot (white color) and cold (black color) balls which we put them in a basket. After sometimes (until equilibrium) you can separate them by their color but not by their heat (simulation of entanglement and not it is). After that I tell them some odd things about this phenomena and when the students are excited, I tell them that math can interprets this odd phenomena simply. I prefer the books by the order in my answer. – Shahrooz Janbaz Mar 6 at 9:28
• @SharoozJanbaz I am not sure what you want to say (to your students) but what you are describing is entirely classical and has nothing to do with entanglement. Did you mean it to say what entanglement is not? – lcv Mar 6 at 17:12
• @Icv I pointed out that it is not quantum entanglement, but it is a kind of simulation. Also, I want to explain the indistinguishable nature of entangled states until you do some measurement! The detail is much more and I can not explain it in the comment. – Shahrooz Janbaz Mar 6 at 19:23

A concise and a bit more mathy review is that by M Keyl Fundamentals of quantum information theory which is based on a $$C^*$$ algebra approach.