For intro to QIT, thoroughly understanding the C^2 space and tensor products is critical. This book appears to be useful for that. Unfortunately, it can be hard to make the jump to the former way of looking at things, without someone explicitly pointing out how to think of L2 as a Hilbert space. Maybe this is just intuitive to some, but it wasn't for me.
And the converse seems to be true, too: you can apparently master the former without having a clue about the latter. As Scott Aaronson points out:
> Today, in the quantum information age, the fact that all the physicists had to learn quantum this way seems increasingly humorous. For example, I've had experts in quantum field theory -- people who've spent years calculating path integrals of mind-boggling complexity -- ask me to explain the Bell inequality to them. That's like Andrew Wiles asking me to explain the Pythagorean Theorem.
By the way, this example illustrates the point I made in another comment: you can have a long career in physics and even become professor of theoretical physics without ever caring about the interpretation of QM.
Edit: I can't resist copying the opening of a recent comment in that thread.
John Baez says:
September 18, 2018 at 4:05 pm
Peter Woit wrote:
"The state of the world is described at a fixed time by a state vector, which evolves unitarily by the Schrodinger equation. No probability here."
And perhaps no physics here, either, unless we say how the state of the world is described by that vector: that is, how we can use the vector to make predictions of experimental results.
(A long and insightful comment by Baez follows)
To say it another way, the configuration of the system remains rooted in position in physical space and it is the value space of the wave function where spin resides. This is made most clear in Bohmian mechanics, where the position of the particle is always defined in the theory, but whether a particle will end up in the spin-z up or down regions does depend on the experimental setup. One can arrange experiments where the particle, due to symmetry for example, will always go up given the same starting point even if one flips the magnetic field in the Stern-Gerlach device so that the experimental conclusion would be spin up in one scenario and spin down in the other. Same exact path happens in both cases, but the spin value conclusion is the opposite. That is to say, while position is determined ahead of time, spin is not. Spin is not a real property of the particle; it is a property of the wave function.
I've read some articles and seen some video courses on Quantum Mechanics, and it seems that there's always way more focus on the mathematics and abstract/simplified physics, and not much description of the actual experiments. I get that the main challenge of teaching it is understanding the mathematics, but it's hard to stay focused when - even though I understand the equations - I'm not sure what the mathematics describes physically. Or when I do understand it, that I have no idea how one would go about preparing the state.
I think I've gotten a half decent understanding of the Stern–Gerlach experiment, but that's about it.
As for interpetation of QM, I don't think that's the problem since one can just set up an experiment based on prediction from the mathematical models. Sure, you will have make multiple runs to show the predicted distribution of the non-deterministic, umm, stuff, but that isn't a problem.
A few articles about the experimental violation of Bell’s inequalities:
If one wants to get into some of the philosophy about it, then try:
Foundations of Quantum Mechanics: An Exploration of the Physical Meaning of Quantum Theory
You might have a look at https://www.amazon.com/Exploring-Quantum-Physics-through-Pro... . This book stops short of covering molecular beams, though.
You might also like John Bell's original papers. Feynman doesn't do entanglement. Conway's "free will" paper is good aswell.
I'm not sure why do you mean by "introductory discussion" but I think "popular science" books focus mostly on the controversial parts. At the university level, at least in my experience, you study a lot of classical physics, early quantum physics (1900-1925), and mathematical physics before taking a first look at Quantum Mechanics (Schroedinger/Heisenberg). And you may never be exposed to the relativistic extensions.
In contrast a textbook about thermodynamics will be all steam engines at the start, or maybe talk about a thermometer. Solid, sensible things and no talk about Gibbs free energy or any of the deep theory that exists.
You could start a QM book talking about the double slit experiment, but would have to write a lot of "one interpretation" and "some views say this". Or you could just brazenly pretend there is no controversy and make every reader that knows of it very suspicious indeed.
In any case introductory books that discuss the physics and not just the mathematics do also exist, like vol 3 in Feynman’s lectures.
Edit: I’m looking at Griffiths’ book and after defining Schroedinger’s equation in section 1.1 he devotes three pages to discussing its statistical interpretation in section 1.2 and mentions different interpretations, the Bell inequalities and the measurement problem. He comes back to these philosophical issues at the end of the book in a 12-page afterword.
That is indeed how my favorite QM book starts: Cohen-Tannoudji. See also Landau for one that starts from experiment and not theory.
For a recent head-scratcher, though, see https://www.nature.com/articles/s41467-018-05739-8
I think it's much less interesting than it seems at first sight.
Related (and clearer): In Defense of a "Single-World" Interpretation of Quantum Mechanics (Jeffrey Bub) https://arxiv.org/abs/1804.03267
It's a graduate level text, so probably somewhat dense if you haven't been exposed to the material in some way before, but as a mostly refresher I found it very good.
And, it's a Dover book, so it's very cheap.
If you get to the end, it is worth noting the similarity of the tensor product and the join operation in a relational database.
Check out the breezy writing in this paper: https://arxiv.org/abs/1602.07618
Also, the Rosetta stone paper (more advanced): https://arxiv.org/abs/0903.0340
I'm not sure if this is what the parent comment is getting at, and I'm not sure I see anything deep lurking there, but that could just be me.
Hope, you'll have to figure it out for yourself. It's not hard. If you just look at the details of how these two operations are defined it should be obvious.
I know tensor products. I know SQL joins. I see superficial similarities between the two, but no unifying underlying principle.
The world has had quite enough of nonsense thrown around because it's "obvious". Put your money where your runaway mouth is.
In the cross join case, all the elements are pairs of elements from the original tables. In the Hilbert space of a composite system only the separable states can be written as the tensor product of states of the subsystems.
"Observation 3.6.5: Interesting two-qubit states. Not every 4-dimensional vector can be written as a Kronecker product of two 2-dimension vectors, e.g., you can have a two-qubit state |Ψ⟩ such that:
|Ψ⟩ = ̸= |ψ⟩|φ⟩, for any one-qubit state |ψ⟩ and |φ⟩
These types of states (called entangled states) are very intriguing and play a
fundamental role in quantum mechanics."
But I have to stress that I do not intend this to be a deep insight, just an interesting (IMHO) observation.
Focusing on superficial similarities is a two-edged sword: anchoring on a familiar concept can help or make things more difficult. And it's guaranteed to annoy people! (I'm mostly getting over it, but I still dislike the use of the word "tensor" to refer to multi-dimensional arrays. Tensor has a meaning in geometry and tensor algebra is not about doing linear algebra on 2-d slices of a larger-dimensional object.)
You are now beyond the limits of my understanding. I never grokked category theory.
SQL join is a Cartesian product. The Tensor product is analogous to a Cartesian product except that the inputs and outputs are ordered tuples instead of sets. That's all I was getting at.
As for HN, a lot of people are going to be self-educated in advanced topics. That means there's going to be holes in people's knowledge, especially as it relates to alternative theoretical treatments.
No infinite dimensional spaces.
No discussion of self-adjoint operators
These are essential mathematical components of QM.
use special mathematics – complex numbers and linear algebra (vectors and
matrices). Unfortunately, most high school mathematics curricula around the world do not teach linear algebra. It’s not very complicated. It’s really just a different and
clever way to add and multiply numbers together, but it’s a very powerful tool."
"The Quantum Cryptography School for Young Students (QCSYS) is a unique, eight-day enrichment program for students hosted by the Institute for Quantum Computing (IQC) at the University of Waterloo. QCSYS will run August 10-17, 2018 with students arriving August 9 and departing August 18.
The school offers an interesting blend of lectures, hands-on experiments and group work focused on quantum cryptography"
>Can I still apply if I am currently in grade 12?
>Yes. If you are currently in grade 12 and you are attending university or college in the fall 2018 you are still eligible to attend the Quantum Cryptography School for Young Students (QCSYS).
So, high school students. Material looks accessible enough and well presented.
If you don't know what quantum computation is or why it's important, then you have a lot of catching up to do.
Absolutely beautiful way to present quantum mechanics.