Tutorial on resourcefulness (CI fix) #1577
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Overall its quite clear! The introduction feels a bit rocky: There are lots of technical terms, but none of them are yet defined. I do not have a good idea on how to deal with that, compared to what you have written here. Otherwise the rest of the demo is conceptually very rewarding to a reader who pushes through, and I think that you build up an intuition for these objects so nicely.
Just two high-level comments
- I don't fully agree with "complex" being interchangeable with "resource" (actually, i don't like the "complex" is used here at all). Equating "complex"="many resource" implies that f(x)=sin(1000*x) is significantly more complex than a function of the form
f(x) = \sum_{\omega < 100} \hat{f}(\omega) sin(\omega*x). I don't see why that is. The former is much more compressible, much simpler to write down. There is not much followup on what 'complex' means (beyond the comparison "complex"="many resource" in the very first paragraph) so I do not think that the word 'complex' adds very much here, and could just be replaced with some language related to resource theory (expensive, not-smooth?) - It was jarring when the representation theory for states in H ended up using representations acting on density matrices in B(H) rather than states in H. It seemed like the introduction was working towards a generalization of the Z_n example, and then suddenly we use some representation that involves conjugating density matrices. What went wrong? Why doesn't the naive generalization of what you already built up work for SU(2)\times SU(2)\times... ?? Everything you wrote was crystal clear, but the reasoning behind the choices is completely opaque...
| # This, of course, is not a very informative fingerprint -- one that is much too coarse! | ||
| # | ||
| # However, we have freedom to define the vector spaces for the fingerprint. | ||
| # It turns out that considering the space :math:`B(H)` of bounded linear operators, in which density matrices live, leads to |
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It turns out that considering the space :math:
B(H)of bounded linear operators, in which density matrices live, leads to a much more nuanced generalised Fourier spectrum. Although this space contains matrices, it is a vector space in the technical sense. This requires some mental gymnastics, but we can think of flattening all density matrices, and turning operators acting on density matrices into other operators that act on the flattened 1-dimensional vectors.
I am struggling with this. What is going on? Is there a deeper way to understand or explain why the rep G \rightarrow GL(H) is an irrep and the above steps don't work out, but that using GL(B(H)) - or GL(H \otimes H*)?? - is somehow the right way to do things??
(this is very relevant for something i am trying to work out on our project!)
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I have no clue. It's something I also feel is very arbitrary! Essentially, some representations just happens to have a trivial decomposition into irreps, and they are poor fingerprints. We somehow need to go to higher-dimensional spaces to get a better picture. I wonder if the highest-dimensional space then ends up to be the regular rep?
| # The point here is that while multipartite entanglement can get complex very quickly as we | ||
| # add qubits (due to the exponential increase in possible correlations), | ||
| # our generalized Fourier analysis still provides a clear, intuitive fingerprint of a state's entanglement structure. | ||
| # Even more intriguingly, just as smoothness guides how we compress classical |
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does "suggests ways of how we might compress quantum states..." refer to specific protocols that have been developed, or is this meant to be open ended? It sounds vaguely like how tensor-network simulations of quantum circuits works...
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I think it refers to an idea in the paper that uses this framework for something...
Co-authored-by: Paul Finlay <50180049+doctorperceptron@users.noreply.github.com>
…o resourcefulness_demo
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@peterse has given me an amazing review, and I implemented almost all suggestions. (In fact, all except the main two points; I think calling a resource a specific "measure of complexity" is fine for didactic purposes and it doesn't have to coincide with other such measures; the complication in the presentation is something that also baffles me still, but comes from the authors). This demo is ready for product review and publication. @daniela-angulo feel free to directly implement all changes, unless you're worried they might change the technical content. |
… into resourcefulness_demo_fix
Not a lot of changes. I made it half through the demo.
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Reading this demo was very pleasant. Another great content contribution.
Thanks, Maria!
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Thanks everyone. I have reviewed it as well and the content was approved by the vetting process. We just need the PR approval from @peterse to merge. |
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Maria has addressed all the main issues I had, lgtm!
This is a direct copy of PR #1462. That PR was open before the switch to the V2 pipeline which has caused CI to stall.