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comment Example of using the Hadamard's matrix to determine the superposition
@quapka: I've always been fond of the first chapter of Sakurai's Modern Quantum Mechanics for providing the physical motivation for the matrix formulation of quantum mechanics. Not sure whether that would help with what you're trying to do.
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2d
comment Example of using the Hadamard's matrix to determine the superposition
@quapka: no problem. Let me know if you have any questions. Your question was somewhat open-ended, so I'm not sure I hit the right points.
May
26
revised Example of using the Hadamard's matrix to determine the superposition
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May
26
comment Example of using the Hadamard's matrix to determine the superposition
I have corrected an inaccuracy in my first paragraph and added some explanation of this point at the end of the post. It is not the case that you would never be able to determine the original state in your second example, which is what I originally said.
May
26
revised Example of using the Hadamard's matrix to determine the superposition
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May
25
answered Example of using the Hadamard's matrix to determine the superposition
May
24
awarded  Revival
May
24
comment How to get from Chebyshev to Ihara?
@draks...: you should feel no obligation. My post doesn't actually tell you how to connect the generating function for $p_n(x)$ to the Ihara zeta function, which, I'm guessing, is not a trivial matter. The point of my comments - and they really should have been comments, but were much too long - was to try to clarify the question.
May
24
revised How to get from Chebyshev to Ihara?
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May
23
awarded  Necromancer
May
21
revised How to get from Chebyshev to Ihara?
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May
21
comment How to get from Chebyshev to Ihara?
@draks...: I'm not sure. The matrix $$\begin{bmatrix}1 & 2\\ 2 & 3\end{bmatrix}$$ is symmetric, but its determinant is not the product of its diagonal elements. Is it possible you mean product of eigenvalues rather than diagonal elements? I'm sorry about my hasty remark regarding whether the two functions are connected. I agree that it's likely they are somehow related.
May
21
revised How to get from Chebyshev to Ihara?
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May
20
comment How to get from Chebyshev to Ihara?
@draks...: $\det$ gives the product of diagonal elements if the matrix is upper or lower triangular, but not in general. Anyway, I think your argument needs to be fleshed out more. It's still not clear to me why one should expect a connection between the Chebyshev generating function and the Ihara zeta function. By the way, the Horton, Stark, and Terras article contains a proof in Section 3 of the determinant formula for Ihara zeta. I haven't gone through it myself.
May
20
comment How to get from Chebyshev to Ihara?
@draks...: Both Wikipedia and the Horton, Stark, and Terras article state that $L(p)$ is the length of the path $p$. It is not the number of paths. (That wouldn't, in fact, make sense: the number of paths wouldn't be a function of a particular path $p$.) The number of paths of a given length does, however, play a role: the product will contain the factor $(1-t^\ell)^{-N_\ell}$, where $N_\ell$ is the number of equivalence classes of closed backtrackless paths of length $\ell$.
May
20
comment How to get from Chebyshev to Ihara?
@draks...: The definition comes from the Wikipedia article you linked to, although that article is not at all clear that the product is indexed by equivalence classes of paths, and not paths themselves. That bit is explained in an article of Horton, Stark, and Terras that I found by Googling.
May
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revised How to get from Chebyshev to Ihara?
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May
19
answered How to get from Chebyshev to Ihara?
May
19
comment Returning Paths on Cubic Graphs Without Backtracking
@draks...: I feel that I have now largely done this to my own satisfaction. See my revised answer to your other post. If it's not clear, please let me know so I can try to improve the presentation.