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In the Wikipedia's Quantum channel article, it is said that a purely quantum channel $\phi$ (it's not exactly the same phi calligraphy but it's close), in the Schrodinger picture, is a linear map between density matrices acting on two Hilbert spaces. Then properties of this mapping are given and we find the following notation :

$$I_n \otimes \Phi$$ We find again this difference of notation in the Heisenberg picture paragraph. However I believe $\phi$ and $\Phi$ are the same and I don't understand the change of notation between the twos. Is there a difference justifying this change or is it a lack of coherence of the notation ?

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    $\begingroup$ I think it's just notational inconsistency. If you look at the page code, the symbols are generated in two different ways: in the text, someone has just inserted the greek letter symbol (presumably a unicode character) whereas in the equation, they've used LaTeX. They're clearly both supposed to be capital phi. $\endgroup$ – DaftWullie Mar 18 at 14:42
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    $\begingroup$ Thank you DaftWullie, it's good enough for me as an answer, I'll mark it as accepted if you put it in an answer. $\endgroup$ – nathan raynal Mar 18 at 14:46
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I think it's just notational inconsistency. If you look at the page code, the symbols are generated in two different ways: in the text, someone has just inserted the greek letter symbol (presumably a unicode character) whereas in the equation, they've used LaTeX. They're clearly both supposed to be capital phi.

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    $\begingroup$ thanks, I'm modifying the Wikipedia page to get consistent notation $\endgroup$ – nathan raynal Mar 18 at 14:57

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