So how can I verify that the Pauli Group is a Group? Then furthermore, Abelian? And then to sum it up, the order of the group. Trying to do some research into the group but I can't find much about it.
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1$\begingroup$ What makes you think it’s abelian? $\endgroup$– Mark SpinelliNov 24, 2020 at 4:35
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1$\begingroup$ en.wikipedia.org/wiki/Pauli_group $\endgroup$– QuestionEverythingNov 24, 2020 at 5:57
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1$\begingroup$ Hasan how is that useful in anyway to what was asked? Anyone can copy and paste a link from a wiki that literally explains nothing. $\endgroup$– RandomGhostNov 24, 2020 at 5:59
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$\begingroup$ And Mark that's what I was asking, is it Abelian haha $\endgroup$– RandomGhostNov 24, 2020 at 6:00
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3$\begingroup$ Using the link @HasanIqbal gave you, you could count the number of elements in the set and multiply a few of them together to check if it were abelian or not. The link pretty much answers your question. $\endgroup$– RammusNov 24, 2020 at 8:49
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1 Answer
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For the three Pauli matrices, {${\sigma_1,\sigma_2}=0$}, so certainly this can not form an abelian group. The Pauli group is an isomorphism with $D_4$.
The elements of the Pauli group are {${I, \sigma_x, \sigma_y, i\sigma_z, -I, -\sigma_x, -\sigma_y, -i\sigma_z}$}, so the order of this group is 8. In variant subgroups are {${I, -I}$}, {${I, -I, \sigma_{x/y}, -\sigma_{x/y}}$} and {${I, -I, i\sigma_z, -i\sigma_z}$}. $I$ and $-I$ is the only emelemt in its class. {$\pm\sigma_{x/y}$} and {$\pm i\sigma_z$} forms a class( totally 5 classes). Order 1 element: $I$. Order 2 elements: $\sigma_x, \sigma_2, -I, -\sigma_x, -\sigma_y$. And Order 3 elements: $\pm i\sigma_z$.
To check that the upper set actually forms a group, you need to prove that the set satisfies the four properties of a group: closure, associativity, identity element, inverse element, see Wikipedia for more detail.
Here comes the group table of the Pauli group.
You can also see Wikipedia and check it is identical to the group table of $D_4$(in fact the tables are different because the ordering of elements is different).
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1$\begingroup$ Note that this is not always the definition of the Pauli group. E.g. Wiki and N&C gives a different definition. See this QCSE answer. $\endgroup$– RammusNov 24, 2020 at 8:46
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$\begingroup$ Yes, construct a Pauli group is my homework about 1 year ago and that time, I just did it by my self. But this is a group and the property should be quite alike(but when reading papers with the standard definition this surly might cause trouble). $\endgroup$ Nov 25, 2020 at 11:30