How can I verify that the Pauli group is a group? And is it abelian?

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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.

RandomGhost

Posted 2020-11-24T03:47:38.653

Reputation: 19

Question was closed 2020-11-29T15:00:47.100

1What makes you think it’s abelian? – Mark S – 2020-11-24T04:35:13.937

1https://en.wikipedia.org/wiki/Pauli_group – Hasan Iqbal – 2020-11-24T05:57:19.607

1Hasan how is that useful in anyway to what was asked? Anyone can copy and paste a link from a wiki that literally explains nothing. – RandomGhost – 2020-11-24T05:59:01.870

And Mark that's what I was asking, is it Abelian haha – RandomGhost – 2020-11-24T06:00:08.697

2Using 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. – Rammus – 2020-11-24T08:49:00.010

please stick to a single question per post. You can create multiple posts to ask multiple questions – glS – 2020-11-25T09:09:46.237

Answers

3

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.

Group table

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).

Yitian Wang

Posted 2020-11-24T03:47:38.653

Reputation: 808

1

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.

– Rammus – 2020-11-24T08:46:58.523

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). – Yitian Wang – 2020-11-25T11:30:27.180