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I'm working to have a greater understanding of the arbitrary unitary transformation matrix when working in the context of the Bloch sphere. At this time I have found several equivalent representations of this arbitrary unitary, but I'm trying to bridge the gap between a version that I feel I understand well, and another that I'm not yet clear on why it is different.

The one that I feel I understand well is $$U_1=\exp(i\gamma)\exp(-i\alpha/2 \hat{n}\cdot\vec{\sigma})$$ which I found and worked through with the help of this set of notes from Ian Glendinning.

The second version is from Audretch's Entangled Systems; New Directions in Quantum Physics. I do not have a link for this book, but on Page 56 he gives the following version of the unitary with little explanation as to its derivation. $$U_2=\exp(i\kappa)\exp(-i\lambda/2\sigma_z)\exp(-i\mu/2\sigma_y)\exp(-i\nu/2\sigma_z)$$

I know that $\kappa$ and $\gamma$ are just different names for the global phase factor.

I believe that $\alpha$ and $\mu$ represent the arbitrary angle of rotation around the $\hat{n}$ axis.

I believe the differences relate primarily to the fact that Audretsch's representation is representing the pairwise orthonormal columns and rows as discussed in Audretsch's book as well as in this post: General parametrisation of an arbitrary $2\times 2$ unitary matrix?)

Any help I could get in better understanding how these are equivalent representations would be appreciated.

With the help of Wikipedia and the stack exchange discussion about general parametrisation linked above, I have been able to fully understand the origin of $U_2$ and how to derive it from an arbitrary 2 x 2 matrix. I am still trying to equate this with $U_1$ above.

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  • $\begingroup$ Both can be found in Nielsen and Chuang's book:) Theorem 4.1 of the book might answer your question. I just quoted: $\endgroup$
    – narip
    Aug 31, 2022 at 14:09
  • $\begingroup$ Since $U$ is unitary, the rows and columns of $U$ are orthonormal, from which it follows that there exist real numbers $\alpha, \beta, \gamma$, and $\delta$ such that $$ U=\left[\begin{array}{cc} e^{i(\alpha-\beta / 2-\delta / 2)} \cos \frac{\gamma}{2} & -e^{i(\alpha-\beta / 2+\delta / 2)} \sin \frac{\gamma}{2} \\ e^{i(\alpha+\beta / 2-\delta / 2)} \sin \frac{\gamma}{2} & e^{i(\alpha+\beta / 2+\delta / 2)} \cos \frac{\gamma}{2} \end{array}\right] . $$ $\endgroup$
    – narip
    Aug 31, 2022 at 14:09
  • $\begingroup$ I saw this in Nielson and Chuang's book and noted that it was the same as my $U_2$ once the rotations they discuss are applied. However, $U_2$ makes use of three rotation operators; $R_z, R_y, R_z$ while $U_1$ uses 5 rotations; $R_z, R_y, R_z, R_y\dagger, R_z\dagger$. Also, the number of angles used differs as $U_2$ uses 4 angles while $U_1$ uses two angles and a point. $\endgroup$
    – PGibbon
    Aug 31, 2022 at 15:34

2 Answers 2

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The simplest way to see this, for me, is like this. If you ignore the global phase, which doesn't have any physical effect, $U_1$ represents a rotation in 3D around an arbitrary axis - this is clear in the Bloch sphere parameterization of a qubit's state. $U_2$, on the other hand, represents 3 successive rotations around the $z, y, z$ axes. There is a theorem stating that you can decompose any 3D rotation $U_1$ into such a sequence of rotations around two other axes, see for example https://math.stackexchange.com/questions/1021727/is-it-true-that-a-arbitrary-3d-rotation-can-be-composed-with-two-rotations-const So, if you pick the correct angles for the 3 successive rotations, you will recover the original rotation exactly.

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With further reading, I believe I have an answer to my own question. This is a tentative answer for now, but if I find a better understanding, I will update.

I made the assumption that with these both being arbitrary unitary transformations that they were equivalent unitary transformations. However, $U_1$ is a unitary that can be used to rotate a given state around a specific axis, $\hat{n}$, while $U_2$ is used to arbitrarily change one state into another using rotations.

Since their purposes distinctly differ, I am no longer convinced that they can be equated.

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