$ \newcommand{\coefcos}[0]{c_1 c_2 - s_1 s_2 \hat{n}_1 \cdot \hat{n}_2} \newcommand{\coefsin}[0]{s_1 c_2 \hat{n}_1 + c_1 s_2 \hat{n}_2 - s_1 s_2 \hat{n}_2 \times \hat{n}_1}$This question relates to Exercise 4.15 from Nielsen & Chuang:

(Composition of single qubit operations) The Bloch representation gives a nice way to visualize the effect of composing two rotations.

(1) Prove that if a rotation through an angle $\beta_1$ about the axis $\hat{n}_1$ is followed by a rotation through an angle $\beta_2$ about an axis $\hat{n}_2$, then the overall rotation is through an angle $\beta_{12}$ about an axis $\hat{n}_{12}$ given by

$$c_{12} = \coefcos \tag{4.19}\label{4.19}$$ $$ s_{12} \hat{n}_{12} = \coefsin, \tag{4.20}\label{4.20} $$ where $c_i = \cos \left( \beta_i/2 \right), s_i = \sin \left( \beta_i/2 \right), c_{12} = \cos \left( \beta_{12}/2 \right),$ and $s_{12}= \sin \left( \beta_{12}/2 \right)$.

Note that $\hat{n}_1$ and $\hat{n}_2$ are real unit vectors in three dimensions (as stated earlier in the book).

I managed to get the expressions on the RHS of \eqref{4.19} and \eqref{4.20} by expanding the rotation matrices into the form $R_{\hat{n}_i} \left( \beta_i \right) = \cos \left( \beta_i/2 \right) I - i \sin \left( \beta_i/2 \right) \hat{n}_i \cdot \vec{\sigma}$ (where $\vec{\sigma}$ denotes the three component vector $(X, Y, Z)$ of Pauli matrices) and taking the product.

Now, assume that $c_{12} \ne 1$. Then I'd expect to recover the unit vector $\hat{n}_{12}$ if I divide \eqref{4.20} through by $s_{12} = \pm \sqrt{1 - c_{12}^2}$:

$$ \hat{n}_{12} = \frac{\coefsin}{ \pm \sqrt{1 - c_{12}^2} } \tag{1} $$

To check that $\hat{n}_{12}$ is of unit length, I take the dot product with itself:

$$ \hat{n}_{12} \cdot \hat{n}_{12} = \frac{s_1^2 c_2^2 + c_1^2 s_2^2 + s_1^2 s_2^2 + 2 c_1 c_2 s_1 s_2 \cos (\theta) }{1 - c_{12}^2} \tag{2} $$

where $\cos (\theta) = \hat{n}_1 \cdot \hat{n}_2 $. Expanding $c_{12}$ in the denominator, I get:

$$ \hat{n}_{12} \cdot \hat{n}_{12} = \frac{s_1^2 c_2^2 + c_1^2 s_2^2 + s_1^2 s_2^2 + 2 c_1 c_2 s_1 s_2 \cos (\theta) }{1 - c_1^2 c_2^2 - s_1^2 s_2^2 \cos^2 (\theta) + 2 c_1 c_2 s_1 s_2 \cos (\theta) } \tag{3} $$

which tells me that the following equation should hold in order for the numerator and denominator to balance to 1:

$$ s_1^2 c_2^2 + c_1^2 s_2^2 + s_1^2 s_2^2 \stackrel{?}{=} 1 - c_1^2 c_2^2 - s_1^2 s_2^2 \cos^2 (\theta) \tag{4}\label{4} $$

However, I don't think \eqref{4} holds in general: $c_i$ and $s_i$ are functions of $\beta_i$. $\theta$ however is a function of $\hat{n}_i$ which I'm free to tune independently of $\beta_i$. So I should be able to conjure up values that violate \eqref{4} easily, which tells me that $\hat{n}_{12}$ is not a unit vector. This seems implausible to me. Where am I making a mistake?


You just missed the fact that $ \hspace{0.3em}||n_1 \times n_2|| = |\text{sin}(\theta)| $.

  • $\begingroup$ Dang, you are right! $\endgroup$ – Attila Kun Oct 4 '20 at 21:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.