# Does composition of two single qubit rotations yield a single rotation around a unit vector?

$$\newcommand{\coefcos}{c_1 c_2 - s_1 s_2 \hat{n}_1 \cdot \hat{n}_2} \newcommand{\coefsin}{s_1 c_2 \hat{n}_1 + c_1 s_2 \hat{n}_2 - s_1 s_2 \hat{n}_2 \times \hat{n}_1}$$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?

• How do we obtain $\sin(\beta_1/2)\sin(\beta_2/2)\hat{n}_2\times \hat{n}_1$ in the expansion of $R_{\hat{n}_2}(\beta_2)R_{\hat{n}_1}(\beta_1)$ ? Sep 22, 2021 at 19:56

You just missed the fact that $$\hspace{0.3em}||n_1 \times n_2|| = |\text{sin}(\theta)|$$.
• Where does the term $\color{red}{\sin(\beta_1/2)\sin(\beta_2/2)\hat{n}_2\times \hat{n}_1}$ comes in the expression for $\sin(\beta_{12}/2)\hat{n}_{12}$ ? Sep 23, 2021 at 5:56