3.7Remarks and exercises

3.7.1 Orthonormal Pauli basis

Show that \{\mathbf{1},\sigma_x,\sigma_y,\sigma_z\} is an orthonormal basis with respect to the Hilbert-Schmidt product in the space of complex (2\times 2) matrices.

3.7.2 Pauli matrix expansion coefficients

Recall that any (2\times 2) complex matrix A has a unique expansion in the form \begin{aligned} A &= \begin{bmatrix} a_0 + a_z & a_x - i a_y \\a_x +i a_y & a_0 - a_z \end{bmatrix} \\&= a_0\mathbf{1}+ a_x \sigma_x + a_y \sigma_y + a_z \sigma_z \\&= a_0\mathbf{1}+ \vec{a}\cdot\vec{\sigma}. \end{aligned} \tag{$\star$} for some complex numbers a_0, a_x, a_y, and a_z.

1. Show that the coefficients a_k (for k=x,y,z) are given by the inner product a_k = (\sigma_k|A) = \frac12\operatorname{tr}\sigma_k A.

In these notes, we usually deal with matrices that are Hermitian (A=A^\dagger) or unitary (AA^\dagger=\mathbf{1}). It is easy to see that, if A is Hermitian, then a_0 and the three components of \vec{a} are all real. The (2\times 2) unitaries are usually parametrised as U = e^{i\gamma}\Big(u_0\mathbf{1}+ i(u_x\sigma_x + u_y\sigma_y + u_z\sigma_z)\Big) where e^{i\gamma} is an overall multiplicative phase factor, with \gamma real, and u_0 and the three components u_x, u_y, u_z are all real numbers.

1. Show that the unitarity condition implies that u_0^2 + u_x^2 + u_y^2 + u_z^2 = 1 and show, using this parametrisation, that the determinant of U is e^{i2\gamma}.

3.7.3 Linear algebra of the Pauli vector

1. Show that \frac12\operatorname{tr}(\vec{a}\cdot\vec{\sigma})(\vec{b}\cdot\vec{\sigma}) = \vec{a}\cdot\vec{b}. (You may find a certain identity involving the cross product helpful.)

2. Show that any \vec{n}\cdot\vec{\sigma} has eigenvalues \pm|\vec{n}|.

3. Show that, if \vec{n}\cdot\vec{m}=0, then the operators \vec{n}\cdot\vec{\sigma} and \vec{m}\cdot\vec{\sigma} anticommute.

3.7.4 Matrix Euler formula

1. Show that, if A^2=\mathbf{1}, then we can manipulate the power series expansion of e^{iA} into a simple expression: for any real \alpha, e^{i\alpha A} = (\cos\alpha)\mathbf{1}+ (i\sin\alpha)A.

2. Show that any (2\times 2) unitary matrix U can be written, up to an overall multiplicative phase factor, as U = e^{i \theta \vec{n}\cdot\vec{\sigma}} = (\cos\theta)\mathbf{1}+ (i\sin\theta)\vec{n}\cdot\vec{\sigma}. (The argument here is the same as the argument that e^{i\theta}=\cos\theta +i\sin\theta).

3.7.5 Special orthogonal matrix calculations

1. Show that \operatorname{tr}\sigma_x\sigma_y\sigma_z = 2i.

2. Consider U(\vec e_k\cdot\sigma_k)U^\dagger=U\sigma_kU^\dagger={\vec f_k}\cdot\vec\sigma. So U maps the unit vectors \vec e_x, \vec e_y, and \vec z_z, (along the x-, y-, and z-axis, respectively), to new unit vectors \vec f_x, \vec f_y, and \vec f_z. We already know that, in Euclidean space, this transformation is described by a (3\times 3) orthogonal matrix R_U. How are the three vectors \vec f_x, \vec f_y, and \vec f_z related to the entries in matrix R_U?

3. Show that \begin{aligned} \operatorname{tr}\sigma_x\sigma_y\sigma_z &= \operatorname{tr}({\vec f_x}\cdot\vec\sigma)( {\vec f_y}\cdot\vec\sigma)({\vec f_z}\cdot\vec\sigma) \\&= 2i\det R_U \end{aligned} (which implies that \det R_U=1).

4. Use the orthonormality of the Pauli basis along with Equation (\ddagger) to show that the elements of the matrix R=R_U can be expressed in terms of those of the matrix U, in the form R_{ij}=\frac12\operatorname{tr}\left(\sigma_i U\sigma_j U^\dagger\right). Here, i and j take values in \{1,2,3\}, and \sigma_1\equiv\sigma_x, \sigma_2\equiv\sigma_y, \sigma_3\equiv\sigma_z.

3.7.6 Phase as rotation

Show that the phase gate P_\varphi = \begin{bmatrix}1&0\\0&e^{i\varphi}\end{bmatrix} represents an anticlockwise rotation about the z-axis through the angle \varphi.

Hint. It might be helpful to start with the \mathrm{SU}(2) version of the phase gate: \begin{aligned} P_\varphi &= e^{-i\frac{\varphi}{2}\sigma_z} \\&= \begin{bmatrix} e^{-i \frac{\varphi}{2}}& 0 \\0 & e^{i \frac{\varphi}{2}} \end{bmatrix} \quad\longrightarrow\quad R \\&= \begin{bmatrix} \cos \varphi & -\sin \varphi & 0 \\\sin \varphi & \cos \varphi & 0 \\0 & 0 & 1 \end{bmatrix} \end{aligned}

1. Express the Hadamard gate H in terms of \vec{n}\cdot\vec{\sigma}, and show that \begin{aligned} HZH&=X \\HXH&=Z \\HYH&=-Y. \end{aligned}

2. Show that the Hadamard gate H turns rotations about the x-axis into rotations about the z-axis, and vice versa. That is, \begin{aligned} H \left( e^{-i\frac{\varphi}{2}Z} \right) H &= e^{-i\frac{\varphi}{2}X} \\H \left( e^{-i\frac{\varphi}{2}X} \right) H &= e^{-i\frac{\varphi}{2}Z}. \end{aligned}

3.7.8 Swiss Granite Fountain

In the Singapore Botanic Gardens, there is a sculpture by Ueli Fausch called “Swiss Granite Fountain”. It is a spherical granite ball which measures 80cm in diameter and weighs 700kg, and is kept afloat by strong water pressure directed through the basal block. It is easy to set the ball in motion, and it keeps rotating in whatever way you start for a long time. Suppose you are given access to this ball only near the top, so that you can push it to make it rotate around any horizontal axis, but you don’t have enough of a grip to make it turn around the vertical axis. Can you make it rotate around the vertical axis anyway?

3.7.9 Dynamics in a magnetic field

A qubit (spin one-half particle) initially in state |0\rangle (spin up) is placed in a uniform magnetic field. The interaction between the field and the qubit is described by the Hamiltonian H = \omega \begin{bmatrix} 0 & - i \\i & 0 \end{bmatrix} where \omega is proportional to the strength of the field.61 What is the state of the qubit after time t=\pi/4\omega?

1. In Earth’s magnetic field, which is about 0.5 gauss, the value of \omega is of the order of 10^6 cycles per second.↩︎