13.8.2: Qubits and Symmetries

The Bloch sphere depicts each operation on a qubit as a trajectory on the sphere. However, any trajectory on the sphere can be represented by means of a sequence of rotations about the three axis. So, one way to address the definition of the operations on the qubit is to study rotations about the axis of the bloch sphere. This is intimately connected with the study of symmetries. Thomas Bohr was the first to suggest to use symmetries to interpret quantum mechanics.

We say that an object has a certain symmetry if after applying the corresponding symmetry operation (for example a rotation) the object appears to not have changed; then we say the object is invariant to that symmetry operation. In general, symmetry operations are rotations, reflections and inversions; and invariant means that start and end position of the object are indistinguishable. For example, Figure 13.3 shows a

Figure 13.3: Concept of invariance. Symmetry operation: rotation of \(\pi\)/2 about the vertical axis.

cube, invariant to a rotation of \(\pi\)/2 about an axis in the center of any of its faces. To distinguish start and end position, you would have to paint one face of the cube, as in the picture to the right of Figure 13.3. Then the cube is no longer invariant to a rotation of \(\pi\)/2. We would then say that the group of symmetries of the cube to the left of Figure 13.3, contains, among others, the group of rotations of \(\pi\)/2 about the axis drawn in the figure.

The way physicists use symmetries is to characterize objects by studying the operations that best describe how to transform the object and what are its invariances. Then the representation of the qubit in the Bloch sphere is particularly useful, since it tells us to focus on the group of spatial rotations. In the remainder of this section we will reconcile both views, our perspective of operators as matrices, and the symmetries of the Bloch sphere.

We have already seen that operators on qubits are 2×2 unitary matrices, the additional technical requirement we have to impose to have the group of spatial rotations is that the determinant of the matrices is +1 (as opposed to -1). This group has a name, it is called SU(2)\(^6\). We can build all of the matrices of SU(2) combining the following four matrices:

\[\mathbb{I} \stackrel{ def }{=}\left(\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right) \quad \sigma_{x} \stackrel{ def }{=}\left(\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right) \quad \sigma_{y} \stackrel{ def }{=}\left(\begin{array}{cc}
0 & -i \\
i & 0
\end{array}\right) \quad \sigma_{z} \stackrel{ def }{=}\left(\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right) \tag{13.54} \]

these matrices are known as the Pauli matrices in honor of Wolfgang Pauli. Note that technically they do not belong to SU(2), to be mathematically rigorous we need to multiply each of them by \(i\), the imaginary number (to verify that this is so, compute their determinant with and without multiplying by \(i\)).

Action of Pauli matrices on an arbitrary qubit

The best way to capture the intuition behind Pauli matrices, is to apply each of them to a qubit in an arbitrary superposition state

\[|\;\psi\rangle=\alpha\;|\;0\rangle+\beta\;|\;1\rangle=\cos \frac{\theta}{2}|\;0\rangle+\sin \frac{\theta}{2} e^{i \varphi}|\;1\rangle \tag{13.55} \]

and interpret the result

\[\begin{align*}
\sigma_{x}\;|\;0\rangle &=\left(\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right)\left(\begin{array}{l}
\alpha \\
\beta
\end{array}\right) \\
&=\left(\begin{array}{c}
\beta \\
\alpha
\end{array}\right) \longrightarrow \text { Rotation of } \pi \text { about } \mathrm{x} \text { axis } \tag{13.56}\\
\sigma_{y}\;|\;0\rangle &=\left(\begin{array}{cc}
0 & -i \\
i & 0
\end{array}\right)\left(\begin{array}{l}
\alpha \\
\beta
\end{array}\right) \\
&=i\left(\begin{array}{c}
-\beta \\
\alpha
\end{array}\right) \longrightarrow \text { Rotation of } \pi \text { about } \mathrm{y} \text { axis } \tag{13.57}\\
\sigma_{z}\;|\;0\rangle &=\left(\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right)\left(\begin{array}{l}
\alpha \\
\beta
\end{array}\right) \\
&=\left(\begin{array}{c}
\alpha \\
-\beta
\end{array}\right) \longrightarrow \text { Rotation of } \pi \text { about } \mathrm{z} \text { axis } \tag{13.58}
\end{align*} \nonumber \]

Figure 13.4 illustrates the operation of \(\sigma_y\) on a qubit in an arbitrary superposition on the Bloch sphere

Figure 13.4: Operation of \(\sigma_y\), on the Bloch sphere

Hence Pauli matrices are rotations of \(\pi\) about each of the axes of the bloch sphere (this motivates the names we gave them). However, to fully explore the surface of the Bloch sphere we need to be able to define arbitrary rotations (not just multiples of \(\pi\)). To do so we use the neat trick of exponentiating Pauli Matrices. Recall Euler’s formula relating the exponential function to sine and cosine,

\[e^{i x}=\cos x+i \sin x. \tag{13.59} \]

Euler’s formula applies when x is a real number. But we are interested in obtaining a similar result for Pauli matrices. We can prove the equivalent to Euler’s formula for Pauli matrices by replacing \(x\) by \(\frac{\theta}{2} \sigma_x\), and expanding the exponential as a Taylor series (note that \(\sigma_x\sigma_x = \mathbb{I})

\[\begin{align*}
e^{i \sigma_{x} \theta / 2} &=1+i \frac{\theta}{2} \sigma_{x}-\frac{1}{2}\left(\frac{\theta}{2}\right)^{2} \mathbb{I}-i \frac{1}{3}\left(\frac{\theta}{2}\right)^{3} \sigma_{x}+\frac{1}{4}\left(\frac{\theta}{2}\right)^{4} \mathbb{I}+\cdots \tag{13.60}\\
&=\left(1-\frac{1}{2}\left(\frac{\theta}{2}\right)^{2}+\frac{1}{4}\left(\frac{\theta}{2}\right)^{4}+\cdots\right) \mathbb{I}+i\left(0+\left(\frac{\theta}{2}\right)-\frac{1}{3}\left(\frac{\theta}{2}\right)^{3}+\cdots\right) \sigma_{x} \tag{13.61}\\
&=\cos \frac{\theta}{2} \mathbb{I}+i \sin \frac{\theta}{2} \sigma_{x} \tag{13.62}
\end{align*} \nonumber \]

This result shows us how to do arbitrary rotations of an angle \(\theta\) about the \(x\) axis, the resulting operator is often called \(R_x(\theta) = e^{i\sigma_x\theta/2}\). The cases of \(R_y\) and \(R_z\) are completely analogous.

Summing up, we have shown how to represent any qubit as a point in the Bloch sphere and we have learnt how to navigate the bloch sphere doing arbitrary rotations about any of the three axis. It follows that we have obtained an expression for the group of operations of symmetry that allow us to write the form of any operator acting on a single qubit.

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\(^6\)Groups are conventionally named with letters like O,U, SU, SO, etc. Each of this letters has a meaning. SU(2) stands for the special (S) group of unitary (U) matrices of dimension 2. Special means that the determinant of the matrices is +1, and unitary has here the same meaning it had in the discussion of the operators.