2.3 Bloch sphere and projective representation

Let us take a close look at the case of a qubit, i.e. \(\mathcal{H} = \mathbb{C}^2\). Pick a basis \(\ket{0}\) and \(\ket{1}\) and expand a unit state \(\ket{\psi}\) as \[\begin{equation} \tag{2.9} \ket{\psi} = \cos(\theta/2) \ket{0} + \sin(\theta/2)e^{\mathrm{i}\phi} \ket{1}, \end{equation}\] where \(0\le \theta \le \pi\) and \(0 \le \phi \le 2\pi\). One can always bring a unit state to this form using the phase ambiguity \(\ket{\psi}\sim e^{\mathrm{i}\alpha}\ket{\psi}\). Note that when \(\theta = 0\) and \(\theta =\pi\), the the correspoinding point in \(\mathbb{P}\mathcal{H}\) is independent of \(\phi\). Therefore \((\theta,\phi)\) is the coordinates (polar and azimuthal angles) on \(S^2\), called the Bloch sphere: \[\begin{equation} \tag{2.10} \mathbb{P}\mathcal{H} \cong \mathbb{CP}^1 \cong S^2. \end{equation}\] The relationship between the polar coordinates on \(S^2\) and the Cartesian coordinates on \(\mathbb{R}^3\) is \((x,y,z) = (\sin\theta\,\cos\phi,\sin\theta\,\sin\phi,\cos\theta)\).

Given two states \(\ket{\psi_1}\) and \(\ket{\psi_2}\), the transition probability is \(|\braket{\psi_2|\psi_1}|^2 = \cos^2(\theta_{12}/2)\), where \(\theta_{12}\) is the angle between the two corresponding point in the Bloch sphere. Symmetry transformation \(T\) is a bijection on Bloch sphere preserving this quantity. Such transformations are one-to-one corresponded to the orthogonal group \[\begin{equation} \tag{2.11} \{\text{Symmetry transformations on a qubit}\} \cong O(3). \end{equation}\] We can also identify the composition of symmetry transformations as the multiplication of the group \(O(3)\). What are the corresponding (anti)unitaries?

2.3.1 Projective phase of \(SO(3)\)

Let us first study the part \(SO(3) \subset O(3)\) that preserves the orientation of the sphere. In particular, the rotation \(R_z(\lambda)\) around the \(z\) (\(\theta =0\) direction) axis sends \[\begin{equation} \tag{2.12} R_z(\lambda) : (\theta ,\phi) \mapsto (\theta, \phi + \lambda). \end{equation}\] A unitary compatible with this transformation is \[\begin{equation} \tag{2.13} U_{R_z(\lambda)}' = \begin{pmatrix} 1 & 0 \\ 0 & e^{\mathrm{i}\lambda}\\ \end{pmatrix}. \end{equation}\] Or, if we can demand that the unitary is also special (\(\mathrm{det}U =1\)) by using the phase ambiguity, achieving \[\begin{equation} \tag{2.14} U_{R_z(\lambda)} = \begin{pmatrix} e^{\mathrm{i}\lambda/2} & 0 \\ 0 & e^{\mathrm{i}\lambda/2}\\ \end{pmatrix}. \end{equation}\] However, this expression has a peculiar feature: that the \(2\pi\) rotation is mapped to \(U_{R_z(\lambda = 2\pi)} = -\mathbf{I}_{2}\), not to the identity matrix \(\mathbf{I}_2\)! In other words, if we use the special unitary \(U_{R_z(\lambda)}\) (\(0\le \lambda < 2\pi\)), the composition law of the unitary operators is \[\begin{equation} \tag{2.15} U_{R_z(\lambda)} U_{R_z(\lambda')} = e^{\mathrm{i}\alpha(\lambda,\lambda')}U_{R_z(\lambda+\lambda' \mathrm{mod}\, 2\pi)} \end{equation}\] which is not quite straightforward. The function \(\alpha(\lambda,\lambda')\) is called the projective phase and in this case it is \[\begin{equation} \tag{2.16} \alpha(\lambda,\lambda')= \begin{cases} 0 & (\lambda+\lambda' < 2\pi)\\ \pi & (\lambda+\lambda' \ge 2\pi) \end{cases}. \end{equation}\]

If one only focus on \(R_z(\lambda)\), one can use \(U'\) in (2.13), which has the trivial projective phase. However, it is known that we cannot find a good unitaries \(U_g\) for all of the group \(g\in SO(3)\) acting on the Block sphere. One choice of the general expression is \[\begin{equation} \tag{2.17} U_{R_{\mathbf{n}}(\lambda)} = \exp(-i\lambda \mathbf{n}\cdot \vec{\sigma}/2). \end{equation}\] The claim is that you cannot come up with a function \(\beta: SO(3) \to \mathbb{R}/2\pi\mathbb{Z}\) such that \(e^{\mathrm{i}\beta(g)}U_{R_{\mathbf{n}}(\lambda)}\) exactly satisfies the group multiplication law of \(SO(3)\) without projective phase. We will prove this in the next section.

When a group \(G\) acts on a vector space, but the represented matrices \(U_g\) cannot avoid the projective phase, the pair of the vector space and the action is called a projective representation. The qubit space \(\mathbb{C}^2\) is a projective representation of \(SO(3)\). The general presentation mat

2.3.2 \(SO(3)\) or \(SU(2)\)?

You might have confused; you might been taught that a qubit, or a spin, is acted by \(SU(2)\), and wondering why I am emphasizing \(SO(3)\) instead. The Lie groups \(SU(2)\) and \(SO(3)\) are closely related: \(SU(2)\) is a double cover of \(SO(3)\). In other words, \[\begin{equation} \tag{2.18} SO(3) = SU(2)/Z(SU(2)), \end{equation}\] where \(Z(SU(2))\cong \mathbb{Z}_2\) is the center of \(SU(2)\) generated by \(-\mathbf{I}_2\). The \(SU(2)\) can naturally act on \(\mathbb{C}^2\). However, \(-\mathbf{I}_2 \in SU(2)\) does not change the ray of the state, which is the physical entity. Therefore, more precise statement is that a symmetry of a qubit/spin is \(SO(3)\), as Wigner defined it, but to describe the Hilbert space \(SU(2)\), that does not suffer from the projective phase, is more convenient. And the projective phase is the quantum anomaly: something is not quite right about the symmetry.

Another way of saying the same this is to focus on the algebra \(\mathcal{A}\) of observables. The symmetry acts on the algebra of operators by conjugation ((2.7)), and thus the center \(-\mathbf{I}_2\) of \(SU(2)\) acts trivially. So the symmetry acting on the observables is \(SO(3)\), not \(SU(2)\). And the projective phase, or the double cover \(SU(2)\) arises only when we consider the vector space of the states.

2.3.3 Anti-Unitary symmetry*

What happens for the orientation reversing map in \(O(3)\), specifically for the reflection \[\begin{equation} \tag{2.19} \mathrm{Ref}: (\theta,\phi) \mapsto (\theta,-\phi). \end{equation}\] For this we have to assign an anti-unitary operator, which is \[\begin{equation} \tag{2.20} U_{\mathrm{Ref}} = K \mathbf{I}_2, \end{equation}\] where \(K\) is the complex conjugation map acting on \(\mathbb{C}\). A general element in \(O(3)\setminus SO(3)\) can be obtained by multiplying a rotation to the reflection. There is no projective phase regarding \(\mathrm{Ref}\).

2.3.4 General finite states model*

For general \(n\) state system, \(\mathcal{H} \cong \mathbb{C}^n\), the projective space is the complex projective space \(\mathbb{CP}^n\). The symmetry transformations are \[\begin{equation} \tag{2.21} \{\text{Symmetry transformations on $\mathbb{C}^n$}\} \cong PSU(n)\rtimes \mathbb{Z}_2, \end{equation}\] where \(PSU(n) = SU(n)/\mathbb{Z}_n\) and \(\mathbb{Z}_2\) acts as the charge conjugation. \(\mathbb{C}^n\) is a projective representation of \(PSU(n)\), while the \(\mathbb{Z}_2\) part is realized as an anti-unitary.