7.5 The Bloch sphere

7.5 The Bloch sphere

Concept links · terms present in this machine draft; source roles are unverified: vector space · complex inner product · eigenvalue

For another point of view on the relation between the two-state system with $\mathcal{H}={\mathbf{C}}^2$ and the geometry of the sphere (known to physicists as the “Bloch sphere” description of states), the unit sphere $S^2\subset {\mathbf{R}}^3$ can be mapped to operators by

$$\mathbf{x}\to \sigma \cdot \mathbf{x}$$

For each point $\mathbf{x}\in S^2$ 2 $\sigma \cdot \mathbf{x}$ has eigenvalues ±1. Eigenvectors with eigenvalue +1 are the solutions to the equation

$$\sigma \cdot \mathbf{x}|\psi ⟩=|\psi ⟩\text{\hspace{1em}(7.4)}$$

and give a subspace $\mathbf{C}\subset \mathcal{H}$ , giving another parametrization of the points in $\mathbf{CP}(1)$ . Note that one could equivalently consider the operators

$$P_{\mathbf{x}}=\frac{1}{2}(1-\sigma \cdot \mathbf{x})$$

and look at the space of solutions to

$$P_{\mathbf{x}}|\psi ⟩=0$$

It can easily be checked that $P_{\mathbf{x}}$ satisfies $P_{\mathbf{x}}^2=P_{\mathbf{x}}$ and is a projection operator.

For a more physical interpretation of this in terms of the spin operators, one can multiply 7.4 by $\frac{1}{2}$ and characterize the $\mathbf{C}\subset \mathcal{H}$ corresponding to $\mathbf{x}\in S^2$ as the solutions to 1

$$\mathbf{S}\cdot \mathbf{x}|\psi ⟩=\frac{1}{2}|\psi ⟩$$

Then the North pole of the sphere is a ${}^6\mathrm{spin}{-\mathrm{up}}^7$ state, and the South pole is a “spin-down” state. Along the equator one finds two points corresponding to states with definite values for $S_1$ , as well as two for states that have definite values for $S_2$

Figure 7.4: The Bloch sphere.

For later applications of the spin representation, we would like to make for each x a choice of solution to equation 7.4, getting a map

$$u_{+}:\mathbf{x}\in S^2\to |\psi ⟩=u_{+}(\mathbf{x})\in \mathcal{H}={\mathbf{C}}^2$$

such that

$$(\mathbf{\sigma }\cdot \mathbf{x})u_{+}(\mathbf{x})=u_{+}(\mathbf{x})\text{\hspace{1em}(7.5)}$$

This equation determines $u_{+}$ only up to multiplication by an x-dependent scalar. A standard choice is

$$u_{+}(\mathbf{x})=\frac{1}{\sqrt{2(1+x_3)}}\left(\genfrac{}{}{0ex}{}{1+x_3}{x_1+i{x}_2}\right)=\left(\genfrac{}{}{0ex}{}{\cos \frac{\theta }{2}}{e^{i\varphi }\sin \frac{\theta }{2}}\right)\text{\hspace{1em}(7.6)}$$

where $\theta ,\varphi$ are standard spherical coordinates (which will be discussed in section 8.3). This particular choice has two noteworthy characteristics:

• One can check that it satisfies

$$u_{+}(R\mathbf{x})=\Omega u_{+}(\mathbf{x})$$

where $R=\Phi (\Omega )$ is the rotation corresponding to an $SU(2)$ element

$$\Omega =\left(\begin{array}{cc}\cos \frac{\theta }{2}& -e^{-i\varphi }\sin \frac{\theta }{2}\\ e^{i\varphi }\sin \frac{\theta }{2}& \cos \frac{\theta }{2}\end{array}\right)$$

$u_{+}(\mathbf{x})$ is determined by setting it to be $\left(\genfrac{}{}{0ex}{}{1}{0}\right)$ at the North pole, and defining it at other points $x$ on the sphere by acting on it by the element Ω which, acting on vectors by conjugation (as usual using the identification of vectors and complex matrices), would take the North pole to x.

• With the specific choices made, $u_{+}(\mathbf{x})$ is discontinuous at the South pole, where $x_3=-1$ , and $\varphi$ is not uniquely defined. For topological reasons, there cannot be a continuous choice of $u_{+}(\mathbf{x})$ with unit length for all x. In applications one generally will be computing quantities that are independent of the specific choice of $u_{+}(\mathbf{x})$ , so the discontinuity (which is choice-dependent) should not cause problems.

One can similarly pick a solution $u_{-}(\mathbf{x})$ to the equation

$$(\mathbf{\sigma }\cdot \mathbf{x})u_{-}(\mathbf{x})=-u_{-}(\mathbf{x})$$

for eigenvectors with eigenvalue −1, with a standard choice

$$u_{-}(\mathbf{x})=\frac{1}{\sqrt{2(1+x_3)}}\left(\genfrac{}{}{0ex}{}{-(x_1-i{x}_2)}{1+x_3}\right)=\left(\genfrac{}{}{0ex}{}{-e^{-i\varphi }\sin \frac{\theta }{2}}{\cos \frac{\theta }{2}}\right)$$

For each $\mathbf{x},u_{+}(\mathbf{x})$ and $u_{-}(\mathbf{x})$ satisfy

$$u_{+}(\mathbf{x}{)}^{†}{u}_{-}(\mathbf{x})=0$$

so provide an orthonormal (for the Hermitian inner product) complex basis for ${\mathbf{C}}^2$

Digression. The association of a diferent vector space $\mathbf{C}\subset \mathcal{H}={\mathbf{C}}^2$ to each point $x$ by taking the solutions to equation $\gamma .5$ is an example of something called a “vector bundle” over the sphere of $\mathbf{x}\in S^2$ . A specific choice for each x of a solution $u_{+}(\mathbf{x})$ is called a “section” of the vector bundle. It can be thought of as a sort of “twisted” complex-valued function on the sphere, taking values not in the same C for each x as would a usual function, but in copies of C that vary with $x$.

These copies of C move around in ${\mathbf{C}}^2$ in a topologically non-trivial way: they cannot all be identified with each other in a continuous manner. The vector bundle that appears here is perhaps the most fundamental example of a topologically non-trivial vector bundle. A discontinuity such as that found in the section $u_{+}$ of equation 7.6 is required because of this topological non-triviality. For a non-trivial bundle like this one, there cannot be continuous non-zero sections.

While the Bloch sphere provides a simple geometrical interpretation of the states of the two-state system, it should be noted that this association of points on the sphere with states does not at all preserve the notion of inner product. For example, the North and South poles of the sphere correspond to orthogonal vectors in $\mathcal{H},$ but of course $(0,0,1)$ and $(0,0,-1)$ are not at all orthogonal as vectors in ${\mathbf{R}}^3$

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