3 Two-state Systems and SU(2)

The simplest truly non-trivial quantum systems have state spaces that are inherently two-complex dimensional. This provides a great deal more structure than that seen in chapter 2, which could be analyzed by breaking up the space of states into one dimensional subspaces of given charge. We’ll study these two-state systems in this section, encountering for the first time the implications of working with representations of a non-commutative group. Since they give the simplest non-trivial realization of many quantum phenomena, such systems are the fundamental objects of quantum information theory (the “qubit”) and the focus of attempts to build a quantum computer (which would be built out of multiple copies of this sort of fundamental object). Many different possible two-state quantum systems could potentially be used as the physical implementation of a qubit.

One of the simplest possibilities to take would be the idealized situation of a single electron, somehow fixed so that its spatial motion could be ignored, leaving its quantum state described solely by its so-called “spin degree of freedom”, which takes values in $H = C^{2}$ . The term “spin” is supposed to call to mind the angular momentum of an object spinning about some axis, but such classical physics has nothing to do with the qubit, which is a purely quantum system.

In this chapter we will analyze what happens for general quantum systems with $H = C^{2}$ by first finding the possible observables. Exponentiating these will give the group $U (2)$ of unitary 2 by 2 matrices acting on $H = C^{2}$ . This is a specific representation of $U (2)$ , the “defining” representation. By restricting to the subgroup $S U (2) \subset U (2)$ of elements of determinant one, one gets a representation of $S U (2)$ on $C^{2}$ often called the “spin $\frac{1}{2}$ ” representation.

Later on, in chapter 8, we will find all the irreducible representations of $S U (2)$ . These are labeled by a natural number

N = 0, 1, 2, 3, \dots

and have dimension $N + 1$ . The corresponding quantum systems are said to have “spin $N /2$ ”. The case $N = 0$ is the trivial representation on $C$ and the case $N = 1$ is the case of this chapter. In the limit $N \to \infty$ one can make contact with classical notions of spinning objects and angular momentum, but the spin $\frac{1}{2}$ case is at the other limit, where the behavior is purely quantum-mechanical.

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3 Two-state Systems and SU(2)