Chapter 3

Two-state Systems and

SU(2)

The simplest truly non-trivial quantum systems have state spaces that are in-
herently 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, leav-
ing its quantum state described solely by its so-called “spin degree of freedom”,
which takes values inH=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
withH=C^2 by first finding the possible observables. Exponentiating these
will give the groupU(2) of unitary 2 by 2 matrices acting onH=C^2. This is
a specific representation ofU(2), the “defining” representation. By restricting
to the subgroupSU(2) ⊂U(2) of elements of determinant one, one gets a
representation ofSU(2) onC^2 often called the “spin^12 ” representation.
Later on, in chapter 8, we will find all the irreducible representations of
SU(2). These are labeled by a natural number

N= 0, 1 , 2 , 3 ,...