The formalism for quantum systems developed so far applies to what are
called pure states. A system in a pure state is completely specified by the
state vector. A complete set of experimental tests will determine the system
state fully: we have maximal knowledge of the system. For example, for a spin
system, we can find a particular orientation of an SG machine such that the
state is in its + or − port. This also means that the state is an eigenvector of
some operator, or is always a linear combination of the computational basis
states.
As opposed to this, as in most practical cases, we only have incomplete
knowledge of the state. This means that the state is in practice not an eigen-
state of an observable, but consists of a mixture of eigenstates with classical
probabilities of being in each state. Such a state is called a mixed state
and CANNOT be represented by a state vector. The most convenient way of
representing and dealing with such systems, is through the density operator
formulation, as proposed first by von Neumann [70] in 1927.
1
For instance, how do we describe the state of an unpolarized beam of spins,
such as those emitted from the oven in the original Stern–Gerlach experiment?
We will find that on analyzing such a beam using an SG machine in any
orientation, it is split into up-spin and down-spin beams of equal intensities.
The state of this beam can be regarded as a 50-50 mixture of basis states of
any representation. This is an example of a mixed state. We cannot represent
it as a superposition of any basis states. However, the output of an SG
z
↑
filter, which splits into up-spin and down-spin beams of equal intensities when
passed through SG
x
or SG
y
machines is a pure state that can be represented
by the state vectors
|0i ≡ |↑
z
i =
1
√
2
[|↑
x
i + |↓
x
i] =
1
√
2
[|↑
y
i + i|↓
y
i…
Another point to keep in mind is that we have so far been describing closed
quantum systems, that are isolated from the environment or not affected by it.
More realistic systems are open to the environment, the effect of which must
be taken into account in some fashion, though one may not have complete
1
The density operator was also independently proposed by Lev Landau [45] and by Felix
Bloch.
77
78 Introduction to Quantum Physics and Information Processing
information as to how the environment affects the system. One way of dealing
with such situations is to regard the system along with the environment as a
big super-system that is closed. So when we concentrate only on the system,
we have to average out the effect of the environment. The resulting system
state typically is mixed, and one needs the density operator formulation to
describe it. The material in this chapter is of a slightly advanced character
and may be skipped at first reading
Leave a Reply