Inner product of vectors gives the component of one vector in the direction
of the other. Similarly for quantum states, the inner product hψ|φi is the
probability amplitude that one state is along the other. For example,
|ψi =
1
√
3
|0i +
r
2
3
|1i.
Then the inner product
h0|ψi =
1
√
3
is the probability amplitude that the state |ψi has spin up.
As another example, the Hilbert space of position states of a particle along
the x-axis would have an infinite set of basis states |xi. A general state |ψi
has a probability amplitude hx|ψi = ψ(x) of being found at the location x.
This probability amplitude as a function of position is better known as the
wave function of the particle.
The inner product also comes in when describing the outcome of a process
that transforms a system from initial state |ψ
i
i to final state |ψ
f
i. The mod-
square of the probability amplitude for this process is then the probability
that such an event can occur:
P(|ψ
i
i → |ψ
f
i) = |hψ
f
|ψ
i
i|
2
. (3.3)
This statement, one of the underpinnings of quantum mechanics, is known as
the Born rule after Max Born,
3
who first postulated it.
3.1.3 Phases
The coefficients in the expansion of a state in terms of the basis states are
complex numbers in general. We saw one reason for this in the last chapter:
3
In a 1926 paper in a German journal, Born mentioned the probability interpretation in
a footnote.
40 Introduction to Quantum Physics and Information Processing
we need to account for interference when probability amplitudes are added.
Now a complex number has a modulus and a phase: z = x + iy has magnitude
r =
p
x
2
+ y
2
and a phase φ = tan
−1
y/x. r and φ are real numbers and we
express the same complex number in modular form as z = re
iφ
. Suppose we
write
|ψi = r
1
e
iθ
1
|0i + r
2
e
iθ
2
|1i. (3.4)
Different values of r
1
θ
1
and r
2
θ
2
give different vectors. For a given vector |ψi,
we can factor out one of the phases to write
|ψi = e
iθ
1
r
1
|0i + r
2
e
i(θ
2
−θ
1
)
|1i
(3.5)
The factored phase θ
1
is called a global phase. This cannot be measured by
any experiment since experiments only measure probabilities. In other words
the above state is experimentally indistinguishable from the state
|ψ
0
i = r
1
|0i + r
2
e
i(θ
2
−θ
1
)
|1i,
since |hψ|ψ
0
i|
2
= 1. What is measurable, however, is the relative phase
(θ
2
− θ
1
), which will show up in an interference experiment. The set of all
states differing by a global phase is called a ray in Hilbert space.
Thus the space of quantum states of a system is the space of rays in Hilbert
space, also called the projective Hilbert space. We will not emphasize this
difference in what follows, but it is a point to be kept in mind.
The fact that relative phases between components in a superposition state
are very important will become more relevant when we consider operations
on quantum systems that impart selective phases to one basis state, say |1i.
For instance, consider an operation
|0i → |0i; |1i → e
iφ
|1i.
Though such an operation produces indistinguishable states out of basis states,
the effect will be non-trivial on superposition states, since it would introduce
a relative phase between the |0i and |1i components:
|ψi = c
1
|0i + c
2
|1i → c
1
|0i + e
iφ
c
2
|1i
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