Someone gives you an electron and asks you: what is the spin? How will
you answer? If you measure S
x
, S
y
, or S
z
you will get one of two answers, at
random. Any observable you measure gives one of its eigenvalues at random.
The state has probabilistic information about each eigenvalue. The meaning
of this is statistical: (i) take a very large number of identical copies of the
state |ψi: a statistical ensemble, (ii) perform the measurement of
ˆ
A on each
copy, then if you expand ψ in the basis of eigenvectors a of
ˆ
A,
|ψi =
X
a
c
a
|ai
then a fraction |c
a
|
2
times you will obtain a as the result of the measure-
ment. The value of the physical observable A is the statistical average value
obtained by all these measurements. This is called the expectation value of
the operator
ˆ
A in the state |ψi denoted by
h
ˆ
Ai
ψ
=
X
a
P(a)a =
X
a
|c
a
|
2
a
=
X
a
hψ|aiha|ψia =
X
a
|ψi
ˆ
A|aiha|ψi
= hψ|
ˆ
A|ψi,
where in the last step we have used the resolution of the identity.
We can thus define statistically the mean value of an observable. The
statistics of measurement is incomplete without the notion of the variance
about the mean. We define the variance ∆
2
A as
∆
2
A = hA
2
i − hAi
2
The square root of the variance, the standard deviation, is called the error or
uncertainty in the value of A.
54 Introduction to Quantum Physics and Information Processing
Example 3.3.1. Expectation value of
ˆ
S
z
and
ˆ
S
x
in the state |0i:
h
ˆ
S
z
i
0
= h0|
ˆ
S
z
|0i = h0|
~
2
|0i
=
~
2
h
ˆ
S
x
i
0
= h0|
ˆ
S
x
|0i = h0|
~
2
|1i
= 0.
Box 3.7: The Uncertainty Principle
This principle is one of the foundation pillars of quantum mechanics, first
enunciated by Werner Heisenberg. More accurately called the indeterminacy
principle, this states that some physical observables are “incompatible” with
each other, in the sense that on measurement in a given state, it is not pos-
sible to get sharp values of both. In fact, the uncertainty in one observable
is inversely related to that in the other. Classic examples are position and
momentum, and also the three components of the spin vector.
Mathematically, compatibility is related to the commutation of the opera-
tors: whether the order of operation of two operators matters or not. For two
operators
ˆ
A and
ˆ
B the commutator is defined as the operator expressing this
difference in ordering:
ˆ
C = [
ˆ
A,
ˆ
B] ≡
ˆ
A
ˆ
B −
ˆ
B
ˆ
A.
It can be shown that the product of uncertainties of two operators measured
in a state |ψi is related to their commutator:
∆
ˆ
A∆
ˆ
B
ψ
≥
1
2
[
ˆ
A,
ˆ
B]
ψ
. (3.22)
Note that experimentally uncertainty refers to the standard deviation from
the mean of a statistically large set of measurements of the observable, made
on identically prepared states. Physically the meaning of the uncertainty prin-
ciple is that if we perform a set of measurements of observable A and B in
an ensemble prepared in a state |ψi, then the products of the uncertainties
of the two observables is limited by the expression on the right, related to
their commutator. Experimental uncertainties would add to this limit. Thus
in principle, the uncertainty in either of a pair of observables that do not
commute can never be zero.
The Essentials of Quantum Mechanics 55
3.4 Evolution
An isolated system is said to evolve when its state changes with time. The
change in state would take place by the action of an operator on it. This action
cannot take it out of the Hilbert space, and must preserve its norm. Therefore
the evolution operator
ˆ
U has to satisfy some conditions.
|ψi
ˆ
U
−→ |ψ
0
i =
ˆ
U|ψi
hψ
0
|ψ
0
i = hψ|
ˆ
U
†
ˆ
U|ψi
If hψ
0
|ψ
0
i = hψ|ψi
then
ˆ
U
†
=
ˆ
U
−1
Such an operator is called unitary:
ˆ
U
†
ˆ
U = .
In quantum computation, any operation we wish to perform on a qubit must
be represented by such an operator. One of the important consequences of
this is that since any unitary operator is invertible, any quantum operation is
reversible.
For example, the Pauli spin operators are unitary and are valid evolution
operators. The operation |0i→ |1iand |1i→ |0iis achieved by the σ
1
operator.
This operation flips the bits 0 and 1, and is therefore also called the NOT
operator X.
Thus evolution is another application of unitary operators in quantum
mechanics. The first one we encountered of course was while implementing
basis change.
3.4.1 Continuous time evolution
From the physical viewpoint, evolution in time occurs due to interaction
of the system with an external force. A characteristic of this “force” is the
energy the system has in its presence. This energy is represented by a func-
tion called the Hamiltonian function H. In a given situation it has to be
determined experimentally. The quantum version of the Hamiltonian is the
Hamiltonian operator
ˆ
H. This operator, being an observable, must be Hermi-
tian. Now it turns out that when the Hamiltonian acts on a state vector, it
creates an infinitesimal time evolution. This gives a differential version of the
time evolution postulate of which there are two (experimentally equivalent)
viewpoints or “pictures”:
56 Introduction to Quantum Physics and Information Processing
3.4.1.1 Schr¨odinger viewpoint
Postulate 4. The evolution in time of a quantum state vector |ψ(t)i is given
by the Schr¨odinger equation:
i~
d|ψi
dt
=
ˆ
H|ψi. (3.23)
We can try to understand what this implies by formally integrating this
equation to solve for |ψ(t)i from |ψ(t
0
)i. Assuming that the Hamiltonian func-
tion is itself explicitly independent of time, we would get
|ψ(t)i = exp
−
i
~
ˆ
H(t − t
0
)
|ψ(t
0
)i.
So the unitary operator for time evolution is just
ˆ
U(t
0
, t) ≡ exp
−
i
~
ˆ
H(t − t
0
)
. (3.24)
We can set t
0
= 0 and write
ˆ
U(t) = e
−i
ˆ
Ht/~
.
Here, the exponential of the operator
ˆ
H is understood as the infinite sum of
powers of
ˆ
H:
e
−i
ˆ
Ht/~
≡ −
it
~
ˆ
H +
1
2
it
~
2
ˆ
H
2
+ ··· ,
itself an operator that can be expressed as a matrix. You can verify that since
ˆ
H is Hermitian,
ˆ
U(t) is indeed unitary.
3.4.1.2 Heisenberg viewpoint
One can focus on the observables being measured instead of the state in
which they are measured, and think of evolution as affecting the observable
(operator) instead of the state vector. In this picture, the evolution of an
observable
ˆ
A(t) is given by
ˆ
A(t) =
ˆ
U(t)
ˆ
A(0)
ˆ
U
†
(t) (3.25)
=⇒
d
ˆ
A
dt
=
d
dt
ˆ
U
ˆ
A(0)
ˆ
U
†
+
ˆ
U
ˆ
A(0)
d
dt
ˆ
U
†
=
i
~
−
ˆ
H
ˆ
U
ˆ
A(0)
ˆ
U
†
+
ˆ
U
ˆ
A(0)
ˆ
H
ˆ
U
†
d
ˆ
A
dt
=
i
~
[
ˆ
A(t),
ˆ
H], (3.26)
where the square brackets indicate the commutator AH −HA. Here we have
assumed that the observable A itself has no explicit time-dependence; that is,
t does not occur in its form. If it did then we would have to add the partial
derivative of
ˆ
A(t) with respect to t. It is straightforward to see that both
pictures give the same value for the experimentally observed quantities: the
expectation values of observables
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