Definition 3.1. The tensor product of two matrices A of dimensions m × n
and B of any dimensions is given by
A ⊗ B =
A
11
B A
12
B . . . A
1n
B
A
21
B A
22
B . . . A
2n
B
.
.
.
.
.
.
.
.
.
A
m1
B A
m2
B . . . A
mn
B
. (3.31)
Thus we have
|00i =
“
1
0
#
⊗
“
1
0
#
=
1
0
0
0
; |01i =
“
1
0
#
⊗
“
0
1
#
=
0
1
0
0
;
|10i =
“
0
1
#
⊗
“
1
0
#
=
0
0
1
0
; |11i =
“
0
1
#
⊗
“
0
1
#
=
0
0
0
1
.
(3.32)
Thus we have the natural basis for the 4-dimensional vector space from those
of two 2-dimensional spaces.
Example 3.5.1. Direct products: to express σ
x
on a 2-qubit state as a matrix,
we take the direct product of two σ
x
s acting on each single qubit state:
σ
x
⊗ σ
x
=
“
0 1
1 0
#
⊗
“
0 1
1 0
#
=
0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
.
We can generalize to n qubits: the natural basis of the n-qubit Hilbert
space H
⊗n
consists of 2
n
orthogonal vectors
{|0i, |1i, |2i, …, |2
n
− 1i}. (3.33)
The interpretation as an n-bit register is straightforward when the labels are
written in binary. For example, the 8
th
basis vector for a 4-qubit Hilbert space
will be
|7i = |0111i = |0i ⊗ |1i ⊗ |1i ⊗ |1i.
The algebra of multi-qubit states generalizes in a natural manner from
that of single qubits.
Problems 59
Box 3.8: Algebra of Tensor Product States
Consider two distinct physical systems A and B with Hilbert spaces H
A
of dimensions 2
n
and H
B
of dimensions 2
m
. Let the basis vectors of these two
spaces be denoted {|i
A
i}, i
A
= 0, 1, …2
n
−1, and {|µ
B
i}, µ
B
= 0, 1, …2
m
−1.
If I pick a state |φ
A
i from A and a state |ψ
B
i from B, I can form a state in
the tensor product Hilbert space H
AB
= H
A
⊗ H
B
as
|Φ
AB
i = |φ
A
i|ψ
B
i.
• Probability amplitude hi
A
, µ
B
|Φi = hi
A
|φ
A
ihµ
B
|ψ
B
i
• Inner product hΦ
1
|Φ
2
i = hφ
A
1
|φ
A
2
ihψ
B
1
|ψ
B
2
i
• Basis states for H
AB
is the set of product basis vectors {|υ
iµ
i =
|i
A
i|µ
B
i}
• The most general state in H
AB
is a linear combination of these basis
states:
|Ψi
AB
=
X
iµ
C
iµ
|υ
iµ
i.
• If two operators
ˆ
A and
ˆ
B act on each space independently then the
action on the product space is given by the operator
ˆ
C =
ˆ
A ⊗
ˆ
B.
Summary: The Math and the Physics
The arena of quantum mechanics is the Hilbert space H, the state vectors
live here, and transformations of the state vector are operators in H. To be able
to work efficiently with the maths, we summarize the correspondence between
the mathematical concept and the physical quantities in Table 3.2. Note that
this is for “pure” states of isolated quantum systems. (We will discuss mixed
states of systems that are influenced by some environment in a later chapter.)
Problems
3.1. Prove that a Hermitian matrix has real eigenvalues and its eigenvectors
corresponding to distinct eigenvalues are orthogonal to each other.
3.2. Prove that a unitary matrix has complex eigenvalues of unit magnitude, and
that its eigenvectors corresponding to distinct eigenvalues are orthogonal.
60 Introduction to Quantum Physics and Information Processing
TABLE 3.2: Correspondence between the math and the physics of quantum
mechanics.
Math Physics
Normalized vector |ψi ∈ H pure state
Hermitian operator
ˆ
A on H physical observable
Eigenvalues {a
i
} of
ˆ
A set of all possible values obtainable on measur-
ing the observable A
Eigenvector |a
i
i of
ˆ
A state in which measuring A gives a value a
i
Computational basis {|ii},
i = 0, 1, 2…
eigenstates of a suitable fiducial observable
Inner product hi|ψi probability amplitude for the state |ψi to be in
the basis state |ii
Amplitude squared |ha
i
|ψi|
2
probability of obtaining the value a
i
on mea-
suring A in the state ψ
Matrix element A
ij
= hi|A|ji amplitude for producing a transition from |ji
to |ii by the action of A (No assumption is
made here about the nature of the operation)
Diagonal element hψ|A|ψi average value of the observable A in the state
|ψi
Unitary operator
ˆ
U possible evolution operator that changes the
state reversibly
3.3. Show that if
ˆ
H is a Hermitian operator then e
i
ˆ
H
is a unitary operator.
3.4. Given a unitary operator
ˆ
U, show that the operator i( +
ˆ
U)( −
ˆ
U) is
Hermitian.
3.5. For a Hermitian or unitary matrix, show that the sum of diagonal elements
(the trace) equals the sum of the eigenvalues, and the determinant equals
the product of the eigenvalues.
3.6. For each of the following matrices, find if they are unitary or Hermitian or
neither. Find their eigenvalues and eigenvectors. Find if their eigenvectors
are orthogonal.
(a)
“
1 i
i −1
#
(b)
“
0 1
0 0
#
Problems 61
3.7. For the three Pauli matrices σ
x
, σ
y
, and σ
z
,
(a) Show that σ
2
i
= .
(b) Show that σ
i
’s are Hermitian as well as unitary.
(c) Find the commutator [σ
i
, σ
j
] = σ
i
σ
j
− σ
j
σ
i
.
(d) Find the anti-commutator {σ
i
, σ
j
} = σ
i
σ
j
+ σ
j
σ
i
.
3.8. Show that all the eigenvalues of any projection operator are either 1 or 0.
3.9. Show that the operator which performs a transformation from the Z basis
to the X basis has the following matrix representation:
H =
1
√
2
1 1
1 −1
!
.
This operator is also known as the Hadamard operator and is very useful in
quantum computation.
Verify that this operator is Hermitian. Show that it can be expressed as a
linear combination of the Pauli matrices.
3.10. Show that for any two operators A and B,
AB =
1
2
[A, B] +
1
2
{A, B}.
3.11. Given a unit vector ˆe = (e
x
, e
y
, e
z
) in an arbitrary direction, we can define
the component of spin along ˆe by
σ
e
= e
x
σ
x
+ e
y
σ
y
+ e
z
σ
z
.
(a) Show that σ
2
e
= .
(b) Find the eigenvalues and eigenvectors of σ
e
.
3.12. Define a “vector matrix” ~σ =
ˆ
iσ
x
+
ˆ
jσ
y
+
ˆ
kσ
z
. Show that
(~a. · ~σ)(
~
b · ~σ) = (~a ·
~
b) + i(~a ×
~
b) · ~σ (3.34)
for vectors ~a and
~
b.
3.13. Find the expectation value of σ
e
in the state |0i. Generalize this result
to find the expectation value of σ
e
in a state |
ˆ
f+i where
ˆ
f is a general
direction making angle θ with the ˆz axis.
Leave a Reply