1. No category

Problems
in
Quantum Computing
by
Willi-Hans Steeb
International School for Scientific Computing
at
University of Johannesburg, South Africa
Yorick Hardy
Department of Mathematical Sciences
at
University of South Africa
updated: July 2, 2015
Preface
The purpose of this book is to supply a collection of problems in quantum
computing.
Prescribed books for problems.
1) Problems and Solutions in Quantum Computing and Quantum Information (third edition)
by Willi-Hans Steeb and Yorick Hardy
World Scientific, Singapore, 2011
ISBN-13 978-981-4366-32-8
http://www.worldscibooks.com/physics/8249.html
2) Classical and Quantum Computing with C++ and Java Simulations
by Yorick Hardy and Willi-Hans Steeb
Birkhauser Verlag, Boston, 2002
ISBN 376-436-610-0
3) Matrix Calculus and Kronecker Product
by Willi-Hans Steeb
World Scientific Publishing, Singapore 2010
ISBN 978-981-4335-31-7
http://www.worldscibooks.com/mathematics/8030.html
4) Problems and Solutions in Introductory and Advanced Matrix Calculus
by Willi-Hans Steeb
World Scientific Publishing, Singapore 2006
ISBN 981 256 916 2
http://www.worldscibooks.com/mathematics/6202.html
5) Continous Symmetries, Lie Algebras, Differential Equations and Computer Algebra, second edition
by Willi-Hans Steeb
World Scientific Publishing, Singapore 2007
ISBN 981-256-916-2
http://www.worldscibooks.com/physics/6515.html
v
The International School for Scientific Computing (ISSC) provides certificate courses for this subject. Please contact the authors if you want to do
this course or other courses of the ISSC.
e-mail addresses of the author:
[email protected]
[email protected]
Home page of the author:
http://issc.uj.ac.za
vi
vii
Contents
Preface
v
Notation
x
1 Qubits
1
2 Kronecker and Tensor Product
12
3 Matrix Properties
24
4 Density Operators
42
5 Partial Trace
59
6 Reversible Logic Gates
61
7 Unitary Transformations and Quantum Gates
68
8 Entropy
76
9 Measurement
80
10 Entanglement
82
11 Bell Inequality
98
12 Quantum Channels
100
13 Miscellaneous
103
Bibliography
114
Index
119
viii
x
Notation
:=
∈
∈
/
∩
∪
∅
N
Z
Q
R
R+
C
Rn
Cn
H
i
<z
=z
|z|
T ⊂S
S∩T
S∪T
f (S)
f ◦g
x
xT
0
k.k
x · y ≡ x∗ y
x×y
A, B, C
det(A)
tr(A)
rank(A)
AT
is defined as
belongs to (a set)
does not belong to (a set)
intersection of sets
union of sets
empty set
set of natural numbers
set of integers
set of rational numbers
set of real numbers
set of nonnegative real numbers
set of complex numbers
n-dimensional Euclidean space
space of column vectors with n real components
n-dimensional complex linear space
space of column vectors with n complex components
Hilbert
space
√
−1
real part of the complex number z
imaginary part of the complex number z
modulus of complex number z
|x + iy| = (x2 + y 2 )1/2 , x, y ∈ R
subset T of set S
the intersection of the sets S and T
the union of the sets S and T
image of set S under mapping f
composition of two mappings (f ◦ g)(x) = f (g(x))
column vector in Cn
transpose of x (row vector)
zero (column) vector
norm
scalar product (inner product) in Cn
vector product in R3
m × n matrices
determinant of a square matrix A
trace of a square matrix A
rank of matrix A
transpose of matrix A
xi
A
A∗
A†
A−1
In
I
0n
AB
A•B
[A, B] := AB − BA
[A, B]+ := AB + BA
A⊗B
A⊕B
δjk
λ
t
ˆ
H
conjugate of matrix A
conjugate transpose of matrix A
conjugate transpose of matrix A
(notation used in physics)
inverse of square matrix A (if it exists)
n × n unit matrix
unit operator
n × n zero matrix
matrix product of m × n matrix A
and n × p matrix B
Hadamard product (entry-wise product)
of m × n matrices A and B
commutator for square matrices A and B
anticommutator for square matrices A and B
Kronecker product of matrices A and B
Direct sum of matrices A and B
Kronecker delta with δjk = 1 for j = k
and δjk = 0 for j 6= k
eigenvalue
real parameter
time variable
Hamilton operator
The Pauli spin matrices are used extensively in the book. They are given
by
0 1
0 −i
1 0
σx :=
, σy :=
, σz :=
.
1 0
i 0
0 −1
In some cases we will also use σ1 , σ2 and σ3 to denote σx , σy and σz .
Chapter 1
Qubits
Problem 1.
i.e.
Let (0 ≤ θ <
Let |0i, |1i be the standard basis in the Hilbert space C2 ,
1
0
|0i =
,
|1i =
.
0
1
π
4)
|Ψ+ (θ)i := cos θ|0i + sin θ|1i,
|Ψ− (θ)i := cos θ|0i − sin θ|1i.
(i) Find the scalar product hΨ− (θ)|Ψ+ (θ)i. Discuss.
(ii) Consider the states
1
|+i := √ (|0i + |1i),
2
1
|−i := √ (|0i − |1i)
2
and the projection operators (projection matrices)
Π+ := |+ih+|,
Π− := |−ih−|.
Find
hΨ+ (θ)|Π+ |Ψ+ (θ)i, hΨ+ (θ)|Π− |Ψ+ (θ)i, hΨ− (θ)|Π+ |Ψ− (θ)i, hΨ− (θ)|Π− |Ψ− (θ)i
and the 2 × 2 matrices Π+ + Π− and Π+ Π− . Discuss.
Problem 2.
(i) Consider the normalized vector in the Hilbert space C3


sin(θ) cos(φ)
n =  sin(θ) sin(φ)  .
cos(θ)
1
2 Problems and Solutions
Show that the vector is normalized.
(ii) Calculate the 2 × 2 matrix
U (θ, φ) = n · σ ≡ n1 σ1 + n2 σ2 + n3 σ3
where σ1 , σ2 , σ3 are the Pauli spin matrices.
(iii) Is the matrix U (θ, φ) unitary? Find the trace and the determinant. Is
the matrix U (θ, φ) hermitian?
(iv) Find the eigenvalues and normalized eigenvectors of U (θ, φ).
Problem 3.
Consider the states
cos(φ/2)
ψ1 (φ) =
,
sin(φ/2)
ψ2 (φ) =
− sin(φ/2)
cos(φ/2)
.
in the Hilbert space C2 .
(i) Show that these states can be generated from the standard basis using
the exponential function and the Pauli matrix σ2 , i.e. calculate
φ
φ
1
0
,
exp −i σ2
.
exp −i σ2
0
1
2
2
(ii) Find the states after the transformation φ → φ + 2π.
Problem 4. Let σ1 , σ2 , σ3 be the Pauli spin matrices and I2 the 2 × 2
identity matrix. Find the eigenvalues and normalized eigenvectors of the
Hamilton operator
ˆ = ε0 I2 + ~ωσ3 + ∆1 σ1 + ∆2 σ2
H
where ε0 > 0. Are the normalized eigenvectors orthonormal to each other?
ˆ be a 2 × 2 hermitian matrix. Consider the normalized
Problem 5. Let H
state
iφ
e cos θ
|ψi =
sin θ
in the Hilbert space C2 . Assume that
ˆ
hψ|H|ψi
= ~ω cos(φ) sin(2θ),
ˆ 2 |ψi = ~2 ω 2 .
hψ|H
ˆ from these three assumptions. Note
Reconstruct the hermitian matrix H
that
cos(θ) sin(θ) ≡
1
sin(2θ),
2
eiφ = cos(φ)+i sin(φ),
e−iφ = cos(φ)−i sin(φ).
Qubits
Problem 6.
3
Consider the Hadamard matrix
1
1 1
H=√
2 1 −1
and the state
|ψi =
cos(θ)
sin(θ)
.
Calculate the variance
VH (ψ) := hψ|H 2 |ψi − (hψ|H|ψi)2
and discuss the dependence on θ.
ˆ be n × n hermitian matrices. Let |ψi be a
Problem 7. Let Aˆ and B
normalized state in the Hilbert space Cn . Then we have the inequality
ˆ B]i|
ˆ
ˆ
ˆ ≥ 1 |h[A,
(∆A)(∆
B)
2
where
∆Aˆ :=
q
ˆ 2,
hAˆ2 i − hAi
ˆ :=
∆B
q
ˆ 2 i − hBi
ˆ 2
hB
and
ˆ := hψ|A|ψi,
ˆ
hAi
Consider the hermitian spin matrices
1 0
1 0 1
,
s2 =
s1 =
2 1 0
2 i
ˆ := hψ|B|ψi.
ˆ
hBi
−i
0
1
s3 =
2
,
1
0
0
−1
.
ˆ = s2 . Find states |ψi such that
Let Aˆ = s1 and B
ˆ
ˆ = 1 |h[A,
ˆ B]i|
ˆ
(∆A)(∆
B)
2
i.e. the inequality given above should be an equality.
Problem 8.
Given the two normalized states
1
1
1
1
|ψi = √
,
|φi = √
.
2 1
2 −1
Find a unitary matrix U such that |ψi = U |φi. Give the eigenvalues of U .
Problem 9.
Let
A=
3
X
k=0
ak σk ,
B=
3
X
`=0
b` σ`
4 Problems and Solutions
where σ0 = I2 and ak , b` ∈ R with a3 6= 0 and b1 = a1 b3 /a3 , b2 = a2 b3 /a3 .
Calculate the commutator [A, B].
Problem 10.
Consider the symmetric matrix over R
h11 h12
H=
h12 h22
and the state
|ψi =
cos θ
sin θ
.
Calculate the variance
VH (|ψi) = hψ|H 2 |ψi − (hψ|H|ψi)2 .
Problem 11.
Let σ1 , σ2 , σ3 be the Pauli spin matrices. Show that
cos(ωt)σ1 − sin(ωt)σ2 = eiωt σ+ + e−iωt σ− ,
e±ωt σ± eiωtσ3 /2 = eiωtσ3 /2 σ±
where σ± := (σ1 ± iσ2 )/2.
Problem 12. Consider the Pauli spin matrices σ1 , σ2 , σ3 . Can one find
a 2 × 2 invertible matrix K with K = K −1 and
Kσ1 K = σ1 ,
Kσ2 K = −σ2 ,
Kσ3 K = σ3 ?
Problem 13. Let σ1 , σ2 , σ3 be the Pauli spin matrices and α ∈ R.
(i) Calculate the 2 × 2 matrices
exp(−iασ1 /2),
exp(−iασ2 /2),
exp(−iασ3 /2).
Are the matrices unitary?
(ii) Let
1
|ψi =
.
0
Find the state exp(−iασ1 /2)|ψi and calculate
hψ| exp(−iασ1 /2)|ψi and |hψ| exp(−iασ1 /2)|ψi|2 .
Problem 14. Let |0i, |1i be an orthonormal basis in a two-dimensional
Hilbert space. Consider the Hamilton operator
ˆ = − 1 ~ω(e−iφ |1ih0| + eiφ |0ih1|).
H
2
Qubits
5
ˆ
Find exp(−iHt/~).
Problem 15.
Consider the Hamilton operator
~ω λa12
ˆ
H(λ) =
λa12 −~ω
where a12 ∈ R. Let I2 be the 2 × 2 identity matrix and E a real parameter.
Solve the system of equations
ˆ
det(H(λ)
− I2 E) = 0
d
ˆ
det(H(λ)
− I2 E) = 0
dE
with respect to E and λ.
Problem 16.
Consider the Hamilton operator
~ω
∆
ˆ
H(λ)
=
.
∆ −~ω
ˆ
(i) Find the eigenvalues and normalized eigenvectors of H.
(ii) Consider the unitary matrix
cos φ
−e−iθ sin φ
U=
.
eiθ sin φ
cos φ
ˆ is a diagonal matrix?
Can one find φ, θ such that U ∗ HU
Problem 17. Consider the Pauli spin matrices σ1 , σ2 and σ3 . Can one
find an α ∈ R such that
exp(iασ3 )σ1 exp(−iασ3 ) = σ2 ?
Problem 18. Let σ1 , σ2 , σ3 be the Pauli spin matrices. Let α1 , α2 , α3 ∈
C. Find the conditions on α1 , α2 , α3 such that
U = α1 σ1 + α2 σ2 + α3 σ3
is a unitary matrix.
Problem 19.
Consider the NOT-gate
0 1
U=
.
1 0
6 Problems and Solutions
Find a 2 × 2 matrix V such that V 2 = U .
Problem 20.
Consider the map f : C2 → R3 defined by


sin(2θ) cos φ
cos θ
f :
7→  sin(2θ) sin φ  .
eiφ sin θ
cos(2θ)
Are the vectors in C2 and R3 normalized? Consider the four normalized
vectors in C2
1
1
1
1
1
1
1
1
√
, √
, √
, √
.
2 1
2 −1
2 i
2 −i
Find the vectors in R3 .
Problem 21.
Let σ1 , σ2 and σ3 be the Pauli spin matrices. Calculate
U (α, β, γ) = e−iασ3 /2 e−iβσ2 /2 e−iγσ3 /2
where α, β, γ are the Euler angles with the range 0 ≤ α < 2π, 0 ≤ β ≤ π
and 0 ≤ γ < 2π.
ˆ 0 and H
ˆ 1 be a pair of real symmetric n × n matrices,
Problem 22. Let H
ˆ
where H0 is a diagonal matrix. Let
ˆ
ˆ 0 + H
ˆ 1.
H()
:= H
(1)
ˆ
When is real, H()
is diagonalizable with eigenvalues E1 (), . . ., En ().
The eigenvalues are given by the characteristic polynomial
ˆ
P (E, ) := det(H()
− EIn ) = 0
(2)
where In is the n × n unit matrix. When is complex, the eigenvalues
may be viewed as the n values of a single function E() of , analytic on a
Riemann surface with N sheets joined at branch point singularities in the
complex plane. The exceptional points in the complex plane are defined
by the solution of (2) together with
d
ˆ
det(H()
− EIn ) = 0.
dE
(i) Consider the two-level system
0
ˆ
H()
=
0
0
1
+
0
1
1
0
(3)
.
Qubits
7
ˆ
Find the exceptional points of H().
(ii) Let 1 and 2 be the two exceptional points. Find the eigenvalues and
ˆ 1 ) and H(
ˆ 2 ). Discuss.
eigenvectors of the matrices H(
Problem 23.
Study the eigenvalue problem for the matrix
σ3 + eiφ σ1
for φ ∈ [0, π/2].
Problem 24.
(i) Let φ ∈ R. Is the matrix
0 e−iφ
A=
eiφ
0
hermitian, unitary?
(ii) Find the rank of the matrix.
(iii) Find the eigenvalues and eigenvectors of A.
(iv) Let I2 be the 2 × 2 unit matrix. Find the eigenvalues of A ⊗ I2 .
Problem 25. Let |φ1 i, |φ2 i be two normalized vectors in the Hilbert
space R2 . Assume that
1
hφ1 |φ2 i = .
2
Give a geometric interpretation of this equation.
Problem 26.
Consider the vectors
i i i
i
,
|ψ
i
=
|ψ1 i =
2
sin(i)
eiπ
in the Hilbert space C2 . Are the vectors normalized? If not normalize the
vectors.
Problem 27. Let H be an arbitrary Hilbert space. Let |ψi and |φi be
arbitrary normalized states in H. Find all the solutions of the equation
hφ|ψihψ|φi = i.
Problem 28.
What is the condition on φ1 , φ2 , φ3 (all real) such that
1
1
eiφ1
V =√
iφ
iφ
2 e 2 e 3
8 Problems and Solutions
is a unitary matrix?
Problem 29. Consider the matrices
1 0
1 1
0
A=
, B=
, C=
1 0
0 0
0
1
1
,
D=
0
1
0
1
.
Find unitary matrices U1 , U2 , U3 , U4 such that
U1−1 AU1 = B,
Problem 30.
U2−1 BU2 = C,
U3−1 CU3 = D,
U4−1 DU4 = A.
(i) Consider the normalized state
−iφ/2
e
cos(θ/2)
|ψi =
.
eiφ/2 sin(θ/2)
in the Hilbert space C2 . Let σ1 , σ2 , σ3 be the Pauli spin matrices. Calculate
nj := hψ|σj |ψi,
Is the vector
j = 1, 2, 3


n1
n =  n2 
n3
in R3 normalized?
(ii) Consider the Hamilton operator
µ~
µ~
ˆ
H(t)
= − B(t) · σ ≡ − (B1 (t)σ1 + B2 (t)σ2 + B3 (t)σ3 )
2
2
where B(t) is a time-dependent homogeneous magnetic field. Show that
the Schr¨
odinger equation
i~
d
ˆ
|ψ(t)i = H(t)|ψ(t)i
dt
can be written as
d
n(t) = −µB(t) × n
dt
where × denotes the vector product.
Problem 31. Find the square roots of the Pauli spin matrices
1 0
0 1
0 −i
1
σ0 =
, σ1 =
, σ2 =
, σ3 =
0 1
1 0
i 0
0
0
−1
Qubits
9
i.e. find the matrices Rj such that Rj2 = σj (j = 0, 1, 2, 3).
Problem 32. Consider a d-dimensional Hilbert space with two orthonormal bases
|b11 i, |b12 i, . . . |b1d i ∈ B1
|b21 i,
|b22 i,
...
|b2d i ∈ B2 .
The two bases are said to be mutually unbiased bases if
1
|hb2j |b1k i| = √
d
for all j, k = 1, . . . , d and h | i denotes the scalar product in the Hilbert
space. Consider the Hilbert space M2 (C) of 2 × 2 matrices over C, where
the scalar product is defined as
hA|Bi = tr(AB ∗ ),
A, B ∈ M2 (C)
Thus d = dim(M2 (C)) = 4. The standard basis in this Hilbert space is
given by
1 0
0 1
0 0
0 0
E11 =
, E12 =
, E21 =
, E22 =
.
0 0
0 0
1 0
0 1
Let UH be the Hadamard matrix
1
1
UH = √
2 1
1
−1
∗
UH
= UH .
ejk (j, k = 1, 2)
(i) Show that the matrices E
∗
ejk = UH Ejk UH
E
,
j, k = 1, 2
and the standard basis form mutually unbiased bases.
ejk (j, k = 1, 2) to find
(ii) Apply the vec-operator to the matrices Ejk and E
4
mutually unbiased bases in the Hilbert space C .
Problem 33. Let σ1 , σ2 , σ3 be the Pauli spin matrices. Find all 2 × 2
matrices A such that
[σ1 , A] = [σ2 , A] = [σ3 , A] = 02 .
Problem 34. Let φ1 , φ2 , φ3 , φ4 ∈ R. The 2 × 2 matrix U = (v1 v2 )
contains the two column vectors
iφ iφ 1
1
e 1
e 3
√
,
v
=
.
v1 = √
2
iφ2
iφ
2 e
2 e 4
10 Problems and Solutions
Find the conditions on φ1 , φ2 , φ3 , φ4 such that
hv1 |v2 i = 0.
Is the matrix U unitary if this condition is satisfied.
Problem 35.
(i) Find the norms of the vectors in the Hilbert space C2
iα e
sin(i)
|ψi =
,
|φi
=
e−iα
cos(i)
where α ∈ R.
(ii) Normalize the vectors |ψi and |φi.
(iii) After normalizing the vectors calculate the probability
p(α) = |hψ(α)|φi|2 .
Discuss p as a function of α.
Problem 36.
Let α ∈ R. Consider the vector in C2
cosh(α)
v=
.
sinh(α)
Normalize the vector and then study the cases α → +∞ and α → −∞.
Can one find a non-zero (column) vector u in C2 such that
u∗ v = 0 ?
Problem 37.
Consider the normalized state
cos(θ/2)
|ψi =
eiφ sin(θ/2)
in the Hilbert space C2 . Let σ1 , σ2 , σ3 be the Pauli spin matrices. Find
the expectation values
hψ|σ1 |ψi,
Problem 38.
hψ|σ2 |ψi,
hψ|σ3 |ψi.
Is the state
|ψi =
cos(θ/2)eiφ/2
sin(θ/2)e−iφ/2
normalized? Find a normalized vector which is orthogonal to this vector.
If so calculate the density matrix ρ = |ψihi.
Qubits
Problem 39.
11
Consider the normalized state |ψi in the Hilbert space C2
cos(θ/2)
|ψi =
eiφ sin(θ/2)
and the Pauli spin matrices σ1 , σ2 , σ3 . Find
hψ|σ1 |ψi,
hψ|σ2 |ψi,
hψ|σ3 |ψi.
Chapter 2
Kronecker and Tensor
Product
Problem 1.
in C8
√
Let := e2πi/3 ≡ (−1 + i 3)/2. Consider the eight states
1
1
1
0
0
0
⊗
⊗
, |ψ2 i =
⊗
⊗
0
0
0
1
1
1
1
1
1
0
1
0
1
0
1
1
√
⊗
⊗
+
⊗
⊗
+
⊗
⊗
0
0
1
0
1
0
1
0
0
3
1
0
0
1
0
1
0
1
0
0
√
⊗
⊗
+
⊗
⊗
+
⊗
⊗
1
1
0
1
0
1
0
1
1
3
1
0
1
1
1
1
0
1
0
1
√
⊗
⊗
⊗
⊗
+
⊗
⊗
+
1
0
0
0
0
1
0
1
0
3
1
1
1
0
1
0
1
0
1
1
√
⊗
⊗
+
⊗
⊗
+
⊗
⊗
0
0
1
0
1
0
1
0
0
3
1
0
0
1
0
1
0
1
0
0
√
⊗
⊗
+
⊗
⊗
+
⊗
⊗
1
1
0
1
0
1
0
1
1
3
1
0
0
1
0
1
0
1
0
0
√
⊗
⊗
+
⊗
⊗
+
⊗
⊗
1
1
0
1
0
1
0
1
1
3
|ψ1 i =
|ψ3 i =
|ψ4 i =
|ψ5 i =
|ψ6 i =
|ψ7 i =
|ψ8 i =
(i) Calculate the scalar product hψj |ψk i for j, k = 1, 2, . . . , 8.
(ii) Which of the vectors are entangled?
12
Kronecker and Tensor Product
13
Problem 2. (i) Can we find 2 × 2 matrices A and B with det A =
a11 a22 − a12 a21 6= 0, det B = b11 b22 − b12 b21 6= 0 (i.e. we assume that A
and B are invertible) such that
 
 
1
0
1 1
1 0
√   = (A ⊗ B) √   ?
2 0
2 1
1
0
On the left-hand side we have the Bell state |Φ+ i and on the right-hand
side we have the Bell state |Ψ+ i. Since A and B are invertible we find that
A ⊗ B is also invertible with (A ⊗ B)−1 = A−1 ⊗ B −1 .
(ii) Can we also find 2 × 2 matrices A, B such that det A = det B = 1, i.e.,
A, B ∈ SL(2, R)?
Problem 3. Can we find 2×2 matrices A, B, C with det A = 1, det B = 1
and det C = 1 such that
 
 
0
1
1
0
 
 
1
0



1 
1 0
0
√   = (A ⊗ B ⊗ C) √   ?
1
0
3
2
 
 
0
0
 
 
0
0
0
1
On the left-hand side we have the W-state and on the right-hand side we
have the GHZ-state.
Problem 4. Let A, B be n × n hermitian matrices over C. Let K be an
n×n hermitian matrix over C and H = ~ωK be a Hamilton operator, where
~ is the Planck constant and ω the frequency. The Heisenberg equation of
motion for the operators A and B are given by
i~
dA
= [A, H](t),
dt
i~
dB
= [B, H](t).
dt
The solutions can be given as
ˆ
ˆ
A(t) = eitH/~ Ae−itH/~ ,
ˆ
ˆ
B(t) = eitH/~ Be−itH/~ .
(i) Find the time evolution of A ⊗ B, B ⊗ A, A ⊗ A and B ⊗ B.
(ii) Assume that [A, H] = 2i~ωB and [B, H] = −2i~ωA. Simplify the
Heisenberg equation of motion with these conditions.
14 Problems and Solutions
Problem 5. Let A be an m × m hermitian matrix and let B be an n × n
hermitian matrix. Then A ⊗ B, A ⊗ In , Im ⊗ B are also hermitian matrices,
where Im is the m×m identity matrix. Let 1 , 2 and 3 be real parameters.
Consider the Hamilton operator
H = ~ω(1 A ⊗ B + 2 A ⊗ In + 3 Im ⊗ B).
The partition function Z(β) is given by Z(β) = tr exp(−βH), where H
is the (hermitian) Hamilton operator and tr denotes the trace. From the
partition function we obtain the Helmholtz free energy, entropy and specific
heat.
(i) Calculate Z(β) for the Hamilton operator given above.
(ii) Consider the special case that n = m = 2 and A, B are any of the Pauli
spin matrices σ1 , σ2 , σ3 .
Problem 6.
Let A, B be n × n matrices over C. Is
tr(eA ⊗ eB ) = tr(eA⊗B ) ?
Prove or disprove.
Problem 7.
(i) Let A, B be n × n matrices. Show that
(A ⊗ In )(In ⊗ B)eA⊗In +In ⊗B = (AeA ) ⊗ (BeB ).
(ii) Let λ be an eigenvalue of A and µ be an eigenvalue of B. Provide an
eigenvalue of (AeA ) ⊗ (BeB ).
Problem 8.
matrix of
(i) Let A be an invertible n × n matrix. Find the inverse
(A−1 ⊗ In )(In ⊗ A).
(ii) Let B be an invertible n × n matrix. Calculate
(A−1 ⊗ In )(In ⊗ A)(B −1 ⊗ In )(In ⊗ B).
Problem 9.
The two-qubit Pauli group P2 can be generated as
P2 = hσ1 ⊗ σ1 , σ3 ⊗ σ3 , σ1 ⊗ σ2 , σ2 ⊗ σ3 , σ3 ⊗ σ1 i.
It is of order 64. Generate the element
i(I2 ⊗ I2 ).
Kronecker and Tensor Product
15
Problem 10. Consider the hermitian matrices of the three dipole operators






0 1 0
0 −i 0
1 0 0
1 
1
L1 = √
1 0 1  , L2 = √  i 0 −i  , L3 =  0 0 0 
2 0 1 0
2 0 i
0
0 0 −1
and the hermitian matrices

0 0
U1 =  0 0
1 0


0 1
0
1 
1 0 −1  ,
V1 = √
2 0 −1 0
of five quadrupole operators



1
0 0 −i
0,
U2 =  0 0 0  ,
0
i 0 0



0 −i 0
1
1 
1
V2 = √
i 0 i  , Q0 = √  0
2 0 −i 0
3 0
0
−2
0
Multiplying these eight hermitian matrices by i we obtain a basis for the
semi-simple Lie algebra su(3). Consider the Hamilton operator
ˆ = κ0 Q0 ⊗ Q0 + κ1 (V1 ⊗ V1 + V2 ⊗ V2 ) + κ2 (U1 ⊗ U1 + U2 ⊗ U2 ).
H
ˆ
Find the eigenvalues and eigenvectors of H.
Problem 11. Consider the Pauli spin matrices σ3 , σ1 , σ2 . The eigenvalues are given by +1 and −1 with the corresponding normalized eigenvectors
1
1
1
1
1
1
i
−i
1
0
√
√
, √
,
, √
.
,
,
1
−1
1
1
0
1
2
2
2
2
Consider the three 4 × 4 matrices
σ1 ⊗ σ1 ,
σ2 ⊗ σ2 ,
σ3 ⊗ σ3 .
(i) Find the eigenvalues.
(ii) Show that the eigenvectors can be given as product states (unentangled
states), but also as entangled states (i.e. they cannot be written as product
states). Explain.
Problem 12. (i) Consider the two 4 × 4 matrices σ1 ⊗ σ3 , σ3 ⊗ σ1 . Find
the eigenvalues.
(ii) Show that the eigenvectors can be given as product states (unentangled
states), but also as entangled states (i.e. they cannot be written as product
states). Explain.
Problem 13. Consider the Pauli spin matrix σ2 . Find the eigenvalues
and eigenvectors for σ2 and σ2 ⊗ σ2 . For σ2 ⊗ σ2 show that we find two sets

0
0.
1
16 Problems and Solutions
of entangled states for the eigenvectors and set of unentangled eigenvectors
(product states).
Problem 14.
operator
Find the eigenvalues and eigenvectors of the Hamilton
ˆ = ~ω1 σ3 ⊗ I2 + ~ω2 I2 ⊗ σ3 + σ2 ⊗ σ2 .
H
Problem 15.
(i) Find the eigenvalues and eigenvectors of
σ1 ⊗ σ2 ⊗ σ3 .
Can one find entangled eigenvectors?
(ii) Find the eigenvalues and eigenvectors of the Hamilton operator
ˆ = 1 (σ1 ⊗ I2 ⊗ I2 ) + 2 (I2 ⊗ σ2 ⊗ I2 ) + 3 (I2 ⊗ I2 ⊗ σ3 ) + γ(σ1 ⊗ σ2 ⊗ σ3 )
H
where 1 , 2 , 3 , γ ∈ R.
Problem 16. (i) Let U1 , U2 be unitary 2 × 2 matrices and Π1 , Π2 be
2 × 2 projection matrices with Π1 Π2 = 0 and Π1 + Π2 = I2 . Show that
U 1 ⊗ Π1 + U 2 ⊗ Π2
is unitary.
(ii) Let U1 = σ1 , U2 = σ3 and
1 1 1
,
Π1 =
2 1 1
Π2 =
1
2
1
−1
−1
1
.
Find the normalized state
(U1 ⊗ Π1 + U2 ⊗ Π2 )
1
0
⊗
.
0
1
Show that this state is entangled, i.e. it can not be written as a product
state.
Problem 17.
Consider the n × n unitary matrices U1 , . . . , Un and
the n × n projection matrices Π1 , . . . , Πn such that Πj Πk = δjk In and
Π1 + · · · + Πn = In . Show that the n2 × n2 matrix
n
X
j=1
is unitary.
Uj ⊗ Πj
Kronecker and Tensor Product
17
Problem 18. (i) Let A be an n × n matrix over C and Π be an m × m
projection matrix. Let z ∈ C. Calculate
exp(z(A ⊗ Π)).
(ii) Let A1 , A2 be n × n matrices over C. Let Π1 , Π2 be m × m projection
matrices with Π1 Π2 = 0. Calculate
exp(z(A1 ⊗ Π1 + A2 ⊗ Π2 )).
(iii) Use the result from (ii) to find the unitary matrix
ˆ
U (t) = exp(−iHt/~)
ˆ = ~ω(A1 ⊗ Π1 + A2 ⊗ Π2 ) and we assume that A1 and A2 are
where H
hermitian matrices.
(iv) Apply the result of (iii) to
1
1 1 1
1 −1
,
A2 = σ3 , Π2 =
.
A1 = σ 1 , Π1 =
2 1 1
2 −1 1
Problem 19.
Every 4 × 4 unitary matrix U can be written as
U = (U1 ⊗ U2 ) exp(i(ασ1 ⊗ σ1 + βσ2 ⊗ σ2 + γσ3 ⊗ σ3 ))(U3 ⊗ U4 )
where Uj ∈ U (2) (j = 1, 2, 3, 4) and α, β, γ ∈ R. Calculate
exp(i(ασ1 ⊗ σ1 + βσ2 ⊗ σ2 + γσ3 ⊗ σ3 )).
Problem 20.
Consider the Hilbert space C16 and the normalized state
1
|ψi = √ (| ↑i ⊗ | ↑i ⊗ | ↑i ⊗ | ↑i + | ↓i ⊗ | ↓i ⊗ | ↓i ⊗ | ↓i
2
where
| ↑i =
1
,
0
| ↓i =
0
.
1
Give a computer algebra implementation that calculates the 256 expectation values
Tjk`m = hψ|σj ⊗ σk ⊗ σ` ⊗ σn |ψi,
j, k, `, m = 0, 1, 2, 3
where σ0 , σ1 , σ2 , σ3 are the Pauli spin matrices with σ0 = I2 (2×2) identity
matrix.
18 Problems and Solutions
Problem 21.
Consider the unitary matrices
1
1
1 1
1
√
√
H=
, A=
1
−1
2
2 i
1
B=√
2
1
i
1
−i
,
1
C=
2
0
0
1
0

0
0 
.
0
−1
1−i
1−i
i
1
,
1+i
−1 − i
and
1
0
R=
0
0

0
1
0
0
Find (B ⊗ C)(R(I2 ⊗ A)R)(H ⊗ H).
Problem 22.
Consider the spin matrix for spin- 12
s1 =
1
1
σ1 =
2
2
0
1
1
0
with the eigenvalues 1/2 and −1/2 and the corresponding normalized eigenvectors
1
1
1
1
,
e−1/2 = √
.
e1/2 = √
2 1
2 −1
Are the four vectors in C4
1
√ (e1/2 ⊗ e1/2 + e−1/2 ⊗ e−1/2 ),
2
1
√ (e1/2 ⊗ e−1/2 + e−1/2 ⊗ e−1/2 ),
2
1
√ (e1/2 ⊗ e1/2 − e−1/2 ⊗ e−1/2 ),
2
1
√ (e1/2 ⊗ e−1/2 − e−1/2 ⊗ e−1/2 ),
2
Do the four vector form a basis in C4 ?
N
Problem 23. Let N ≥ 1. Consider the Hilbert space C2 . The (N + 1)
Dicke states are defined by
N
N
1
(|0i ⊗ · · · ⊗ |0i ⊗ |1i ⊗ · · · ⊗ |1i +permutations)
, ` − i := p
N
2
{z
} |
{z
}
2
C` |
`
N −`
where ` = 0, 1, . . . , N and N C` = N !/(`!(N − `)!). Write down the Dicke
states for N = 2 and N = 3. Which of the states are entangled?
Kronecker and Tensor Product
Problem 24.
19
Consider the 2 × 2 permutation matrices
1 0
0 1
P1 = I2 =
,
P2 =
.
0 1
1 0
(i) Show that
Π1 =
1
(P1 + P2 ),
2
Π2 =
1
(P1 − P2 )
2
are projection matrices. Find Π1 Π2 . Discuss.
(ii) Show that
Π1 =
1
(P1 ⊗ P1 + P2 ⊗ P2 ),
2
Π2 =
1
(P1 ⊗ P1 − P2 ⊗ P2 )
2
are projection matrices. Find Π1 Π2 . Discuss.
Problem 25. Consider the six 3 × 3 permutation matrices






0 1 0
1 0 0
1 0 0
P1 = I3 =  0 1 0  , P2 =  0 0 1  , P3 =  1 0 0  ,
0 0 1
0 1 0
0 0 1






0 0 1
0 0 1
0 1 0
P4 =  0 0 1  , P5 =  1 0 0  , P6 =  0 1 0 
1 0 0
0 1 0
1 0 0
with the signatures of the permutation P1 → +1, P2 → −1, P3 → −1,
P4 → +1, P5 → +1, P6 → −1.
(i) Is
1
Π1 = (P1 + P2 + P3 + P4 + P5 + P6 )
6
a projection matrix?
(ii) Is
1
Π2 = (P1 − P2 − P3 + P4 + P5 − P6 )
6
a projection matrix? Calculate Π1 Π2 . Discuss.
(iii) Is
Π1 =
1
(P1 ⊗ P1 + P2 ⊗ P2 + P3 ⊗ P3 + P4 ⊗ P4 + P5 ⊗ P5 + P6 ⊗ P6 )
6
a projection matrix?
(iv) Is
Π2 =
1
(P1 ⊗ P1 − P2 ⊗ P2 − P3 ⊗ P3 + P4 ⊗ P4 + P5 ⊗ P5 − P6 ⊗ P6 )
6
20 Problems and Solutions
a projection matrix? Find Π1 Π2 . Discuss.
Consider the Hilbert space C9 and the three normalized
       
1
0
0
1
1        
|ψ12 i = √
0 ⊗ 1 − 1 ⊗ 0
2
0
0
0
0
       
0
0
0
0
1
|ψ23 i = √  1  ⊗  0  −  0  ⊗  1 
2
0
1
1
0
       
0
1
1
0
1
|ψ31 i = √  0  ⊗  0  −  0  ⊗  0  .
2
1
0
0
1
Problem 26.
states
(i) Are the states entangled?
(ii) Find the density matrices.
(iii) Form a mixed state from the three density matrices.
Problem 27. Consider the two Hilbert spaces H1 = H2 = Cd and the
product Hilbert space H = H1 ⊗ H2 . A state |ψi ∈ H is called maximally
entangled if
1
trH1 (|ψihψ|) = trH2 (|ψihψ|) = .
d
Apply this definition to the Bell states in H = C4 , i.e. d = 2

 

1
1
1  0 
1 0
|ψ1 i = √   , |ψ2 i = √ 
,
0
2 0
2
1
−1
 


0
0
1 1
1  1 
|ψ3 i = √   , |ψ4 i = √ 
.
1
2
2 −1
0
0
Problem 28.
(i) Let
|1i =
1
,
0
|2i =
0
1
be the standard basis in C2 . Calculate the 4 × 4 matrix
P :=
2
X
j=1
|jihk| ⊗ |kihj|.
Kronecker and Tensor Product
What type of matrix is this?
(ii) Calculate P 2 . Discuss.
(iii) Let
1
1
,
|1i = √
2 1
1
|2i = √
2
1
−1
21
be the Hadamard basis in C2 . Calculate the 4 × 4 matrix
P :=
2
X
|jihk| ⊗ |kihj|.
j=1
What type of matrix is this?
(iv) Calculate P 2 . Discuss.
Problem 29.
Can the normalized state
1
√ (1 1 0 0 0 0
2
0
0)
T
in the Hilbert space C8 be written as a product state of three normalized
vectors in C2 ?
Problem 30. (i) Let σ1 , σ2 , σ3 be the Pauli spin matrices. Find the
commutators and anticommutators
[σ1 , σ2 ],
[σ1 , σ2 ]+ ,
[σ2 , σ3 ],
[σ2 , σ3 ]+ ,
[σ3 , σ1 ]
[σ3 , σ1 ]+
(ii) Consider the 4 × 4 matrices σ1 ⊗ σ2 , σ2 ⊗ σ3 , σ3 ⊗ σ1 . Find the commutators and anticommutators
[σ1 ⊗ σ2 , σ2 ⊗ σ3 ],
[σ1 ⊗ σ2 , σ2 ⊗ σ3 ]+ ,
[σ2 ⊗ σ3 ⊗ σ3 ⊗ σ1 ],
[σ3 ⊗ σ1 , σ1 ⊗ σ2 ]
[σ2 ⊗ σ3 ⊗ σ3 ⊗ σ1 ]+ ,
[σ3 ⊗ σ1 , σ1 ⊗ σ2 ]+
(iii) Consider the 8 × 8 matrices σ1 ⊗ σ2 ⊗ σ3 , σ3 ⊗ σ1 ⊗ σ2 , σ2 ⊗ σ3 ⊗ σ1 .
Find the commutators and anticommutators
[σ1 ⊗σ2 ⊗σ3 , σ3 ⊗σ1 ⊗σ2 ],
[σ1 ⊗σ2 ⊗σ3 , σ3 ⊗σ1 ⊗σ2 ]+ ,
[σ3 ⊗σ1 ⊗σ2 , σ2 ⊗σ3 ⊗σ1 ],
[σ2 ⊗σ3 ⊗σ1 , σ1 ⊗σ2 ⊗σ3 ]
[σ3 ⊗σ1 ⊗σ2 , σ2 ⊗σ3 ⊗σ1 ]+ ,
[σ2 ⊗σ3 ⊗σ1 , σ1 ⊗σ2 ⊗σ3 ]+ .
Problem 31. Let σ0 , σ1 , σ2 , σ3 be the Pauli spin matrices, where σ0 = I2
is the 2 × 2 unit matrix. Is
3
P =
1X
σj ⊗ σj
2 j=0
22 Problems and Solutions
a permutation matrix?
Problem 32. (i) Let σ0 , σ1 , σ2 , σ3 be the Pauli spin matrices, where
σ0 = I2 is the 2 × 2 unit matrix. Let
 
v1
v =  v2 
v3
be a vector in R3 with kvk ≤ 1. Show that
ρv =
1
(σ0 + v1 σ1 + v2 σ2 + v3 σ3 )
2
is a density matrix.
(ii) Is
ρ=
3
X
1
(σ0 ⊗ σ0 +
vj σ j ⊗ σ j )
4
j=1
a density matrix?
(iii) Is
ρ=
3
X
1
vj σj ⊗ σj ⊗ σj )
(σ
⊗
σ
⊗
σ
+
0
0
0
23
j=1
a density matrix? Extend the result to n Kronecker products.
Problem 33.
Consider the invertible

1 0
1 0 1
U=√ 
2 0 1
1 0
matrix
0
1
−1
0

1
0 
.
0
−1
Can the matrix be written as the Kronecker product of two 2 × 2 matrices?
Are the two state in C9
       
   
1
1
0
0
0
0
1        
|ψ1 i = √
0 ⊗ 0 + 1 ⊗ 1 − 2  0  ⊗  0 
6
0
0
0
0
1
1
           
1
1
0
0
0
0
1
|ψ2 i = − √  0  ⊗  0  +  1  ⊗  1  +  0  ⊗  0 
3
0
0
0
0
1
1
Problem 34.
orthogonal to each other?
Kronecker and Tensor Product
23
Problem 35. Let σ1 , σ3 be the Pauli spin matrices. Find the 4 × 4
permutation matrix P such that
P (σ1 ⊗ σ3 )P −1 = σ3 ⊗ σ1 .
Problem 36.
Consider the two normalized states
 
1
iα
iδ
1
1 0
1
e cos(β)
e cos(γ)
|ψi = √   , |φi = √
⊗√
sin(β)
sin(γ)
2 0
2
2
1
with α, β, γ, δ ∈ [0, 2π). Find
maxα,β,γ,δ |hψ|φi|2 .
Chapter 3
Matrix Properties
Problem 1. Let H be a hermitian n × n matrix. Show that exp(H) is a
positive definite matrix.
Problem 2.
Can the unitary matrix (permutation matrix)
1
0
U =
0
0

0
1
0
0
0
0
0
1

0
0

1
0
be written as the Kronecker product of two 2 × 2 matrices, i.e. U = A ⊗ B?
Problem 3. Let A, B, C be n × n matrices. Let In be the n × n identity
matrix.
(i) What can be said about the eigenvalues and eigenvectors of
A ⊗ In ⊗ In + In ⊗ B ⊗ In + In ⊗ In ⊗ C
if we know the eigenvalues and eigenvectors of A, B, C?
(ii) Is
eA⊗In ⊗In +In ⊗B⊗In +In ⊗In ⊗C = eA ⊗ eB ⊗ eC ?
Problem 4.
Let σ1 , σ3 be the Pauli spin matrices. Calculate (θ ∈ R)
√
R(θ) = exp(−i(θ/2)(σ1 + σ3 )/ 2)
24
Matrix Properties
25
Is the matrix R(θ) unitary?
Problem 5. Let σ1 , σ2 , σ3 be the Pauli spin matrices. Does the set of
4 × 4 matrices
{ I2 ⊗ I2 ,
σ1 ⊗ σ1 ,
−σ2 ⊗ σ2 ,
σ3 ⊗ σ3 }
form a group under matrix multiplication?
Problem 6.
The spin matrices for spin- 23 particles are given by
√


3 0
0
√0
~ 3 0
2 √0 
J1 = 

0
2 √0
3
2
3 0
0
0
√


0
−i
3
0
0
√
~ i 3
0
−2i
0√ 
J2 = 

0
2i
0
−i
3
2
√
0
0
0
i 3


3 0 0
0
~ 0 1 0
0 
J3 = 
.
2 0 0 −1 0
0 0 0 −3
(i) Show that the matrices are hermitian.
(ii) Find the eigenvalues and eigenvectors of these matrices.
(iii) Calculate the commutation relations.
(iv) Are the matrices unitary?
Problem 7. Two orthonormal bases in an n-dimensional complex Hilbert
space
{ |uj i : j = 1, 2, . . . , n },
{ |vj i : j = 1, 2, . . . , n }
are called mutually unbiased if inner products (scalar products) between all
possible
pairs of vectors taken from distinct bases have the same magnitude
√
1/ n, i.e.
1
|huj |vk i| = √
n
for all
j, k ∈ { 1, 2, . . . , n }.
(i) Find such bases for the Hilbert space C2 . Start of with the standard
basis
1
0
u1 =
,
u2 =
.
0
1
26 Problems and Solutions
(ii) Find such bases for the Hilbert space C3 . Start of with the standard
basis
 
 
 
1
0
0
u1 =  0  ,
u2 =  1  ,
u3 =  0  .
0
0
1
(iii) Find such bases for the Hilbert space C4 using the result from C2 and
the Kronecker product.
Problem 8. (i) Let A, B be n × n matrices over C such that A2 = In
and B 2 = In . Furthermore assume that
[A, B]+ ≡ AB + BA = 0n
i.e. the anticommutator vanishes. Let α, β ∈ C. Calculate eαA+βB using
eαA+βB =
∞
X
(αA + βB)j
j!
j=0
.
(ii) Consider the case that n = 2 and
α = −iωt,
β = −i∆t/~,
1 0
A = σ3 =
0 −1
0 1
B = σ1 =
.
1 0
(iii) Consider the case that n = 8 and
α = −iωt,
A = σ3 ⊗ σ3 ⊗ σ3
β = −i∆t/~,
B = σ1 ⊗ σ1 ⊗ σ1 .
Problem 9. Let A, B be n × n matrices with A2 = In and B 2 = In .
Assume that the commutator of A and B vanishes, i.e.
[A, B] = AB − BA = 0n .
Let a, b ∈ C. Calculate
eaA+bB .
(ii) Let a = −iωt, b = −i∆t/~ (∆ real) and
A = σ3 ⊗ σ3 ⊗ · · · ⊗ σ3 ,
B = σ1 ⊗ σ1 ⊗ · · · ⊗ σ1
with n (even) factors of the Kronecker products. Then the conditions given
above are satisfied. Simplify the result from (i) with this assumption.
Matrix Properties
27
Problem 10. Let A, B be n × n matrices with A2 = In and B 2 = In .
Assume that the anticommutator of A and B vanishes, i.e.
[A, B]+ = AB + BA = 0n .
(i) Let a, b ∈ C. Calculate
eaA+bB .
(ii) Let a = −iωt, b = −i∆t/~ (∆ real) and
A = σ3 ⊗ σ3 ⊗ · · · ⊗ σ3 ,
B = σ1 ⊗ σ1 ⊗ · · · ⊗ σ1
with n (odd) factors of the Kronecker products. Then the conditions given
above are satisfied. Simplify the result from (i) with this assumption.
Problem 11.
Consider the Hilbert space C3 and the standard basis
 
 
 
0
0
1
|0i =  0  , |1i =  1  , |2i =  0  .
1
0
0
Consider the unitary matrices


1 0 0
R = 0 ω 0 ,
0 0 ω2

0
T = 1
0
0
0
1

1
0
0
where ω = e2πi/3 .
(i) Calculate the state R|ji, T |ji, where j = 0, 1, 2.
(ii) Find the commutator [R, T ].
(iii) Consider the normalized state
1
|ψi = √ (|0i ⊗ |0i + |1i ⊗ |1i + |2i ⊗ 2i).
3
Calculate the state (R ⊗ T )|ψi and discuss.
Problem 12.
Let σ1 , σ2 , σ3 be the Pauli spin matrices. Show that
[σm ⊗ σn , σk ⊗ I2 ] ≡ [σm , σk ] ⊗ σn
[σm ⊗ σn , I2 ⊗ σk ] ≡ σm ⊗ [σn , σk ]
where k, m, n ∈ { 1, 2, 3 }.
Problem 13. Given two arbitrary normalized states |ψi and |φi in C2 .
Find a 2 × 2 unitary matrix U such that |ψi = U |φi, i.e. U must be
expressed in terms of the compenents of the states |ψi and φi.
28 Problems and Solutions
Problem 14.
Consider the Hamilton operator in C4
ˆ = −t(|00ih11| + |11ih00|) + v(|00ih00| + |11ih11|.
H
The kinetic parameter is t ≥ 0 and v is the potential parameter. Find
ˆ Keep t = 1 fixed and disucss the
the eigenvalues and eigenvectors of H.
ˆ as a function of v.
dependence of the eigenvalues of H
Problem 15. Let H be an n × n hermitian matrix. Let λ1 , . . . , λn be
the eigenvalues with the pairwise orthogonal normalized eigenvectors v1 ,
. . . , vn . Then we can write
H=
n
X
λ` v` v`∗ .
`=1
Let
P = In − vj vj∗ − vk vk + vj vk∗ + vk vj∗ ,
j 6= k.
(i) What is condition on the eigenvalues of H such that P HP ∗ = H.
(ii) Find P 2 .
Problem 16.
Let B be an n × n matrix with B 2 = In . Show that
1
exp − iπ(B − In ) ≡ B.
2
Problem 17.
Consider the vector


sin(φ1 ) sin(φ2 ) sin(θ)
 sin(φ1 ) sin(φ2 ) cos(θ) 
|ψi = 

sin(φ1 ) cos(φ2 )
cos(φ1 )
in the Hilbert space R4 with φ1 , φ2 , θ ∈ R. Find the norm of this vector.
For which values of φ1 , φ2 , θ is the norm a mimimum? What is the use of
this vector?
Problem 18. Let R be a nonsingular n × n matrix. Let A and B be
n × n matrices. Assume that R−1 AR and R−1 BR are diagonal matrices.
Calculate the commutator [A, B].
Problem 19.
Let A, B be two n × n matrices. Assume that
trA = 0,
trB = 0
Matrix Properties
29
Can we conclude that tr(AB) = 0? Prove or disprove.
Problem 20. We know that any n × n hermitian matrix has only real
eigenvalues. Assume that a given n × n matrix has only real eigenvalues.
Can we conclude that the matrix is hermitian? Prove or disprove.
Problem 21. Consider the Hilbert space Cn . Let e1 , e2 , . . . , en be the
standard basis in Cn , Sn be the symmetric group of order n! and Uσ be the
unitary matrix on ⊗n Cn such that
Uσ (e1 ⊗ · · · ⊗ en ) := eσ(1) ⊗ · · · ⊗ eσ(n)
where σ ∈ Sn . We define the matrix (“antisymmetrization operator”) in
the Hilbert space ⊗n Cn by
1 X
sgn(σ)Uσ
Πn :=
n!
σ∈Sn
where sgn is the signature of the permutation σ ∈ Sn . The matrices Πn
are projection matrices.
(i) Find Π2 .
(ii) Find Π3 .
Problem 22. (i) The four-dimensional face-centered hypercubic lattice
plays a central role in simulating three-dimensional hydrodynamics on a
cellular automata machine. Consider the four-dimensional face-centered
hypercubic lattice in connection with entanglement. This lattice is generated from the four basis vectors
(±1, ±1, 0, 0).
(1)
Permuting the components of these four vectors in R4 we find 20 additional
vectors. Show that the 24 vectors can be classified as follows. In class A
we have eight vectors
 






 




1
1
−1
−1
0
0
0
0
 0 
 0 
 0 
1
 1 
 −1 
 , 
, 
, 
,  , 
, 
,
0
0
0
0
1
−1
1
1
−1
1
−1
0
0
0
√
The normalization factor would be 1/ 2. In class B we have the eight
vectors
 






 




0
0
0
0
1
−1
1
1
 1 
 −1 
 −1 
0
 0 
 0 
 , 
, 
, 
,  , 
, 
,
0
0
0
0
1
1
−1
1
−1
1
−1
0
0
0

0
 −1 

.
−1
0


−1
 0 

.
−1
0

30 Problems and Solutions
√
The normalization factor is also 1/ 2. In class C we have the eight vectors
 
0
0
 ,
1
1

0
 0 

,
1
−1


0
 0 

,
−1
1


0
 0 

,
−1
−1

 
1
1
 ,
0
0

1
 −1 

,
0
0


−1
 1 

,
0
0

√
Again the normalization factor is 1/ 2. Show that if α(nA , m) is the angle
between the nth vector of class A and the mth vector of class B, then
α(nA , mB ) = α(nB , mC ) = α(nC , mA )
and
α(nA , mA ) = α(nB , mB ) = α(nC , mC ).
Each class contains four oppositely oriented pairs of vectors. This means
that the ordering of the vectors is such that class B is related to class A in
exactly the same way as class C is related to B and A is related to C.
(ii) Show that the normalized vectors in class A are maximally entangled.
(iii) Show that the vectors in class B and class C can be written as the
Kronecker product of two vectors from R2 .
(iv) The Hadamard gate given by the unitary matrix
1
1 1
H=√
2 1 −1
plays a central role in quantum computing. Consider now the 4 × 4 matrix
R = I2 ⊗ H
where ⊗ denotes the Kronecker product and I2 is 2×2 unit matrix. Thus R
itself is a unitary matrix. Applying this matrix to the 24 vectors. Discuss.
(v) Show that the construction given above can be extended to higher
dimensional cases. For example in R8 we would start with
1
√ (±1, ±1, 0, 0, 0, 0, 0, 0)T
2
and all permutations. Show that this provides us with the GHZ-state
1
√ (1 0 0 0 0 0 0 1)T
2
which is fully entangled when we use the three tangle (based on the hyperdeterminant) as measure of entanglement. Show that we find a set of

−1
 −1 

.
0
0

Matrix Properties
31
unentangled states, for example
 
1
1
 
0

1 
1
1
1
1
0
= √  .
⊗
⊗√
0
0
0
2 1
2
 
0
 
0
0
Show that using the three-tangle as measure for entanglement we find 0 for
these vectors.
Problem 23. The associative algebra Md (C) of d × d matrices can be
considered as a C ∗ algebra with the square of the norm k . k defined by
(A ∈ Md (C))
kAk2 := largest eigenvalue of the (normal) matrixA∗ A.
Let d = 2 and
A=
1
i
i
−1
.
Find the norm.
Problem 24. Consider the Hilbert space Cd . Let |ji (j = 1, . . . , d) be
an orthonormal basis in Cd . Then a d × d matrix A acting in Cd can be
written as
d
X
A=
ajk |jihk|
j,k=1
with ajk ∈ C. Obviously A depends on the underlying orthonormal basis.
If we have the standard basis, then A reduces to the matrix A = (ajk ). We
2
can associate a vector |ψA i in the Hilbert space Cd with the matrix A via
|ψA i =
d
X
ajk |ji ⊗ |ki.
j,k=1
(i) Let d = 2 and consider the standard basis
1
0
|1i =
,
|2i =
.
0
1
Find A and |ψA i.
32 Problems and Solutions
(ii) Let d = 2 and consider the Hadamard basis
1
1
1
1
|1i = √
,
|2i = √
.
2 1
2 −1
Find A and |ψA i.
(iii) Let d = 3 and consider the basis
 
 
1
0
1  
0 , |2i =  1  ,
|1i = √
2 1
0


1
1 
|3i = √
0 .
2 −1
Find A and |ψA i.
(iv) Describe the connection of the map A 7→ |ψA i with the vec-operator.
Problem 25. Let φ1 , φ2 ∈ R. From the Bell basis
 iφ1 




e
0
0
1  eiφ2 
1  eiφ2 
1  0 
√ 
 , √  iφ2  , √  iφ2  ,
0
2
2 e
2 −e
eiφ1
0
0

eiφ1
1  0 
√ 

0
2
−eiφ1

we form the matrix
eiφ1
1  0
M (φ1 , φ2 ) = √ 
0
2
eiφ1

0
0
iφ2
iφ2
e
eiφ2
0
e
−eiφ2
0

eiφ1
0 
.
0
−eiφ1
Is M (φ1 , φ2 ) an element of the Lie group SU (4)?
Problem 26. (i) Let x1 , x2 , x3 ∈ R. Let σ1 , σ2 , σ3 be the Pauli spin
matrices. Show that
sin(r)
ei(x1 σ1 +x2 σ2 +x3 σ3 ) = cos(r)I2 +
i(x1 σ1 + x2 σ2 + x3 σ3 )
r
cos(r) + ix3 sin(r)/r i(x1 − ix2 ) sin(r)/r
=
i(x1 + ix2 ) sin(r)/r cos(r) − ix3 sin(r)/r
p
where r := x21 + x22 + x23 .
(ii) Let y1 , y2 , y3 ∈ R and
X := x1 σ1 + x2 σ2 + x3 σ3 ,
Y := y1 σ1 + y2 σ2 + y3 σ3 .
Consider the maps


x1
X ↔ x =  x2  ,
x3


y1
Y ↔ y =  y2  .
y3
Matrix Properties
33
Let x · y := x1 y1 + x2 y2 + x3 y3 (scalar product). Show that
x·y =
1
tr(XY ).
2
(iii) Show that


x2 y3 − x3 y2
i
− [X, Y ] ↔ x × y =  x3 y1 − x1 y3  .
2
x1 y2 − x2 y1
Problem 27. Let s be a spin with a fixed total angular momentum
quantum number
s ∈ {1/2, 1, 3/2, 2, . . .}.
The (normalized) eigenstates of x3 -angular momentum |s, mi form a ladder
with
m = −s, −s + 1, . . . , s − 1, s.
The eigenstates |s, mi form an orthonormal basis in a 2s + 1 dimensional
Hilbert space. For example if s = 1/2 we have the two states |1/2, −1/2i,
|1/2, 1/2i and can identify
1
0
|1/2, 1/2i 7→
, |1/2, −1/2i 7→
.
0
1
Thus we have the Hilbert space C2 . For s = 1 we have the three states
|1, −1i, |1, 0i, |1, 1i and can identify
 
 
 
0
0
1
|1, −1i 7→  0  , |1, 0i 7→  1  , |1, 1i 7→  0  .
0
1
0
A spin coherent state |s, θ, φi for s = 1/2, 1, 3/2, . . . can be given by
s
m=s
X
(2s)!
(cos(θ/2))s+m (sin(θ/2))s−m e−imφ |s, mi.
|s, θ, φi =
(s
+
m)!(s
−
m)!
m=−s
(i) Find |1/2, θ, φi and write it as a vector in C2 .
(ii) Find |1, θ, φi and write it as a vector in C3 .
(iii) For a given s find the scalar product hs, m|s, θ, φi.
Problem 28. (i) Consider the Pauli spin matrix σ2 and the Lie group
SL(2, C). Let S ∈ SL(2, C). Show that
Sσ2 S T = σ2
34 Problems and Solutions
where T denotes the transpose.
(ii) Show that
(S ⊗ S)(σ2 ⊗ σ2 )(S T ⊗ S T ) = σ2 ⊗ σ2 .
Problem 29. Let |1i, |2i, . . . , |di be an orthonormal basis in the Hilbert
space Cd . Consider the matrix
S=
d
X
(|jihk| ⊗ |kihj|).
j,k=1
Is S independent of the chosen orthonormal basis?
Problem 30.
(i) Let φ1 , φ2 ∈ R. Show that
1
U (φ1 , φ2 ) = √
2
eiφ1
eiφ2
−e−iφ2
e−iφ1
is unitary. Is U (φ1 , φ2 ) an element of SU (2)? Find the eigenvalues of
U (φ1 , φ2 ).
(ii) Let φ1 , φ2 , φ3 , φ4 ∈ R. Show that
eiφ1
1  0
U (φ1 , φ2 , φ3 , φ4 ) = √ 
0
2
eiφ2

0
0
eiφ3
eiφ4
0
−e−iφ4
e−iφ3
0

−e−iφ2
0 

0
e−iφ1
is unitary. Is U (φ1 , φ2 , φ3 , φ4 ) an element of SU (4)? Find the eigenvalues
of U (φ1 , φ2 , φ3 , φ4 ).
(iii) Let φ1 , φ2 ∈ R. Show that

eiφ1
1 
U (φ1 , φ2 ) = √
0
2 eiφ2
√0
2
0

−e−iφ2
0 
e−iφ1
is unitary. Is U (φ1 , φ2 ) an element of SU (2)? Find the eigenvalues of
U (φ1 , φ2 ).
Problem 31.
Consider the Hadamard matrix
1
1 1
.
U=√
2 1 −1
Matrix Properties
35
The eigenvalues of the Hadamard matrix are given by +1 and −1 with the
corresponding normalized eigenvectors
p
p
√ √ 1
1
4+2 2
4−2 2
p
p
√
√
√
√
,
.
4−2 2
8
8 − 4+2 2
How can this information be used to find the eigenvalues and eigenvectors
of the Bell matrix


1 0 0
1
1 0 1 1
0 
B=√ 
.
0
1
−1
0
2
1 0 0 −1
Problem 32.
 
1
1 1
v1 =   ,
2 1
1
(i) Consider the Hilbert space C4 . Do the vectors

1
1  0 
v2 = √ 
,
2 −1
0


0
1  1 
v3 = √ 
,
0
2
−1


1
1  −1 
v4 = 

1
2
−1

form an orthonormal basis in C4 . Prove or disprove.
(ii) Can the vectors v1 , v2 , v3 , v4 be written as Kronecker products of
vectors in C2 . Prove or disprove.
(iii) Consider the 4 × 4 matrices
0
0
S=
1
0

0
1
0
0
1
0
0
0

0
0
,
0
1
1
0
T =
0
0

0
0
0
1
0
0
1
0

0
1
.
0
0
Find the eigenvalues and normalized eigenvectors of the two matrices. Compare to (i). Disucss.
(iv) Find the commutator of S and T , i.e. [T, S]. What can be said about
eigenvectors of such a pair of matrices? Discuss. Hint. Look at your result
from (iii).
Problem 33.
Consider the matrix
√ 
 √
1/ 2 0 1/ 2
U =  0√
1
0√  .
1/ 2 0 −1/ 2
(i) Is the matrix unitary?
36 Problems and Solutions
(ii) Find the eigenvalues and nonnormalized eigenvectors of U . Use this
information to write down the spectral decomposition of U .
(iii) Find a skew-hermitian matrix K such that U = exp(K). One can
utilize the results from (ii).
(iv) Apply the unitary matrix to the normalized state
 
1
1
|ψi = √  1  .
3 1
Find the state U |ψi and calculate the probability |hψ|K|ψi|2 .
Problem 34.
Let φ1 , φ2 , φ3 , φ4 ∈ R. Consider the 2 × 2 matrix
iφ
1
e 1 eiφ2
U (φ1 , φ2 , φ3 , φ4 ) = √
.
iφ
iφ
2 e 3 e 4
The matrix contains the two column vector
iφ iφ 1
1
e 1
e 3
v1 = √
,
v2 = √
.
iφ2
iφ
e
2
2 e 4
Find the conditions on φ1 , φ2 , φ3 , φ4 such that
hv1 |v2 i = 0.
Is the matrix unitary if this condition is satisfied?
Problem 35. (i) An n × n matrix H = (hjk ) over C is called a complex
Hadamard matrix if |hjk | = 1 for j, k = 1, . . . , n and HH ∗ = nIn . Note
that √1n H is then a unitary matrix. Let φ ∈ [0, π). Show that
1
1
H(φ) = 
1
1

1
ieiφ
−1
−ieiφ
1
−1
1
−1

1
−ieiφ 

−1
ieiφ
is a complex Hadamard matrix.
(ii) Given two complex Hadamard matrices H1 and H2 . Is H1 ⊗ H2 a
complex Hadamard matrix?
Problem 36.
Consider the Hamilton operator


0 0 1 0
 0 0 0 −1 
γˆ3 = i~ωA,
A=

−1 0 0 0
0 1 0 0
Matrix Properties
37
where ~ and ω (frequency) are constants.
(i) Find
exp(−iˆ
γ3 t/~).
(ii) Let
 
1
1 0
|ψ(t = 0)i = √  
2 0
1
be the initial state in the Hilbert space C4 . Calculate
|ψ(t)i = exp(−iˆ
γ3 t/~)|ψ(t = 0)i
and thus solve the Schr¨
odinger equation.
(iii) If we know the eigenvalues of γ3 what can be said about the eigenvalues
of exp(−iˆ
γ3 t/~)?
Problem 37.
Let α, θ, φ ∈ R. Consider the vector in C4


sinh(α) sin(θ) cos(φ)
u =  sinh(α) sin(θ) sin(φ)  .
cosh(α) cos(θ)
(i) Normalize the vector.
(ii) Apply the Bell matrix
1
1 0
B=√ 
2 0
1

0
1
1
0
0
1
−1
0

1
0 

0
−1
to the normalized vector. Calculate u∗ Bu. Discuss.
Problem 38. Let σ1 , σ2 , σ3 be the Pauli spin matrices. For the Dirac
equation the following 4 × 4 matrices play a central role. We define
I2 0 2
02 σk
β :=
,
αk =
k = 1, 2, 3.
02 −I2
σk 0 2
Let γk = iβαk for k = 1, 2, 3, γ0 = −iβ and γ5 = iγ1 γ2 γ3 γ0 . Find the
gamma matrices and calculate their anticommutators.
Problem 39. (i) Consider the finite-dimensional Hilbert space Cd . A
symmetric informatially complete positive operator valued measure (SICPOVM) consists of d2 outcomes that are subnormalized projection matrices
Πj onto pure states
1
Πj = |ψj ihψj |
d
38 Problems and Solutions
for j, k = 1, . . . , d2 such that
1 + dδjk
.
d+1
Consider the case d = 2. Show that the normalized vectors
 q

√
(3 + 3)/6

q
|ψ1 i = 
√
iπ/4
e
(3 − 3)/6
q


√
(3 + 3)/6

q
|ψ2 i = 
√
−eiπ/4 (3 − 3)/6
q


√
eiπ/4 (3 − 3)/6

|ψ3 i =  q
√
(3 + 3)/6
q


√
−eiπ/4 (3 − 3)/6

q
|ψ4 i = 
√
(3 + 3)/6
|hψk |ψk i|2 =
satisfy this condition.
(ii) Consider the matrices σ1 , −iσ2 , σ3 . Find
σ1 |ψ1 i,
−iσ2 |ψ1 i,
σ3 |ψ1 i.
(iii) Let d = 2. Let
Sd :=
d
X
|ji⊗|ji⊗hj|⊗hj|+
j=1
1
1
√ (|ji⊗|ki+|ki⊗|ji)⊗ √ (hj|⊗hk|+hk|⊗hj|)
2
2
k>j=1
X
where |1i, |2i denotes the standard basis in C2 , i.e.
1
0
|1i =
,
|2i =
.
0
1
Show that
2
d
X
j=1
|ψj i ⊗ |ψj ihψj | ⊗ hψj | =
2d
.
d+1
(iv) Can one find a SIC-POVM in C4 using the states from (i) and the
Kronecker product?
Problem 40. Let a1 , a2 , b1 , b2 be real. Find the normalization factors
for the vector in C4


a1 cos(φ/2) + b1 sin(φ/2)
 a cos(φ/2) + b2 sin(φ/2) 
|ψi =  2
.
ia1 sin(φ/2) − ib1 cos(φ/2)
ia2 sin(φ/2) − ib2 cos(φ/2)
Matrix Properties
39
Problem 41. Any 2 × 2 matrix can be written as a linear combination
of the Pauli spin matrices and the 2 × 2 identity matrix
A = aI2 + bσ1 + cσ2 + dσ3
where a, b, c, d ∈ C.
(i) Find A2 and A3 .
(ii) Use the result from (i) to find all matrices A such that A3 = σ1 .
Problem 42. Let r, s, θ ∈ R. Consider the Hamilton operator given by
the 2 × 2 matrix
iθ
ˆ
H
re
s
ˆ
=
.
K=
s
re−iθ
~ω
(i) Is the matrix a normal matrix?
(ii) Is the matrix hermitian?
ˆ
(iii) Find the eigenvalues and eigenvectors of K.
Problem 43. Consider the vector space of n × n matrices over C. Let
B1 , B2 , . . . , Bn2 be a basis. Assume that all B’s are invertible. Is B1−1 ,
B2−1 , . . . , Bn−1
2 also a basis for the vector space?
Problem 44. What can be said about the eigenvalues of an n × n matrix
which is unitary and skew-hermitian? Give an example of such a matrix.
Problem 45.
Let φ11 , φ12 , φ21 , φ22 ∈ R. Consider the matrix
iφ
1
e 11 eiφ12
V =√
.
iφ
iφ
2 e 21 e 22
(i) What are the conditions on φ11 , φ12 , φ21 , φ22 such that the matrix is
unitary?
(ii) What are the conditions on φ11 , φ12 , φ21 , φ22 such that the matrix is
hermitian?
What are the conditions on φ11 , φ12 , φ21 , φ22 such that V = V −1 ?
Problem 46.
unitary?
Is the 3 × 3 matrix
√
√ 
 √
1/ 2
−1/ √6
1/
p 3
3) − 2/3 
V =  1/2 −1/(2
√
3/2
0
1/2
40 Problems and Solutions
Problem 47. Let A be an n × n matrix over C. An n × n matrix B is
called a square root of A if B 2 = A. Find the square roots of the 2 × 2
identity matrix applying the spectral theorem. The eigenvalues of I2 are
λ1 = 1 and λ2 = 1. As normalized eigenvectors choose
iφ
iφ
e 1 cos(θ)
e 1 sin(θ)
,
eiφ2 sin(θ)
−eiφ2 cos(θ)
√ √
which
form an orthonormal
in C2 . Four√cases√( λ1 , λ2 ) = (1, 1),
√ basis
√
√ √
( λ1 , λ2 ) = (1, −1), ( λ1 , λ2 ) = (−1, 1), ( λ1 , λ2 ) = (−1, −1) have
to
The first and last case are trivial. So study the second case
√
√be studied.
( λ1 , λ2 ) = (1, −1). The second case and the third case are “equivalent”.
Problem 48. Let |ji (j = 1, . . . , d) be an orthonormal basis in Cd and
hk| (k = 1, . . . , d) be the dual basis. We define
Rjk = |jihk|,
j, k = 1, . . . , d.
Show that
Rjk R`m = Rjm δ`k ,
[Rjk , R`m ] = Rjm δ`k − R`k δjm ,
d
X
Rjj = Id .
j=1
Hint. Utilize
Problem 49. Let |0i, |1i, . . . |d − 1i be an orthonormal basis in Cd . Let
Tjk ∈ C with j, k = 0, 1, . . . , d − 1. Consider the linear operator
T =
d−1 X
d−1
X
Tjk |jihk|.
j=0 k=0
(i) Let d = 2 and
|0i =
1
,
0
|1i =
0
.
1
Find T .
(ii) Let d = 2 and
1
|0i = √
2
1
,
1
1
|1i = √
2
1
−1
.
Find T .
ˆ be a hermitian n×n matrix describing the Hamilton
Problem 50. Let H
operator and acting in the Hilbert space Cn . Let A, B be n × n hermitian
Matrix Properties
41
matrices and |ψi ∈ Cn . One defines (quantum correlation function)
Q(|ψi) :=
where
1
hψ|(A(t)B − AB(t) + BA(t) − B(t)A)|ψi
2
ˆ
ˆ
A(t) = eiHt/~ Ae−iHt/~ ,
ˆ
ˆ
B(t) = eiHt/~ Be−iHt/~ .
(i) Let
ˆ = ~ωσ2 ,
H
A = σ1 ,
B = σ3 ,
|ψi =
cos(θ)
sin(θ)
.
(ii) Let

cos(φ1 )
sin(φ1 ) cos(φ2 )


|ψi = 

sin(φ1 ) sin(φ2 ) cos(φ3 )
sin(φ1 ) sin(φ2 ) sin(φ3 )

ˆ = ~ωσ2 ⊗σ2 ,
H
A = σ1 ⊗σ1 ,
B = σ3 ⊗σ3 ,
Problem 51. Let S1 , S2 , S3 be the spin matrices for spin = 1/2, 1, 3/2, 2, . . ..
The matrices are (2s + 1) × (2s + 1) hermitian matrices with trace equal to
0 satisfying the commutation relations
[S1 , S2 ] = iS3 ,
[S2 , S3 ] = iS1 ,
[S3 , S1 ] = iS2 .
(i) Study the spectrum for the (hermitian) Hamilton operator
ˆ
ˆ = H = S1 ⊗ S2 + S2 ⊗ S3 + S3 ⊗ S1
K
~ω
ˆ = 0. Then extend to arbitrary
for s = 1/2 and s = 1. Note that tr(K)
spin.
(ii) Find the commutators
[S1 ⊗ S2 , S2 ⊗ S3 ],
[S2 ⊗ S3 , S3 ⊗ S1 ],
[S3 ⊗ S1 , S1 ⊗ S2 ].
for s = 1/2 and s = 1.
(iii) Find the anticommutators
[S1 ⊗ S2 , S2 ⊗ S3 ]+ ,
[S2 ⊗ S3 , S3 ⊗ S1 ]+ ,
[S3 ⊗ S1 , S1 ⊗ S2 ]+
for s = 1/2 and s = 1.
ˆ for s = 1/2 and s = 1, where z ∈ C.
(iv) Calculate exp(z K)
ˆ with the spectra of
(v) Compare the spectra for K
S1 ⊗ S1 + S2 ⊗ S2 + S3 ⊗ S3
for s = 1/2 and s = 1.
Chapter 4
Density Operators
Problem 1.
Consider the 2 × 2 matrix
√ −iφ 3/4
2e /4
ρ = √ iφ
.
2e /4
1/4
(i) Is the matrix a density matrix?
(ii) If so do we have a pure state or a mixed state?
(iii) Find the eigenvalues of ρ.
(iv) Find tr(σ1 ρ), where σ1 is the first Pauli matrix.
Problem 2.
Let ∈ [0, 1]. Is
ρ = p
(1 − )eiφ
p
(1 − )e−iφ
1−
with 0 ≤ φ < 2π at density matrix?
Problem 3. (i) Find a normalized state |φi in the Hilbert space C2 such
that we have the density matrix
1
1
I2 + √ (σ1 + σ3 ) .
|φihφ| =
2
2
(ii) Find a normalized state |ψi in the Hilbert space C2 such that we have
the density matrix
1
1
|ψihψ| =
I2 + √ (σ1 + σ2 + σ3 ) .
2
3
42
Density Operators
43
Problem 4. Let σ1 , σ2 , σ3 be the Pauli spin matrices. Find the conditions
on the coefficients aj , bj and cjk such that ρ
ρ=
3
3
3
X
X
X
1
(I4 + (
aj σj ) ⊗ I2 + I2 ⊗ (
bj σ j ) +
cjk σj ⊗ σk )
4
j=1
j=1
j,k=1
is a density matrix.
Let m, n ∈ R3 and kmk = knk = 1. Is the 4 × 4 matrix
Problem 5.
ρ(m, n) =
1
(I4 + (n · σ) ⊗ I2 + I2 ⊗ (m · σ) + (n · σ) ⊗ (m · σ))
4
a density matrix?
Problem 6.
Consider the 3 × 3 matrix


1/2 0 1/4
ρ =  0 1/4 0  .
1/4 0 1/4
(i) Find the eigenvalues of ρ.
(ii) Is ρ a density matrix? Prove or disprove. If so, is ρ a mixed or pure
state?
Problem 7.
Consider the normalized state

 i(α+γ)
e
cos(β) sin(θ)
|ψi = e−iφ  e−i(α−γ) sin(β) sin(θ)  .
cos(θ)
Find the density matrix ρ = |ψihψ| and the eigenvalues of ρ.
Problem 8.
Let ∈ R and || < 1. Is the 4 × 4 matrix
1
1 0
ρ() = 
0
2
1−


0 0 1−
0 0
0 

0 0
0
0 0
1
a density matrix?
Problem 9.
Show that the 4 × 4 matrices
ρ− =
1
(I2 ⊗ I2 − σ1 ⊗ σ1 − σ2 ⊗ σ2 − σ3 ⊗ σ3 )
4
44 Problems and Solutions
1
(I2 ⊗ I2 − σ1 ⊗ σ1 + σ2 ⊗ σ2 + σ3 ⊗ σ3 )
4
1
ω + = (I2 ⊗ I2 + σ1 ⊗ σ1 − σ2 ⊗ σ2 + σ3 ⊗ σ3 )
4
1
+
ρ = (I2 ⊗ I2 + σ1 ⊗ σ1 + σ2 ⊗ σ2 − σ3 ⊗ σ3 )
4
ω− =
are density matrices. How they are related to the 4 Bell states

0
1  1 
|ψ − i = √ 
,
2 −1
0


1
1  0 
|φ− i = √ 
,
0
2
−1

 
1
1
0
|φ+ i = √   ,
2 0
1
 
0
1
1
|ψ + i = √   ?
2 1
0
Problem 10. Let ρ1 and ρ2 be density matrices is a finite-dimensional
Hilbert space. Let λ ∈ [0, 1]. Is
λρ1 + (1 − λ)ρ2
a density matrix?
Problem 11.
Show that

ε1
 0
ρ=
0
√
ε1 ε2
0
0
0
0
0
0
1 − ε1 − ε2
0ε2
√

ε1 ε2
0 

0
where 0 ≤ ε1 , ε2 ≤ 1 and ε1 + ε2 ≤ 1 is a density matrix.
Problem 12.
Consider the density matrix
ρ=
4
X
pj |ψj ihψj |,
0 ≤ pj ≤ 1,
j=1
4
X
pj = 1
j=1
where the |ψj i are the Bell states
1
|ψ1 i = √ (|0i ⊗ |0i + |1i ⊗ |1i),
2
1
|ψ2 i = √ (|0i ⊗ |0i − |1i ⊗ |1i)
2
1
|ψ3 i = √ (|0i ⊗ |1i + |1i ⊗ |0i),
2
1
|ψ2 i = √ (|0i ⊗ |1i − |1i ⊗ |0i).
2
Write ρ using the Pauli spin matrices σ1 , σ2 , σ3 , the 2 × 2 identity matrix
I2 and the Kronecker product.
Density Operators
45
Problem 13. Consider the Hilbert space Cn . Let ρ be a density matrix,
i.e. ρ ≥ 0 and tr(ρ) = 1. The mean value of an observable A (hermitian
n × n matrix) is given by
hAi = tr(ρA).
If the density ρ is unkown, then it may be determined using n2 mean values
hA(k) i (k = 1, 2, . . . , n2 ) obtained from measurement if the set {A(k) } is a
basis in the space of all hermitian n × n matrices.
(i) Let n = 2,
0 −i
A = σ2 =
i 0
and
tr(ρA) = 0,
tr(ρA2 ) = 1,
tr(ρA3 ) = 0,
tr(ρA4 ) = 1.
Find the density matrix.
(ii) Let n = 2 and
tr(ρI2 ) = 1,
tr(ρσ1 ) = −1,
tr(ρσ2 ) = 0,
tr(ρσ3 ) = 0.
Find ρ.
Problem 14.
(i) Let x1 , x2 , x3 ∈ R. Consider the hermitian matrix
1
1 + x3 x1 − ix2
.
ρ=
2 x1 + ix2 1 − x3
Find the condition on x1 , x2 , x3 such that ρ2 = ρ. Is this matrix then a
density matrix?
(ii) Let ∈ [0, 1]. Consider the hermitian matrix


+ x3
0
x1 − ix2
1
.
ρ= 
0
2 − 2
0
2
x1 + ix2
0
− x3
Find the condition on x1 , x2 , x3 and such that ρ2 = ρ.
Problem 15.
Consider the density matrix
ρ=
4
X
pj |ψj ihψj |,
0 ≤ pj ≤ 1,
j=1
4
X
pj = 1
j=1
where the |ψj i are the Bell states
1
|ψ1 i = √ (|0i ⊗ |0i + |1i ⊗ |1i),
2
1
|ψ2 i = √ (|0i ⊗ |0i − |1i ⊗ |1i)
2
46 Problems and Solutions
1
|ψ3 i = √ (|0i ⊗ |1i + |1i ⊗ |0i),
2
1
|ψ2 i = √ (|0i ⊗ |1i − |1i ⊗ |0i).
2
Write ρ using the Pauli spin matrices and the 2 × 2 identity matrix I2 .
Problem 16. Let A be a nonzero n × n matrix over C. Consider the map
A→ρ=
AA∗
.
tr(AA∗ )
(i) Show that ρ is a density matrix.
(ii) Show that ρ is invariant under the map A → AU , where U is an n × n
unitary matrix.
(iii) Is AA∗ = A∗ A in general?
(iv) Consider the map
A∗ A
A→σ=
.
tr(A∗ A)
Is σ = ρ? Prove or disprove.
Problem 17.
Consider the state
cos θ
|ψi =
eiφ sin θ
and the density matrix
ρ = |ψihψ|.
Given the Hamilton operator
ˆ = ~ωσ1 .
H
ˆ The von
Solve the von Neumann equation for the given ρ and the given H.
Neumann equation is given by
i~
with the solution
dρ
ˆ ρ](t)
= [H,
dt
ˆ
ˆ
ρ(t) = e−iHt/~ ρ(0)eiHt/~ .
Problem 18.
Consider the Bell state
 
1
1 0
|ψi = √  
2 0
1
Density Operators
47
and the density matrix
ρ = |ψihψ|.
Given the Hamilton operator
ˆ = ~ωσ1 ⊗ σ1 .
H
ˆ The von
Solve the von Neumann equation for given ρ and the given H.
Neumann equation is given by
i~
dρ
ˆ ρ](t)
= [H,
dt
with the solution
ˆ
ˆ
ρ(t) = e−iHt/~ ρ(0)eiHt/~ .
Problem 19.
(i) Is the 2 × 2 matrix
1/2 −i/2
ρ=
i/2 1/2
a density matrix?
(ii) Can one find a state |ψi in C2 such that
ρ = |ψihψ| ?
(iii) Are the 4 × 4 matrices
ρ ⊗ ρ,
ρ ⊕ ρ,
ρ?ρ
density matrices? Here ⊗ denotes the Kronecker product, ⊕ the direct sum
and ? operation which is defined for two 2 × 2 matrices A and B as
a11
 0
A?B =
0
a21

0
b11
b21
0
0
b12
b22
0

a12
0 
.
0
a22
Problem 20. Let |0i, |1i be the standard basis in C2 . Consider the
entangled state
1
|ψi = √ (|0i ⊗ |1i − |1i ⊗ |0i)
2
with the density matrix ρ = |ψihψ|. Find the reduced density matrix ρ1 .
Discuss.
48 Problems and Solutions
Problem 21.
Is the 2 × 2 matrix
1 1 + r cos θ
ρ=
2 r sin θeiφ
r sin θe−iφ
1 − r cos θ
a density matrix? What are the conditions on r, θ, φ?
Problem 22. Consider a finite dimensional Hilbert space of dimension
d on which the density matrix ρ acts. A quantum operation is represented
by a completely positive and trace preserving map Λ which takes the form
2
Λ(ρ) =
d
X
Vj ρVj∗ .
j=1
Show that the trace preserving condition tr(Λ(ρ)) = tr(ρ) is equivalent to
the equality
d2
X
Vj∗ Vj = I.
j=1
Problem 23. Let S be the set of unit vectors in the Hilbert space Cn .
Let u ∈ S. A function µ(u) from S to R is called a generalized probability
measure if the following two conditions hold: (i) for u ∈ S, 0 ≤ µ(u) ≤ 1,
n
(ii)
Pn if u1 , . . . , un form an orthonormal basis in the Hilbert space C , then
j=1 µ(uj ) = 1.
Let n ≥ 3. Then any generalized probability measure µ on Cn has the form
µ(ρ) = tr(ρuu∗ )
for a uniquely defined density matrix ρ. (Gleason 1957)
(i) Consider the Hilbert space C3 , the orthonormal basis
 
 


1
0
1
1
1  
0 , u2 =  1  , u3 = √  0 
u1 = √
2 1
2 −1
0
and the density matrix

1
1
ρ=
1
3
1
1
1
1

1
1.
1
Density Operators
49
Find µ(u1 ), µ(u2 ), µ(u3 ).
(ii) Consider the Hilbert space C4 , the orthonormal basis
 iφ 
 iφ 


e
e
0
iφ
1  0 
1  0 
1 e 
u1 = √ 
 , u2 = √ 
 , u3 = √  iφ  ,
0
0
2
2
2 e
eiφ
−eiφ
0

0
iφ
1  e 
u4 = √  iφ 
2 −e
0

and the density matrix
1
1 1
ρ= 
4 1
1

1
1
1
1
1
1
1
1

1
1
.
1
1
Find µ(u1 ), µ(u2 ), µ(u3 ), µ(u4 ).
Problem 24.
Consider the two 2 × 2 density matrices
ρ11 ρ12
σ11 σ12
ρ=
,
σ=
.
ρ21 ρ22
σ21 σ22
Is the 4 × 4 matrix
ρ11
1 0
ρ?σ = 
0
2
ρ21

0
σ11
σ21
0
0
σ12
σ22
0

ρ12
0 

0
ρ22
a density matrix?
Problem 25.
Consider the density matrix
1 1 1
ρ=
2 1 1
and let σ1 , σ2 , σ3 be the Pauli spin matrices. Calculate the commutators
[ρ, σ1 ], [ρ, σ2 ], [ρ, σ3 ] and discuss.
Problem 26.
Consider the density matrix
1 1 1
ρ=
.
2 1 1
Find the Cayley transform
U = (ρ − iI2 )(ρ + iI2 )−1
50 Problems and Solutions
and then the commutator [ρ, U ]. Discuss
Problem 27. Consider the Pauli spin matrices σ1 , σ2 , σ3 . Find the
normalized eigenvectors
v11 , v12 ,
v21 , v22 ,
v31 , v32
and construct the six density matrices (pure states)
∗
ρjk = vjk vjk
where j = 1, 2, 3 and k = 1, 2. Calculate commutators [ρjk , ρj 0 k0 ] and anticommutators [ρjk , ρj 0 k0 ]+ and compare to the commutators [σj , σk ] and
anti-commutators [σj , σk ]+ .
Problem 28.
Does the density matrix

1 0
1 0 1
ρ= 
4 0 1
1 0
0
1
1
0

1
0

0
1
represent a pure or mixed state?
Problem 29. Consider the Hamilton operator acting in the Hilbert space
C4
ˆ = ~ω1 (σ3 ⊗ I2 + I2 ⊗ σ3 ) + ~ω2 (σ1 ⊗ σ1 )
H
where ω1 , ω2 > 0.
ˆ is hermitian) E0 , E1 , E2 , E3
(i) Find the (real) eigenvalues (the matrix H
with the ordering E0 ≤ E1 ≤ E2 ≤ E3 .
(ii) Find the corresponding normalized eigenvectors |E0 i, |E1 i, |E2 i, |E3 i.
Are the eigenvectors separable?
(iii) Calculate the partition function Z(β) (β = 1/(kB T )) defined by
Z(β) :=
3
X
exp (−βEj ) .
j=0
(iv) We define
pj (β) :=
e−βEj
,
Z(β)
j = 0, 1, 2, 3.
Calculate the density matrix
ρ(β) =
3
X
j=0
pj (β)|Ej ihEj |.
Density Operators
51
Do we have a mixed or pure state? Study the cases ρ(∞) and ρ(0).
Problem 30.
Let 1 , 2 , 3 ∈ R. Consider the hermitian matrix
1 + 3
1 0
ρ(1 , 2 , 3 ) = 
0
4
1 + 2

0
1 − 3
1 − 2
0
0
1 − 2
1 − 3
0

1 + 2
0 
.
0
1 + 3
What is the condition such that ρ(1 , 2 , 3 ) is a density matrix?
Problem 31. Consider the Hilbert space Cn . Let ρ be a density matrix
in this Hilbert space and H and K be two hermitian n × n matrices. One
defines
hHi := tr(ρH), hH 2 i := tr(ρH 2 )
and analogously for K. Let
p
∆H := hH 2 i − hHi2 ,
∆K :=
p
hK 2 i − hKi2 .
Then we have the uncertainty relation
(∆H)(∆K) ≥
1
|hi[H, K]i| .
2
Let

1
1
0
ρ=
2
0
0
0
0

0
0
1
and

0
H = 1
0
1
2
0

0
0,
0


0 i 0
K =  −i 0 0  .
0 0 0
Show that the uncertainty relation becomes an equality for the given ρ, H
and K.
Problem 32.
Let Id be the d × d identity matrix. Consider the matrix
ρ=
1
(Id + K)
d
where K is a hermitian d × d matrix with all diagonal entries equal to 0.
What is the condition on such a K such that ρ is a density matrix?
52 Problems and Solutions
Problem 33.
Let α ∈ [0, 1]. Show that

1−α
0
0
1 0
1 + α −2α
ρ(α) = 
0
−2α 1 + α
4
0
0
0

0
0 

0
1−α
is a density matrix (so-called Werner state). Find the eigenvalues and
eigenvectors of ρ.
Problem 34.
Are the matrices

2 sin2 θ
1 0
ρ(θ) = 
0
2
0
0
cos2 θ
cos2 θ
0
0
cos2 θ
cos2 θ
0

0
0

0
0
and

2 cos2 θ
0
0
0
2
2
1 0
sin θ sin θ 0 
ρ(θ) = 

0
sin2 θ sin2 θ 0
2
0
0
0
0
density matrices? Prove or disprove. If so, do we have a mixed or pure
state?

Problem 35.
Is the matrix

1 0 0
1 0 1 1
ρ= 
4 0 1 1
1 0 0

1
0 1
 ≡ (I2 ⊗ I2 + σ1 ⊗ σ1 )
0
4
1
a density matrix?
Problem 36.
Can the density matrix

1 0
1 0 1
ρ= 
4 1 0
0 1
1
0
1
0

0
1

0
1
be written as a Kronecker product of two 2 × 2 density matrices?
Problem 37. Let ρ be a density matrix given as an n × n matrix and U
be an n × n unitary matrix. Then U ρU −1 is again a density matrix. Let




1 0 0 1
1 √0
0 1
1 0 1 1 0
1  0
2 √0 0 
ρ= 
, U = √ 
.
0
0
2 0
4 0 1 1 0
2
1 0 0 1
−1 0
0 1
Density Operators
53
Find the density matrix U ρU −1 .
Problem 38.
Let

0 0
1
0 1

ρ∓

1 =
2 0 ∓1
0 0
0
∓1
1
0

0
0
,
0
0
1
1
0

ρ∓

2 =
0
2
∓1


0 0 ∓1
0 0 0 

0 0 0
0 0 1
be the four density matrices for the Bell states.
(i) Let t ∈ [0, 1]. Is the convex combination
ρ = tρ1 + (1 − t)ρ2
a density matrix?
(ii) The Hilbert-Schmidt distance d(ρ1 , ρ2 ) is given by
p
d(ρ1 , ρ2 ) := tr((ρ1 − ρ2 )2 ).
Find d(ρ1 , ρ2 ) for the given density matrices.
Problem 39. Let σ1 , σ2 , σ3 be the Pauli spin matrices. Find the conditions on the real coefficients rj , uj , tjk (j, k = 1, 2, 3) such that
ρ=
3
3
3 X
3
X
X
X
1
(I2 ⊗ I2 +
rj σj ⊗ I2 +
uj I2 ⊗ σj +
tjk σj ⊗ σk )
4
j=1
j=1
j=1
k=1
is a density matrix. Note that since tr(σj ) = 0 for j = 1, 2, 3 we have
tr(ρ) = 1.
Problem 40. The variance of an observable A and a density operator ρ
in a Hilbert space H is defined as
V (ρ, A) := tr(ρA2 ) − (tr(ρA))2 .
Let |ψi be a normalized state in the Hilbert space H. Show that if ρ =
|ψihψ| (pure state) we obtain
V (|ψihψ|, A) = hψ|A2 |ψi − hψ|A|ψi2 .
Problem 41. (i) Consider the spin-1 matrices




0 1 0
0 −1 0
1 
i
S1 = √
1 0 1  , S2 = √  1 0 −1  ,
2 0 1 0
2 0 1
0

1
S3 =  0
0
0
0
0

0
0 
−1
54 Problems and Solutions
which are hermitian and traceless. Let I3 be the 3 × 3 unit matrix. Let
 
v1
v =  v2 
v3
be a vector in R3 with kvk ≤ 1. Is the matrix
ρ=
3
X
1
(I3 +
vj Sj )
3
j=1
a density matrix. Obviously this matrix is hemitian and has trace 1, but
are all the eigenvalues are non-zero?
(ii) Is the matrix
3
X
1
vj Sj ⊗ Sj )
ρ = (I3 ⊗ I3 +
9
j=1
a density matrix?
Problem 42. (i) Consider the three



1
1 0 0
1
ρ1 =  0 0 0  , ρ2 =  1
3
1
0 0 0
3 × 3 matrices


1 1
1
1
1 1  , ρ3 =  0
3
1 1
0
0
1
0

0
0
1
Which of these matrices are density matrices?
(ii) For the matrices which represent density matrices found out whether it
represents of pure state or mixed state. If it is pure state find the state |ψi
in the Hilbert space C3 such that ρ = |ψihψ|.
Problem 43. Consider a mixture of 25% of the pure state (1, 0)T , 25%
of the pure state (0, 1)T and 50% of the pure state √12 (1, 1)T described by
the density matrix
1 1
1 0
1 1
1
1
√ (1 1).
ρ=
(1 0) +
(0 1) + √
4 0
4 1
2 2 1
2
Find the spectral representation of ρ. Use the spectral representation of ρ to
find another mixture of pure states with the same (measurement) statistical
properties as ρ.
Problem 44.
Consider the state
cos(θ)
|ψi =
eiφ sin(θ)
Density Operators
55
in the Hilbert space C2 , where φ, θ ∈ R. Let ρ(t = 0) = ρ(0) = |ψihψ| be
ˆ = ~ωσ1 .
a density matrix at time t = 0. Given the Hamilton operator H
Solve the von Neumann equation to find ρ(t).
Problem 45. Let H1 and H2 be two Hilbert spaces and H1 ⊗ H2 be the
product Hilbert space. Let ρ be a density operators of the Hilbert space
H1 ⊗ H2 . Show that if one of the reduced density operators trH2 (ρ) = ρ1
or trH1 (ρ) = ρ2 is pure, then ρ = ρ1 ⊗ ρ2 . If both ρ1 and ρ2 are pure, then
ρ is pure too.
Problem 46.
Let α ∈ [0, 1] and φ ∈ R. Is

α
0
1
0 2 − 2α
ρ(α, φ) =
2
eiφ
0

e−iφ
0 
α
a density matrix?
Problem 47. Let |φj i (j = 1, . . . , d) be an orthonormal basis in the
Hilbert space Cd . Is
d
1 X
ρ=
|φj ihφk |
d
j,k=1
a density matrix.
Problem 48.
(i) Consider the density matrix (pure state)
1 0
1
ρ=
=
(1 0).
0 0
0
Apply the Cayley transform to find the corresponding unitary matrix. Discuss.
(ii) Consider the density matrix (pure state)
1
1 1 −i
1
1
√ ( 1 −i ) .
ρ=
=√
2 i 1
2 i
2
Apply the Cayley transform to find the corresponding unitary matrix. Discuss.
(iii) Consider the n × n density matrix (pure state)


1 1 ··· 1
1 1 1 ··· 1
ρ= 
. . ..
. .
n  .. ..
. .. 
1 1 ··· 1
56 Problems and Solutions
Apply the Cayley transform to find the corresponding unitary matrix. Discuss.
(iv) Consider the mixed state
1/2 0
ρ=
.
0 1/2
Apply the Cayley transform to find the corresponding unitary matrix. Discuss.
Problem 49. Let |ni (n = 0, 1, . . . , N ) be the standard basis in CN +1 .
Consider the states
N 1/2
X
N
|θ, φi =
(cos(θ/2))N −n (sin(θ/2))n e−inφ |ni.
n
n=0
Consider the density matrix
ρ(t) =
N
N X
X
∗
Cm
(t)Cn (t)|nihm|.
n=0 m=0
Show that
Q(θ, φ, t) :=
=
N +1
hθ, φ|ρ(t)|θ, φi
4π
N
N 1/2 1/2
N +1 X X N
N
4π
m
m=0 n=0
2N −m−n
×(cos(θ/2))
n
∗
Cm
(t)Cn (t)
(sin(θ/2))m+n e−i(m−n)φ .
Problem 50. A quantum system is described by the density matrix ρ a
positive semi-definite operator with tr(ρ) = 1. The observable is described
by self-adjoint operators A and their expectation values are given by tr(Aρ).
Consider the Hilbert space C2 , the density matrices
1/2 0
1/2 1/2
ρ1 =
,
ρ2 =
0
1/2
1/2 1/2
and the hermitian 2 × 2 matrix
σ2 =
0
i
−i
0
.
Find
tr(ρ1 σ2 ),
tr(ρ2 σ2 ),
tr((ρ1 ⊗ ρ2 )(σ2 ⊗ σ2 )).
Density Operators
57
Problem 51. Let t ∈ [0, 1]. Let ρ1 , ρ2 be two density matrices.
(i) Is the convex combination
ρ = tρ1 + (1 − t)ρ2
a density matrix.
(ii) If so apply it to the density matrices which are related to the Bell states
0
1 0
ρ1 = 
2 0
0

0
1
1
0
0
1
1
0

0
0
,
0
0
1
1 0
ρ2 = 
2 0
1

0
0
0
0
0
0
0
0

1
0
.
0
1
(iii) The Hilbert-Schmidt distance d(ρ1 , ρ2 ) is given by
p
d(ρ1 , ρ2 ) = tr((ρ1 − ρ2 )2 ).
Find the distance for the two density matrices given in (ii).
Problem 52. Let σ1 , σ2 , σ3 be the Pauli spin matrices and I2 the 2 × 2
identity matrix.
(i) Show that the four matrices
ρ1 =
1
(I2 + σ3 ),
2
ρ2 =
1
(I2 − σ3 ),
2
ρ3 =
1
(I2 + σ1 ),
2
ρ4 =
1
(I2 + σ2 )
2
are density matrices.
(ii) Show that the four matrices ρ1 , ρ2 , ρ3 , ρ4 form a basis in the Hilbert
space M2 (C) with the scalar product hA, Bi := tr(AB ∗ ).
Problem 53.
Consider the Hilbert space C2 and the projection matrices
1 1 1
1
1 −1
Π1 =
,
Π2 =
.
2 1 1
2 −1 1
Find Π1 Π2 and Π1 + Π2 . Let
cos(θ)
ρ(θ) =
( cos(θ)
sin(θ)
sin(θ) ) .
Find
tr(ρΠ1 ),
tr(ρΠ2 ).
Problem 54. Let σ1 , σ2 , σ3 be the Pauli spin matrices and I2 be the
2 × 2 identity matrix.
58 Problems and Solutions
(i) Show that the four 2 × 2 matrices
ρ1 =
1
(I2 + σ3 ),
2
ρ2 =
1
(I2 − σ3 ),
2
ρ3 =
1
(I2 + σ1 ),
2
ρ4 =
1
(I2 + σ2 )
2
are density matrices in the Hilbert space C2 .
(ii) Show that the four matrices form a basis in the Hilbert space M2 (C)
with the scalar product hA, Bi := tr(AB ∗ ).
(iii) Are the matrices ρ1 ⊗ρ1 , ρ2 ⊗ρ2 , ρ3 ⊗ρ3 density matrices in the Hilbert
space M2 (C).
Chapter 5
Partial Trace
Problem 1. Consider the finite-dimensional Hilbert spaces H1 = Cn1
and H2 = Cn2 . Let H1 ⊗ H2 be the product Hilbert space. Let |ψi and |φi
be states in the product Hilbert space H1 ⊗ H2 . Show that if
trH2 (|ψihψ|) = trH2 (|φihφ|)
then there exists a unitary matrix U acting in the Hilbert space H2 such
that
|ψi = (In1 ⊗ U )|φi
where In1 is the identity matrix in the Hilbert space H1 .
Problem 2. Consider the GHZ-state in the Hilbert space C8 (C8 ∼
=
C2 ⊗ C2 ⊗ C 2 )
1
1
1
1
0
0
0
|GHZi = √
⊗
⊗
+
⊗
⊗
.
0
0
0
1
1
1
2
Then the density matrix is given by the 8 × 8 matrix

1 0 0 0 0 0
0 0 0 0 0 0

0 0 0 0 0 0
1
0 0 0 0 0 0
ρ = |GHZihGHZ| = 
2 0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0
1 0 0 0 0 0
59
0
0
0
0
0
0
0
0

1
0

0

0
.
0

0

0
1
60 Problems and Solutions
(i) Calculate the partial trace ρAB = trC (ρ) with the basis
1
0
I4 ⊗
,
I4 ⊗
.
0
1
(ii) Calculate the partial trace ρA = trB (ρAB ) with the basis
1
0
I2 ⊗
,
I2 ⊗
.
0
1
Chapter 6
Reversible Logic Gates
Problem 1. For reversible gates the following boolean expression plays
an important role
(a11 · a22 ) ⊕ (a12 · a21 )
where a11 , a12 , a21 , a22 ∈ { 0, 1 }. It could be considered as the determinant
of the 2 × 2 binary matrix
a11 a12
.
a21 a22
Find the inverse of the matrix when it exists.
Problem 2.
Find the truth table for the boolean function
f (a, a0 , b, b0 ) = (a · b0 ) ⊕ (a0 · b).
Problem 3.
given by
Consider a two input gate (x, y) / two output gate (x0 , y 0 )
x0 = a · x ⊕ b · y ⊕ c
y 0 = a0 · x ⊕ b0 · y ⊕ c0
where a, b, a0 , b0 , c, c0 ∈ { 0, 1 }.
(i) Let a = 0, b = 1, a0 = 1, b0 = 0 and c = c0 = 0. Find the output (x0 , y 0 )
for all possible inputs (x, y). Is the transformation invertible?
61
62 Problems and Solutions
(ii) Let a = 1, b = 1, a0 = 1, b0 = 1 and c = c0 = 0. Find the output (x0 , y 0 )
for all possible inputs (x, y). Is the transformation invertible?
Problem 4.
Consider the Toffoli gate
T : {0, 1}3 → {0, 1}3 ,
T (a, b, c) := (a, b, (a · b) ⊕ c)
where a
¯ is the NOT operation, + is the OR operation, · is the AND operation and ⊕ is the XOR operation.
1. Express N OT (a) exclusively in terms of the TOFFOLI gate.
2. Express AN D(a, b) exclusively in terms of the TOFFOLI gate.
3. Express OR(a, b) exclusively in terms of the TOFFOLI gate.
4. Show that the TOFFOLI gate is invertible.
Thus the TOFFOLI gate is universal and reversible (invertible).
Problem 5.
Consider the Fredkin gate
F : {0, 1}3 → {0, 1}3 ,
F (a, b, c) := (a, a · b + a
¯ · c, a · c + a
¯ · b)
where a
¯ is the NOT operation, + is the OR operation, · is the AND operation and ⊕ is the XOR operation.
1. Express N OT (a) exclusively in terms of the FREDKIN gate.
2. Express AN D(a, b) exclusively in terms of the FREDKIN gate.
3. Express OR(a, b) exclusively in terms of the FREDKIN gate.
4. Show that the FREDKIN gate is invertible.
Thus the FREDKIN gate is universal and reversible (invertible).
Problem 6. The Toffoli gate T(x1 , x2 ; x3 ) has 3 inputs (x1 , x2 , x3 ) and
three outputs (y1 , y2 , y3 ) and is given by
(x1 , x2 , x3 ) → (x1 , x2 , x3 ⊕ (x1 · x2 ))
where x1 , x2 , x3 ∈ { 0, 1 }, ⊕ is the XOR-operation and · the AND-operation.
Give the truth table.
Problem 7. A generalized Toffoli gate T(x1 , x2 , . . . , xn ; xn+1 ) is a gate
that maps a boolean pattern (x1 , x2 , . . . , xn , xn+1 ) to
(x1 , x2 , . . . , xn , xn+1 ⊕ (x1 · x2 · . . . · xn ))
Reversible Logic Gates
63
where ⊕ is the XOR-operation and · the AND-operation. Show that the
generalized Toffoli gate includes the NOT-gate, CNOT-gate and the original
Toffoli gate.
Problem 8. The Fredkin gate F(x1 ; x2 , x3 ) has 3 inputs (x1 , x2 , x3 ) and
three outputs (y1 , y2 , y3 ). It maps boolean patterns
(x1 , x2 , x3 ) → (x1 , x3 , x2 )
if and only if x1 = 1, otherwise it passes the boolean pattern unchanged.
Give the truth table.
Problem 9. The generalized Fredkin gate F(x1 , x2 , . . . , xn ; xn+1 , xn+2 ) is
a gate is the mapping of the boolean pattern
(x1 , x2 , . . . , xn , xn+1 , xn+2 ) → (x1 , x2 , . . . , xn , xn+2 , xn+1 )
if and only if the boolean product x1 · x2 · . . . · xn = 1 (· is the bitwise AND
operation), otherwise the boolean pattern passes unchanged. Let n = 2
and (x1 , x2 , x3 , x4 ) = (1, 1, 0, 1). Find the output.
Problem 10.
Is the gate (a, b, c ∈ { 0, 1 })
(a, b, c) → (a, a · b ⊕ c, a · c ⊕ b)
reversible?
Problem 11. Prove that the Fredkin gate is universal. A set of gates is
called universal if we can build any logic circuits using these gates assuming
bit setting gates are given.
Problem 12.
The half-adder is given by
S=A⊕B
C = A · B.
Construct a half-adder using two Toffoli gates.
Problem 13.
The Feynman gate is a 2 input/2 output gate given by
x01 = x1
x02 = x1 ⊕ x2
(i) Give the truth table for the Feynman gate.
64 Problems and Solutions
(ii) Show that copying can be implemented using the Feynman gate.
(iii) Show that the complement can be implemented using the Feynman
gate.
(iv) Is the Feynman gate invertible?
Problem 14.
Consider the 3-input/3-output gate given by
x01 = x1
x02 = x1 ⊕ x2
x03 = x1 ⊕ x2 ⊕ x3 .
(i) Give the truth table.
(ii) Is the transformation invertible.
Problem 15.
Consider the 3-input/3-output gate given by
x01 = x1
x02 = x1 ⊕ x2
x03 = x3 ⊕ (x1 · x2 ).
(i) Give the truth table.
(ii) Is the gate invertible?
Problem 16.
Consider the 3-input/3-output gate given by
x01 = x1 ⊕ x3
x02 = x1 ⊕ x2
x03 = (x1 · x2 ) ⊕ (x1 · x3 ) ⊕ (x2 · x3 ).
(i) Give the truth table.
(ii) Is the gate invertible?
Problem 17.
Consider the 3-input/3-output gate given by
x01 = x1 ⊕ x3
x02 = x1 ⊕ x2
x03 = (x1 + x2 ) ⊕ (x1 + x3 ) ⊕ (x2 + x3 ).
(i) Give the truth table.
(ii) Is the gate invertible?
Reversible Logic Gates
Problem 18.
65
Consider the 4-input/4-output gate given by
x01 = x1
x02 = x2
x03 = x3
x04 = x4 ⊕ x1 ⊕ x2 ⊕ x3 .
(i) Give the truth table.
(ii) Is the gate invertible?
Problem 19.
Consider the 4-input/4-output gate given by
x01 = x1 ⊕ x3
x02 = x2 ⊕ x3 ⊕ (x1 · x2 ) ⊕ (x2 · x3 )
x03 = x1 ⊕ x2 ⊕ x3
x04 = x4 ⊕ x3 ⊕ (x1 · x2 ) ⊕ (x2 · x3 ) .
(i) Give the truth table.
(ii) Is the gate invertible?
Problem 20.
Show that one Fredkin gate
(a, b, c) → (a, a · b + a · c, a · c + a · b)
is sufficient to implement the XOR gate. Assume that either b or c are
available.
Problem 21.
abc
000
100
010
110
001
101
011
111
->
->
->
->
->
->
->
->
Show that the map f : {0, 1}3 → {0, 1}3
xyz
000
100
101
011
001
010
110
111
is invertible. The map describes a reversible half-adder. If c = 0, then x is
the first digit of the sum a + b and y is the carry bit. If c = 1, then z is the
first digit of the sum a + b + c and y is the carry bit.
66 Problems and Solutions
Problem 22.
Show that the Toffoli gate which maps
|ai ⊗ |bi ⊗ |ci 7→ |ai ⊗ |bi ⊗ |c ⊕ (a · b)i
can simulate the FANOUT and the NAND gate.
Problem 23. (i) Let x1 , x2 ∈ {0, 1}. Let ⊕ be the XOR operation. Show
that
(x1 , x2 ) 7→ (x1 ⊕ 1, x1 ⊕ x2 )
is a 2-bit reversible gate.
(ii) Let
1
|0i =
,
0
0
|1i =
.
1
Find the 4 × 4 permutation matrix P such that
P (|x1 i ⊗ |x2 i) = |x1 ⊕ 1i ⊗ |x1 ⊕ x2 i .
(iii) Show that
(x1 , x2 ) 7→ (x1 ⊕ x2 , x2 ⊕ 1)
is a 2-bit reversible gate.
(iv) Find the 4 × 4 permutation matrix P such that
P (|x1 i ⊗ |x2 i) = |x1 ⊕ x2 i ⊗ |x2 ⊕ 1i.
(v) Given a 4 × 4 permutation matrix (as a quantum gate). How can one
construct a corresponding 2-bit reversible gate? Apply it to the permutation matrix


0 1 0 0
0 0 1 0
P =
.
0 0 0 1
1 0 0 0
Problem 24. The NOT, AND and OR gate form a universal set of operations (gates) for boolean algebra. The NAND operation is also universal
for boolean algebra. However these sets of operations are not reversible sets
of operations. Consider the Toffoli and Fredkin gates
T OF F OLI : {0, 1}3 → {0, 1}3 ,
F REDKIN : {0, 1}3 → {0, 1}3 ,
T OF F OLI(a, b, c) = (a, b, (a · b) ⊕ c)
F REDKIN (a, b, c) = (a, a·c+¯
a·b, a·b+¯
a·c)
Reversible Logic Gates
67
where a
¯ is the NOT operation, + is the OR operation, · is the AND operation and ⊕ is the XOR operations.
1. Express NOT(a) exclusively in terms of the TOFFOLI gate.
2. Express NOT(a) exclusively in terms of the FREDKIN gate.
3. Express AND(a,b) exclusively in terms of the TOFFOLI gate.
4. Express AND(a,b) exclusively in terms of the FREDKIN gate.
5. Express OR(a,b) exclusively in terms of the TOFFOLI gate.
6. Express OR(a,b) exclusively in terms of the FREDKIN gate.
7. Show that the TOFFOLI gate is reversible.
8. Show that the FREDKIN gate is reversible.
Thus the TOFFPLI and FREDKIN gates are eachuniversal and reversible
(invertible).
Chapter 7
Unitary Transformations
and Quantum Gates
Problem 1. Consider the compact Lie group SU (4). Let U ∈ SU (4).
Then the 4 × 4 matrix U can be factorized as follows

3
X
i
θj σj ⊗ σj  (V3 ⊗ V4 )
U = (V1 ⊗ V2 ) exp 
2 j=1

where V1 , V2 , V3 , V4 ∈ SU (2) and θj ∈ R. Let
0
0
S=
0
1

0
0
1
0
0
1
0
0

1
0
.
0
0
Show that S ∈ SU (4). Find the factorization given above for S.
Hint. Since [σj ⊗ σj , σk ⊗ σk ] = 04 we can write


3
X
i
iθ1
iθ2
iθ3


exp
θj σj ⊗ σj ≡ exp
σ1 ⊗ σ1 exp
σ2 ⊗ σ2 exp
σ3 ⊗ σ3 .
2 j=1
2
2
2
68
Unitary Transformations and Quantum Gates
Problem 2.
Consider the Bell matrix

1 0 0
1  0 1 −1
B=√ 
0 1 1
2
−1 0 0
69

1
0
.
0
1
(i) Show that B is invertible and find B −1 . Is B unitary?
(ii) Express B 2 using the Pauli spin matrices and the Kronecker product.
(iii) Find a 4 × 4 matrix A such that B = exp(iA).
(iv) Can one find a positive integer n such that B n = I4 ?
(v) Show that
1
B = √ (I4 + B 2 ).
2
Consider the state |ψi in the Hilbert space C9
           
0
0
0
0
1
1
1            
.
|ψi = √
0 ⊗ 0 + 1 ⊗ 1 + 0 ⊗ 0
3
1
1
0
0
0
0
Problem 3.
Is the state invariant under U ⊗ U , where U is the 3 × 3 unitary matrix


0 0 1
U = 0 1 0.
1 0 0
Problem 4.
(i) The Schr¨
odinger equation is given by
i~
dψ
= Hψ(t)
dt
(1)
with ψ(0) the initial value. The evolution of ψ(t) is determined by
ψ(t) = U (t)ψ(0)
(2)
where U (t) is a unitary evolution operator and U (0) = I. Show that
i~
dU (t)
ψ(0) = HU (t)ψ(0).
dt
(3)
(ii) Assume that H = ~ωσ3 . Find U (t).
Problem 5. Let I2 be the 2 × 2 identitity matrix and σ1 be the Pauli
spin matrix and |0i, |1i be the standard basis. The CNOT-gate can be
represented as
UCN OT = |0ih0| ⊗ I2 + |1ih1| ⊗ σ1 .
70 Problems and Solutions
2
Is UCN OT hermitian? Is UCN
OT = I4 ?
Problem 6. Let U be an n × n unitary matrix. Show that if the bipartite
states |ψi, |φi ∈ Cn ⊗ Cm satisfy
|φi = (U ⊗ Im )|ψi
then the ranks of the corresponding reduced density matrices satisfy
φ
r(ρψ
1 ) ≥ r(ρ1 ),
φ
r(ρψ
2 ) ≥ r(ρ2 ).
Problem 7.
Consider the unitary matrices
1
1 1
0 1
VH = √
⊗ I2 ,
VM =
⊗ I2
1 0
2 1 −1
iχ
0 0
e
0
VC =
⊗ U2 +
⊗ I2
0 1
0 0
where U2 is an arbitrary 2 × 2 unitary matrix and χ ∈ R. Consider the
4 × 4 density matrix
1
1 0
ρin =
( 1 0 ) ⊗ ρ2 ≡
⊗ ρ2
0
0 0
where ρ2 is an arbitrary 2 × 2 density matrix. Find the density matrix
∗
ρout = VH VM VC VH ρin VH∗ VC∗ VM
VH∗ .
Problem 8.
The n-qubit Pauli group is defined by
Pn := { I2 , σ1 , σ2 , σ3 }⊗n ⊗ { ±1, ±i }
where σ1 , σ2 , σ3 are the 2 × 2 Pauli matrices and I2 is the 2 × 2 identity matrix. The dimension of the Hilbert space under consideration is dim H = 2n .
Thus each element of the Pauli group Pn is (up to an overall phase ±1, ±i)
a Kronecker product of Pauli matrices and 2 × 2 identity matrices acting
on n qubits.
The n-qubit Clifford group Cn is the normalizer of the Pauli group. A
2n × 2n unitary matrix U acting on n qubits is an element of the Clifford
group iff
U M U ∗ ∈ Pn for each M ∈ Pn .
Unitary Transformations and Quantum Gates
71
This means the unitary matrix U acting by conjugation takes a Kronecker
product of Pauli matrices to Kronecker product of Pauli matrices. An
element of the Clifford group is defined as this action by conjugation, so
that the overall phase of the unitary matrix U is not relevant. In other
words the Clifford group is the group of all matrices that leave the Pauli
group invariant.
(i) What is order of the n-qubit Pauli group?
(ii) Show that the single-qubit Hadamard gate
1
1 1
UH = √
2 1 −1
and the single-qubit phase gate
UP =
1
0
0
i
are elements of the Clifford group C1 .
(iii) Show that the CNOT-gate
1
0
=
0
0

UCN OT
0
1
0
0
0
0
0
1

0
0

1
0
is an element of C2 .
(iv) Is the Fredkin gate an element of C3 ?
Problem 9.
Find the 4 × 4 matrix
U = e−iπ(σ1 ⊗I2 )/4 e−iπ(σ3 ⊗σ3 )/4 e−iπ(σ2 ⊗I2 )/4 .
Is the matrix U unitary?
Problem 10.
and the state
Consider the four unary gates (2 × 2 unitary matrices)
1
1 1
0 1
,
N=
, H=√
1 0
2 1 −1
1
0
1
0
V =
,
W
=
0 eiπ/2
0 eiπ/4
1
|ψi = √
2
1
.
1
Calculate the state N HV W |ψi and the expection value hψ|N HV W |ψi.
72 Problems and Solutions
Problem 11. (i) Let U be an n × n unitary matrix. Then the eigenvalues
take the form eiφ , where φ ∈ R. Let eiφ1 , . . . , eiφn be the eigenvalues of U
with the corresponding normalized eigenvectors u1 , . . . , un which form an
orthonormal basis in Cn . Then one has (spectral decomposition)
U=
n
X
eiφj uj u∗j .
j=1
Then the unitary matrix
V =
n
X
eiφj /2 uj u∗j
j=1
satisfies V 2 = U and can be viewed as the square root of U . Show that
[U, V ] = 0n .
(ii) Let U1 , U2 be two unitary matrices with the spectral representation
U1 =
n
X
eiφ1j u1j u∗1j
j=1
U2 =
n
X
eiφ2j u2j u∗2j
j=1
where eiφ1j , eiφ2j (j = 1, . . . , n) are the eigenvalues of U1 and U2 , respectively and u1j , u2j (j = 1, . . . , n) are the corresponding normalized
eigenvectors of U1 and U2 , respectively. Let the unitary matrices
V1 =
V2 =
n
X
j=1
n
X
eiφ1j /2 u1j u∗1j
eiφ2j /2 u2j u∗2j
j=1
be the square roots of U1 and U2 , respectively. Find the commutators
[U1 , U2 ] and [V1 , V2 ].
(iii) Study the question from (ii) under the condition that the bases u1j
and u2j (j = 1, . . . , n) are mutually unbiased bases, i.e.
|hu1j |u2k i|2 =
Problem 12.
1
,
n
j, k = 1, . . . , n.
(i) Consider the Hadamard gate
1
1
1 1
≡ √ (σ3 + σ1 )
UH = √
2 1 −1
2
Unitary Transformations and Quantum Gates
73
with the eigenvalues +1 and −1. Find a square root of the Hadamard gate.
(ii) The star product of the Hadamard gate with itself provides the Bell
matrix


1 0 0
1
1 0 1 1
0 
B = UH ? UH = √ 
.
2 0 1 −1 0
1 0 0 −1
Use the result from (i) to find a square root of the Bell matrix. Note that
the eigenvalues of the Bell matrix are +1 (twice) and −1 (twice).
In the Hilbert space C4 the Bell states
 
 




1
0
0
1
1 0
1 1
1  1 
1  0 
√  , √  , √ 
, √ 

0
2 0
2 1
2 −1
2
1
0
0
−1
Problem 13.
(i) Let ω = e2πi/4 . Apply the Fourier transformation
1
1 1
UF = 
2 1
1
1
ω
ω2
ω3

1
ω2
1
ω2

1
3
ω 

ω2
ω
to the Bell states and study the entanglement of these states.
(ii) Apply the Haar wavelet transformation
1
1  √1
= 
2
2
0
1
1
√

UH
− 2
0
1
−1
√0
2

1
−1 

0
√
− 2
to the Bell states and study the entanglement of these states.
(iii) Apply the Walsh-Hadamard transformation
1
1  −1
= 
2 −1
1

UW

1 1 1
−1 1 1 

1 1 −1
−1 1 −1
to the Bell states and study the entanglement of these states.
n
Extend to the Hilbert space C2 with the first Bell state given by
1
√ (1
2
0 ···
0
T
1)
74 Problems and Solutions
Problem 14. Consider the Bell matrix B and the normalized vector v


 
1 0 0
1
1
1 1
1 0 1 1
1
1
0 
1
1
B=√ 
⊗√
.
, v =   ≡ √
2 1
2 0 1 −1 0
2 1
2 1
1 0 0 −1
1
Is the normalized vector Bv entangled?
Find a 4 × 4 unitary matrix U such that
 
 
 
 
1
0
0
1
1 1
1 0
1
0
U   = √  , U   = √  ,
0
0
2 0
2 1
1
0
0
0


 


 
0
0
1
0
1  0 
1  1 
0
0
U = √ 
, U   = √ 
.
0
0
1
2 −1
2
0
1
−1
0
Problem 15.
Problem 16.
Find a 4 × 4 matrix U such that
 
 
1
1
1 1
1 0
U   = √  .
2 1
2 0
1
1
Problem 17.
Apply the quantum Fourier transform to the state
7
1X
cos(2πj/8)|ji
2 j=0
where the quantum Fourier transform is given by
7
1 X −i2πkj/8
e
|kihj|.
UQF T = √
2 2 j,k=0
Is the operator UQF T unitary? Prove or disprove. Remember that
eiθ = cos(θ) + i sin(θ)
N
−1
X
k=0
ei2πk(n−m)/N = N δnm .
Unitary Transformations and Quantum Gates
We use
{|ji, j = 0, 1, . . . , 7}
as an orthonormal basis in C8 .
Problem 18.
Write the Bell matrix

1 0
1 0 1
UB = √ 
2 0 1
1 0
0
1
−1
0

1
0 

0
−1
as a linear combination of Kronecker products of Pauli spin matrices.
75
Chapter 8
Entropy
Problem 1. An n × n density matrix ρ is a positive semidefinite matrix
such that tr(ρ) = 1. The nonnegative eigenvalues of ρ are the probabilities
of the physical states described by the corresponding eigenvectors. The
entropy of the statistical state described by the density matrix ρ is defined
by
S(ρ) := −tr(ρ ln ρ)
with the convention 0 ln 0 = 0. For the n × n hermitian matrix H (energy
operator) the statistical average of the energy E is defined by
E := tr(Hρ).
Let
ψ(ρ) := tr(Hρ) − tr(ρ ln ρ).
(i) Show that
ln tr(eH ) = max { tr(Hρ) + S(ρ) }.
(ii) Show that
−S(ρ) = max { tr(Hρ) − ln tr(eH ) }.
Problem 2. The von Neumann entropy, the standard measure of randomness of a statistical ensemble described by a n × n density matrix ρ, is
defined by
n
X
S(ρ) = −tr(ρ log ρ) = −
λj log λj
j=1
76
Entropy
77
where λj (j = 1, 2, . . . , n) are the eigenvalues of the density matrix ρ and
the log is taken to base n, the dimension of the Hilbert space Cn . Consider
the density matrix in C4


1/3 0
0 1/6
0 
 0 1/6 0
ρ=
.
0
0 1/6 0
1/6 0
0 1/3
Find the eigenvalues of ρ and then the von Neumann entropy S(ρ).
Problem 3. Consider the normalized states |ψk i, k = 0, 1, . . . , N − 1
in the Hilbert space CN . A positive operator valued measure is specified
by a decomposition of the identity matrix IN into M positive semidefinite
matrices Pm , i.e.
M
−1
X
IN =
Pm .
m=0
The mutual information is defined by
I=
−1
N
−1 M
X
X
pnm logN
n=0 m=0
pnm
pn· p·m
where
pnm := hψn |Pm |ψn i
are the joint probabilities and
pn· :=
M
−1
X
pnm ,
m=0
p·m :=
N
−1
X
pnm
n=0
are their marginals. Let M = N = 2 and
1 1 1
1
1 −1
P0 =
,
P1 =
2 1 1
2 −1 1
1
1
0
|ψ0 i = √
,
|ψ1 i =
.
1
2 −1
Find pnm , pn· , p·m and then I.
Problem 4. Let A, B be n × n hermitian matrices acting in the Hilbert
space Cn . Assume that the eigenvalues of A are pairwise different and
analogously for B. Then the normalized eigenvectors |αj i (j = 1, . . . , n) of
A form an orthonormal basis in Cn and analogously for B the normalized
78 Problems and Solutions
eigenvectors |βj i (j = 1, . . . , n) form an orthonormal basis in Cn . Let
|ψi be a normalized state in Cn . Then there are n possible outcomes for
measurements of each observable and the probabilties pj (A, |ψi), pj (B, |ψi)
(j = 1, . . . , n) are given by
pj (A, |ψi) := |hψ|αj i|2 ,
pj (B, |ψi) := |hψ|βj i|2 .
Let H|ψi (X) be the Shannon information entropy
H|ψi (X) := −
n
X
pj (X, |ψi) ln pj (X, |ψi)
j=1
corresponding to the probability distribution {pj (X, |ψi)} (j = 1, . . . , n).
The (Maassen-Uffink) entropic uncertainty relation is given by
H|ψi (A) + H|ψi (B) ≥ −2 ln( max |hαj |βk i|) > 0.
1≤j,k≤n
Note that the right-hand side does not involve the state |ψi.
(i) Let
0 1
1 0
cos θ
A = σ1 =
, B = σ3 =
, |ψi =
.
1 0
0 −1
sin θ
Calculate the left and right-hand side of the entropic uncertainty relation.
Is the entropic uncertainty relation tight for this case?
(ii) The (Landau-Pollak) uncertainty relation states that
p
p
arccos( PA ) + arccos( PB ) ≥ arccos( max |hαj |βk i|)
1≤j,k≤n
where
PA := max pj (A, |ψi),
1≤j≤n
PB := max pj (B, |ψi).
1≤j≤n
Calculate the left-hand and right-hand side of this uncertainty relation for
A and B given in (i).
Problem 5. Consider the Hilbert space Cn . Let A, B be two hermitian
n × n matrices (observable). Assume that A and B have non-degenerate
eigenvalues with the corresponding normalized eigenvectors |a1 i, |a2 i, . . . ,
|an i and |b1 i, |b2 i, . . . , |bn i, respectively. The entropic uncertainty relation
is an inequality given by
S (A) + S (B) ≥ S (AB)
where
S (A) = −
n
X
j=1
|hψ|aj i|2 ln(|hψ|aj i|2 ),
S (B) = −
n
X
j=1
|hψ|bj i|2 ln(|hψ|bj i|2 ),
Entropy
79
and S (AB) is a positive constant which gives the lower bound of the righthand side of the inequality. Consider the Hilbert space C2 . Let
cos θ
A = σ1 , B = σ2 , |ψi =
.
sin θ
Find S (A) , S (B) and S (A) + S (B) .
Problem 6. Consider the Hilbert space Cn and |ψi ∈ Cn . Let A and
B n × n hermitian matrices (observable) with non-degenerate eigenvalues
and corresponding normalized eigenvectors |uj i, |vj i (j = 1, . . . , n). The
entropic uncertainty relation is an inequality of the form
S (A) + S (B) ≥ SAB
where
S (A) = −
n
X
|hψ|uj i|2 ln(|hψ|uj i|2 ),
S (A) = −
n
X
|hψ|vj i|2 ln(|hψ|vj i|2 )
j=1
j=1
and SAB is a positive constant providing the lower bound of the right-hand
side of the inequality. Let
0 1
1 0
A = σ1 =
, B = σ3 =
1 0
0 −1
and
|ψi =
Calculate S (A) and S (B) .
cos(θ)
sin(θ)
.
Chapter 9
Measurement
Problem 1.
Consider the tripartite states
1
|W i = √ (|001i + |010i + |100i).
3
1
|GHZi = √ (|000i + |111i),
2
Find the probability
p = |hW |GHZi|2 .
Problem 2.
Consider the W -state
1
|W i = √ (|0i ⊗ |0i ⊗ |1i + |0i ⊗ |1i ⊗ |0i + |1i ⊗ |0i ⊗ |0i).
3
Apply the invertible local operator
√ √
√
a √d
3
ˆ
L=
⊗
0
c
0
√ 0√
3b/ a
⊗
1
0
0
1
to the W -state, where a, b, c > 0 and d = 1 − (a + b + c) ≥ 0. Calculate the
probability |hW |LW i|2 .
Problem 3.
Consider the single qubit state
|ψi := a|0i + b|1i,
|a|2 + |b|2 = 1.
Rewrite the first two qubits of the state
1
|ψi ⊗ √ (|01i + |10i)
2
80
Measurement
81
in terms of the Bell basis
1
|Φ+ i = √ (|00i + |11i),
2
1
|Φ− i = √ (|00i − |11i),
2
1
|Ψ+ i = √ (|01i + |10i),
2
1
−
|Ψ i = √ (|01i − |10i).
2
Describe how to obtain |ψi as the state of the last qubit by measuring the
first two qubits in the Bell basis. Suppose that the only errors which can
occur to three qubits are described by the transforms
{I ⊗ I ⊗ I, I ⊗ UN OT ⊗ UN OT , I ⊗ UP ⊗ UP , I ⊗ (UP UN OT ) ⊗ (UP UN OT )}.
Describe how an arbitrary error
αI ⊗I ⊗I +βI ⊗UN OT ⊗UN OT +δI ⊗UP ⊗UP +γI ⊗(UP UN OT )⊗(UP UN OT )
on the state
1
√ (|01i + |10i) ⊗ |ψi
2
can be corrected to obtain the correct |ψi as the last qubit.
Chapter 10
Entanglement
Problem 1.
Consider the singlet state (Bell state)

0
1  1 
|ψi = √ 
.
2 −1
0

Let σ1 , σ2 , σ3 be the Pauli spin matrices. Show that the matrices I2 ⊗ I2 ,
−σ1 ⊗ σ1 , −σ2 ⊗ σ2 , −σ3 ⊗ σ3 leave the state |ψi invariant.
Problem 2.
commutators
Let σ1 , σ2 , σ3 be the Pauli spin matrices. Calculate the
[σ1 ⊗ σ1 , σ2 ⊗ σ2 ],
[σ1 ⊗ σ1 , σ3 ⊗ σ3 ],
[σ2 ⊗ σ2 , σ3 ⊗ σ3 ].
Problem 3. Let |0i, |1i be the standard basis in the Hilbert space C2 .
Consider the GHZ-state
1
|ψi = √ (|0i ⊗ |0|i ⊗ |0i + |1i ⊗ |1i ⊗ |1i).
2
Find the expectation values
hψ|σ1 ⊗ σ2 ⊗ σ2 |ψi,
hψ|σ2 ⊗ σ1 ⊗ σ2 |ψi,
hψ|σ2 ⊗ σ2 ⊗ σ1 |ψi,
hψ|σ1 ⊗ σ1 ⊗ σ1 |ψi.
82
Entanglement
Problem 4.
83
Consider the state in the Hilbert space C4

 

n0 (τ, φ, θ)
sin((τ − φ)/2) sin(θ/2)
 n1 (τ, φ, θ)   sin((τ + φ)/2) cos(θ/2) 

=
.
n2 (τ, φ, θ)
cos((τ − φ)/2) sin(θ/2)
n3 (τ, φ, θ)
cos((τ + φ)/2) cos(θ/2)
The state is obviously normalized, i.e. n20 + n21 + n22 + n23 = 1. Find the
conditions on φ, τ , θ such that n0 n3 = n1 n2 (separability condition). Show
that in this case the state can be written as product state.
Problem 5. Let |0i, |1i be an arbitrary orthonormal basis. Can the state
1
1
1
1
|ψi = √ |0i ⊗ |0i + √ |0i ⊗ |1i + √ |1i ⊗ |0i + √ |1i ⊗ |1i
2
8
8
4
be written as a product state?
Problem 6.
Consider the Hamilton operator
H = ~ω(σ1 ⊗ σ1 + σ2 ⊗ σ2 + σ3 ⊗ σ3 ).
(i) Is the 4 × 4 matrix H hermitian? Find the trace of H. What can be
said about the eigenvalues of H.
(ii) Find the eigenvalues and normalized eigenvectors of H.
(iii) Calculate exp(−iHt/~).
Problem 7.
Consider the unitary matrices
U1 = eiπσ1 /4 ⊗ eiπσ1 /4 ,
U2 = eiπσ2 /4 ⊗ eiπσ2 /4 .
Calculate
U1∗ (σ3 ⊗ σ3 )U1 ,
Problem 8.
|ψi =
U2∗ (σ3 ⊗ σ3 )U2 .
Consider the state
1
(|0i ⊗ |0i + eiφ1 |0i ⊗ |1i + eiφ2 |1i ⊗ |0i + eiφ3 |1i ⊗ |1i).
2
(i) Let φ3 = φ1 + φ2 . Is the state |ψi a product state?
(ii) Let φ3 = φ1 + φ2 + π. Is the state |ψi a product state?
Problem 9. There are six different types of quark known as flavor: up,
down, charm, strange, top, bottom. Consider the two equations for states
1
1
1
cos θ √ (|uui + |ddi + |ssi) +sin θ √ (|uui + |ddi − 2|ssi) = √ (|uui+|ddi)
3
6
2
84 Problems and Solutions
cos θ
1
1
√ (|uui + |ddi − 2|ssi) − sin θ √ (|uui + |ddi + |ssi) = −|ssi
6
3
where |uui ≡ |ui ⊗ |ui etc. Find cos θ and sin θ from this two equations.
Problem 10.
Let
 
1
|0i =  0  ,
0
 
0
|1i =  1  ,
0
 
0
|2i =  0  .
1
Consider the normalized state (Aharonov state)
1
|ψi = √ (|012i − |021i + |120i − |102i + |201i − |210i)
6
where |012i = |0i ⊗ |1i ⊗ |2i etc and

1 0
S3 =  0 0
0 0

0
0 .
−1
Is |ψi an eigenstate of Sz ⊗ Sz ⊗ Sz ?
Problem 11.
Consider the 4 × 4 matrix
H=
1
(σ1 ⊗ σ1 + σ2 ⊗ σ2 + σ3 ⊗ σ3 + I2 ⊗ I2 ).
2
(i) Is H hermitian? Find the trace of H.
(ii) Calculate H 2 and tr(H 2 ).
(iii) Using the result from (ii) calculate exp(iθH), exp(−iπH/4) and exp(−iπH/2).
(iv) Using the results from (i) and (ii) find the eigenvalues of H.
(v) Find the normalized eigenstates of H.
Problem 12.
Consider the normalized state
1
|ψi = √ (|0000i−|0011i−|0101i+|0110i+|1001i+|1010i+|1100i+|1111i)
2 2
where we used the notation |0000i ≡ |0i ⊗ |0i ⊗ |0i ⊗ |0i etc. and
1
0
|0i =
,
|1i =
.
0
1
Calculate the states
(σ1 ⊗ σ3 ⊗ σ3 ⊗ σ1 )|ψi,
(σ1 ⊗ σ1 ⊗ I2 ⊗ σ3 )|ψi,
(I2 ⊗ σ1 ⊗ σ1 ⊗ I2 )|ψi
Entanglement
85
and
(I2 ⊗ σ2 ⊗ σ3 ⊗ σ2 )|ψi,
(σ1 ⊗ σ2 ⊗ σ2 ⊗ σ1 )|ψi,
(I2 ⊗ σ3 ⊗ σ2 ⊗ σ2 )|ψi.
Problem 13. The hyperdeterminant of a 2×2×2 hypermatrix C = (cijk )
(i, j, k ∈ { 0, 1 }) is defined by
DetC := −
1
2
1
X
1
X
ii0 jj 0 kk0 mm0 nn0 pp0 cijk ai0 j 0 m cnpk0 cn0 p0 m0
i,j,k,m,n,p=0 i0 ,j 0 ,k0 ,m0 ,n0 ,p0 =0
where 00 = 11 = 0, 01 = 1, 10 = −1.
(i) Calculate DetC.
(ii) Consider the three qubit state
|ψi =
1
X
cijk |ii ⊗ |ji ⊗ |ki.
i,j,k=0
The three tangle τ3 is a measure of entanglement and is defined for the three
qubit state |ψi as
τ123 := 4|DetC|
where C = (cijk ). Find the three tangle for the GHZ-state
1
|GHZi = √ (|0i ⊗ |0i ⊗ |0i + |1i ⊗ |1i ⊗ |1i)
2
and the W -state
1
|W i = √ (|0i ⊗ |0i ⊗ |1i + |0i ⊗ |1i ⊗ |0i + |1i ⊗ |0i ⊗ |0i).
3
Problem 14.
Calculate the product of the unitary matrices
exp(iπ(σ2 ⊗ I2 )/4) exp(−iπ(σ3 ⊗ σ3 )/4) exp(−iπ(σ1 ⊗ I2 )/4).
Problem 15. Let |0i, |1i be an orthonormal basis in C2 . Consider the
normalized state
1
X
|ψi =
cjk |ji ⊗ |ki
j,k=0
4
in the Hilbert space C and the 2 × 2 matrix C = (cjk ). Using the 4
coefficients cjk , j, k ∈ {0, 1}) we form a multilinear polynomial p in two
variables x1 , x2
p(x1 , x2 ) = c00 + c01 x1 + c10 x2 + c11 x1 x2 .
(1)
86 Problems and Solutions
Show that determinant det C = c00 c11 − c01 c10 is the unique irreducible
polynomial (up to sign) of content one in the 4 unkowns cjk that vanishes
whenever the system of equations
p=
∂p
∂p
=
=0
∂x1
∂x2
(2)
has a solution (x∗1 , x∗2 ) in C2 .
Problem 16. Let |0i, |1i be an orthonormal basis in C2 . Consider the
normalized state
1
X
|ψi =
cjk` |ji ⊗ |ki ⊗ |`i
j,k,`=0
8
in the Hilbert space C and the 2 × 2 × 2 array C = (cjk` ) (j, k, ` ∈ {0, 1}.
Using the 8 coefficients cjk` we form a multilinear polynomial in three
variables x1 , x2 , x3
p(x1 , x2 , x3 ) = c000 +c001 x1 +c010 x2 +c100 x3 +c011 x1 x2 +c101 x1 x3 +c110 x2 x3 +c111 x1 x2 x3 .
(1)
Show that the hyperdeterminant
DetC = c2000 c2111 + c2001 c2110 + c2010 c2101 + c2100 c2011
−2(c000 c001 c110 c111 + c000 c010 c101 c111
+c000 c100 c011 c111 + c001 c010 c101 c110
+c001 c100 c011 c110 + c010 c100 c011 c101 )
+4(c000 c011 c101 c110 + c001 c010 c100 c111 )
is the unique irreducible polynomial (up to sign) of content one in the 8
unkowns cjk` that vanishes whenever the system of equations
p=
∂p
∂p
∂p
=
=
=0
∂x1
∂x2
∂x3
(2)
has a solution (x∗1 , x∗2 , x∗3 ) in C3 .
Problem 17.
Consider the state
1
|ψi = √ (|Hi ⊗ |V i − |V i ⊗ |Hi).
2
We define a polarization state that is rotated by an angle α from the horizontal axis as
|αi = cos(α)|Hi + sin(α)|V i
Entanglement
87
and analogously
|βi = cos(β)|Hi + sin(β)|V i.
Calculate the probability
2
p(α, β) = |(hα| ⊗ hβ|)|ψi| .
Problem 18.
Consider the state
 
1
0
|ψi =  
0
0
and the unitary operator (4 × 4 matrix)
U = e−iπσ2 /4 ⊗ I2 .
Find the state U |ψi.
Problem 19. Let σ1 , σ2 , σ3 be the Pauli spin matrices. Consider the
Hamilton operator
ˆ = J(σ1 ⊗ σ1 + σ2 ⊗ σ2 + σ3 ⊗ σ3 ).
H
(i) Let = Jβ ≡ J/(kB T ), where kB is the Boltzmann constant and T the
absolute temperature and J > 0. Calculate
ρ() =
1
ˆ ≡ 1 exp(−(σ1 ⊗ σ1 + σ2 ⊗ σ2 + σ3 ⊗ σ3 ))
exp(−β H)
Z()
Z()
where Z() is the partition function
Z() = tr exp(−(σ1 ⊗ σ1 + σ2 ⊗ σ2 + σ3 ⊗ σ3 )).
(ii) The concurrence C(ρ()) is defined by
C(ρ()) = max(0, µ1 () − µ2 () − µ3 () − µ4 ())
where the µj ’s are the square roots of the eigenvalues of the 4 × 4 matrix
ρ(σ2 ⊗ σ2 )ρ∗ (σ2 ⊗ σ2 )
in decreasing order. Calculate C(ρ()) and discuss the result as function of
≡ Jβ.
88 Problems and Solutions
Problem 20.
Problem 21.
state
Can we find 2 × 2 matrices S1 and S2 such that
 
1
1 0
1
1
(S1 ⊗ S2 )
⊗
= √  .
0
0
2 0
1
(1)
Let N be an integer larger than 5. Consider the following
N −1
1 X
|ψi = √
|j mod N i ⊗ |3j mod N i ⊗ |5j mod N i.
N j=0
Let U be the quantum Fourier transform. Calculate (U ⊗ U ⊗ U )|ψi. Write
the answer in the basis { |0i, |1i, . . . , |N − 1i}⊗3 . Show that it is the superposition of equally probable states. Find the probability.
Problem 22.
Consider the Hamilton operator
ˆ = ~ωσ1 ⊗ σ3 ⊗ σ1
H
and the corresponding unitary operator
ˆ
U (t) = e−iHt/~ = e−iωtσ1 ⊗σ3 ⊗σ1 .
ˆ 2 and U (t).
(i) Calculate H
(ii) Show that U (t) can be written as, i.e. we decompose U (t) into elementary gates of one qubit rotations and two qubits interactions,
U (t) = e−iπI⊗I⊗σ2 /4 eiπσ3 ⊗I⊗σ3 /4 eiπσ1 ⊗I⊗I/4 e−iωtσ3 ⊗σ3 ⊗I e−iπσ1 ⊗I⊗I/4 e−iπσ3 ⊗I⊗σ3 /4 eiπI⊗I⊗σ2 /4
where I is the 2 × 2 unit matrix.
Problem 23.
matrix
Let σ1 , σ2 , σ3 be the Pauli spin matrices. Find the 4 × 4
U = e−iπ(σ1 ⊗I2 )/4 e−iπ(σ3 ⊗σ3 )/4 e−iπ(σ2 ⊗I2 )/4 .
Is the matrix unitary?
Problem 24.
Consider the Hamilton operator
H=
1
(−~ω1 σ3 ⊗ I2 − ~ω2 I2 ⊗ σ3 + ~γσ3 ⊗ σ3 ) .
2
Find
U = e−iπ(σ1 ⊗I2 )/2 e−iHt/~ e−iπ(σ1 ⊗I2 )/2 e−iHt/~ .
Entanglement
89
Give an interpretation of the result.
Problem 25.
Consider the normalized state
|ψi = cos(α)|00i + sin(α)|11i,
0 < α < π/4
where α is called the Schmidt angle.
(i) Find the eigenvalues of the density matrix |ψihψ|.
(ii) Find the partically traced density matrix (we find when we trace over
one of the subsystems).
(iii) Show that the partically traced has two unequal and non-zero eigenvalues λ1 = cos2 (α) and λ2 = sin2 (α).
(iv) Calculate the von Neumann entropy for the corresponding density matrix. Show that the entropy grows monotonically with the Schmidt angle.
Problem 26.
Consider the Hadamard matrix
1
1 1
.
UH = √
2 1 −1
Is UH ∈ SU (2)? Is iUH ∈ SU (2)?
Problem 27.
Consider the finite-dimensional Hilbert space
HN := span{ |ni : n = 0, 1, . . . , N − 1 }
i.e. dim(H)N = N and hn|mi = δnm with m = 0, 1, . . . , N − 1. Let
N −1
1 X
|φ` i := √
exp(inφ` )|ni,
N n=0
φ` := φ0 + 2π
`
N
for ` ∈ ZN . We define a self-adjoint phase operator as
φˆN :=
N
−1
X
φ` |φ` ihφ` |.
`=0
Find the matrix elements of the phase operator φˆN in the occupation number basis |ni with n = 0, 1, . . . , N − 1.
Problem 28. Calculate the three-tangle for the W -state
1
1
1
0
1
0
1
0
1
1
⊗
⊗
+
⊗
⊗
+
⊗
⊗
.
|W i = √
0
0
1
0
1
0
1
0
0
3
90 Problems and Solutions
Problem 29.
Summarize the requirements for quantum computation.
Problem 30. Find the eigenvalues and normalized eigenvectors of the
Hamilton operator
ˆ = ~ω(σ3 ⊗ σ3 ) + ∆(σ1 ⊗ σ1 ).
H
ˆ
Calculate exp(−iHt/~).
Problem 31.
ator
Consider the XX-model described by the Hamilton operHXX =
N
X
(J(σx,j σx,j+1 ) + Bσz,j )
j=1
with the periodic boundary conditions σ1,N +1 = σ1,1 , σ3,N +1 = σ3,1 . We
have
σ1,j = I2 ⊗ · · · ⊗ I2 ⊗ σ1 ⊗ I2 ⊗ · · · ⊗ I2
where σ1 is at the j-position with j = 1, 2, . . . , N . Let
Σ3 :=
N
X
σ3,j .
j=1
Calculate the commutator [HXX , Σ3 ]. Discuss.
Problem 32. Consider the Hamilton operator (so-called transverse XY model in one dimension)
ˆ = −g
H
L−1
X
j=0
x
y
σjz
−
L−1
X
j=0
1−γ y y
1+γ x x
σj σj+1 +
σj σj+1
2
2
z
where σ , σ , σ are the Pauli spin matrices, 0 ≤ γ ≤ 1, g is a constant
y
x
and we impose cyclic boundary conditions. This means σL
= σ0x , σL
= σ0y ,
z
z
σL = σ0 .
ˆ C],
ˆ where
(i) Find the commutator [H,
Cˆ :=
L−1
Y
σjz .
j=0
(ii) Calculate Cˆ 2 . Show that Cˆ and Cˆ 2 form a group under matrix mulˆ We
tiplication. Give the character table. What are the eigenvalues of C?
define
ˆ := 1 (I − C)
ˆ
Q
2
Entanglement
91
where I is the unit operator (2L × 2L identity matrix). Calculate the
ˆ
eigenvalues of Q.
ˆ
(iii) Let L = 4. Calculate the eigenvalues of H.
ˆ
ˆ
(iv) Let γ = 0. Calculate [Z, H], where
Zˆ =
L−1
X
σjz .
j=0
Discuss.
Problem 33. Let |0i, |1i be an orthonormal basis in C2 . Consider the
normalized state
1
X
|ψi =
cjk |ji ⊗ |ki.
j,k=0
Using the four coefficients cjk we form the polynomial p in the two variables
x1 , x2
p(x1 , x2 ) = c00 + c01 x1 + c10 x2 + c11 x1 x2 .
Consider the three equations p = 0, ∂p/∂x1 = 0, ∂p/∂x2 = 0, i.e.
p(x1 , x2 ) = c00 + c01 x1 + c10 x2 + c11 x1 x2 = 0
and
∂p
= c01 + c11 x2 = 0
∂x1
∂p
= c10 + c11 x1 = 0.
∂x2
Show that this system of three equations with two unkowns x1 , x2 only
admits solutions if
det(C) ≡ c00 c11 − c01 c10 = 0
where C is the 2 × 2 matrix
C=
Problem 34.
c00
c10
c01
c11
.
Consider the finite dimensional Hilbert space
HN := span{ |ni : n = 0, 1, . . . , N − 1 }
92 Problems and Solutions
where hn0 |ni = δnn0 . Thus dimHN = N . We define the state
N −1
1 X
|φ` i := √
exp(inφ` )|ni,
N n=0
φ` := φ0 + 2π
`
N
for ` ∈ ZN . We define the linear operator
φˆN :=
N
−1
X
φ` |φ` ihφ` |.
n=0
Find the matrix elements of this linear operator in the occupation number
basis |ni.
n
Problem 35. We consider the finite-dimensional Hilbert space H = C2
and the normalized state
|ψi =
1
X
cj1 ,j2 ,...,jn |j1 i ⊗ |j2 i ⊗ · · · ⊗ |jn i
j1 ,j2 ,...,jn =0
in this Hilbert space. Here |0i, |1i denotes the standard basis. Let jk
(j, k = 0, 1) be defined by 00 = 11 = 0, 01 = 1, 10 = −1. Let n be even
or n = 3. Then an n-tangle can be introduced by
1
X
cα1 ...αn cβ1 ...βn cγ1 ...γn cδ1 ...δn
τ1...n = 2 α1 ,...,α
n =0
...
δ ,...,δ
1
n =0
×α1 β1 α2 β2 · · · αn−1 βn−1 γ1 δ1 γ2 δ2 · · · γn−1 δn−1 αn γn βn δn .
√
√
(i) Consider the case n = 4 and a state |ψi with c0000 = 1/ 2, c1111 = 1/ 2
and all other coefficients are 0. Find τ1234 .
√
(ii) Consider
the case n = 4 and a state |ψi with c0000 = 1/ 2, c1111 =
√
−1/ 2 and all other coefficients are 0. Find τ1234 .
√
(iii)
√ Consider the case n = 4 and a state |ψi with c0001 = 1/ 2, c1110 =
1/ 2 and all other coefficients are 0. Find τ1234 .
√
(iv) √
Consider the case n = 4 and a state |ψi with c0001 = 1/ 2, c1110 =
−1/ 2 and all other coefficients are 0. Find τ1234 .
Problem 36.
The n-qubit Pauli group is defined by
Pn := { I2 , σ1 , σ2 , σ3 }⊗n ⊗ { ±1, ±i }
where σ1 , σ2 , σ3 are the 2 × 2 Pauli matrices and I2 is the 2 × 2 identity
matrix. The dimension of the Hilbert space under consideration is dim H =
Entanglement
93
2n . Thus each element of the Pauli group Pn is (up to an overall phase
±1, ±i) a Kronecker product of Pauli matrices and 2 × 2 identity matrices
acting on n qubits. What is the order of the n-qubit Pauli group?
Problem 37.
Consider the Hamilton operator
ˆ = ~ω(σ3 ⊗ σ3 ) + ∆1 σ1 ⊗ σ1 + ∆2 σ2 ⊗ σ2 .
H
(i) Find the eigenvalues. Discuss energy level crossing. Find the normalized
eigenvectors.
(ii) Calculate the commutators
[σ1 ⊗ σ1 , σ2 ⊗ σ2 ],
[σ2 ⊗ σ2 , σ3 ⊗ σ3 ],
[σ3 ⊗ σ3 , σ2 ⊗ σ2 ]
ˆ
(iii) Use the result from (ii) to calculate exp(−iHt/~).
Problem 38. Let σ1 = σ1 , σ2 = σ2 , σ3 = σ3 be the Pauli spin matrices.
We form the nine 4 × 4 matrices
Σjk := σj ⊗ σk ,
j, k = 1, 2, 3.
ˆ and a
Note that [Σjk , Σmn ] = 0. The variance of an hermitian operator O
wave vector |φi is defined by
2
ˆ 2 |φi − (hφ|O|φi)
ˆ
VOˆ (|φi) := hφ|(O)
.
The remoteness for a given normalized state |ψi in C4 is defined by
R(|ψi) =
3 X
3
X
hψ|(Σjk )2 |ψi − (hψ|Σjk |ψi)2 .
j=1 k=1
Find the remoteness for the Bell states
1
|φ+ i = √ (|0i ⊗ |0i + |1i ⊗ |1i),
2
1
|φ− i = √ (|0i ⊗ |0i − |1i ⊗ |1i)
2
1
|ψ + i = √ (|0i ⊗ |1i + |1i ⊗ |0i),
2
1
|ψ − i = √ (|0i ⊗ |1i − |1i ⊗ |0i).
2
Problem 39. Let e1 , e2 , e3 be the standard basis in the Hilbert space
C3 . Are the states in the Hilbert space C27 are entangled
1
√ (e1 ⊗e2 ⊗e3 +e2 ⊗e3 ⊗e1 +e3 ⊗e1 ⊗e2 +e1 ⊗e3 ⊗e2 +e3 ⊗e2 ⊗e1 +e2 ⊗e1 ⊗e3 )
6
94 Problems and Solutions
1
√ (e1 ⊗e2 ⊗e3 +e2 ⊗e3 ⊗e1 +e3 ⊗e1 ⊗e2 −e1 ⊗e3 ⊗e2 −e3 ⊗e2 ⊗e1 −e2 ⊗e1 ⊗e3 )
6
1
√ ((e1 ⊗e2 ⊗e3 +e2 ⊗e1 ⊗e3 )+ε(e2 ⊗e3 ⊗e1 +e1 ⊗e3 ⊗e2 )+ε∗ (e3 ⊗e1 ⊗e2 +e3 ⊗e2 ⊗e1 ).
6
Problem 40. Find the entanglement (three tangle) as a function of θ of
the normalized state in C8
|ψi = cos(θ)e1 ⊗ e1 ⊗ e1 − i sin(θ)e2 ⊗ e2 ⊗ e2
where
1
e1 =
,
0
0
e2 =
1
and 0 < θ < π/4.
Problem 41. Given the eigenvalue equations Ax = λx, Ay = λy and
x∗ y = 0. Then A(x + y) = λ(x + y). Thus x + y is also an eigenvector
with eigenvalue λ. Consider the 4 × 4 matrix
0
0
σ1 ⊗ σ1 = 
0
1

0
0
1
0
0
1
0
0

1
0
.
0
0
The eigenvalues are +1 (twice) and −1 (twice). The normalized eigenvectors for +1 are

 

1
1
1
1
1 1
1
1  −1 
1
1
1
1
1
√
⊗√
=  , √
⊗√
= 
.
1
1
1
−1
−1
2
2 −1
2
2
2
2
1
1
These two states are orthonormal to each other and obviously not entangled.
The normalized eigenvectors for the eigenvalue −1 are


1
1
1
1  −1 
1
1
√
⊗√
= 
,
1
2
2 1
2 −1
−1
1
√
2


1
1
1 1 
1
1
⊗√
= 
.
−1
2 −1
2 1
−1
These two states are orthonormal to each other and obviously not entangled.
All four vectors form an orthonomal basis in C4 . Find linear combinations
of the two cases so that the eigenvectors are entangled and still form an
orthonormal basis in C4 .
Entanglement
95
Are the states in C4
1
1 0
1 1
1
1
1
0
|±i = √
⊗
±
⊗
⊗
±
0
0
1
0
0
1
2
2
2
Problem 42.
entangled?
Problem 43.
(i) Consider the two states in C4
 
1
1 0
|ψi = √   ,
2 0
1

cos(α) cos(β)
 cos(α) sin(β) 
≡
.
sin(α)cos(β)
sin(α) sin(β)

|φi =
cos α
sin α
⊗
cos β
sin β
One defines
G(|ψi) = max |hφ|ψi|
α,β
as the maximum overlap between |ψi and the product state |φi. Find
G(|ψi).
(ii) Given the state
 
1
1 1
|χi =   .
2 1
1
Find G(|χi) with the product state given at (i). Discuss.
Problem 44. Consider a bipartite system and the product Hilbert space
H = H1 ⊗ H2 . Let |ψi ∈ H and normalized. Then a density matrix (pure
state)
ρ12 := |ψihψ|
is entangled when the density matrices
j, k = 1, 2, j 6= k
ρj = trk (ρ12 ),
provided by partial tracing as non-zero von Neumann entropy, i.e.
S(ρj ) = −tr(ρj log(ρj )) 6= 0,
j = 1, 2.
There is no entanglement if S(ρj ) = 0. Consider the Hilbert spaces H1 =
H2 = C3 and H = C9 . Is the normalized state in C9
1
|ψi = √ ( 1
3
entangled?
0
0
0
1
0
0
0
1)
T
96 Problems and Solutions
Problem 45. An entanglement measure is the relative entropy of entanglement. It is defined for a density matrix σ as
ER (σ) := min S(σkρ)
ρ∈D
where D is the set of density matrices with positive partial transpose (PPT
states) and
S(σkρ) := tr(σ log2 (σ) − σ log2 (ρ)).
Find S(σkρ) for the density matrix (one of the Werner states)
2
1 0
σ= 
6 0
0

Problem 46.
0
1
1
0
0
1
1
0

0
0
.
0
2
Let
|1i =
1
,
0
|0i =
0
.
1
Show that the normalized state
1
(|0i⊗|0i⊗|0i⊗|0i+|1i⊗|0i⊗|0i⊗|1i+|0i⊗|0i⊗|1i⊗|0i+|1i⊗|0i⊗|1i⊗|1i)
2
in the Hilbert space C16 is three-separable and thus biseparable.
Problem 47.
Are the vectors


1
 cos(π/4) 
v1 = 
,
cos(π/2)
cos(3π/4)

0
 sin(π/4) 
v2 = 

sin(π/2)
sin(3π/4)

entangled?
Problem 48.
Can the normalized vector in C16
1
(|0i⊗|0i⊗|0i⊗|0i+|0i⊗|1i⊗|0i⊗|1i+|1i⊗|0i⊗|1i⊗|0i+|1i⊗|1i⊗|1i⊗|1i)
2
be written as Kronecker product of lower dimensional vectors?
Problem 49. Let H be the finite dimensional Hilbert space Cd . Let Id
be the d × d identity matrix and A an arbitrary d × d matrix over C. We
Entanglement
97
call a vector |Ψi ∈ H ⊗ H maximally entangled, if it normalized, and its
reduced density matrix is maximally mixed, i.e., a multiple of Id
hΨ(A ⊗ Id )|Ψi =
1
tr(A).
d
(i) Let d = 2. Consider the normalized state
 
1
1 0
|Ψi = √   .
2 0
1
Calculate hΨ|(A ⊗ I2 )|Ψi and d1 tr(A).
(ii) Let d = 2. Consider the normalized state
 
1
1 1
|Ψi =   .
2 1
1
Calculate hΨ|(A ⊗ I2 )|Ψi and d1 tr(A).
Problem 50.
(i) Is the state in C4
 
1
1 0
|ψi = √  
2 1
0
entangled?
(ii) Is the state in C4

0
1 
|ψi = √
−i
2
0
entangled?


i
Chapter 11
Bell Inequality
Problem 1. Consider four observers: Alice (A), Bob (B), Charlie (C)
and Dora (D) each having one of the qubits. Every observer is allowed
to choose between two dichotomic observables. Denote the outcome of
observer X’s measurement by Xi (X = A, B, C, D) with i = 1, 2. Under
the assumption of local realism, each outcome can either take the value
+1 or −1. The correlations between the measurement outcomes of all four
observers can be represented by the product Ai Bj Ck Dl , where i, j, k, l =
1, 2. In a local realistic theory, the correlation function of the measurement
performed by all four observers is the average of Ai Bj Ck Dl over many runs
of the experiment
Q(Ai Bj Ck Dl ) := hψ|Ai Bj Ck Dl |ψi
The Mermin-Ardehali-Belinskii-Klyshko inequality is given by
Q(A1 B1 C1 D1 ) − Q(A1 B1 C1 D2 ) − Q(A1 B1 C2 D1 ) − Q(A1 B2 C1 D1 )
−Q(A2 B1 C1 D1 ) − Q(A1 B1 C2 D2 ) − Q(A1 B2 C1 D2 ) − Q(A2 B1 C1 D2 )
−Q(A1 B2 C2 D1 ) − Q(A2 B1 C2 D1 ) − Q(A2 B2 C1 D1 ) + Q(A2 B2 C2 D2 )
+Q(A2 B2 C2 D1 ) + Q(A2 B2 C1 D2 ) + Q(A2 B1 C2 D2 ) + Q(A1 B2 C2 D2 ) ≤ 4 .
Each observer X measures the spin of each qubit by projecting it either
X
along nX
1 or n2 . Every observer can independently choose between two
arbitrary directions. For a four qubit state |ψi, the correlation functions
are thus given by
B
C
D
Q(Ai Bj Ck Dl ) = hψ|(nA
i · σ) ⊗ (nj · σ) ⊗ (nk · σ) ⊗ (nl · σ)|ψi .
98
Bell Inequality
99
X
X
X
where · denotes the scalar product, i.e. nX
j · σ := nj1 σ1 + nj2 σ2 + nj3 σ3 .
Let
 
 
 
 
1
0
0
0
A
B
B







nA
=
0
,
n
=
0
,
n
=
1
,
n
=
0
1
2
1
2
1
0
1
0


 
 
 
−1
1
0
0
1
1
C
D
D







√
√
0
,
n
0 .
nC
=
1
,
n
=
0
,
n
=
=
1
2
1
2
2
2
1
1
0
1
Show that the Mermin-Ardehali-Belinskii-Klyshko inequality is violated for
the state
1
|ψi = √ (|0000i−|0011i−|0101i+|0110i+|1001i+|1010i+|1100i+|1111i)
2 2
where
|0i =
1
,
0
and |0000i ≡ |0i ⊗ |0i ⊗ |0i ⊗ |0i etc..
|1i =
0
.
1
Chapter 12
Quantum Channels
We consider the Hilbert space H of n × n matrices over C with the scalar
product (Frobenius inner product)
hA, Bi := tr(AB ∗ )
with A, B ∈ H. A state is described using n × n density matrices ρ, i.e.
tr(ρ) = 1 and ρ ≥ 0 (positive semidefinite). The space of trace-class operators acting in this Hilbert space is denoted by S(H). A quantum channel
from a Hilbert space HA to a Hilbert space HB is represented by a completely positive trace-preserving map Φ : S(HA ) → S(HB ). Such a positive trace-preserving map can be represented in Stinespring representation,
Kraus operator representation and Choi-Jamiolkowski representation.
Problem 1. Let Hn be the vector space of n × n hermitian matrices.
The adjoint (conjugate transpose) of a matrix A ∈ Cn×n is denoted by A∗ ,
Consider a family V1 , V2 , . . . , Vm of n × n matrices over C. We associate
with this family the completely positive map ψ : Hn → Hn defined by
ψ(X) =
m
X
Vj XVj∗ .
j=1
The map ψ is said to be a Kraus map if ψ(In ) = In , i.e.
m
X
Vj Vj∗ = In
j=1
100
Quantum Channels
101
and the matrices V1 , V2 , . . . , Vm are called Kraus operators.
Let m = n = 2 and
V1 =
0
0
1
0
, V2 =
0
1
0
0
.
Show that V1 and V2 are Kraus operators and find the associated Kraus
map.
Problem 2. Let ψ : Hn → Hn be a Kraus map. Thus ψ is linear. Show
that there exists Ψ ∈ Cn×n such that for all X ∈ Hn
vec(ψ(X)) = Ψ vec(X)
where 1 is an eigenvalue of Ψ. What is a corresponding eigenvector?
Problem 3. Find all Kraus maps ψ : H2 → H2 , associated with families
of 2 Kraus operators (V1 and V2 ), which provide the transformation
1 0
0 0
ψ
=
.
0 0
0 1
Calculate
ψ
0
0
0
1
.
Is there a Kraus map associated with a single Kraus operator which also
provides this transformation?
Problem 4. Let p ∈ [0, 1] and σ1 , σ2 , σ3 , σ0 = I2 be the Pauli spin
matrices.
(i) Show that the four 2 × 2 matrices
√
√
√
√
1 + 3p
1−p
1−p
1−p
σ0 , K1 =
σ1 , K2 =
σ2 , K3 =
σ3
K0 =
2
2
2
2
are Kraus operators.
(ii) Show that the sixteen 4 × 4 matrices
Kj ⊗ K` ,
j, ` = 0, 1, 2, 3
are Kraus operators, where ⊗ denotes the Kronecker product.
(iii) Show that the sixteen 4 × 4 matrices
Kj ? K ` ,
j, ` = 0, 1, 2, 3
102 Problems and Solutions
are Kraus operators, where ? denotes the star product.
Problem 5.
Let Kj (j = 1, . . . , m) be n × n matrices over C with
m
X
Kj Kj∗ = In .
j=1
Show that
m X
m
X
(Kj ⊗ K` )(Kj∗ ⊗ K`∗ ) = In ⊗ In ≡ In2 .
j=1 `=1
Problem 6.
Let Kj (j = 1, . . . , m) be 2 × 2 matrices over C with
m
X
Kj Kj∗ = I2 .
j=1
Show that
m
m X
X
(Kj ? K` )(Kj∗ ? K`∗ ) = I2 ⊗ I2 = I4 .
j=1 `=1
Problem 7. (i) Let A be an n × n matrix over C. Let G be a finite group
given by n × n matrices over C and g ∈ G. Consider the linear map
e=
A 7→ A
1 X
gAg −1
|G|
g∈G
where |G| denotes the number of elements in the finite group G. Show that
e
tr(A) = tr(A).
(ii) Is the determinant preserved under the linear map?
e positive semi-definite?
(iii) Let A be positive semi-definite. Is A
(iv) Apply it to the case of 4 × 4 matrices with
1
1 0
A=ρ= 
2 0
1

0
0
0
0
0
0
0
0

1
0

0
1
and the group is given by the 4×4 permutation matrices with |G| = 4! = 24.
Chapter 13
Miscellaneous
Problem 1. Let H0 and V be n × n hermitian matrices and ∈ R.
Consider the hermitian matrix H = H0 + V . Let
U (β) = e−β(H0 +V )
with β ≥ 0. Then
dU (β)
= −(H0 + V )e−β(H0 +V ) = −(H0 + V )U (β)
dβ
where U (β = 0) = In . Let
U (β) = e−βH0 W (β) .
(i) Show that W (β) is given by
Z β Z β1
Z
∞
X
k k
W (β) =
(−1) ···
k=0
0
0
where Ve (β) := eβH0 V e−βH0 .
(ii) Apply (i) to
1
H0 = ~ω
0
Problem 2.
βk−1
dβ1 dβ2 · · · dβk Ve (β1 )Ve (β2 ) · · · Ve (βk ) .
0
0
1
,
V =∆
0
1
1
0
.
Let H, A, B be hermitian matrices. Let
A(t) := eiHt Ae−iHt ,
B(s) := eiHs Be−iHs
103
104 Problems and Solutions
where s, t ∈ R. Show that
tr(A(t)B(s)e−βH ) = tr(eiH(t−s) Ae−iH(t−s) Be−βH ) .
ˆ
Problem 3. Let H(t)
be a given time-dependent hermitian Hamilton
ˆ
operator given as an n × n matrix. We assume that H(t)
depends smoothly
on t. Find the solution of the initial value problem of the matrix differential
equation
i ˆ
dU (t)
= − H(t)U
(t),
U (0) = In
dt
~
where In is the n×n identity matrix. Apply the ansatz (Magnus expansion)
U (t) = exp(Ω(t))
P∞
and Ω(t) = k=1 Ωk (t). Find the first two terms in the expansion, i.e. find
Ω1 (t) and Ω2 (t).
Problem 4.
Consider the Hamilton operator
ˆ = ~ωσ 1 · σ 2 ≡ ~ω(σ1 ⊗ σ1 + σ2 ⊗ σ2 + σ3 ⊗ σ3 ).
H
ˆ
Find the eigenvalues and normalized eigenvectors of H.
Problem 5. Let A, H be n × n hermitian matrices, where H plays the
role of the Hamilton operator. The Heisenberg equations of motion is given
by
dA(t)
i
= [H, A(t)].
dt
~
with A = A(t = 0) = A(0). Let Ej (j = 1, 2, . . . , n2 ) be an orthonormal
basis in the Hilbert space H of the n × n matrices with scalar product
hX, Y i := tr(XY ∗ ),
X, Y ∈ H.
Now A(t) can be expanded using this orthonormal basis as
2
A(t) =
n
X
cj (t)Ej
j=1
and H can be expanded as
2
H=
n
X
j=1
hj Ej .
Miscellaneous
105
Find the time evolution for the coefficients cj (t), i.e. dcj /dt, where j =
1, 2, . . . , n2 .
Consider the standard basis in the Hilbert space C9
Problem 6.
|00i,
|01i,
|02i,
|10i,
|11i,
|11i,
|12i,
|20i,
|21i,
|22i
where |00i ≡ |0i ⊗ |0i, and |0i, |1i, |2i is the standard basis in C3 . Show
that the normalized states
2
1 X 2πijn/3
|ψinm = √
e
|ji ⊗ |(j + m) mod 3i
3 j=0
i.e.
1
|ψi00 = √ (|00i + |11i + |22i)
3
1
|ψi10 = √ (|00i + e2πi/3 |11i + e4πi/3 |22i)
3
1
|ψi20 = √ (|00i + e4πi/3 |11i + e2πi/3 |22i)
3
1
|ψi01 = √ (|01i + |12i + |20i)
3
1
|ψi11 = √ (|01i + e2πi/3 |12i + e4πi/3 |20i)
3
1
|ψi21 = √ (|01i + e4πi/3 |12i + e2πi/3 |20i)
3
1
|ψi02 = √ (|02i + |10i + |21i)
3
1
|ψi12 = √ (|02i + e2πi/3 |10i + e4πi/3 |21i)
3
1
|ψi22 = √ (|02i + e4πi/3 |10i + e2πi/3 |21i)
3
form an orthonormal basis in the Hilbert space C9 .
Problem 7.
Consider the state
0
0
0
0
|ψi = −E1 E2 ei(k1 ·r+k2 ·r )−i(ω1 t+ω2 t ) + ei(k2 ·r+k1 ·r )−i(ω2 t+ω1 t ) |0i ⊗ |0i.
Find
w ∝ hψ|ψi.
106 Problems and Solutions
Problem 8. Let Aˆ be a nonzero bounded linear operator in a Hilbert
space H. Let |ni, |mi be normalized states in the Hilbert space H. We
define
hm|AˆAˆ† |ni
ˆ := q
q
.
S(|mi, |ni, A)
hm|AˆAˆ† |mi hn|AˆAˆ† |ni
Consider the Hilbert space C2 . Calculate S for Aˆ = σ1 and the normalized
states
1
1
1
|ui =
,
|vi = √
.
0
2 −1
Problem 9.
Consider the state |ψi
|ψi =
1
X
cj0 ,j1 ,...,jN −1 |j0 i ⊗ |j1 i ⊗ · · · ⊗ |jN −1 i
j0 ,j1 ,...,jn =0
n
in the Hilbert space C2 . The bitstring j0 j1 . . . jN −1 can be mapped oneto-one into a non-negative integer j
j=
N
−1
X
jk 2k
k=0
where jk ∈ {0, 1}. Thus we can write the state as
|ψi =
N −1
2X
cj |ji.
j=0
We can associate a polynomial with the state |ψi via
p(|ψi, x) =
N −1
2X
cj x j .
j=0
(i) Consider the Bell state (N = 2)
1
|ψi = √ (|0i ⊗ |0i + |1i ⊗ |1i).
2
Find the polynomial of |ψi and calculate the roots.
(ii) Consider the state (N = 2)
|φi =
1
(|0i ⊗ |0i + |1i ⊗ |0i − |0i ⊗ |1i − |1i ⊗ |1i).
2
Miscellaneous
107
Find the polynomial of |φi and calculate the roots.
Problem 10. Let A, B be observable, i.e. hermitian matrices. Then the
uncertainty relation is given by
∆2 A · ∆2 B ≥
1
|h[A, B]i|2 + cov(A, B)
4
where [ , ] denotes the commutator,
cov(A, B) :=
1
(hABi + hBAi) − hAihBi
2
and
∆2 A := cov(A, A).
This inequality can generalized to 2n observable A1 , A2 , . . . , A2n . We have
det(Σ) ≥ det(C)
where
Σk` = cov(Ak , A` ),
Let
A = σ1 ,
B = σ2 ,
i
Ck` = − h[Ak , A` ]i.
2
1
|ψi = √
2
1
−1
.
Find the left-hand side and right-hand side of the inequality.
Problem 11.
The most general real three-qubit state can be written as
|ψi = −c3 cos2 θ|0i ⊗ |0i ⊗ |1i − c2 |0i ⊗ |1i ⊗ |0i + c3 sin(θ) cos(θ)|0i ⊗ |1i ⊗ |1i
−c1 |1i ⊗ |0i ⊗ |0i − c3 sin(θ) cos(θ)|1i ⊗ |0i ⊗ |1i + (c0 + c3 sin2 (θ))|1i ⊗ |1i ⊗ |1i
where c0 , c1 , c2 , c3 , θ are real parameters. Classify the state with respect to
entanglement.
Problem 12. Let A, B be n × n matrices acting in the Hilbert space Cn .
Then A, B can be considered as observable. The two overvable A and B
are called complementary if their eigenvalues are non-degenerate and any
two normalized eigenvectors aj of A and bj of B satisfy
1
|a∗j bk | = √
n
where ∗ means transpose and conjugate complex. Give an example for such
hermitian matrices in C2 .
108 Problems and Solutions
Problem 13.
Two orthonormal bases
{ uj : j = 1, 2, . . . , n },
{ vk : k = 1, 2, . . . , n }
in the Hilbert space Cn are called mutually unbiased if
1
u∗j vk = √
n
j, k ∈ { 1, 2, . . . , n }.
for all
(i) Give an example in C2 .
(ii) Give an example in C4 .
Problem 14. Solve the initial value problem of the optical Bloch equations.
dρ22
b
dρ11
=−
= i (e−i(ω−α)t ρ12 − ei(ω−α)t ρ21 )
dt
dt
2
dρ12
dρ∗
b
= 21 = i ei(ω−α)t (ρ11 − ρ22 )
dt
dt
2
with the initial conditions
ρ11 (0) = ρ12 (0) = ρ21 (0) = 0,
ρ22 (0) = 1.
ˆ be a
Problem 15. Consider a finite dimensional Hilbert space. Let H
hermitian Hamilton operator. Let |ψi be the normalized ground state of
ˆ
the system, i.e. H|ψi
= E0 |ψi. Let |φi be another normalized state. Let
Fˆ be a positive semidefinite operator and hφ|Fˆ |φi > 0. Then we have the
inequality
hψ|Fˆ |ψi ≥
(hφ|ψihφ|Fˆ |φi − (∆Fˆ )(1 − hφ|ψi2 )1/2 )2
hφ|Fˆ |φi
where
(∆F )2 := hφ|Fˆ 2 |φi − hφ|Fˆ |φi2 .
The inequality follows from the non-negativity of the Gramian determinant
of the vectors |ψi, |φi, and Fˆ |φi. Consider the Hilbert space C2 and the
Hamilton operator
0 1
ˆ
H = ~ω
1 0
the positive semidefinite operator
Fˆ =
1
1
1
1
Miscellaneous
109
and the normalized states
1
|ψi = √
2
1
−1
,
1
|φi =
0
ˆ
with H|ψi
= E0 |ψi and E0 = −~ω. Apply the inequality to these operators
and states, i.e. calculate the left and right-hand side of the inequality.
Consider the normalized states in the Hilbert space C3
 
 


1
0
1
1  
1
|ψ1 i = √
0 , |ψ2 i =  1  , |ψ3 i = √  0  .
2 1
2 −1
0
Problem 16.
Find the unitary matrices U12 , U23 , U31 such that
|ψ2 i = U12 |ψ1 i,
Problem 17.
|ψ3 i = U23 |ψ2 i,
|ψ1 i = U31 |ψ3 i.
Consider the Hadamard matrix
1
1 1
.
UH = √
2 1 −1
Is UH ∈ SU (2)? Is iUH ∈ SU (2)?
Problem 18. The available uncertainty relations in finite-dimensional
Hilbert spaces are those of Robertson and Schr¨odinger. Let ρ be the state
of the quantum system (density matrix), i.e. a positive semi-definite, selfadjoint linear operator with tr(ρ) = 1. The mean value functional is
h · i := tr(ρ·).
Then for two self-adjoint operators, A and B, the variance is defined by
(∆A)2 := hA2 i − hAi2 ,
(∆B)2 := hB 2 i − hBi2 .
We have the inequalities
(∆A)(∆B) ≥ |hABi − hAihBi| ≥
1
|h[A, B]i|.
2
Note that we have the identity
|hABi|2 =
1
1
1
|h[A, B]+ i + h[A, B]i|2 = |h[A, B]+ i|2 + |h[A, B]i|2 .
4
4
4
110 Problems and Solutions
(i) Let
A = σ1 ,
B = σ2 ,
ρ=
1/2
0
0
1/2
.
Calculate the left-hand side of the inequalty and the right-hand sides of the
inequality. Discuss.
(ii) Let
1/4
 0
ρ=
0
0

A = σ1 ⊗ σ1 ,
B = σ2 ⊗ σ2 ,
0
0
1/4 0
0 1/4
0
0

0
0 
.
0
1/4
Calculate the left-hand side of the inequalty and the right-hand sides of the
inequality. Discuss.
Problem 19. A spin- 21 system in a time-dependent magnetic fields S(t)
is described by the Hamilton operator
1
1
ˆ
H(t)
= ~ωσ3 + ~γS(t)σ1
2
2
where σ1 and σ3 are the Pauli matrices
0 1
1
σ1 =
,
σ3 =
1 0
0
0
−1
.
Then the Schr¨
odinger equation
i~
∂ψ
ˆ
= H(t)ψ
∂t
for the spinor ψ = (ψ1 , ψ2 )T takes the form
i
dψ1
1
1
= − ωψ1 + γS(t)ψ2 ,
dt
2
2
i
dψ2
1
1
= ωψ2 + γS(t)ψ1 .
dt
2
2
Rewrite this system in terms of the observable Bloch variables
A(t) := |ψ2 |2 − |ψ1 |2 ,
Problem 20.
B(t) := i(ψ2 ψ1∗ − ψ1 ψ2∗ ),
C(t) := ψ2 ψ1∗ + ψ1 ψ2∗ .
Consider the operators
H1 = 1,
H2 = x,
H3 =
∂2
,
∂x2
H4 = i
∂
.
∂x
(i) Show that we have a nilpotent Lie algebra under the commutator.
Miscellaneous
111
(ii) Let
α1 (t) = cf (t),
α2 (t) = c,
1
α3 (t) = − ,
2
α4 (t) =
df
.
dt
Consider the Hamilton operator
K=
4
X
αj (t)Hj
j=1
and the Schr¨
odinger equation
i
∂ψ
= Kψ.
∂t
We write the solution of the Schr¨odinger equation in the form
ψ(x, t) = U (t, 0)ψ(x, 0)
where the unitary time evolution operator is give by
U (t, 0) = exp(β1 (t)H1 ) exp(β2 (t)H2 ) exp(β3 (t)H3 ) exp(β4 (t)H4 ).
Find the system of ordinary differential equations for βj (t) (j = 1, 2, 3, 4)
and solve them.
Problem 21.
Consider the spin Hamilton operator
ˆ = ∆1 σ1 ⊗ σ1 + ∆2 σ2 ⊗ σ2 + ∆3 σ3 ⊗ σ3 .
H
ˆ = β H.
ˆ The partition function is
Let K
Z(β) = tr exp(−K).
The logarithm of the partition function is given by the cumulant expansion
ln(Z(β)) = ln tr(I) − hKi +
1
(hK 2 i − hKi2 )
2!
1
(hK 3 i − 3hK 2 ihKi + 2hKi3 )
3!
1
+ (hK 4 i − 4hK 3 ihKi − 3hK 2 i2 + 12hK 2 ihKi2 − 6hKi4 )
4!
−···
−
Here I is the identity operator given by In ⊗ In with n = 2 and
h· · ·i :=
tr(· · ·)
.
trI
112 Problems and Solutions
Calculate the function ln(Z(β)) up to this order.
Problem 22. Let |ψi, |φi be normalized states in the Hilbert space Cn .
Let K be a positive semi-definite matrix in Cn . Show that


1
hφ|ψi
hφ|K|ψi
G := det  hφ|ψi
1
hφ|K|φi  ≥ 0.
hφ|K|ψi hφ|K|φi hφ|K 2 |φi
G is called the Gramian. Apply it to the Hilbert space C2 and
1
1
1
1 1
|ψi =
, |φi = √
, K=
.
1 1
0
2 1
Problem 23. Let C = (cjk ) (j, k = 1, . . . , n) be an n × n matrix with
real entries. Then C is called a quantum correlation matrix if there are
self-adjoint operators Aj , Bk (j, k = 1, . . . , n) on a Hilbert space H with
kAj k ≤ 1, kBk k ≤ 1 and u in the unit sphere of H ⊗ H such that
cjk = h(Aj ⊗ Bk )u, ui
where h , i denotes the scalar product. If the self-adjoint operators Aj ,
Bk (j, k = 1, . . . , n) commute the matrix C is called a classical correlation
matrix. Consider the case with n = 3, the Hilbert space C2 , A1 = B1 = σ1 ,
A2 = B2 = σ2 , A3 = B3 = σ3 and the Bell state
1
uT = √ ( 1
2
0
0
1).
Find the correlation matrix C. What is the significance of this matrix?
Miscellaneous
113
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Index
W -state, 85
Optical Bloch equations, 108
Aharonov state, 84
Partition function, 14, 111
Pauli group, 70, 92
Phase operator, 89
Bell matrix, 69, 74
Bell states, 44
Bloch variables, 110
Cayley transform, 49
Clifford group, 70
Complementary, 107
Determinant, 61
Dicke states, 18
Entropic uncertainty relation, 78
Exceptional points, 6
Feynman gate, 63
Fredkin gate, 62, 63
Generalized Fredkin gate, 63
Generalized Toffoli gate, 62
GHZ-state, 13, 85
Gramian, 112
Gramian determinant, 108
Quantum correlation function, 41
Quantum correlation matrix, 112
Remoteness, 93
Schr¨odinger equation, 69
Shannon information entropy, 78
Spin coherent state, 33
Three tangle, 85
Three-tangle, 89
Toffoli gate, 62
Uncertainty relation, 107
Uncertainty relations, 109
Variance, 3, 4, 93
W-state, 13
Werner state, 52
Half-adder, 63
Hyperdeterminat, 85
Kraus map, 100
Kraus operators, 101
Magnus expansion, 104
Mutually unbiased, 25
Mutually unbiased bases, 9
119