MMath Paper, Canlin Zhang

THE OPERATOR THEORY BASIS OF
QUANTUM COMPUTING
by
CANLIN ZHANG
A research essay
presented to the University of Waterloo
in fulﬁlment of the
research essay requirement for the degree of
Master of Mathematics
in
Pure Mathematics
Waterloo, Ontario, Canada, 2014
c CANLIN ZHANG 2014

AUTHOR’S DECLARATION
I hereby declare that I am the sole author of this research essay. This is
a true copy of the research essay, including any required ﬁnal revisions, as
accepted by my examiners.
I understand that my research essay may be made electronically available to
the public.
CANLIN ZHANG
ii

Abstract
This paper will introduce some operator theory which has connec-
tions with quantum computation and quantum information. We ﬁrst
introduce some basic ideas and notations in operator theory. Then
we will discuss quantum algorithms. We mainly focus on quantum
channels, which are also called completely positive trace preserving
maps. After that, we will outline the main theorems for quantum er-
ror detection and correction, which is also the most interesting part of
this paper. Finally, we conclude with a discussion of a special passive
method in quantum error detection.
iii

Acknowledgements
I am grateful so much to Professor Kenneth Davidson, my supervisor, for
his patient instruction and farsighted enlightenment. I would not complete
my paper without the help from him. I am very grateful to Professor David
Kribs. This paper is mainly based on [7], which is a paper of Kribs. I am
also very grateful to Professor Laurent Marcoux for his helpful suggestion
and guiding. Moreover, I am also grateful to Cameron Williams and Michael
Hartz for their help on Latex. Thanks also to members of Department of
Pure Mathematics at University of Waterloo for kind supporting during the
preparation of this paper.
iv

Contents
1 Introduction 1
2 Quantum Computing Basics 2
3 Quantum Algorithms 5
4 Quantum Channels 9
5 Quantum Error Correction 25
6 Noiseless Subsystems via The Noise Commutant 34
v

1 Introduction
In the past few decades, researchers have being devoting themselves to es-
tablishing the theoretical basis for quantum computation and quantum in-
formation. Although many important results have been found and proved,
there are still more problems, theoretical and experimental, that need to be
overcome. And for those theoretical ones, many are related to deep mathe-
matical problems. This paper mainly tries to provide a basic idea of quantum
computing for researchers interested in operator theory or operator algebras.
But we note that in order to understand this paper, a reader only needs to
have knowledge of linear algebra and basic functional analysis.
Following standard practice in quantum computing, we use the physical
notation for operators on a Hilbert space.
This paper is organized as follows:
Section 1 is the background introduction.
Section 2 mainly introduce the basic concepts, terms and notations in the
ﬁeld of quantum computing;
Section 3 will provide the basic ideas in quantum algorithms by describ-
ing Deutsch’s algorithm [1, 2], which is an elementary example of quantum
algorithm, to show the power of quantum computation;
Section 4 primarily provides the mathematical basis required for the de-
scription of evolution within a quantum system. This section would mostly
focus on quantum channels, or namely the completely positive and trace pre-
serving maps [8] in pure mathematics.
Section 5 would be the most interesting part of this paper. Although, this
paper only provides basic ideas, we will provide a quite detailed discussion of
quantum error correction methods in section 5. The fundamental theorems
for quantum error detection and correction would be presented in the ‘stan-
dard model’;
In section 6 (the last section), we would describe the ‘noiseless subsystem
via noise commutant’ [3], which is a very basic and simple method of quan-
tum error prevention.
1

2 Quantum Computing Basics
The properties of operators on Hilbert space is a central issue in the study
of mathematical basis for quantum information and quantum computing.
Let H be a (complex) Hilbert space. For vectors and vector duals in H,
we use the Dirac notation: A typical vector in H will be denoted as a ‘ket’
|ψ , and the linear functional on H determined by this |ψ will be denoted as
a ‘bra’ ψ|. (Here the “linear function determined by ψ|” means the inner
product produced by ψ| and other vectors.)
Notice that a bra and a ket yield a inner product ψ1||ψ2 , while a ket and
a bra yield a rank one operator |ψ2 ψ1|. Particularly, for a unit vector |ψ ,
the rank one projection from H to the subspace {λ|ψ : λ ∈ C} is written as
|ψ ψ|. Let B(H) denote the set of bounded operators (or equivalently, the
set of continuous operators) acting on H. We use the physics symbol U†
to
represent the adjoint of the operator U.
Now, we are ready to show the postulate of quantum mechanics:
(i) Typically, a quantum system can be regarded as a portion of the
physical universe chosen for analysis. And the portion outside our chosen
quantum system is called the environment. A quantum system is called
closed if it has no mass or energy exchange with the environment.
(ii) We use a Hilbert space H to represent a closed quantum system, and
use a mapping |ψ ψ| (where |ψ is a unit vector) to represent a certain state
of the quantum system at a precise moment of time. Also, we usually only
write |ψ rather than |ψ ψ| for simplicity. However, the state of a quantum
system is usually not certain. In this case, we use a probability density func-
tion to describe the state space of the quantum system. In quantum physics,
this density function is a positive operator ρ on H with trace equals to one.
It is called a density operator.
(iii) We use a unitary transformation to describe an evolution in a closed
quantum system. This means: Let the initial state of the quantum system
be ρ. Then after an evolution, the state would become UρU†
.
(iv) We use ﬁnitely many operators to describe a measurement of a quan-
tum system. That is, a set of operators Mk, 1 ≤ k ≤ r, such that
r
k=1
M†
kMk = 1,
2

where 1 is the identity operator.
We call a measurement projective if all the Mk are projections. Then the
Mk have mutually orthogonal ranges. (Mk is a projection implies M†
k = Mk.
Then r
k=1 M†
kMk = r
k=1 M2
k = r
k=1 Mk. So, r
k=1 Mk = 1. Then ob-
viously their ranges should be mutually orthogonal.) Moreover, we call a
positive measurement classical when each of the Mk is rank one. In ex-
periments, if the initial state of a quantum system is certain (for example
|ψ ), then the probability of the occurrence of the evolution Mk|ψ ψ|M†
k is
p(k) = ψ|M†
kMk|ψ (See [5], page 13-16). Notice that:
k
p(k) =
k
ψ|M†
kMk|ψ = ψ|(
k
M†
kMk)|ψ = ψ||ψ = 1.
So p(k) indeed deﬁnes a probability distribution.
(v) Let H1 , ... , Hm be the Hilbert spaces associated with m quantum
systems. We call the quantum system associated with the Hilbert space
H1 ⊗ ... ⊗ Hm the composite quantum system.
If H1 and H2 have dimensions n1 and n2 respectively, the dimension of
H1 ⊗ H2 will then be n1n2. Note that a typical vector in H1 ⊗ H2 is a linear
combination of the form
i
hi ⊗ ki. If {e1, · · · , en1 } is an orthogonalized
basis of H1 and {f1, · · · , fn2 } is an orthogonalized basis of H2, then {ei ⊗fj :
1 ≤ i ≤ n1, 1 ≤ j ≤ n2} is an orthogonalized basis of H1 ⊗ H2. Also, the
composite inner product is:
h1 ⊗ k1, h2 ⊗ k2 = h1, h2 k1, k2 ,
for any h1 ⊗ k1, h2 ⊗ k2 ∈ H1 ⊗ H2.
So, given the (pure) state |ψi of Hi, we obtain the (pure) state
m
i=1
⊗|ψi
of the composite system.
These concepts are basic in quantum mechanics. After introducing them,
we are ready to show some basic concepts and conventions in quantum com-
putation.
The most frequently discussed Hilbert spaces in quantum computation
and quantum information are those with dimension N = dn
for any positive
integers n ≥ 1 and d ≥ 2. For simplicity, we always suppose d = 2 in this
3

paper. So, we write the Hilbert space as HN = H2n = C2
⊗...⊗C2
= (C2
)⊗n
,
which is a n-fold tensor product. We drop the N when convenient.
Now consider H2 = C2
: let {|0 , |1 } be an orthonormal basis for H2.
This is a typical representation of the classical base in many two level quan-
tum systems, such as the ground and excited states of an electron captured
by an atomic nucleus, the ‘spin-up’ and ‘spin-down’ of an electron, and the
two polarizations of a photon of light. To describe the standard orthonormal
basis of H2n = (C2
)⊗n
C2n
, we use the abbreviated form in quantum me-
chanics. For example, the standard orthonormal basis of H4 shall be written
as {|ij : i, j ∈ Z2} or {|i |j : i, j ∈ Z2}, where |ij = |i |j = |i ⊗ |j is
the tensor product of two vectors. In other words, the basis of H4 can be
written as {|00 , |01 , |10 , |11 } or {|0 |0 , |0 |1 , |1 |0 , |1 |1 }.
In quantum information, we call a vector |ψ in H2 a quantum bit, or a
‘qubit’. Notice that any vector |ψ in H2 can be written as |ψ = a|0 + b|1 .
The vector |ψ is called a superposition of the classical states |0 and |1 , if
both a and b are not zero. A ‘qudit’ is a unit vector in Cd
. In the composite
space HN , a vector (state) |ψ is called entangled if it cannot be written as
a tensor product of vectors from the component systems. For instance, the
‘EPR pairs’ |ψ = |00 +|11
√
2
in H4 is an entangled vector.
In the following part, we do not distinguish a quantum system from its
corresponding Hilbert space. We would directly say “ the quantum system
HN ”.
For an evolution on a quantum system HN , let U be the corresponding
unitary operator. Since HN is ﬁnite dimensional, U is actually a unitary
matrix. Then, decoherence can be regarded as the process to vanish the
oﬀ-diagonal entries of U.
The following discussions are about several classical unitary matrices
(evolutions) on a quantum system HN .
In the 1-qubit case (N = 21
), the Pauli matrices are given by:
X =
0 1
1 0
, Y =
0 −i
i 0
, Z =
1 0
0 −1
.
Let 12 be the 2 × 2 identity matrix. These matrices can be regarded as
operators acting on H2 under the basis {|0 , |1 }. And in the n-qubit case
(N = 2n
), the set of ‘single qubit unitary gates’ generated by the Pauli
matrices is {Xk, Yk, Zk : 1 ≤ k ≤ n}, where Xk = 1
⊗(k−1)
2 ⊗ X ⊗ 1
⊗(n−k)
2 and
Yk, Zk are similar.
4

Let UCN be an operator on H4 such that UCN (|ij ) = |i |(i+j) mod 2 .
Then it is easy to see that UCN is unitary. We call this UCN the ‘controlled-
NOT gate’, or CNOT gate for short. The CNOT gate has natural extensions
{U
(k,l)
CN : 1 ≤ k = l ≤ n} to unitary operators (gates) on HN : If N = 2n
,
let the basis of HN be {|i1 |i2 ...|in : ik = 0 or 1 , k = 1, 2...n}. Then
we have: U
(k,l)
CN : |i1 ...|in −→ |i1 ...|ik ...|(ik + il) mod 2 ...|in . Note that
UCN = U
(1,2)
CN .
Note that all the N × N unitary matrices form a group under matrix
multiplication. Denote this group as U(N). Then, it is easy to see that
{Xk, Yk, Zk, U
(k,l)
CN : 1 ≤ k = l ≤ n} generates U(N).
At the end of this section, we introduce the Hadamard gate H and the
spin-1
2
Pauli matrices σk on H2:
H =
1
√
2
1 1
1 −1
,
and:
σk =
1
2
K , for k = x, y, z and K = X, Y, Z ,
where X, Y, Z are the Pauli matrices.
The Hadamard gate was first used by J. Hadamard in the year 1893,
while the Pauli matrices and Pauli-1
2
spin matrices were first studied by W.
Pauli in 1932. These operators were widely used in the research of quantum
information.
3 Quantum Algorithms
Generally speaking, a quantum algorithm is a collection of initial states ρ to-
gether with their evolutions UρU†
under a unitary matrix U. Many quantum
algorithms, such as the factoring algorithm of Shor [9, 10] and the search al-
gorithm of Grover [4], have being taking into useage recently. In this section,
we will first introduce the Deutsch algorithm (or Deutsch-Josza algorithm)
[1, 2], which is a simple but useful quantum algorithms. After that, we
will provide some examples to show how powerful the quantum computation
could be comparing with classical computations.
Before continuing, we shall use a simple example to show how the basic
operations (such as addition) can be performed by a quantum algorithm: Let
5

dim HN = 2n
. Then we can see that there is a natural bijection between the
basis vectors |i1 ... in and integers 0, 1, ..., 2n
− 1 (the basis has 2n
elements).
Then define x ⊕ y = x + y mod N, and define the unitary operator U on
HN ⊗ HN by U|x |y = |x |x ⊕ y . Then the corresponding quantum algo-
rithm would be associated with an addition operation modulo N. (Note that
when N = 2, U is just the CNOT gate.)
Note that the tensor product of matrix is: Let H =
a b
c d
, then
H⊗n
=
aH⊗n−1
bH⊗n−1
cH⊗n−1
dH⊗n−1 .
Now we are able to discuss the Deutsch algorithm. Let Hn = H⊗n
be the
n-fold tensor product of the Hadamard gate on HN . We have:
Hn|0 ⊗n
=
1
√
2n
2n−1
x=0
|x .
Fix positive integers k, m ≥ 1. Let Hm,k = H2m ⊗ H2k . Then Hm,k has
the basis |x |y = |x ⊗ |y , where x ∈ Zm
2 and y ∈ Zk
2 . Let f: Zm
2 → Zk
2 be
any function. Then, define Uf ∈ B(Hm,k) via:
Uf : |x |y −→ |x |y ⊕ f(x) .
Then it is easy to see that Uf is an unitary operator. Also we can see that
U permutes |y in the basis vectors {|x |y : y ∈ Zk
2} .
Note 3.1. Notice that for any x ∈ Zm
2 , we have Uf (|x |0 ) = |x |f(x) . This
is how Uf simulates the function f on a quantum computer. Therefore, any
classical function can be simulated in this way on a quantum computer.
Then we have:
Uf ((Hm|0 ⊗m
) ⊗ |0 ⊗k
) = Uf (
1
√
2m
2m−1
x=0
|x ⊗ |0 )
=
1
√
2m
2m−1
x=0
Uf (|x ⊗ |0 ) =
1
√
2m
2m−1
x=0
|x ⊗ |f(x)
This means Uf acting on Hm ⊗ 12k would yield a simultaneous compu-
tation (parallel computation) of f on every possible value of x. This is the
6

so-called quantum parallelism. We put its diagram (which is called a ‘circuit-
gate’) here:
|0 ⊗k
|0 ⊗m
Hm
Uf
1√
2m
2m−1
x=0 |x ⊗ |f(x)
In the diagram, the states of every component systems ( |0 ⊗m
and |0 ⊗k
)
are called ‘circuits’ and are drawn at the left hand side. The unitary opera-
tors corresponding to the evolutions of every component systems are called
‘gates’ and are drawn as the boxes in the middle. The result at the right
hand side is the final state of the composite system.
Note 3.2. Comparing with a classical computer, one of the most important
advantages of a quantum computer is the simultaneous computing ability,
which is reflected perfectly by the quantum parallelism in the above example
of the Deutsch algorithm. Hence, we can understand why the Deutsch algo-
rithm can become very powerful on a quantum computer. Moreover, Deutsch
algorithm is widely used in the research of quantum computation since it is
very simple and hence efficient.
We conclude this section by introducing the Deutsch-Josza generalization
[2].
Let f: Zm
2 → Z2 be any function. Then f is called constant if f(x) = f(y)
for any x, y ∈ Zm
2 , and f is called balanced if |f−1
(0)| = |f−1
(1)| = 2m−1
.
Suppose f is either constant or balanced, and we wish to know which situ-
ation f is in. If we work on a classical computer, we have to test 2m−1
+ 1
values to know for sure the situation of f, which means 2m−1
+1 steps of eval-
uation need to be done. On the other hand, the Deutsch-Josza generalization
allowed us to get the result on a quantum computer with a single operation
(algorithm), which contains only four steps of evaluation. This shows that
a quantum computer can have tremendous advantage in calculating if the
algorithms are suitable.
Here is the diagram for this algorithm:
7

|1
|0 ⊗m
Hm
Uf g
Hm
H
The initial state is |0 ⊗m
⊗ |1 on H2m ⊗ H2, where Hm is the tensor
products of Hadamard gate. Let |+ = |0 +|1
√
2
and |− = |0 −|1
√
2
, and let
g = 0|⊗m
⊗ −| be a linear functional on H2m ⊗ H2.
The ﬁrst stage of the algorithm is:
(Hm ⊗ H)(|0 ⊗m
⊗ |1 ) = (Hm|0 ⊗m
) ⊗ (H|1 )
= (
1
√
2m
2m−1
x=0
|x ) ⊗
|0 − |1
√
2
= S ⊗ |−
Recall that Uf (|x |y ) = |x |y ⊕ f(x) for any |x ∈ Zm
2 and |y ∈ Z2.
Then the second stage is:
Uf (S ⊗ |− ) = Uf (S ⊗
|0 − |1
√
2
)
=
1
√
2
Uf (S ⊗ |0 − S ⊗ |1 )
=
1
√
2m+1
2m−1
x=0 Uf (|x ⊗ |0 ) − 2m−1
x=0 Uf (|x ⊗ |1 )
=
1
√
2m+1
2m−1
x=0 (|x ⊗ |f(x) ) − 2m−1
x=0 (|x ⊗ |1 ⊕ f(x) )
= ±
1
√
2m
2m−1
x=0 (−1)f(x)
|x ⊗ |−
We can see that the ﬁrst row of Hm is (1, 1...1), while each of the other
rows consists of 2m−1
“1” and 2m−1
“−1”.
If f is constant, then we have:
Hm
2m−1
x=0 (−1)f(x)
|x = ±Hm
2m−1
x=0 |x = ±|0 ⊗m
.
8

If f is balanced, then the first row of Hm acting on 2m−1
x=0 (−1)f(x)
|x
yields zero. Then we have:
Hm
2m−1
x=0 (−1)f(x)
|x = 2m−1
x=1 kx|x ,
where kx is of the form 2m−2P
2m , for P in {0, 1...2m−1
}.
Hence, after passing the second last gate (Hm ⊗ 12), the state becomes:
(Hm ⊗ 12) 2m−1
x=0 (−1)f(x)
|x ⊗ |−
=
±|0 ⊗m
⊗ |− , if f is constant;
( k2|0...1 ⊗m
+ ... + k2m |1...1 ⊗m
) ⊗ |− , if f is balanced;
Then, after the final gate, we get the resulting state to be:
1 , if f is constant;
0 , if f is balanced;
Notice that the result contains no uncertainty. It would be a certain
number rather than a probability density function: If we get 1 (0), then we
know the probability for f to be constant (balanced) is 1.
Note 3.3. Readers who want to acquire more knowledge about quantum
gate may have a look at chapters six, nine and ten of [5].
4 Quantum Channels
A quantum system is called open if it has mass or energy exchange with the
environment. Mathematically, an open quantum system is represented by a
subset in a larger Hilbert space (or equivalently, a subset in a larger closed
quantum system). And quantum channels are often applied to deal with
open quantum systems.
Before continuing, we shall introduce some basic theories in operator the-
ory:
(i) Let H(k)
= H ⊕ H... ⊕ H. Then there is a natural way to define the
9

norm and inner product on H(k)
in order to make it a Hilbert space. Namely,



h1
...
hk



2
= h1
2
+ · · · + hk
2
and 


h1
...
hk


 ,



l1
...
lk



H(k)
= h1, l1 H + · · · + hk, lk H
where 


h1
...
hk


 and



l1
...
lk



belong to H(k)
.
Let Mk(B(H)) denote the “tensor” of B(H), which is the set of k ×
k matrices with entries from B(H). Let (Tij) denote a typical element of
Mk(B(H)). Deﬁne:
(Tij)



h1
...
hk


 =



k
j=1 T1jhj
...
k
j=1 Tkjhj



Then Tij becomes an operator on H(k)
. It is straightforward to verify that
Mk(B(H)) = B(H(k)
).
(ii) If A is a C*-algebra, then Mk(A) is a C*-algebra as well. And since
every C*-algebra is isometrically ∗-isomorphic to a concrete C*-algebra, all
the theories of the “tensor” of B(H) work for a general C*-algebra A as well.
(iii) Let A, B be two C*-algebras and let E be a linear map from A to B.
Then:
E is called positive if for any a ∈ A, a ≥ 0 implies E(a) ≥ 0.
E is called k-positive if for the integer k (k ≥ 2), the ‘ampliﬁcation
10

map’:
E(k)
: Mk(A) → Mk(B)
defined by E(k)
((aij)) = (E(aij)) is a positive map.
E is called completely positive if E(k)
is a positive map for any integer
k.
Completely positivity is a very strong condition. But W. Forrest Stine-
spring proved that every completely positive map can be regarded as a ‘com-
pression’ of a ∗-homomorphism on a larger Hilbert space containing the for-
mer space. We now give the description and proof of this result according to
[8], page 43-45:
Theorem 4.2. (Stinespring’s dilation theorem). Let A be a unital C*-
algebra, and let φ: A → B(H) be a completely positive map. Then there
exists a Hilbert space K, a unital ∗-homomorphism π: A → B(K), and a
bounded operator V : H → K with φ(1) = V 2
such that
φ(a) = V †
π(a)V .
Proof. Let A⊗ H denote the algebraic tensor product of A and H. Then for
any a ⊗ x, b ⊗ y in H, we define a symmetric product , by
a ⊗ x, b ⊗ y = φ(b†
a)x, y H
and make this product to be bilinear, where , H is the inner product on H.
Define an inner product on the direct sum H(n)
= H ⊕ · · · ⊕ H by



x1
...
xn


 ,



y1
...
yn



H(n)
= x1, y1 H + · · · + xn, yn H ,
and by this definition we can easily prove that H(n)
is still a Hilbert space.
Then, since φ is completely positive, we have that
n
j=1
aj ⊗ xj,
n
i=1
ai ⊗ xi = φn((a†
i aj))



x1
...
xn


 ,



x1
...
xn



H(n)
≥ 0 ,
which means , is positive semidefinite on A ⊗ H.
11

Hence, because of the bilinearity and the positive semidefinite property,
, satisfies the Cauchy-Schwarz inequality: | u, v |2
≤ u, u · v, v . So we
have that
{u ∈ A ⊗ H| u, u = 0} = {u ∈ A ⊗ H| u, v = 0 for any v ∈ A ⊗ H}
is a subspace of A ⊗ H, which we denote as N. Also, the induced bilinear
form on the quotient space A⊗H/N defined by u+N, v +N = u, v will
be an inner product.
Define K to be the completion of the inner product space A⊗H/N. Then
K is a Hilbert space.
For any a ∈ A, we define a linear map π(a): A ⊗ H → A ⊗ H to be
π(a) ai ⊗ xi = (aai) ⊗ xi .
Also, notice that in Mn(A) we have:
(a†
i aj) =



a†
1
...
a†
n


 ·



a1
...
an



T
and
(a†
i a†
aaj) =



(aa1)†
...
(aan)†


 ·



aa1
...
aan



T
So it is not hard to get that (a†
i a†
aaj) ≤ a†
a · (a†
i aj) always holds true in
Mn(A)+
. Therefore, we have
π(a) aj ⊗ xj , π(a) ai ⊗ xi
=
i,j
φ(a†
i a†
aaj)xj, xi H
≤ a†
a ·
i,j
φ(a†
i aj)xj, xi H
= a 2
· aj ⊗ xj, ai ⊗ xi
12

Thus, since π(a) leaves N invariant, it induces a quotient linear transfor-
mation on A ⊗ H/N, which we still denote by π(a). The above inequality
also indicates that π(a) ≤ a . Thus, π(a) extends to a bounded linear
operator on K, which we still denote by π(a).
Also, it is easy to see that the map π: A → B(K) is a unital ∗-homomorphism.
Now deﬁne the mapping V from H to K by
V (x) = 1 ⊗ x + N .
Then since
V x 2
= 1 ⊗ x, 1 ⊗ x = φ(1)x, x H ≤ φ(1) · x 2
,
we know that V is bounded.
Also, it is easy to see that V 2
= sup{ φ(1)x, x H : x ≤ 1} = φ(1) .
Finally, observe that
V †
π(a)V x, y H = π(a)1 ⊗ x, 1 ⊗ y K = φ(a)x, y H
holds for any x and y in H. Therefore, we get V †
π(a)V = φ(a).
We call the triple (π, V, K) a Stinespring representation. Then, a Stine-
spring representation is called minimal if the closure of span{π(A)V H} is
K.
Another result from [8] shows that any two minimal Stinespring rep-
resentations of the same completely positive map φ are actually unitarily
equivalent. We now provide this result according to [8], page 46-47:
Proposition 4.3. Let A be a C*-algebra, let φ: A → B(H) be completely
positive, and let
(πi, Vi, Ki) , i = 1, 2,
be two minimal Stinespring representations for φ. Then there exists a unitary
U: K1 → K2 satisfying UV1 = V2 and Uπ1U†
= π2.
Proof. If U exists, then it has to satisfy
U
i
π1(ai)V1hi =
i
π2(ai)V2hi .
Also note that by the minimal condition, U will have dense range and
13

hence be onto. Hence, if we can show that the above formula yields a well-
defined isometry from K1 to K2, we will complete the proof.
To this end, observe that
i
π1(ai)V1hi
2
=
i,j
V †
1 π1(a†
i aj)V1hj, hi
=
i,j
φ(a†
i aj)hj, hi =
i
π2(ai)V2hi
2
.
So, U is isometric and therefore well defined, as desired.
Researchers in the fields of quantum physics and operator theory have
been devoting themselves to the research of completely positive maps for the
last three decades. However, many important results were obtained indepen-
dently in these two fields without knowing the works of the other. The proof
of Stinespring’s dilation theorem is an example of this. The books of Kraus
[5] and Paulsen [8] discuss the subject from the perspectives of physicists
and mathematicians respectively.
Now we are ready to introduce the main theories of this section:
Definition 4.4. A quantum channel is a map E from B(H) to B(H) which
is completely positive and trace preserving.
Let dim H = N, then we have B(H) ∼= MN , and we may use the nota-
tion HN here. Then under a given basis of HN , the trace of an operator is
just the trace of the corresponding matrix. Mathematically, trace preserving
means: for any ρ ∈ B(HN ), Tr(E(ρ)) = Tr(ρ). In quantum information, it
is equivalent to requiring that the probabilities remain the same as a state
evolve through the channel.
We require a quantum channel to be positive because density operators
must evolve to density operators, and we require a quantum channel to be
completely positive because the tensor product of the initial system and an-
other quantum system should also have this ‘density operator preserving’
property.
The following theorem was proved by Choi [7] and Kraus [5] indepen-
dently. We provide Choi’s operator proof here, which is cited from [7].
14

Theorem 4.5. Let E: B(HN ) → B(HN ) be a completely positive map. Then
there are operators Ek ∈ B(HN ), where k = 1, 2, ..., r and 1 ≤ r ≤ N2
, such
that
E(ρ) =
r
k=1
EkρE†
k for all ρ ∈ B(HN ) . (1)
Proof. Define eij = |i j|, where |i , |j are from the standard basis of HN .
It is straightforward to show that (eij) is a positive matrix in B(H
(N)
N ) =
MN (B(HN )). Then R = E(N)
((eij)) is also a positive matrix by the N-
positivity of E.
And since R is positive, it can be diagonalized. Therefore, there is a
decomposition: R = r
k=1 |ak ak|, where the |ak ’s are the normalized
eigenvectors of R and r ≤ N2
. (It is straightforward to justify that the
normalization can always been done. And obviously |ak are linearly in-
dependent.) It is easy to see that |ak ∈ CN2
. We decompose CN2
into
CN2
= CN
⊕ ... ⊕ CN
. Let {Pi : 1 ≤ i ≤ N} be the set of projections onto
each CN
. Then Pi have mutually orthogonal ranges to each other and satisfy
PiRPj =



0 · · · 0
... E(eij)
...
0 · · · 0


 (zero matrix in every N × N sub-block except an
E(eij) in the block of i’th “row” and j’th “column”). Also, |ak = N
i=1 Pi|ak .
Define operators Ek : CN
→ CN
to be Ek|i ≡ Pi|ak . Then
R =
k i,j
Pi|ak ak|Pj =
i,j
Pi
k
Ek|i j|E†
k Pj .
Therefore,
E(ei,j) = E(|i j|) = PiRPj =
r
k=1
Ek|i j|E†
k .
Finally, we get equation (1) by the linearity of E.
In quantum information, equation (1) is called the operator-sum representation
of E. The operators Ek are referred to as error or the noise operators of the
channel. Moreover, since we only work on finite dimensional Hilbert space,
15

we can regard operators as matrices. Then for any operators A, B on HN ,
we have Tr(AB) = Tr(BA).
Then, for any ρ ∈ B(HN ), we have
Tr(E(ρ)) = Tr(
k
EkρE†
k) =
k
Tr((Ekρ)E†
k) =
k
Tr(E†
k(Ekρ))
= Tr(
k
E†
k(Ekρ)) = Tr((
k
E†
kEk)ρ)
Then, Tr(E(ρ)) = Tr(ρ) if and only if Tr(( k E†
kEk)ρ) = Tr(ρ). Hence
the trace preservation of E is equivalent to requiring its noise operators Ek
to satisfy
k
E†
kEk = 1 ,
where 1 is the identity operator.
Remark 4.6. Sometimes we want to fully recover an unknown quantum
channel from only a small portion of experimental data. Now, with Choi’s
work, we only need to recover the noise operators of a quantum channel in
order to achieve this. Hence, Choi’s work provides us a manner to recover the
unknown, complicated quantum channel from known, simple experimental
data.
The following theorem indicates the connections between diﬀerent sets of
noise operators for the same channel. This theorem is frequently used in the
following part of this paper.
Theorem 4.7. On a Hilbert space HN , let {E1, ... , Er} and {F1, ... , Fs} be
two sets of linearly independent noise operators for channels E and E re-
spectively. Then E = E if and only if r = s and there exists an r × r scalar
unitary matrix U = (uij), such that
Ei =
s
j=1
uijFj for 1 ≤ i ≤ r (2)
Proof. We can write (1) as
E(ρ) =
r
k=1
E†
kρEk , (3)
16

by replacing Ek with E†
k. The form (3) is more convenient for us to use the
Stinespring’s dilation theorem, we replace (1) by (3) in the following proof.
But we still go back to (1) later on, since the form (1) is the convention of
quantum physics.
Suppose E = E . Then we have
E(ρ) =
r
i=1
E†
i ρEi =
s
j=1
F†
j ρFj
= E†
1, ..., E†
r



ρ
...
ρ






E1
...
Er


 = F†
1 , ..., F†
s



ρ
...
ρ






F1
...
Fs



Let K1 =
r times
⊗HN , K2 =
s times
⊗HN . We can see that K1 CNr and
K2 CNs. Moreover:
Deﬁne
V1 =



E1
...
Er


 and V2 =



F1
...
Fs


 .
So, V1: HN → K1 and V2: HN → K2.
Deﬁne
π1(ρ) =



ρ
...
ρ


 = ρ ⊗ Ir and π2(ρ) =



ρ
...
ρ


 = ρ ⊗ Is .
Then it is straightforward to prove π1: MN = B(HN ) → B(K1) and π2:
MN = B(HN ) → B(K2) are two ∗-homomorphism between C*-algebras.
Therefore, E(ρ) = V †
1 π1(ρ)V1 = V †
2 π2(ρ)V2. Then we get that (π1, V1, K1)
and (π2, V2, K2) are two Stinespring representations.
Now look at span{π1(MN )V1HN }:
17

Let hk = (0, 0, ..., 1, ..., 0)T
be the element in the standard basis of HN ,
where the 1 is at the k’th position and 0 at all the others. Let eij =


0 · · · 0
... 1
...
0 · · · 0


 be the element in the standard basis of MN , where the 1 is
in the i’th row and j’th column and 0 at all the other places.
Let fijk = π1(eij)V1(hk), then
fijk =



eij
...
eij






E1
...
Er


 (hk) =



eij
...
eij






E1(hk)
...
Er(hk)



=



eijE1(hk)
...
eijEr(hk)


 =



E
(1)
jk hi
...
E
(r)
jk hi



where E
(τ)
jk is the entry in the j’th row and k’th column of the matrix Eτ . It
is a scalar. (i.e. fijk is the column vector with zero in all positions except
E
(τ)
jk in the (τ + i)’th position for τ = 1, 2, · · · , r.)
Fix i, and deﬁne
Ai = (fi11 · · · fi1N · · · fiN1 · · · fiNN ) =



E
(1)
11 hi · · · E
(1)
NN hi
... · · ·
...
E
(r)
11 hi · · · E
(r)
NN hi


 .
Then, Ai is a r × N2
matrix.
Now, we note the τ’th row of Ai to be gτ . So gτ is a N2
-dimensional
vector. Then for any scalars x1, x2, ..., xr, consider the linear combination:
x1g1 + x2g2 + · · · + xrgr = 0 ,
which is equivalent to



E
(1)
11 x1 + · · · + E
(r)
11 xr = 0
...
E
(1)
NN x1 + · · · + E
(r)
NN xr = 0
,
18

which is equivalent to
x1E1 + · · · + xrEr = 0 .
Then x1 = x2 = · · · = xr = 0 because of the linearly independence of
Ek’s.
So rank Ai = rank{fi11, fi12 , ..., fiNN } = rank{g1, g2, ..., gr} = r. So
rank{A1, A2, ..., AN } = Nr. (fijk and fi j k are linearly independent if i = i ,
since the E
(i)
jk and E
(i )
j k are not in the same positions.)
Then we have:
span{fijk} = span{π1(eij)V1(hk)} = span{π1(MN )V1HN } = CNr = K1 ,
which means (π1, V1, K1) is minimal. Similarly, (π2, V2, K2) is minimal as
well.
Hence, by proposition 4.3, there exists an unitary map U: K1 → K2, such
that V2 = UV1 and π2 = Uπ1U†
. Thus we get r = s from the unitary. So
actually, π1 = π2.
Then for any ρ ∈ B(HN ) and any h ∈ HN , we have
π2(ρ)V2h = π1(ρ)V2h = π1(ρ)UV1h
π2(ρ)V2h = Uπ1(ρ)U†
V2h = Uπ1(ρ)U†
UV1h = Uπ1(ρ)V1h
So Uπ1(ρ)V1h = π1(ρ)UV1h for any ρ ∈ B(HN ), any h ∈ HN . So we have
Uπ1(ρ) = π1(ρ)U for any ρ ∈ B(HN ).
Let U =



U11 · · · U1r
...
...
...
Ur1 · · · Urr


. Then we have



U11 · · · U1r
...
...
...
Ur1 · · · Urr






ρ
...
ρ


 =



ρ
...
ρ






U11 · · · U1r
...
...
...
Ur1 · · · Urr


 .
Or equivalently,



U11ρ · · · U1rρ
...
...
...
Ur1ρ · · · Urrρ


 =



ρU11 · · · ρU1r
...
...
...
ρUr1 · · · ρUrr


 .
19

So, Uijρ = ρUij for i, j ∈ {1, 2, ..., r}. Hence, we get Uij = uijI, where
U0 =



u11 · · · u1r
...
...
...
ur1 · · · urr


 is unitary because U is unitary.
Finally, we have
V2 =



F1
...
Fr


 = UV1 =



u11I · · · u1rI
...
...
...
ur1I · · · urrI






E1
...
Er


 = U0 ⊗ I



E1
...
Er


 .
Hence,



E1
...
Er


 = U†



F1
...
Fr


 = U†
0 ⊗ I



F1
...
Fr


 ,
which is just the form of (2).
Conversily, suppose r = s and there exists a unitary matrix U = (uij)
such that the form (2) holds true. Then for any ρ ∈ B(HN ), we have:
E(ρ) =
r
i=1
EiρE†
i =
r
i=1


r
j=1
uijFj ρ
r
j=1
uijFj
†


=
r
i=1
r
j=1
uijFj ρ
r
j=1
uijF†
j =
r
i=1
r
j=1
uijFj ρ
r
j=1
u†
jiF†
j
=
r
i=1


 F1 · · · Fr



ui1
...
uir


 ρ u†
1i · · · u†
ri



F†
1
...
F†
r






=
r
i=1


 F1 · · · Fr uiju†
ki



ρ
...
ρ






F†
1
...
F†
r






20

= F1 · · · Fr
r
i=1 u†
kiuij



ρ
...
ρ






F†
1
...
F†
r



= F1 · · · Fr U†
U



ρ
...
ρ






F†
1
...
F†
r



= F1 · · · Fr



ρ
...
ρ






F†
1
...
F†
r


 =
r
j=1
FrρF†
r = E (ρ)
Thus, since ρ is arbitrary, we have E = E .
The following corollary show that any set of noise operators can be ex-
pressed by a set of linearly independent noise operators of the same channel.
Corollary 4.8. Let E be a quantum channel with noise operators {E1, ... , Er}
on the Hilbert space HN , where Rank{Ei} = s. Then there exists a set of
linearly independent noise operators {F1, ... , Fs} of E, such that
Ei =
s
j=1
λijFj for 1 ≤ i ≤ r
Proof. Without loss of generality, suppose {E1, ... , Es} is a basis of {E1, ... , Er}.
Then, let Ei = s
j=1 aijEj for s+1 ≤ i ≤ r. And let A =



as+1,1 · · · ar,1
...
...
...
as+1,s · · · ar,s



be the scalar matrix consisting of aij. Then we have
E1 · · · Er = E1 · · · Es · Is A ,
where each matrix Ei is multiplied by a scalar at the right hand side of the
equation.
Let Is A ·
Is
A† = K, then K ≥ Is. So K is positive and invertible
in B(HN ), which means that it is diagonalizable under an orthogonal basis.
21

In other words, K ∼=



µ1
...
µs


 under an orthogonal basis {u1, ..., us}.
Let ξj =
√
µjuj, then we have K = s
j=1 ξjξ†
j .
Then, let Fj = E1 · · · Es



yj1
...
yjs


 = s
l=1 yjlEl for 1 ≤ j ≤ s, where



yj1
...
yjs


 = ξj. It is not hard to show that {F1, ..., Fs} is linearly independent.
And for any ρ ∈ B(H), we have
E(ρ) =
r
i=1
EiρE†
i = E1, ..., Er



ρ
...
ρ






E†
1
...
E†
r



= E1, ..., Es Is A



ρ
...
ρ



Is
A†



E†
1
...
E†
s



= E1, ..., Es (
s
j=1
ξjρξ†
j )



E†
1
...
E†
s


 =
s
j=1
FjρF†
j ,
which means that {F1, ..., Fs} is a set of linearly independent noise operators
of E. And obviously, there exists a s × s scalar matrix (λij) such that Ei =
s
j=1 λijFj for 1 ≤ i ≤ r. Therefore, we complete the proof.
Then, we can immediately get:
Corollary 4.9. For a ﬁxed quantum channel E, the rank of its noise opera-
tors remains invariant under transformations of noise operators.
Hence, we can deﬁne the rank of a quantum channel to be the rank of its
noise operators.
There is a simple but widely used C*-algebra in the study of quantum
computing:
22

Consider the algebra generated by a pair of adjoint operators: A =
Alg{Ei, E†
i }. Then A is the set of polynomials of Ei and E†
i . We call A the
interaction algebra in quantum computation, and obviously A is †-closed.
Since A is a subspace of B(HN ), it is finite dimensional.
The interaction algebra A is widely used in the study of quantum channel.
Note 4.10. The study of quantum channel capacities is highly active in the
research of quantum channel. A series of deep mathematical problems have
been realized to have connections with computing the capacity of a quantum
channel to carry information.
Now let us consider more specific examples.
Examples 4.11. (i) Let the basis of H2 be {|0 , |1 }. For any p ∈ (0, 1),
define two operators on H2 by
E1 = ( 1 − p)12 , E2 = (
√
p)X ,
where X is one of the Pauli matrices described in section 2. Notice that E†
1 =
E1 and E†
2 = E2. Then the bit flip channel is defined as E = E1ρE†
1 +E2ρE†
2.
It flips the vector (state) |0 to |1 with probability p and vice versa. For
instance, for an vector v ∈ H2, let v = a|0 + b|1 , we have
E(|0 0|)(v) = E1|0 0|E1(v) + E2|0 0|E2(v)
= E1|0 0|( 1 − p)(a|0 + b|1 ) + E2|0 0|(
√
p)(b|0 + a|1 )
= ( 1 − p)E1(a|0 ) + (
√
p)E2(b|0 )
= (1 − p)a|0 + p · b|1
= ((1 − p)|0 0| + p|1 1|) · (a|0 + b|1 )
= ((1 − p)|0 0| + p|1 1|) (v)
Hence, since v is arbitrary, we get
E(|0 0|) = (1 − p)|0 0| + p|1 1|
(ii) Now, let E1 = 1
2
12 (again, E†
1 = E1) and define the channel E to be
E(ρ) = E1ρE1 + σxρσx + σyρσy + σzρσz ,
where σk = 1
2
K for k = x, y, z and K = X, Y, Z are the spin -1
2
Pauli matrices
described in section 2.
23

Also, an operator ρ on a separable Hilbert space H is called density,
if ρ = i pi|ψi ψi|, where 0 < pi < 1, i pi = 1 and ψi||ψi = 1. Let
ρ =
a b
c d
be a density operator on H2. Then for any v = r|0 + s|1 , we
have
E(ρ)(v) =(E1ρE1 + σxρσx + σyρσy + σzρσz)(v)
=E1ρE1(v) + σxρσx(v) + σyρσy(v) + σzρσz(v)
=
1
4
[(ar + bs)|0 + (cr + ds)|1 ] +
1
4
[(cs + dr)|0 + (as + br)|1 ]
+
1
4
[(−cs + dr)|0 + (as − br)|1 ] +
1
4
[(ar − bs)|0 + (−cr + ds)|1 ]
=
1
2
[(a + d)r|0 + (a + d)s|1 ]
=
1
2
tr(ρ) · [r|0 + s|1 ]
Since ρ is density, we have
tr(ρ) =
τ=0,1 i
τ||ψi Pi ψi||τ =
i
ψi||ψi Pi = 1
So, E(ρ)(v) = 1
2
[r|0 + s|1 ] = 1
2
12(v). Hence by the arbitrary of v, we
have E(ρ) = 1
2
12, which means E turns every density operator (matrix) into
the same density operator (matrix) 1
2
12.
(iii) Let 0 < r < 1 and deﬁne operators on H2 by
E1 =
1 0
0
√
1 − r
E2 =
0
√
r
0 0
Then E(ρ) = E1ρE†
1 +E2ρE†
2 is called the amplitude damping channel. This
channel is related to the energy dissipation in a quantum system.
(iv) Choose positive real numbers r1, · · · , rd such that i ri = 1, and let
U1, · · · , Ud be unitaries acting on a common Hilbert space H. Then deﬁne
E(ρ) = d
i=1 riUiρU†
i . In section 6 we will work on the ‘unital’ channel and
this one is a prototypical example of the unital channel. Moreover, it is easy
to see that every unital channel on H2 can be written as a (convex) summa-
tion of unitaries. But this is not true to a non-unital channel.
(v) An entanglement breaking channel is the channel which can be writ-
24

ten as
E(ρ) =
k
|ψk ψk| φk|ρ|φk ,
for some vectors |ψk and |φk . Then since tr(ρ|φk φk|) = φk|ρ|φk , the
trace preserving of E is equivalent to k |φk φk| = 1. This channel is called
entanglement breaking because E(d)
(Γ) will never be entangled for any d ≥ 1
and any initial operator Γ.
5 Quantum Error Correction
There is no computing system, including the quantum computing system,
that can fully eliminate errors. Moreover, errors occurred in a quantum
computing system can be more complicated than those occurred in a clas-
sical computing system, since the only kind of errors occurred in classical
computing is bit flips. But fortunately, many precise and delicate methods
have been developed for quantum error detection and correction.
Because of the lack of knowledge, we only discuss the quantum error de-
tection and correction with respect to a quantum channel. We shall first
present the basic theories in quantum error detection and correction. Then
we introduce some simple methods in quantum error correction.
I. Quantum Error Detection. Simply speaking, quantum error de-
tection with respect to a quantum channel means to figure out all the noise
operators of that channel. The theory of quantum error detection with re-
spect to a quantum channel is fundamental and crucial in the entire quantum
error detection theory. This is because the operating of all kinds of quantum
systems is deeply involved with quantum channels. Moreover, we will see
that the manner used here is simple and hence useful.
Before continuing, we present the following settings:
For a given closed quantum system, let H be the corresponding Hilbert
space. Then a quantum code on H can be any subspace C of H. Also, let PC
be the projection of H onto C. Then the projections PC and P⊥
C are applied
to determine whether a given state |ψ ∈ H belongs to the given quantum
code. So we can regard {PC, P⊥
C } as a measurement (See section 2).
Correctly using the Projection PC is important to the quantum error de-
tection with respect to a quantum channel. We need the following definition
25

for later discussion:
Definition 5.1. Let C be a quantum code on some Hilbert space H, and let
E be an error (noise operator) corresponding to a given quantum channel on
H. Suppose that C is fixed. Then we say C can detect the error E, if there
exists a scalar λE which only depends on E such that
PCE|ψ = λE|ψ for any |ψ ∈ C .
We shall note that all the operators that can be detected by a fixed
quantum code C form a subspace in B(H).
The following theorem will provide several equivalent conditions for error
detecting.
Theorem 5.2. Let C be a quantum code on some Hilbert space H, and let
E be an error (noise operator) corresponding to a given quantum channel on
H. Then the following conditions are equivalent:
(i) E can be detected by C, with scaling factor λE.
(ii) PCEPC = λEPC, where PC is the projection from H to C.
(iii) ψ1|E|ψ2 = λE ψ1||ψ2 for any |ψi ∈ C, i = 1, 2.
(iv) For any pair of orthogonal vectors {|ψ1 , |ψ2 } in C, the vectors E|ψ1
and |ψ2 are also orthogonal.
Proof. The most difficult part is (iv) ⇒ (iii). We start from it.
(iv) ⇒ (iii). Obviously we can assume that dim C ≥ 2, otherwise there
will be no orthogonal vectors in C. Define Ψ = {|ψ1 , |ψ2 , · · · } to be an
orthonormal basis for C. Then for any |ψi , |ψj ∈ Ψ and i = j, let |+ =
|ψi + |ψj and |− = |ψi − |ψj . Then |+ and |− are orthogonal to each
other in C. So by (iv) we have
ψi|E|ψi − ψj|E|ψj = ψi|E|ψi − ψi|E|ψj + ψj|E|ψi − ψj|E|ψj
= ψi|(E|ψi − E|ψj ) + ψj|(E|ψi − E|ψj )
= ( ψi| + ψj|)E(|ψi − |ψj )
= +|E|− = 0
So ψi|E|ψi = ψj|E|ψj holds for any |ψi , |ψj ∈ Ψ and i = j.
Therefore, define λE = ψi|E|ψi . So λE is independent of i.
Now, for any two vectors |ψ and |φ in C, let |ψ = α1|ψ1 +α2|ψ2 +· · ·
26

and |φ = β1|ψ1 + β2|ψ2 + · · · . Then we have that
ψ|E|φ =
i,j
αiβj ψi|E|ψj =
i
αiβi ψi|E|ψi
= λE
i
αiβi = λE
i
αiβi ψi||ψi
= λE
i,j
αiβj ψi||ψj
= λE
i
αi ψi|
j
βj|φj
= λE ψ||φ ,
which is the result we want.
(iii) ⇒ (ii). For any |x , |y ∈ H, let |x = |ψx + |ψx
⊥
and |y =
|ψy + |ψy
⊥
, where |ψx and |ψy belong to C while |ψx
⊥
and |ψy
⊥
belong
to C⊥
. Then we can get
y|PCEPC|x = y|PCE|ψx = ψy|E|ψx
= λE ψy||ψx = λE ψy|PC|x
= λE y|PC|x = y|λEPC|x .
Hence, since |x and |y are arbitrary, we get PCEPC = λEPC.
(ii) ⇒ (i). Trivial.
(i) ⇒ (iv). If |ψ1 and |ψ2 are orthogonal, then
ψ2|E|ψ1 = ψ2|PCE|ψ1 = ψ2|λE|ψ1 = λE ψ2||ψ1 = 0
So |ψ2 and E|ψ1 are orthogonal.
Remark 5.3. Since the Hilbert space H we work on is always finite dimen-
sional, we can take a further discussion.
Let C be a fixed quantum code in H. Then we can write PC as
1C 0
0 0
if we appropriately choose the basis of H, and it is a finite matrix since B(H)
27

is ﬁnite dimensional. Also let E =
A B
C D
be an operator in B(H) which
can be detected by C. Then the condition (ii) of Theorem 5.2 shows that
1C 0
0 0
A B
C D
1C 0
0 0
= λE
1C 0
0 0
=
λE1C 0
0 0
.
But since
1C 0
0 0
A B
C D
1C 0
0 0
=
1C 0
0 0
A 0
C 0
=
A 0
0 0
,
we can get that an operator E can be detected by C if and only if
E =
λE1C ∗
∗ ∗
.
Or in other words,
E ∈
λ1C ∗
∗ ∗
: λ ∈ C .
Now we provide a simple example of undetectable errors.
Example 5.4. Deﬁne the quantum code C in H8 to be
C = span{|000 , |111 } .
Then consider the operator E = Z1 = Z ⊗ 12 ⊗ 12, where Z is the Pauli
matrix. Observe that E|000 = |000 and E|111 = −|111 . Assume that E
can be detected by C. Then we have
λE = 000|E|000 = 1 and λE = 111|E|111 = −1 ,
which is a contradiction. Hence, the operator E cannot be detected by the
quantum code C.
We shall mention that the above operator E = Z1 is a noise operator
of the so-called 3-qubit depolarizing channel. And the n-qubit depolarizing
channel is E = {12n , Z1 , ... , Zn}, where
Zi = 12 ⊗ · · · ⊗ 12 ⊗ Z ⊗ 12 ⊗ · · · ⊗ 12
28

with Z in the i’th position.
II. Quantum Error Correction. On a finite dimensional Hilbert space
H, let E be the given quantum channel and let C be the given quantum code.
The basic idea of quantum error detection corresponding to this setting is
trying to find another quantum channel R such that the compounded channel
R◦E can “work well” on C, which means that all operators that are supported
on C will be invariant under R ◦ E. We present the following definition as a
more clear description:
Definition 5.5. Let E be the given quantum channel and let C be the given
quantum code with projection PC. Then C is correctable for E if there exists
a quantum channel R such that
R ◦ E(ρ) = ρ
for any ρ supported on C, that is, any ρ with ρ = PCρPC.
There are several equivalent conditions of correctable quantum codes,
which will be discussed in the following theorem. Note that in the proof of
(iii) ⇒ (i) below, the error correction channel R is explicitly constructed.
This provides us a way to construct R.
Theorem 5.6. Let E be the given quantum channel with noise operators
{Ei : 1 ≤ i ≤ p} and let C be the given quantum code with projection PC.
Then the following conditions are equivalent:
(i) C is correctable for E.
(ii) All the operators in the set {E†
i Ej : 1 ≤ i, j ≤ p} can be detected by
C.
(iii) There exist a scalar matrix Λ = (λij) such that
PCE†
i EjPC = λijPC for all 1 ≤ i, j ≤ p .
Proof. For (i) ⇒ (iii), let R be the corresponding error correction channel
with noise operators {Rs : 1 ≤ s ≤ q}. Define a new channel EC via EC(ρ) ≡
E(PCρPC). Then by condition (i), we have
R(EC(ρ)) = R(E(PCρPC)) =
s,i
RsEiPCρPCE†
i R†
s = PCρPC (4)
29

holds for any ρ ∈ B(H), since PCρPC is always supported on C.
Then by (4), we can see that {RsEiPC : 1 ≤ s ≤ q, 1 ≤ i ≤ p} and {PC}
are two sets of noise operators for the channel R◦EC. Then by Corollary 4.8,
there exists a set of linearly independent noise operators {Fk : 1 ≤ k ≤ r}
of R ◦ EC such that RsEiPC = r
k=1 µsikFk. But by Theorem 4.7, we have
that {Fk : 1 ≤ k ≤ r} and {PC} must have the same cardinality. The
only way this can happen is that Fk = βkPC for some scalars βk, where
r
k=1 βk = 1. Hence, we can see that RsEiPC = αsiPC for some scalars αsi,
where q
s=1
p
i=1 αsi = 1. Hence,
PCE†
i R†
sRsEjPC = αsiαsjPC for all i, j, s. (5)
Then since R is trace preserving, we have that q
s=1 R†
sRs = 1. So sum
over (5) with respect to s, we have
PCE†
i EjPC = λijPC for all i, j,
where λi,j = q
s=1 αsiαsj.
For (iii) ⇒ (i), let A = [E1PC E2PC · · · ]. Then the formula in condition
(iii) can be written as A†
A = (λijPC). Since A†
A is positive, it is straight-
forward to show that Λ = (λij) is positive, and hence diagonalizable. So we
can ﬁnd a unitary scalar matrix U such that U†
ΛU = diag(dkk) = D. Thus,
k
dkk = Tr(D) = Tr(U†
ΛU) = Tr(ΛUU†
) = Tr(Λ) =
i
λii .
But since E is trace preserving, we have i E†
i Ei = 1. Hence,
i
λii PC =
i
λiiPC =
i
PCE†
i EiPC = PC
i
E†
i Ei PC = PC .
So we have that Tr(D) = k dkk = 1.
Let Fk = i uikEi. Then {Fk} is another set of noise operators of E
according to Theorem 4.7. And a simple computation can show that
PCF†
k FlPC = dklPC for all k, l.
30

The polar decomposition of FkPC is
FkPC = Uk PCF†
k FkPC = dkkUkPC ,
where we may assume Uk to be unitary rather than partial isometry as H is
finite dimensional.
Define projections Pk ≡ UkPCU†
k. Recall that U†
ΛU = diag(dkk) = D. So
dlk = 0 if l = k. Then
PlPk = UlPCU†
l UkPCU†
k = Ul(UlPC)†
(UkPC)U†
k
=
Ul(FlPC)†
(FkPC)U†
k
√
dlldkk
=
UlPCF†
l FkPCU†
k
√
dlldkk
=
dlk
√
dlldkk
UlPCU†
k = 0 , if k = l .
So the Pk have mutually orthogonal ranges.
Without loss of generality, we can assume that k Pk = 1 (Otherwise
we can add the projection which is onto the orthogonal complement.) Then
define a new channel via
R(ρ) =
k
U†
kPkρPkUk .
Then for any operator ρ that satisfies ρ = PCρPC, we have
R(E(ρ)) =
k,l
U†
kPkFlρF†
l PkUk =
k,l
U†
kUkPCU†
kFlρF†
l UkPCU†
kUk
=
k,l
PCU†
kFlρF†
l UkPC =
k,l
(UkPC)†
FlρF†
l (UkPC)
=
k,l
FkPC
√
dkk
†
FlρF†
l
FkPC
√
dkk
=
k,l
1
dkk
PCF†
k FlPCρPCF†
l FkPC
=
k,l
dkldlk
dkk
PCρPC =
k
dkkρ = ρ .
31

And hence C is correctable for E.
Condition (ii) and (iii) are equivalent by Theorem 5.2. So finally we
complete the proof.
Now we provide an example to show how to determine whether a quantum
code is correctable for a given channel by using Theorem 5.6.
Example 5.7. Define two orthogonal vectors in H29 via
|0L =
(|000 + |111 ) ⊗ (|000 + |111 ) ⊗ (|000 + |111 )
2
√
2
|1L =
(|000 − |111 ) ⊗ (|000 − |111 ) ⊗ (|000 − |111 )
2
√
2
Then Shor’s 9-qubit code is defined as CS = span{|0L , |1L }.
For a fixed k in {1, 2, ... , 9}, let ES be the quantum channel with noise
operators {Xk, Yk, Zk}, where
Zk = 12 ⊗ · · · ⊗ 12 ⊗ Z ⊗ 12 ⊗ · · · ⊗ 12
and Z is in the k’th position. Xk, Yk are similarly defined.
Let B = {|0...0 , ... , |1...1 } be a basis of H29 , where the basis ele-
ments can be regarded as a list of the first 2n
binary numbers. Then both
|0L and |1L are the linear combination of eight basis elements, which are
|000 |000 |000 , ... , |111 |111 |111 , which are the τ1’th basis element, τ2’th
basis element, ... and τ8’th basis element in B. Note that all the coefficients
of the basis elements in |0L and |1L have the same absolute value 1
2
√
2
. Then
the value of each τi and the corresponding signals for |0L and |1L are:
signal
τi’s value
1 23
26
−23
+1 26
|0L + + + +
|1L + − − +
signal
τi’s value
29
−26
+1 29
−26
+23
29
−23
+1 29
|0L + + + +
|1L − + + −
32

Observe that we can separate the eight basis elements into four groups.
In each group the diﬀerence of the two τ is 7.
And under the basis deﬁned above, we have
Xk =







0 12k−1
12k−1 0
0 12k−1
12k−1 0
...







,
Yk =







0 i12k−1
−i12k−1 0
0 i12k−1
−i12k−1 0
...







,
and
Zk =







12k−1 0
0 −12k−1
12k−1 0
0 −12k−1
...







.
So we can see that Xk = X†
k, Yk = Y †
k and Zk = Z†
k.
Now we consider the position change of a basis element under Xk, Yk and
Zk (Or in other words, the τ’s change under these operators): Suppose that
τ = p · 2k−1
+ q, where p, q ∈ N and q < 2k−1
. Then we have
Zk : τ → τ ,
and
Xk, Yk : τ →
(p − 1)2k−1
+ q , if p is even
(p + 1)2k−1
+ q , if p is odd
.
Then we have XkYk : τ → τ, and so is YkXk, XkXk, and YkYk. Hence,
33

we have
PCX†
kXkPC = PC , PCX†
kYkPC = iPC
PCY †
k XkPC = iPC , PCY †
k YkPC = −i · iPC = PC
(6)
Finally, since τ is invariant under Zk but variant under Xk and Yk, we
have
PCX†
kZkPC = PCY †
k ZkPC = PCZ†
kXkPC = PCZ†
kYkPC = 0 . (7)
Observe that (6) and (7) establish the condition (iii) in Theorem 5.6.
Hence, CS is correctable for ES.
6 Noiseless Subsystems via The Noise Com-
mutant
In section 5, we mainly focused on two problems: whether a given quantum
code is correctable for a given quantum channel, and what kind of quantum
channel a given quantum code is correctable for. But in experiments, we
often need to consider another problem, that is, what kind of quantum code
is correctable for a given quantum channel. In section 6, we will provide a
basic but effective method to show how to solve this problem when the given
channel is unital. We call this method the noiseless subsystem method, or
the noiseless subsystem via noise commutant method.
Let E : B(H) → B(H) be a unital channel with noise operators {E1, ..., Er}.
Recall that E is unital if and only if E(1) = r
i=1 EiE†
i = 1. Define
A = Alg{E1, ..., Er} and A†
= Alg{E†
1, ..., E†
r}. Define the noise commutant
to be
A = {ρ ∈ B(H) : ρA = Aρ for any A ∈ A}
= {ρ ∈ B(H) : ρEi = Eiρ for i = 1, ..., r} .
And define
Fix(E) = {ρ ∈ B(H) : E(ρ) = ρ} .
Also, let C be a quantum code which is correctable for E and let R be the
corresponding correction channel for C as in Definition 5.5. Then we say that
C is a noiseless subsystem if R is the identity channel, that is, R(ρ) = ρ for
34

any ρ ∈ C.
The main result of this section is showing that A is a noiseless subsystem.
We shall note that Fix(E) is †-closed. Indeed, if ρ ∈ Fix(E), then we have
E(ρ†
) =
r
i=1
Eiρ†
E†
i =
r
i=1
(EiρE†
i )†
= (
r
i=1
EiρE†
i )†
= (E(ρ))†
= ρ†
.
Thus Fix(E) is †-closed. Also, it is straightforward to show that Fix(E) is a
subspace of B(H).
Before continuing, we shall provide the supportive theories of this section,
which are Lemma 6.1 to Lemma 6.5. We shall mention that all the proofs in
this section are provided in [6].
Lemma 6.1. Let H be a general Hilbert space and let ρ be a positive and
contractive operator in B(H). Then for any vector |ψ in H, we have that
(ρ|ψ , |ψ ) ≤ (|ψ , |ψ ). Moreover, the equality holds if and only if ρ|ψ =
|ψ .
Proof. Since ρ is positive, we have ρ = δδ†
for some δ ∈ B(H). Since ρ
is contractive and B(H) is a C*-algebra, we have ρ = δδ†
= δ 2
=
δ† 2
≤ 1. So δ and δ†
are contractive as well. Then
(ρ|ψ , |ψ ) = (δδ†
|ψ , |ψ ) = (δ†
|ψ , δ†
|ψ ) = δ†
|ψ 2
≤ |ψ 2
= (|ψ , |ψ ) ,
which proves the inequality.
If (ρ|ψ , |ψ ) = (|ψ , |ψ ), we assume that ρ|ψ = |ψ +|φ . Then we have
(ρ|ψ , |ψ ) = (|ψ + |φ , |ψ ) = (|ψ , |ψ ) + (|φ , |ψ ) = (|ψ , |ψ ) ,
which shows that (|φ , |ψ ) = 0. Then, since ρ is contractive, we have
ρ|ψ 2
= |ψ + |φ 2
= (|ψ + |φ , |ψ + |φ ) = |ψ 2
+ |φ 2
≤ |ψ 2
.
Thus obviously |φ = 0, which means ρ|ψ = |ψ . And the other direction is
trivial. Thus we complete the proof.
Lemma 6.2. Suppose E : B(H) → B(H) is a unital quantum channel with
noise operators {E1, ... , Er}, where H is a ﬁnite dimensional Hilbert space.
Let ρ be an operator in B(H) with 0 ≤ ρ ≤ E(ρ). Then the subspace
ker(ρ − ρ 1) is E†
i -invariant for all 1 ≤ i ≤ r.
35

Proof. Without loss of generality, suppose ρ = 1. Then ρ is contractive.
Let M = ker(ρ − 1). Then according to Lemma 6.1, for any |ψ ∈ M, we
have
|ψ 2
= (ρ|ψ , |ψ ) ≤
r
i=1
(EiρE†
i |ψ , |ψ )
=
r
i=1
(ρE†
i |ψ , E†
i |ψ ) ≤
r
i=1
(E†
i |ψ , E†
i |ψ )
= (
r
i=1
EiE†
i |ψ , |ψ ) = (|ψ , |ψ ) = |ψ 2
Thus, all the inequalities above are actually equalities. In particular, we
have r
i=1(ρE†
i |ψ , E†
i |ψ ) = r
i=1(E†
i |ψ , E†
i |ψ ). Again, by Lemma 6.1,
we get ρE†
i |ψ = E†
i |ψ for all 1 ≤ i ≤ r. So (ρ − 1)(E†
i |ψ ) = 0 for all
1 ≤ i ≤ r. Then, since |ψ is arbitrary in M, we get M is E†
i -invariant for
all 1 ≤ i ≤ r.
Now, we are able to provide the following theorem, which is necessary for
the proof of the main theorem.
Theorem 6.3. Suppose E : B(H) → B(H) is a unital quantum channel with
If P is an orthogonal projection in B(H), then the following results holds
true:
(i) E(P) ≥ P if and only if Ran(P) is E†
(ii) E(P) ≤ P if and only if Ran(P) is Ei-invariant for all 1 ≤ i ≤ r.
(iii) E(P) = P if and only if Ran(P) is Ei-reducing for all 1 ≤ i ≤ r.
Proof. Note that since E is unital, E(P) ≥ P if and only if E(1−P) ≤ 1−P.
In addition, Ran(P) is E†
i -invariant if and only if Ran(1 − P) = Ran(P)⊥
is
Ei-invariant. Thus, we only need to prove (i).
Suppose E(P) ≥ P holds true. Since P is a projection, we have that
0 ≤ P ≤ E(P) and ker(P − P 1) = Ran(P) (actually P = 1). Then by
Lemma 6.2, Ran(P) is E†
To see the converse, consider the decomposition H = PH ⊕ P⊥
H. Then
36

we can write Ei in the form
Ei =
Bi 0
Ci Di
for 1 ≤ i ≤ r .
Then since E(1) = r
i=1 EiE†
i = 1, we have
r
i=1
BiB†
i = 1PH ,
r
i=1
BiC†
i = 0 and
r
i=1
(CiC†
i + DiD†
i ) = 1P⊥H .
Then, writing P =
1 0
0 0
according to the above decomposition, we will get
E(P) =
r
i=1
EiPE†
i =
1 0
0 r
i=1 CiC†
i
≥
1 0
0 0
= P .
This completes the proof of (i).
Then, since E(P) = P means E(P) ≥ P and E(P) ≤ P hold true simul-
taneously, we can easily get (iii).
Now we are close to the main result of this section. But before going on,
we still need the following two lemmas.
Lemma 6.4. Suppose E : B(H) → B(H) is a unital quantum channel with
Then A = B(H) will imply Fix(E) = C1. In other words, Fix(E) consists of
scalars.
Proof. Since A = B(H), we have A†
= B(H). Now, assume that there exists
an operator ρ ∈ Fix(E) which is non-scalar. Since Fix(E) is †-closed and E is
unital, without loss of generality we can suppose ρ to be positive. Then by
Lemma 6.2, M = ker(ρ − ρ 1) are E†
Since B(H) is ﬁnite dimensional, the spectrum σ(ρ) of ρ is a ﬁnite set.
But since ρ is positive, we know that ρ is an extreme point of σ(ρ). So
we can see that ρ ∈ σ(ρ), which means M = ∅. Also, we have M =
B(H) (otherwise, we would get ρ = ρ 1). Then, M is a proper invariant
subspace of A†
= B(H), which is absurd. Hence we complete the proof by
contradiction.
37

Lemma 6.5. Suppose the same conditions in Lemma 6.4 hold true. Then for
any projection P, E(P) = P holds true when either E(P) ≤ P or E(P) ≥ P
holds true. Moreover, every subspace that is invariant for E†
= {E†
i : 1 ≤
i ≤ r} is also reducing for E†
. And the same result holds true for E = {Ei :
1 ≤ i ≤ r}.
Proof. Assume that there is a projection P such that E(P) ≤ P and E(P) =
P. Then P −E(P) is positive. So tr(P −E(P)) = i λi > 0, where λi are the
eigenvalues of P − E(P). But tr(P − E(P)) = 0 since E is trace preserving,
contradiction. Hence, E(P) ≤ P will imply E(P) = P. The same discussion
works for the case E(P) ≥ P.
Then, the rest of this lemma can be obtained directly form Theorem
6.3.
Finally we can provide the main result of this section. The proof is from
[6]. We should say that some simple results in [6] are directly used here since
they can be easily verified according to some basic knowledge in C*-algebra.
Theorem 6.6. Let E be a unital quantum channel. Then A = A†
is a
C*-algebra and Fix(E) coincides with A .
Proof. We first prove that A = A†
is a C*-algebra. Define {Pj} to be
a maximal family of pairwise orthogonal projections, where each PjH is a
minimal reducing subspace for the family {E1, ..., Er}. For simplicity, we
will just say that Pj is minimal reducing for {Ei}. And let Hj = PjH for
each j. It is easy to see that EiPj = PjEi for all i, j, and the maximality
of {Pj} implies 1 = j Pj. Hence, we have that A = j PjAPj and A is
block diagonal with respect to {Pj}. Moreover, for a fixed j, we have that
APj is an algebra and APj = PjA = PjAPj. But the only subspaces of H
that are invariant for APj are the trivial subspaces. (Indeed, if a subspace
PH is Ei-invariant for all 1 ≤ i ≤ r, then PH is Ei-reducing according to
Lemma 6.5. So by the minimality of Pj, we get either P = 0 or P = Pj.) So
applying Burnside’s classical theorem, we can get that APj = B(Hj), which
shows that A = A†
is a finite dimensional C*-algebra.
Now we prove that Fix(E) = A .
For any ρ ∈ A , we have Eiρ = ρEi for i = 1, ..., r. Hence we have
E(ρ) =
r
i=1
EiρE†
i = ρ
r
i=1
EiE†
i = ρ · 1 = ρ .
38

So ρ belongs to Fix(E). Thus A ⊂ Fix(E).
To see the converse inclusion, let Ei,j = EiPj for all j and i = 1, ..., r.
Let Fj = (E1,j, ..., Er,j) ∈ B(H
(r)
j , H). Given any ρ ∈ B(H), build the block
decomposition ρ = (ρjk) according to {Pj}. Then it is not hard to show
that ρjk = PjρPk. If E(ρ) = ρ holds true, then a simple deduction can
show that Ej(ρjj) = ρjj, where Ej : B(Hj) → B(Hj) is defined as Ej(ρ) =
r
i=1 Ei,jρE†
i,j, and hence all the Ej are unital and completely positive. Then
since APj = B(Hj), according to Lemma 6.4 we have that ρjj = λjjPj for
some scalar λjj.
For j = k (j and k are fixed here), we will show that either ρjk = ρkj = 0
for all ρ ∈ Fix(E) with ρ = ρ†
, or there exists a unitary Wjk : Hk → Hj such
that Ei,j = WjkEi,kW†
jk for i = 1, ..., r. Suppose that there is a ρ ∈ Fix(E)
with ρ = ρ†
such that ρjk = 0. Then without loss of generality, we can let
ρjk = 1. Define M = {|ψ ∈ Hk : ρjk|ψ = |ψ } and let N = ρjkM
(Since j, k are fixed, we do not need to use the notation Mk and Nk). Then
for any |ψ ∈ M, we have
ρjk|ψ = (PjE(ρ)Pk)|ψ = E(ρjk)|ψ = (Fjρ
(r)
jk F†
k )|ψ , (8)
where ρ
(r)
jk =



ρjk
...
ρjk


. The form (8) indicates that M is invariant
under each E†
i,k. (Indeed, it is easy to show that Fj , F†
k ≤ 1. Also,
ρjk|ψ = |ψ . Then, we can see that F†
k |ψ ∈ M(r)
, where M(r)
= {ξ ∈
H
(r)
k : ρ
(r)
jk ξ = ξ }. So it is easy to see that M is invariant under each
E†
i,k.) Then by Lemma 6.5, we can see that M is reducing of each Ei. Also,
M is a non-zero subspace contained in Hk. Hence, by the minimality of Pk,
we get M = Hk. And we get N = Hj since ρkj = ρ†
jk. Also, ρjk and ρ†
jk
are partial isometries and hence the operator Wjk = ρjk|Hk
: Hk → Hj is a
unitary operator, which is the one we desire.
Hence, it is not hard to show that Wjk = FjW
(r)
jk F†
k . Then for any
|ψ ∈ Hk, we have
|ψ = Wjk|ψ = FjW
(r)
jk F†
k |ψ ≤ W
(r)
jk F†
k |ψ ≤ |ψ .
Thus Fj is isometric from RanW
(r)
jk F†
k to RanWjk = Hj. (Recall that the
39

domain of Fj is H
(r)
j . Here we only guarantee that Fj is isometric when ξ ∈
RanW
(r)
jk F†
k .) Also, since Fj is a contraction ( Fj ≤ 1), Fj(ξ) can only be
zero if ξ ∈ (RanW
(r)
jk F†
k )⊥
. Hence, it is not hard to show that F†
j is isometric
from Hj to RanW
(r)
jk F†
k (Again, here we guarantee the isometry only when
|ψ ∈ Hj). Therefore, we can get F†
j Wjk = W
(r)
jk F†
k . Then after simplification
it becomes E†
i,j = WjkE†
i,kW†
jk for i = 1, ..., r. Hence Ei,j = WjkEi,kW†
jk, as
desired.
Now suppose E(ρ) = ρ = ρ†
= (ρjk) for an operator ρ ∈ B(H), where the
decomposition (ρjk) is based on {Pj}. Then fix one arbitrary pair (j, k) with
j = k, we have
WjkEk(W†
jkρjk) =
r
i=1
WjkEi,kW†
jkρjkE†
i,k
=
r
i=1
Ei,jρjkE†
i,k
= PjE(ρ)Pk = ρjk .
Thus by Lemma 6.4, we get that W†
jkρjk = λkkPk for some scalar λkk. Hence
we have ρjk = µjkWjk for some scalar µjk, and also ρkj = ρ†
jk = µjkW†
jk. All
the other off-diagonal entries of ρ are either zero or in the same form.
Then decomposing each Ei according to {Pj} as well, we get Ei =
j EiPj = j Ei,j. So, it is easy to see that ρEi = Eiρ for all i = 1, ..., r
when ρ ∈ Fix(E) and ρ = ρ†
.
Hence, all the self-adjoint elements in Fix(E) is contained in A . Then, let
{ρ1, ..., ρn} be a basis of Fix(E). Then {(a1ρ1+a1ρ†
1), ..., (anρn+anρ†
n)}, which
consists of self-adjoint elements, can also be a basis if the scalars a1, ..., an are
suitable. Hence, Fix(E) is spanned by its self-adjoint elements. Then, since
A is a subspace and all the self-adjoint elements of Fix(E) are contained in
A , we get Fix(E) ⊂ A .
Therefore, Fix(E) = A .
Remark 6.7. Note that Fix(E) = (A) = A. But A = A since A is a
subspace in the finite dimensional B(H). Hence we have Fix(E) = (A) = A.
Moreover, since Fix(E) is finite dimensional and †-closed, we can see that
Fix(E) = A is a C*-subalgebra of B(H), where H is a finite dimensional
Hilbert space. This means that A is a C*-algebra of compact operators.
40

Hence, A is unitarily equivalent to a unique direct sum of ampliﬁed matrix
algebras, which means
A
k
⊕k(1mk
⊗ Mnk
) =
k
⊕kM(mk)
nk
=
k
⊕kB(Hnk
)(mk)
,
where Mnk
is the space of nk × nk matrices, and Hnk
is the nk dimensional
Hilbert space. And therefore we can decompose ρ accordingly.
Theorem 6.6 shows that A is a noiseless subsystem. This is why Theorem
6.6 is valuable.
However, when E is not guaranteed to be unital, Theorem 6.6 is not
helpful any more, and we cannot use A for error correction. We show this
by the following proposition.
Proposition 6.8. Let E be a quantum channel with noise operator {E1, ..., Er}
such that AE = E(1) is not invertible. Let PE be the projection onto the sub-
space HE = Ran(AE). If Ei = PEEiPE for any Ei, then H⊥
E is non-zero.
Moreover, for any operator ρ in B(H⊥
E ), we have ρ ∈ A and E(ρ) = 0.
Proof. Since AE is not invertible, we can see that H⊥
E is non-zero. Let ρ ∈
B(H⊥
E ) and ρ ≥ 0, then ρ = P⊥
E ρP⊥
E . Then ρ ∈ A since ρEi = 0 = Eiρ for
any i. At last, we have E(ρ) = E(P⊥
E ρ) = 0.
Remark 6.9. A more general deﬁnition of unital channel is a channel that
make the identity operator evolve to a multiple of a projection. That is,
E(1H) = mP for some projection P. Suppose dimH = N. Then since E is
trace preserving, we have N = tr(1) = tr(E(1)) = tr(mP) = mtr(P). So m
divides the dimension of H. Then, the noise commutant A works for this
kind of ‘general unital channel’. A typical example is the channel E with
noise operators Ai = |0 i| for 1 ≤ i ≤ dimH ≡ d. Then we have
E(1d) =
d
i=1
AiA†
i =
d
i=1
(|0 i|)(|i 0|) = d|0 0| .
At the end of this paper, we provide some special examples of unital
channels.
41

Examples 6.10. (i) For any 0 < p < 1, let E1, E2 be operators defined on
the standard basis of H2 by
E1 = ( 1 − p)12 and E2 = (
√
p)Z .
Then the phase flip channel is the quantum channel E : B(H2) → B(H2)
with noise operators {E1, E2}. Since E1 = E†
1 and E2 = E†
2, we have
E(12) = E112E†
1 + E212E†
2 = E1E1 + E2E2 = (1 − p)12 + p12 = 12 ,
and hence E is unital.
Notice that the only difference between the phase flip channel and the
bit flip channel is that E2 = (
√
p)Z in the former one, while E2 = (
√
p)X in
the later one. Moreover, define |+ = |0 +|1
√
2
and |− = |0 −|1
√
2
as in section 3.
Then by applying the exactly same method in the discussion of the bit flip
channel, we can prove that
E(|+ +|) = (1 − p)|+ +| + p|− −| .
Hence, E flips the phases of |+ +| and |− −| with probability p. (This is
also why E is called the phase flip channel.)
For any operator ρ ∈ B(H2), write ρ as
a b
c d
. Also, let A = Alg{E1, E2}
and let A be the noise commutant of E. Then we have
A = {ρ ∈ B(H2) : ρA = Aρ for any A ∈ A}
= {ρ ∈ B(H2) : ρEi = Eiρ for i = 1, 2}
= {ρ ∈ B(H2) : ρE2 = E2ρ}
=
a b
c d
:
a b
c d
1 0
0 −1
=
1 0
0 −1
a b
c d
=
a b
c d
:
a −b
c −d
=
a b
−c −d
=
a 0
0 d
: a, b ∈ C C1 ⊕ C1 .
Hence, any noiseless subsystem of the phase flip channel is non-trivial. Be-
cause of this nice property, the phase flip channel is widely used in quantum
error correction.
42

(ii) For any 0 < p < 1, let E1, E2 be operators defined on the standard
basis of H4 by
E1 = 1 − p
12 0
0 12
and E2 =
√
p
Z 0
0 −Z
.
Again, E1 = E†
1 and E2 = E†
2.
Similarly as in (i), write ρ in the form of (aij), where 1 ≤ i, j ≤ 4. Since
M4 = B(H4), we have
A = {ρ ∈ B(H4) : ρEi = Eiρ for i = 1, 2}
= {ρ ∈ B(H4) : ρE2 = E2ρ}
= (aij) ∈ M4 : (aij)
Z 0
0 −Z
=
Z 0
0 −Z
(aij)
=



(aij) ∈ M4 :




a11 −a12 −a13 a14
a21 −a22 −a23 a24
a31 −a32 −a33 a34
a41 −a42 −a43 a44



 =




a11 a12 a13 a14
−a21 −a22 −a23 −a24
−a31 −a32 −a33 −a34
a41 a42 a43 a44







=







a11 0 0 a14
0 a22 a23 0
0 a32 a33 0
a41 0 0 a44



 : aij ∈ C



M2 ⊕ M2
(iii) Suppose E1 = λ1U1 , ... , Er = λrUr, where Ui are unitaries on
a n-dimensional Hilbert space H and λi are scalars with r
i=1 |λi|2
= 1.
Let E be the unital channel with noise operators {E1, ..., Er}. Define A =
Alg{E1, ..., Er} as usual, and define A = Alg{E1, E†
1 , ... , Er, E†
r}.
Let f(λ) = |λI − U1| be the characteristic polynomial of U1. Then
U−1
1 = U†
1 since U1 is unital. Then by Cayley-Hamilton theorem, we have
f(U1) = Un
1 + cn−1Un−1
1 + cn−2Un−2
1 + · · · + c1U1 + c0I = 0 . (9)
Now, let U1 = TJ1T−1
, where J1 is the Jordan form of U1. Then
|λI − U1| = |λI − J1| =
s
j=1
(λ − λ
(1)
j )kj
,
43

where λ
(1)
j are the eigenvalues of U1 and J1 (Jordan transformation does
not change the characteristic polynomial and hence does not change the
eigenvalues.) Hence, we have
f(U1) =
s
j=1
(U1 − λ
(1)
j I)kj
= Un
1 + cn−1Un−1
1 + cn−2Un−2
1 + · · · + c1U1 + (−1)n
s
j=1
(λ
(1)
j )kj
I .
But
s
j=1
(λ
(1)
j )kj
= |J1| = |U1|, and |U1| = 1 since U1 is unitary. Hence
c0 = (−1)n
= 0. Taking the value of c0 back into (9), we have
(−1)n+1
I = U1(Un−1
1 + cn−1Un−2
1 + cn−2Un−2
1 + · · · + c1I) .
In other words,
I = (−1)n+1
U1(Un−1
1 + cn−1Un−2
1 + cn−2Un−2
1 + · · · + c1I) .
Multiplying U†
1 (which is equal to U−1
1 ) from left at both side, we get
U†
1 = (−1)n+1
(Un−1
1 + cn−1Un−2
1 + cn−2Un−2
1 + · · · + c1I) .
Thus, U†
1 can be expressed by a polynomial of U1, which is also true for
U†
2 , ... , U†
r . Hence, we have {U†
1 , ... , U†
r } ⊂ Alg{E1, ... , Er} = A, which indi-
cates that A ⊂ A. And obviously we have A ⊂ A. So ﬁnally, we get A = A.
(iv) An important case of the above unital channel is the class of ‘collec-
tive rotation channels’.
Recall that σk = 1/2K for k = x, y, z and K = X, Y, Z, where X, Y, Z
are the Pauli matrices. Then a collective rotation channel deﬁned on B(H2n )
is the unital channel E with noise operators Jk = n
m=1 J
(m)
k for k = x, y, z,
where J
(1)
k = σk ⊗ (12)⊗(n−1)
, J
(2)
k = 12 ⊗ σk ⊗ (12)⊗(n−2)
, etc. The collective
rotation channel is widely used in the research of quantum information.
44

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