This is the first project report at my University. This report describes No Cloning Theorem, an introductory topic of Quantum Computation and Quantum Information Theory. The report also covers the necessary mathematics and physics.

1. Calcutta University
Computer Science and Engineering
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Term Paper - I
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No Cloning Theorem
Ritajit Majumdar
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Roll No: 91/CSE/111006
Registration No: 0029169 of 2008-09
Supervisor:
Supervisor:
Guruprasad Kar
Department of Computer
Science and Engineering
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Physics and Applied
Mathematics Unit
Pritha Banerjee
Indian Statistical Institute,
Kolkata
February 12, 2014
Calcutta University

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Abstract
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In this report, I present the idea of No Cloning Theorem, which
was proposed by Wootters and Zurek. This theorem essentially
states that non-orthogonal states of a closed quantum system
cannot be reliably distinguished, and hence cannot be copied.
The linearity of quantum mechanics prohibits the presence of a
perfect cloning device. Hence, generally speaking, it is not possible to develop a universal cloning apparatus which can clone
any arbitrary quantum state.

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Contents
1 Introduction
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2 Introductory Mathematics for Quantum Computation
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2.1 Hilbert Space . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Linear Vector Space . . . . . . . . . . . . 5
2.1.2 Inner Product Space . . . . . . . . . . . . 6
2.2 Observables and Tensor Product . . . . . . . . . . 8
2.2.1 Observables . . . . . . . . . . . . . . . . . 8
2.2.2 Tensor Products . . . . . . . . . . . . . . 8
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3 Introductory Quantum Mechanics
3.1 Postulates of Quantum Mechanics
3.1.1 State Space . . . . . . . .
3.1.2 Evolution . . . . . . . . .
3.1.3 Measurement . . . . . . .
3.1.4 Composite System . . . .
3.2 Distinguising Quantum States . .
3.2.1 Orthogonal States . . . . .
3.2.2 Non Orthogonal States . .
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4 No Cloning Theorem
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4.1 No Cloning Theorem . . . . . . . . . . . . . . . . 15
1

4. 4.1.1
4.1.2
4.1.3
Photon Emission . . . . . . . . . . . . . . 15
Linearity of Quantum Mechanics . . . . . 16
A single quantum cannot be cloned . . . . 17
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5 Conclusion
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Chapter 1
Introduction
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The modern incarnation of computer science was announced by
the great mathematician Alan Turing. He developed a model
of computation known as Turing Machine. He claimed that if
an algorithm can be performed in a computer, then there is an
equivalent algorithm for the Universal Turing Machine.
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Furthermore, Moore’s law stated that computer power will
double for constant cost roughly once in every two years. In
spite of dramatic miniaturization in computer technology in last
few decades, our basic understanding of how a computer works is
still the same (i.e. the Turing Machine). However, conventional
approaches to the fabrication of computer technology are beginning to run up against fundamental difficulties of size. Quantum
effects are beginning to interfere in the functioning of electronic
devices as they are made smaller and smaller [1].
One possible solution to the problem posed by the eventual failure of Moore’s law is to move to a different computing paradigm. Richard Feynman first proposed the concept of
a computer which exploits quantum laws. He said “Atoms on
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6. small scale behave like nothing on a large scale, for they satisfy
the laws of quantum mechanics. So, as we go down and fiddle
around with the atoms there, we are working with different laws,
and we can expect to do different things”
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A quantum computer is not just a computer following quantum laws. Rather it is a machine which can make explicit use
of certain quantum phenomena which are not present in the
classical realm - e.g. - Superposition [3.1.1 State Space], Entanglement [2] etc.
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Recent studies have shown that quantum algorithms are usually faster than classical ones. Quantum Algorithms are essentially parallel. Polynomial time quantum algorithms for some
NP problems (e.g. prime factorization) have been developed.
Quantum Cryptography holds the promise of better (nearly perfect) secrecy than classical cryptography. And information processing using quantum laws are also more efficient than their
classical counterparts.
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In this report, a basic introductory topic of quantum computation, namely No Cloning Theorem, has been described.
The report starts with the necessary mathematical and physical background and then enters the theorem.
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Chapter 2
2.1.1
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Hilbert Space
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Introductory Mathematics for
Quantum Computation
Linear Vector Space
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For Quantum Computation, the vector space of interest is Cn ,
which is the complex vector space of n dimension. The elements
of the space are called vectors, and are represented by the
column matrix
 
v1
v 
 2
.
.
.
vn
The vectors are written as |v , while their complex conjugate
(i.e. the row matrix) is written as v|.
Let |v , |w and |z be three vectors in a space V and α and β
are two scalars (usually complex numbers). Then V is a Linear
Vector Space if the following conditions are satisfied 5

8. 1. Closure: |v + |w ∈ V .
2. Closure: α |v ∈ V .
3. Commutative: |v + |w = |w + |v .
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4. Associative: α(β |v ) = (αβ) |v
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5. Associative: |v + (|w + |z ) = (|v + |w ) + |z .
6. There is a zero vector such that |v + 0 = |v .
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7. There is an additive inverse which maps a vector to the zero
vector |v + |−v = 0
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8. Distributive: α(|v + |w ) = α |v + α |w
9. Distributive: (α + β) |v = α |v + β |v
Inner Product Space
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2.1.2
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An inner product is a linear function which takes as input two
vectors and outputs a complex number. So mathematically inner product is a function that maps from V × V −→ C.
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Inner product between two vectors  and |w is computed
|v 
 
v1
w1
v 
w 
 2
 2
as w|v . So if |v =  .  and |w =  . 
.
 . 
.
.
vn
wn
then the inner product
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∗
where wi is the complex conjugate of wi .
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∗
∗
w|v = w1 w2
 
v1
v 
∗  2
· · · wn . . 
.
.
vn
Again, let |v and |w be two vectors and λ is a scalar. A
Linear Vector Space is an Inner Product Space if the following
conditions are satisfied -
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1. v|λw = λ v|w
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2. v|w = ( w|v )∗
3. v|v ≥ 0, the value is 0 iff |v = 0
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The length of a vector is defined as v =
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v|v
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And the length is called norm. A vector is said to be unit vector or normalised if its norm is 1.
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Two vectors whose inner product is zero are said to be orthogonal. A collection of mutually orthogonal, normalised
vectors is called an orthonormal set.
αi |αj = δij ,
where δij
= 0, i = j
= 0, i = j
A linear vector space with inner product is called
a Hilbert Space.
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10. 2.2
2.2.1
Observables and Tensor Product
Observables
U †U = U U † = I
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A quantum state is defined as a vector in Hilbert Space. The operators (e.g time evolution operator, quantum gates) are called
observables. Observables in Quantum mechanics are unitary
matrices. A matrix U is said to unitary if
where U † = (U ∗ )T and I is the identity matrix.
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If a state vector is of dimension n, then the observable operating on it is of dimension n × n.
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Operators in quantum mechanics are Hermitian Matrices. A
matrix H is said to be hermitian if
H = H†
Tensor Products
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2.2.2
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Thus, unlike classical gates, quantum gates are reversible.
Operating the same gate twice on the state returns the original
state.
Tensor product is a way of putting vector spaces together to
form a larger vector space. Let V and W be two vector spaces
of dimensions m and n respectively. Then the tensor product
V
W is a vector space of mn dimension [1].
Let A =
a11 a12
a21 a22
and B =
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b11 b12
b21 b22

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
b12
b22 

b12 
b22

a12 b12
a12 b22 

a22 b12 
a22 b12
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Then the tensor product A B is

b11 b12
b11
a11 b21 b22 a12 b21


b b
b
a21 11 12 a22 11
b21 b22
b21

a11 b11 a11 b12 a12 b11
a11 b21 a11 b22 a12 b21
=
a21 b11 a21 b12 a22 b11
a21 b21 a21 b22 a22 b11

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Chapter 3
3.1.1
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Postulates of Quantum Mechanics
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Introductory Quantum
Mechanics
State Space
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Postulate 1: Associated to any isolated system is a
Hilbert Space called the state space. The system is
completely defined by the state vector, which is a
unit vector in the state space [1].
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In classical computer, a bit can be either 0 or 1. In quantum
mechanics, the quantum bit or qubit is a vector in the state
space. And a qubit can be in any linear superposition of |0 or
|1 . In general, a qubit is mathematically represented as |ψ = α0 |0 + α1 |1
where α0 and α1 are complex numbers. α0 and α1 are called
the amplitudes of |0 and |1 respectively. The square of the
amplitude, |αi |2 , gives the probability that the system collapses
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13. to state |i (this will be further elaborated in the 3rd postulate).
Since the total probability is always 1, hence
|α0 |2 + |α1 |2 = 1
3.1.2
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This is called the normalisation condition. Hence, a qubit is
a unit vector in the state space.
Evolution
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Postulate 2: The evolution of a closed quantum system is described by a unitary transformation [1].
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If |ψ(t1 ) is the state of the quantum system at time t1 and
|ψ(t2 ) is the state of the system at time t2 , then
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|ψ(t2 ) = U (t1 , t2 ) |ψ(t1 )
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where U (t1 , t2 ) is a unitary matrix.
Schr¨dinger Equation
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The time evolution of a closed quantum system is given by the
Schr¨dinger equation o
i
d|ψ
dt
= H |ψ
where H is the total enery (kinetic + potential) of the system
and is called the Hamiltonian.
Integrating the equation within the time limits t1 and t2 , we
have |ψ2 = exp −i
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(t2 −t1 )
H
|ψ1

14. Now taking U ≡ exp −i
picture of time evolution.
we get back the unitary matrix
Measurement
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3.1.3
(t2 −t1 )
H
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Unlike classical physics, measurement in quantum mechanics is
not deterministic. Even if we have the complete knowledge of
a system, we can at most predict the probability of a certain
outcome from a set of possible outcomes. If we have a quantum
state |ψ = α |0 + β |1 , then the probability of getting outcome
|0 is |α|2 and that of |1 is |β|2 . After measurement, the state of
the system collapses to either |0 or |1 with the said probability.
†
p(m) = ψ| Mm Mm |ψ
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Postulate 3: Quantum measurements are described
by a collection of measurement operators {Mm }.
These are operators acting on the state space of
the system being measured. The index m refers to
the measurement outcomes that may occur in the experiment. If the state of the quantum system is |ψ
immediately before the measurement then the probability that result m occurs is [1]
and the state of the system after measurement is
Mm |ψ
†
ψ| Mm Mm |ψ
The measurement operators satisfy the completeness relation
†
m Mm Mm
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15. 3.1.4
Composite System
Postulate 4: The state space of a composite physical
system is the tensor product of the state spaces of
the component physical systems [1].
|ψ2
···
|ψn
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|ψ1
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If we have n systems, numbered 1 through n, and the ith
system is prepared in state |ψi , then the composite state of the
total system is
3.2.1
Distinguising Quantum States
Orthogonal States
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3.2
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In a classical computer, if we have a RAM of size m and
add another RAM of size n, then the composite size is m + n.
However, from this postulate, it is obvious that in a quantum
computer, if we have a qubit of dimension m and another qubit
of dimension n, then the dimension of the composite system is
mn.
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An observable M can be written in the form |m m|. This is
called Spectral Decomposition 1 .
Consider a set of projectors Mi = |ψi ψi | for i = 1, · · · , n
and M0 = I −
1
i=0 |ψi
ψi | (from Completeness Relation)
For further information, see Nielsen, Chuang Page 72
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16. So if an orthonormal state |ψi is prepared, then
p(i) = ψi | Mi |ψi = ψi |ψi ψi |ψi = 1
3.2.2
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Thus result i occurs with certainty and hence it is possible
to reliably distinguish orthonormal states |ψi .
Non Orthogonal States
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Consider two quantum states
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|ψ2
|ψ1 = |0
= α |0 + β |1
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If a measurement is performed, then |ψ1 is projected to |0
with probability 1. However, |ψ2 is also projected to |0 with
probability |α|2 .
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So, if the outcome is |0 , then it is not possible to say whether
the state was |ψ1 or |ψ2 .
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Hence, non orthogonal states cannot be reliably distinguished.
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Chapter 4
4.1
No Cloning Theorem
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No Cloning Theorem
Photon Emission
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4.1.1
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Possibly the most prominent feature that distinguishes between
classical and quantum information theory is the “no cloning theorem” which prevents in producing perfect copies of an arbitrary
quantum mechanical state [4].
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When a photon having a definite polarization encounters an excited atom, there is some probability that the atom will emit a
photon due to stimulation. If there is such an emission, then the
second photon is guaranteed to have the same polarization as
the original photon [3].
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img : en.wikipedia.org/wiki/Stimulated emission
Linearity of Quantum Mechanics
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This phenomena gave rise to the question whether using this
method (or any other method), it is possible to clone a quantum
state.
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From the 2nd postulate of quantum mechanics, we have
|ψ(t2 ) = U (t1 , t2 ) |ψ(t1 )
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Now, from Schr¨dinger equation, it is evident that quantum
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laws are linear. So if we have any arbitrary quantum state,
|ψ = α |ψ1 + β |ψ2 , and if we take,
U |ψ1 = |φ1 and U |ψ2 = |φ2
then operating U on the state |ψ U |ψ = U (α |ψ1 + β |ψ2 )
= αU |ψ1 + βU |ψ2
= α |φ1 + β |φ2
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19. 4.1.3
A single quantum cannot be cloned
The proof of this theorem is indirect, i.e. proof by contradiction.
Let us consider |A is a universal quantum cloner. Let |s be
the state of the incident photon. So we should have -
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|A |s = |As |ss
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where |As is the state of the apparatus after operation (we
are not much interested in the state of the apparatus) and |ss
represents the state of the two photons, one original and the
other its copy.
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Now, let us consider two orthogonal states |0 and |1 . So,
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|A |0 = |A0 |00
|A |1 = |A1 |11
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So for an arbitrary quantum state |ψ = α |0 + β |1 , we
expect = |Aψ
|A |ψ = |Aψ |ψψ
(α |0 + β |1 )(α |0 + β |1 )
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= |Aψ (α2 |00 + αβ |01 + αβ |10 + β 2 |11 )
(4.1)
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However, from linearity of quantum mechanics, it is evident
that
|A |ψ = |A (α |0 + β |1 )
= α |A |0 + β |A |1
= α |00 + β |11
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(4.2)

20. Equations (4.1) and (4.2) will be equal only if αβ = 0. However, for αβ to be 0, either α = 0 or β = 0.
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But if α = 0, then the term |00 vanishes in equation (4.1)
and if β = 0, then the term |11 vanishes. Thus the linearity of
quantum mechanics prohibits a perfect copy. Hence, A single
quantum cannot be cloned.
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However, the above argument does not prohibit the copying of orthogonal states. This is because orthogonal states are
distinguishable. Since non-orthogonal states are not reliably distinguishable, we cannot make a perfect copy of them.
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Hence, in general, the theorem states that it is not possible to
have a universal quantum cloner that can create perfect clones
of any arbitrary quantum state.
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Chapter 5
Conclusion
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The linearity of quantum mechanics prohibits the existance of a
universal quantum copy machine. This has both advantages and
disadvantages. The disadvantages are immediately prominent.
Unlike classical gates, fan out is not possible in quantum gates.
If a quantum computer is made, copying, at least perfect copying, will not be possible so easily. However, a huge advantage
is observable in quantum cryptography. Hackers cannot make a
copy of the data being sent.
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An immediate consequence of no-cloning theorem is that information cannot be copied. So for information sending, a protocol named quantum teleportation [5] has been proposed which
states that to send an information, the information at the sender’s
end must be destroyed. Studies are being pursued on performing
partial cloning of a quantum state [4][6][7][8].
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Bibliography
[1] Quantum Computation and Quantum Information
Nielsen, Chuang
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[2] Einstein, Podolsky, Rosen
PRL Vol 47, May 15, 1935
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[3] A single quantum cannot be cloned
Wootters, Zurek
Nature, Vol 299, 1982
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[4] S. Bandyopadhyay, G. Kar
arXiv:quant-ph/9902073v3
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[5] Bennett et. al
PRL Vol 70, number 13, March 1993
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[6] R. F. Werner
PRA Vol 58, number 3, September 1998
[7] Buzek, Hillery
PRA Vol 54, number 3, September 1996
[8] Buzek et. al
PRA vol 55, number 5, May 1997
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