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Calcutta University

Computer Science and Engineering

Term Paper - I

No Cloning Theorem

Ritajit MajumdarRoll No: 91/CSE/111006

Registration No: 0029169 of 2008-09

Supervisor:

Guruprasad Kar

Physics and AppliedMathematics Unit

Indian Statistical Institute,Kolkata

Supervisor:

Pritha Banerjee

Department of ComputerScience and Engineering

Calcutta University

February 12, 2014

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Abstract

In this report, I present the idea of No Cloning Theorem, whichwas proposed by Wootters and Zurek. This theorem essentiallystates that non-orthogonal states of a closed quantum systemcannot be reliably distinguished, and hence cannot be copied.The linearity of quantum mechanics prohibits the presence of aperfect cloning device. Hence, generally speaking, it is not pos-sible to develop a universal cloning apparatus which can cloneany arbitrary quantum state.

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Contents

1 Introduction 3

2 Introductory Mathematics for Quantum Compu-tation 52.1 Hilbert Space . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Linear Vector Space . . . . . . . . . . . . 52.1.2 Inner Product Space . . . . . . . . . . . . 6

2.2 Observables and Tensor Product . . . . . . . . . . 82.2.1 Observables . . . . . . . . . . . . . . . . . 82.2.2 Tensor Products . . . . . . . . . . . . . . 8

3 Introductory Quantum Mechanics 103.1 Postulates of Quantum Mechanics . . . . . . . . . 10

3.1.1 State Space . . . . . . . . . . . . . . . . . 103.1.2 Evolution . . . . . . . . . . . . . . . . . . 113.1.3 Measurement . . . . . . . . . . . . . . . . 123.1.4 Composite System . . . . . . . . . . . . . 13

3.2 Distinguising Quantum States . . . . . . . . . . . 133.2.1 Orthogonal States . . . . . . . . . . . . . . 133.2.2 Non Orthogonal States . . . . . . . . . . . 14

4 No Cloning Theorem 154.1 No Cloning Theorem . . . . . . . . . . . . . . . . 15

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4.1.1 Photon Emission . . . . . . . . . . . . . . 154.1.2 Linearity of Quantum Mechanics . . . . . 164.1.3 A single quantum cannot be cloned . . . . 17

5 Conclusion 19

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Chapter 1

Introduction

The modern incarnation of computer science was announced bythe great mathematician Alan Turing. He developed a modelof computation known as Turing Machine. He claimed that ifan algorithm can be performed in a computer, then there is anequivalent algorithm for the Universal Turing Machine.

Furthermore, Moore’s law stated that computer power willdouble for constant cost roughly once in every two years. Inspite of dramatic miniaturization in computer technology in lastfew decades, our basic understanding of how a computer works isstill the same (i.e. the Turing Machine). However, conventionalapproaches to the fabrication of computer technology are begin-ning to run up against fundamental difficulties of size. Quantumeffects are beginning to interfere in the functioning of electronicdevices as they are made smaller and smaller [1].

One possible solution to the problem posed by the even-tual failure of Moore’s law is to move to a different comput-ing paradigm. Richard Feynman first proposed the concept ofa computer which exploits quantum laws. He said “Atoms on

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small scale behave like nothing on a large scale, for they satisfythe laws of quantum mechanics. So, as we go down and fiddlearound with the atoms there, we are working with different laws,and we can expect to do different things”

A quantum computer is not just a computer following quan-tum laws. Rather it is a machine which can make explicit useof certain quantum phenomena which are not present in theclassical realm - e.g. - Superposition [3.1.1 State Space], Entan-glement [2] etc.

Recent studies have shown that quantum algorithms are usu-ally faster than classical ones. Quantum Algorithms are essen-tially parallel. Polynomial time quantum algorithms for someNP problems (e.g. prime factorization) have been developed.Quantum Cryptography holds the promise of better (nearly per-fect) secrecy than classical cryptography. And information pro-cessing using quantum laws are also more efficient than theirclassical counterparts.

In this report, a basic introductory topic of quantum com-putation, namely No Cloning Theorem, has been described.The report starts with the necessary mathematical and phys-ical background and then enters the theorem.

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Chapter 2

Introductory Mathematics forQuantum Computation

2.1 Hilbert Space

2.1.1 Linear Vector Space

For Quantum Computation, the vector space of interest is Cn,which is the complex vector space of n dimension. The elementsof the space are called vectors, and are represented by thecolumn matrix

v1v2...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 βare two scalars (usually complex numbers). Then V is a LinearVector Space if the following conditions are satisfied -

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1. Closure: |v〉+ |w〉 ∈ V .

2. Closure: α |v〉 ∈ V .

3. Commutative: |v〉+ |w〉 = |w〉+ |v〉.

4. Associative: α(β |v〉) = (αβ) |v〉

5. Associative: |v〉+ (|w〉+ |z〉) = (|v〉+ |w〉) + |z〉.

6. There is a zero vector such that |v〉+ 0 = |v〉.

7. There is an additive inverse which maps a vector to the zerovector |v〉+ |−v〉 = 0

8. Distributive: α(|v〉+ |w〉) = α |v〉+ α |w〉

9. Distributive: (α + β) |v〉 = α |v〉+ β |v〉

2.1.2 Inner Product Space

An inner product is a linear function which takes as input twovectors and outputs a complex number. So mathematically in-ner product is a function that maps from V × V −→ C.

Inner product between two vectors |v〉 and |w〉 is computed

as 〈w|v〉. So if |v〉 =

v1v2...vn

and |w〉 =

w1

w2...wn

then the inner product

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〈w|v〉 =(w∗1 w∗2 · · · w∗n

).

v1v2...vn

where w∗i is the complex conjugate of wi.

Again, let |v〉 and |w〉 be two vectors and λ is a scalar. ALinear Vector Space is an Inner Product Space if the followingconditions are satisfied -

1. 〈v|λw〉 = λ 〈v|w〉

2. 〈v|w〉 = (〈w|v〉)∗

3. 〈v|v〉 ≥ 0, the value is 0 iff |v〉 = 0

The length of a vector is defined as -

‖ v ‖=√〈v|v〉

And the length is called norm. A vector is said to be unit vec-tor or normalised if its norm is 1.

Two vectors whose inner product is zero are said to be or-thogonal. A collection of mutually orthogonal, normalisedvectors is called an orthonormal set.

〈αi|αj〉 = δij, where δij

{= 0, i = j

6= 0, i 6= j

A linear vector space with inner product is called

a Hilbert Space.

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2.2 Observables and Tensor Product

2.2.1 Observables

A quantum state is defined as a vector in Hilbert Space. The op-erators (e.g time evolution operator, quantum gates) are calledobservables. Observables in Quantum mechanics are unitarymatrices. A matrix U is said to unitary if

U †U = UU † = I

where U † = (U ∗)T and I is the identity matrix.

If a state vector is of dimension n, then the observable oper-ating on it is of dimension n× n.

Operators in quantum mechanics are Hermitian Matrices. Amatrix H is said to be hermitian if

H = H†

Thus, unlike classical gates, quantum gates are reversible.Operating the same gate twice on the state returns the originalstate.

2.2.2 Tensor Products

Tensor product is a way of putting vector spaces together toform a larger vector space. Let V and W be two vector spacesof dimensions m and n respectively. Then the tensor productV⊗

W is a vector space of mn dimension [1].

Let A =

(a11 a12a21 a22

)and B =

(b11 b12b21 b22

)

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Then the tensor product A⊗

B isa11(b11 b12b21 b22

)a12

(b11 b12b21 b22

)a21

(b11 b12b21 b22

)a22

(b11 b12b21 b22

)

=

a11b11 a11b12 a12b11 a12b12a11b21 a11b22 a12b21 a12b22a21b11 a21b12 a22b11 a22b12a21b21 a21b22 a22b11 a22b12

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Chapter 3

Introductory QuantumMechanics

3.1 Postulates of Quantum Mechanics

3.1.1 State Space

Postulate 1: Associated to any isolated system is aHilbert Space called the state space. The system is

completely defined by the state vector, which is a

unit vector in the state space [1].

In classical computer, a bit can be either 0 or 1. In quantummechanics, the quantum bit or qubit is a vector in the statespace. 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 calledthe amplitudes of |0〉 and |1〉 respectively. The square of theamplitude, |αi|2, gives the probability that the system collapses

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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

This is called the normalisation condition. Hence, a qubit isa unit vector in the state space.

3.1.2 Evolution

Postulate 2: The evolution of a closed quantum sys-

tem is described by a unitary transformation [1].

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

|ψ(t2)〉 = U(t1, t2) |ψ(t1)〉

where U(t1, t2) is a unitary matrix.

Schrodinger Equation

The time evolution of a closed quantum system is given by theSchrodinger equation -

i~d|ψ〉dt = H |ψ〉

where H is the total enery (kinetic + potential) of the systemand is called the Hamiltonian.

Integrating the equation within the time limits t1 and t2, wehave -

|ψ2〉 = exp −i~(t2−t1)H |ψ1〉

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Now taking U ≡ exp −i~(t2−t1)H we get back the unitary matrixpicture of time evolution.

3.1.3 Measurement

Unlike classical physics, measurement in quantum mechanics isnot deterministic. Even if we have the complete knowledge ofa system, we can at most predict the probability of a certainoutcome from a set of possible outcomes. If we have a quantumstate |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 ofthe system collapses to either |0〉 or |1〉 with the said probability.

Postulate 3: Quantum measurements are described

by a collection of measurement operators {Mm}.These are operators acting on the state space ofthe system being measured. The index m refers to

the measurement outcomes that may occur in the ex-

periment. If the state of the quantum system is |ψ〉immediately before the measurement then the prob-ability that result m occurs is [1]

p(m) = 〈ψ|M †mMm |ψ〉

and the state of the system after measurement is

Mm |ψ〉√〈ψ|M †

mMm |ψ〉

The measurement operators satisfy the completeness relation∑mM

†mMm = I

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3.1.4 Composite System

Postulate 4: The state space of a composite physicalsystem is the tensor product of the state spaces ofthe component physical systems [1].

If we have n systems, numbered 1 through n, and the ith

system is prepared in state |ψi〉, then the composite state of thetotal system is

|ψ1〉⊗|ψ2〉

⊗· · ·⊗|ψn〉

In a classical computer, if we have a RAM of size m andadd another RAM of size n, then the composite size is m + n.However, from this postulate, it is obvious that in a quantumcomputer, if we have a qubit of dimension m and another qubitof dimension n, then the dimension of the composite system ismn.

3.2 Distinguising Quantum States

3.2.1 Orthogonal States

An observable M can be written in the form |m〉 〈m|. This iscalled Spectral Decomposition 1.

Consider a set of projectors -

Mi = |ψi〉 〈ψi| for i = 1, · · · , n

and M0 = I −∑

i6=0 |ψi〉 〈ψi| (from Completeness Relation)

1For further information, see Nielsen, Chuang Page 72

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So if an orthonormal state |ψi〉 is prepared, then

p(i) = 〈ψi|Mi |ψi〉 = 〈ψi|ψi〉 〈ψi|ψi〉 = 1

Thus result i occurs with certainty and hence it is possibleto reliably distinguish orthonormal states |ψi〉.

3.2.2 Non Orthogonal States

Consider two quantum states

|ψ1〉 = |0〉|ψ2〉 = α |0〉+ β |1〉

If a measurement is performed, then |ψ1〉 is projected to |0〉with probability 1. However, |ψ2〉 is also projected to |0〉 withprobability |α|2.

So, if the outcome is |0〉, then it is not possible to say whetherthe state was |ψ1〉 or |ψ2〉.

Hence, non orthogonal states cannot be reliably distinguished.

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Chapter 4

No Cloning Theorem

4.1 No Cloning Theorem

Possibly the most prominent feature that distinguishes betweenclassical and quantum information theory is the “no cloning the-orem” which prevents in producing perfect copies of an arbitraryquantum mechanical state [4].

4.1.1 Photon Emission

When a photon having a definite polarization encounters an ex-cited atom, there is some probability that the atom will emit aphoton due to stimulation. If there is such an emission, then thesecond photon is guaranteed to have the same polarization asthe original photon [3].

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img : en.wikipedia.org/wiki/Stimulated emission

This phenomena gave rise to the question whether using thismethod (or any other method), it is possible to clone a quantumstate.

4.1.2 Linearity of Quantum Mechanics

From the 2nd postulate of quantum mechanics, we have

|ψ(t2)〉 = U(t1, t2) |ψ(t1)〉

Now, from Schrodinger equation, it is evident that quantumlaws 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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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〉 bethe state of the incident photon. So we should have -

|A〉 |s〉 = |As〉 |ss〉

where |As〉 is the state of the apparatus after operation (weare not much interested in the state of the apparatus) and |ss〉represents the state of the two photons, one original and theother its copy.

Now, let us consider two orthogonal states |0〉 and |1〉. So,

|A〉 |0〉 = |A0〉 |00〉|A〉 |1〉 = |A1〉 |11〉

So for an arbitrary quantum state |ψ〉 = α |0〉 + β |1〉, weexpect -

|A〉 |ψ〉 = |Aψ〉 |ψψ〉= |Aψ〉 (α |0〉+ β |1〉)(α |0〉+ β |1〉)

= |Aψ〉 (α2 |00〉+ αβ |01〉+ αβ |10〉+ β2 |11〉) (4.1)

However, from linearity of quantum mechanics, it is evidentthat

|A〉 |ψ〉 = |A〉 (α |0〉+ β |1〉)= α |A〉 |0〉+ β |A〉 |1〉

= α |00〉+ β |11〉 (4.2)

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Equations (4.1) and (4.2) will be equal only if αβ = 0. How-ever, for αβ to be 0, either α = 0 or β = 0.

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 ofquantum mechanics prohibits a perfect copy. Hence, A singlequantum cannot be cloned.

However, the above argument does not prohibit the copy-ing of orthogonal states. This is because orthogonal states aredistinguishable. Since non-orthogonal states are not reliably dis-tinguishable, we cannot make a perfect copy of them.

Hence, in general, the theorem states that it is not possible tohave a universal quantum cloner that can create perfect clonesof any arbitrary quantum state.

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Chapter 5

Conclusion

The linearity of quantum mechanics prohibits the existance of auniversal quantum copy machine. This has both advantages anddisadvantages. 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 copy-ing, will not be possible so easily. However, a huge advantageis observable in quantum cryptography. Hackers cannot make acopy of the data being sent.

An immediate consequence of no-cloning theorem is that in-formation cannot be copied. So for information sending, a pro-tocol named quantum teleportation [5] has been proposed whichstates that to send an information, the information at the sender’send must be destroyed. Studies are being pursued on performingpartial cloning of a quantum state [4][6][7][8].

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Bibliography

[1] Quantum Computation and Quantum InformationNielsen, Chuang

[2] Einstein, Podolsky, RosenPRL Vol 47, May 15, 1935

[3] A single quantum cannot be clonedWootters, ZurekNature, Vol 299, 1982

[4] S. Bandyopadhyay, G. KararXiv:quant-ph/9902073v3

[5] Bennett et. alPRL Vol 70, number 13, March 1993

[6] R. F. WernerPRA Vol 58, number 3, September 1998

[7] Buzek, HilleryPRA Vol 54, number 3, September 1996

[8] Buzek et. alPRA vol 55, number 5, May 1997

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