Aug 28, 2014

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.

- Calcutta University Computer Science and Engineering AJ UM DA R Term Paper - I IT AJ I T M No Cloning Theorem Ritajit Majumdar IG HT -R Roll No: 91/CSE/111006 Registration No: 0029169 of 2008-09 Supervisor: Supervisor: Guruprasad Kar Department of Computer Science and Engineering CO PY R Physics and Applied Mathematics Unit Pritha Banerjee Indian Statistical Institute, Kolkata February 12, 2014 Calcutta University
- R AJ UM DA Abstract CO PY R IG HT -R IT AJ I T M 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.
- AJ UM DA R Contents 1 Introduction 3 IG HT -R IT AJ I T M 2 Introductory Mathematics for Quantum Computation 5 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 CO PY R 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 10 10 11 12 13 13 13 14 4 No Cloning Theorem 15 4.1 No Cloning Theorem . . . . . . . . . . . . . . . . 15 1
- 4.1.1 4.1.2 4.1.3 Photon Emission . . . . . . . . . . . . . . 15 Linearity of Quantum Mechanics . . . . . 16 A single quantum cannot be cloned . . . . 17 19 CO PY R IG HT -R IT AJ I T M AJ UM DA R 5 Conclusion 2
- AJ UM DA R Chapter 1 Introduction -R IT AJ I T M 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. CO PY R IG HT Furthermore, Moores 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 diculties of size. Quantum eects 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 Moores law is to move to a dierent computing paradigm. Richard Feynman rst proposed the concept of a computer which exploits quantum laws. He said Atoms on 3
- small scale behave like nothing on a large scale, for they satisfy the laws of quantum mechanics. So, as we go down and ddle around with the atoms there, we are working with dierent laws, and we can expect to do dierent things AJ UM DA R 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. IG HT -R IT AJ I T M 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 ecient than their classical counterparts. CO PY R 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. 4
- AJ UM DA R Chapter 2 2.1.1 T Hilbert Space IT AJ I 2.1 M Introductory Mathematics for Quantum Computation Linear Vector Space CO PY R IG HT -R 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 and are two scalars (usually complex numbers). Then V is a Linear Vector Space if the following conditions are satised 5
- 1. Closure: |v + |w V . 2. Closure: |v V . 3. Commutative: |v + |w = |w + |v . R 4. Associative: ( |v ) = () |v AJ UM DA 5. Associative: |v + (|w + |z ) = (|v + |w ) + |z . 6. There is a zero vector such that |v + 0 = |v . M 7. There is an additive inverse which maps a vector to the zero vector |v + |v = 0 IT AJ I T 8. Distributive: (|v + |w ) = |v + |w 9. Distributive: ( + ) |v = |v + |v Inner Product Space -R 2.1.2 PY R IG HT 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. CO 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 6
- AJ UM DA where wi is the complex conjugate of wi . R w|v = w1 w2 v1 v 2 wn . . . . vn Again, let |v and |w be two vectors and is a scalar. A Linear Vector Space is an Inner Product Space if the following conditions are satised - M 1. v|w = v|w IT AJ I T 2. v|w = ( w|v ) 3. v|v 0, the value is 0 i |v = 0 -R The length of a vector is dened as v = IG HT v|v PY R And the length is called norm. A vector is said to be unit vector or normalised if its norm is 1. CO Two vectors whose inner product is zero are said to be orthogonal. A collection of mutually orthogonal, normalised vectors is called an orthonormal set. i |j = ij , where ij = 0, i = j = 0, i = j A linear vector space with inner product is called a Hilbert Space. 7
- 2.2 2.2.1 Observables and Tensor Product Observables U U = U U = I AJ UM DA R A quantum state is dened 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. IT AJ I T M If a state vector is of dimension n, then the observable operating on it is of dimension n n. -R Operators in quantum mechanics are Hermitian Matrices. A matrix H is said to be hermitian if H = H Tensor Products CO 2.2.2 PY R IG HT 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 = 8 b11 b12 b21 b22
- T IT AJ I -R IG HT PY R CO 9 R AJ UM DA b12 b22 b12 b22 a12 b12 a12 b22 a22 b12 a22 b12 M 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
- AJ UM DA R Chapter 3 3.1.1 T Postulates of Quantum Mechanics IT AJ I 3.1 M Introductory Quantum Mechanics State Space IG HT -R 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]. CO PY R 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 | = 0 |0 + 1 |1 where 0 and 1 are complex numbers. 0 and 1 are called the amplitudes of |0 and |1 respectively. The square of the amplitude, |i |2 , gives the probability that the system collapses 10
- to state |i (this will be further elaborated in the 3rd postulate). Since the total probability is always 1, hence |0 |2 + |1 |2 = 1 3.1.2 AJ UM DA R This is called the normalisation condition. Hence, a qubit is a unit vector in the state sp

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