Book contents
- Frontmatter
- Contents
- Foreword
- Introduction
- Acknowledgments
- 1 Probability basics
- 2 Probability distributions
- 3 Measuring information
- 4 Entropy
- 5 Mutual information and more entropies
- 6 Differential entropy
- 7 Algorithmic entropy and Kolmogorov complexity
- 8 Information coding
- 9 Optimal coding and compression
- 10 Integer, arithmetic, and adaptive coding
- 11 Error correction
- 12 Channel entropy
- 13 Channel capacity and coding theorem
- 14 Gaussian channel and Shannon–Hartley theorem
- 15 Reversible computation
- 16 Quantum bits and quantum gates
- 17 Quantum measurements
- 18 Qubit measurements, superdense coding, and quantum teleportation
- 19 Deutsch–Jozsa, quantum Fourier transform, and Grover quantum database search algorithms
- 20 Shor's factorization algorithm
- 21 Quantum information theory
- 22 Quantum data compression
- 23 Quantum channel noise and channel capacity
- 24 Quantum error correction
- 25 Classical and quantum cryptography
- Appendix A (Chapter 4) Boltzmann's entropy
- Appendix B (Chapter 4) Shannon's entropy
- Appendix C (Chapter 4) Maximum entropy of discrete sources
- Appendix D (Chapter 5) Markov chains and the second law of thermodynamics
- Appendix E (Chapter 6) From discrete to continuous entropy
- Appendix F (Chapter 8) Kraft–McMillan inequality
- Appendix G (Chapter 9) Overview of data compression standards
- Appendix H (Chapter 10) Arithmetic coding algorithm
- Appendix I (Chapter 10) Lempel–Ziv distinct parsing
- Appendix J (Chapter 11) Error-correction capability of linear block codes
- Appendix K (Chapter 13) Capacity of binary communication channels
- Appendix L (Chapter 13) Converse proof of the channel coding theorem
- Appendix M (Chapter 16) Bloch sphere representation of the qubit
- Appendix N (Chapter 16) Pauli matrices, rotations, and unitary operators
- Appendix O (Chapter 17) Heisenberg uncertainty principle
- Appendix P (Chapter 18) Two-qubit teleportation
- Appendix Q (Chapter 19) Quantum Fourier transform circuit
- Appendix R (Chapter 20) Properties of continued fraction expansion
- Appendix S (Chapter 20) Computation of inverse Fourier transform in the factorization of N = 21 through Shor's algorithm
- Appendix T (Chapter 20) Modular arithmetic and Euler's theorem
- Appendix U (Chapter 21) Klein's inequality
- Appendix V (Chapter 21) Schmidt decomposition of joint pure states
- Appendix W (Chapter 21) State purification
- Appendix X (Chapter 21) Holevo bound
- Appendix Y (Chapter 25) Polynomial byte representation and modular multiplication
- Index
- References
16 - Quantum bits and quantum gates
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Foreword
- Introduction
- Acknowledgments
- 1 Probability basics
- 2 Probability distributions
- 3 Measuring information
- 4 Entropy
- 5 Mutual information and more entropies
- 6 Differential entropy
- 7 Algorithmic entropy and Kolmogorov complexity
- 8 Information coding
- 9 Optimal coding and compression
- 10 Integer, arithmetic, and adaptive coding
- 11 Error correction
- 12 Channel entropy
- 13 Channel capacity and coding theorem
- 14 Gaussian channel and Shannon–Hartley theorem
- 15 Reversible computation
- 16 Quantum bits and quantum gates
- 17 Quantum measurements
- 18 Qubit measurements, superdense coding, and quantum teleportation
- 19 Deutsch–Jozsa, quantum Fourier transform, and Grover quantum database search algorithms
- 20 Shor's factorization algorithm
- 21 Quantum information theory
- 22 Quantum data compression
- 23 Quantum channel noise and channel capacity
- 24 Quantum error correction
- 25 Classical and quantum cryptography
- Appendix A (Chapter 4) Boltzmann's entropy
- Appendix B (Chapter 4) Shannon's entropy
- Appendix C (Chapter 4) Maximum entropy of discrete sources
- Appendix D (Chapter 5) Markov chains and the second law of thermodynamics
- Appendix E (Chapter 6) From discrete to continuous entropy
- Appendix F (Chapter 8) Kraft–McMillan inequality
- Appendix G (Chapter 9) Overview of data compression standards
- Appendix H (Chapter 10) Arithmetic coding algorithm
- Appendix I (Chapter 10) Lempel–Ziv distinct parsing
- Appendix J (Chapter 11) Error-correction capability of linear block codes
- Appendix K (Chapter 13) Capacity of binary communication channels
- Appendix L (Chapter 13) Converse proof of the channel coding theorem
- Appendix M (Chapter 16) Bloch sphere representation of the qubit
- Appendix N (Chapter 16) Pauli matrices, rotations, and unitary operators
- Appendix O (Chapter 17) Heisenberg uncertainty principle
- Appendix P (Chapter 18) Two-qubit teleportation
- Appendix Q (Chapter 19) Quantum Fourier transform circuit
- Appendix R (Chapter 20) Properties of continued fraction expansion
- Appendix S (Chapter 20) Computation of inverse Fourier transform in the factorization of N = 21 through Shor's algorithm
- Appendix T (Chapter 20) Modular arithmetic and Euler's theorem
- Appendix U (Chapter 21) Klein's inequality
- Appendix V (Chapter 21) Schmidt decomposition of joint pure states
- Appendix W (Chapter 21) State purification
- Appendix X (Chapter 21) Holevo bound
- Appendix Y (Chapter 25) Polynomial byte representation and modular multiplication
- Index
- References
Summary
This chapter represents our first step into quantum information theory (QIT). The key to operating such a transition is to become familiar with the concept of the quantum bit, or qubit, which is a probabilistic superposition of the classical 0 and 1 bits. In the quantum world, the classical 0 and 1 bits become the pure states |0〉 and |1〉, respectively. It is as if a coin can be classically in either heads or tails states, but is now allowed to exist in a superposition of both! Then I show that qubits can be physically transformed by the action of unitary matrices, which are also called operators. I show that such qubit transformations, resulting from any qubit manipulation, can be described by rotations on a 2D surface, which is referred to as the Bloch sphere. The Pauli matrices are shown to generate all such unitary transformations. These transformations are reversible, because they are characterized by unitary matrices; this property always makes it possible to trace the input information carried by qubits. I will then describe different types of elementary quantum computations performed by elementary quantum gates, forming a veritable “zoo” of unitary operators, called I, X, Y, Z, H, CNOT, CCNOT, CROSSOVER or SWAP, controlled-U, and controlled-controlled-U. These gates can be used to form quantum circuits, involving any number of qubits, and of which several examples and tools for analysis are provided. Finally, the concept of tensor product, as progressively introduced through the above description, is eventually formalized.
- Type
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- Information
- Classical and Quantum Information TheoryAn Introduction for the Telecom Scientist, pp. 304 - 332Publisher: Cambridge University PressPrint publication year: 2009