Preface

5. Quantum Information Theory

    5.1 The density matrix
           5.1.1 The density matrix for a qubit
           5.1.2 Composite systems
           5.1.3 * The quantum copying machine

   5.2 The Schmidt decomposition

   5.3 Purification

   5.4 The Kraus representation

   5.5 Measurement of the density matrix for a qubit

   5.6 Generalized measurements
           5.6.1 * Weak measurements
     5.6.2 POVM measurements

    5.7 The Shannon entropy

    5.8 Classical data compression
           5.8.1 Shannon's noiseless coding theorem
           5.8.2 Examples of data compression

    5.9 The von Neumann entropy
           5.9.1 Example 1: source of orthogonal pure states
           5.9.2 Example 2: source of non-orthogonal pure states

    5.10 Quantum data compression
           5.10.1 Schumacher's quantum noiseless coding theorem
           5.10.2 Compression of an n-qubit message
   5.10.3 Example 1: two-qubit messages
           5.10.4 Example 2: three-qubit messages

    5.11 Accessible information
           5.11.1 The Holevo bound
           5.11.2 Example 1: two non-orthogonal pure states
           5.11.3 * Example 2: three non-orthogonal pure states

    5.12 Entanglement concentration and von Neumann entropy

    5.13 The Peres separability criterion

    5.14 * Entropies in physics
           5.14.1 * Thermodynamic entropy
           5.14.2 * Statistical entropy
       5.14.3 * Dynamical Kolmogorov-Sinai entropy

    5.15 A guide to the bibliography

6. Decoherence

      6.1 Decoherence models for a single qubit
             6.1.1 The quantum black box
             6.1.2 Measuring a quantum operation acting on a qubit
             6.1.3 Quantum circuits simulating noise channels
             6.1.4 The bit-flip channel
             6.1.5 The phase-flip channel
             6.1.6 The bit-phase-flip channel
             6.1.7 The depolarizing channel
             6.1.8 Amplitude damping
             6.1.9 Phase damping
             6.1.10 De-entanglement

     6.2 The master equation
             6.2.1 * Derivation of the master equation
             6.2.2 * The master equation and quantum operations
             6.2.3 The master equation for a single qubit

     6.3 Quantum to classical transition
             6.3.1 The Schrodinger's cat
     6.3.2 Decoherence and destruction of cat states

     6.4 * Decoherence and quantum measurements

     6.5 * Quantum chaos
             6.5.1 * Dynamical chaos in classical mechanics
             6.5.2 * Quantum chaos and the correspondence principle
             6.5.3 * Time scales of quantum chaos
             6.5.4 * Quantum chaos and Anderson localization
             6.5.5 * The hydrogen atom in a microwave field
             6.5.6 * Quantum chaos and universal spectral fluctuations
             6.5.7 * The chaos border for the quantum computer hardware
             6.5.8 * The quantum Loschmidt echo
             6.5.9 * Dynamical stability of quantum motion
             6.5.10 * Dynamical chaos and dephasing: the double-slit experiment
             6.5.11 * Entanglement and chaos

     6.6 Decoherence and quantum computation
             6.6.1 * Decoherence and quantum trajectories

     6.7 * Quantum computation and quantum chaos
             6.7.1 * Quantum versus classical errors
             6.7.2 * Static imperfections versus noisy gates

     6.8 A guide to the bibliography

7. Quantum Error Correction

     7.1 The three-qubit bit-flip code

     7.2 The three-qubit phase-flip code

     7.3 The nine-qubit Shor code

     7.4 General properties of quantum error correction
             7.4.1 The quantum Hamming bound

     7.5 * The five-qubit code

     7.6 * Classical linear codes
             7.6.1 * The Hamming codes

     7.7 * CSS codes

     7.8 Decoherence-free subspaces
             7.8.1 * Conditions for decoherence-free dynamics
             7.8.2 * The spin-boson model

     7.9 * The Zeno effect

     7.10 Fault-tolerant quantum computation
             7.10.1 Avoidance of error propagation
             7.10.2 Fault-tolerant quantum gates
             7.10.3 The noise threshold for quantum computation

     7.11 * Quantum cryptography over noisy channels

     7.12 * Quantum channels with memory

     7.13 A guide to the bibliography

8. First Experimental Implementations

      8.1 NMR quantum computation
             8.1.1 The system Hamiltonian
             8.1.2 The physical apparatus
             8.1.3 Quantum ensemble computation
             8.1.4 Refocusing
             8.1.5 Demonstration of quantum algorithms

      8.2 Cavity quantum electrodynamics
             8.2.1 Rabi oscillations
             8.2.2 Entanglement generation
             8.2.3 The quantum phase gate

      8.3 The ion-trap quantum computer
             8.3.1 The Paul trap
             8.3.2 Laser pulses
             8.3.3 Realization of the Cirac-Zoller CNOT gate
             8.3.4 Entanglement generation

      8.4 Solid state qubits
             8.4.1 Spins in semiconductors
             8.4.2 Quantum dots
             8.4.3 Superconducting qubit circuits

      8.5 Quantum communication with photons
             8.5.1 Linear optics
             8.5.2 Experimental quantum teleportation
             8.5.3 Experimental quantum-key distribution

      8.6 Problems and prospects

      8.7 A guide to the bibliography

Appendix B. Solutions to the exercises

Bibliography

Index