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Chapter 1 Chapter 2 Preliminaries Measurements To Vera Rae 3 Contents 1 Preliminaries 14 1.1 Elements of Linear Algebra ............................ 17 1.2 Hilbert Spaces and Dirac Notations ........................ 23 1.3 Hermitian and Unitary Operators; Projectors. .................. 27 1.4 Postulates of Quantum Mechanics ......................... 34 1.5 Quantum State Postulate ............................. 36 1.6 Dynamics Postulate ................................. 42 1.7 Measurement Postulate ............................... 47 1.8 Linear Algebra and Systems Dynamics ...................... 50 1.9 Symmetry and Dynamic Evolution ........................ 52 1.10 Uncertainty Principle; Minimum Uncertainty States ............... 54 1.11 Pure and Mixed Quantum States ......................... 55 1.12 Entanglement; Bell States ............................. 57 1.13 Quantum Information ............................... 59 1.14 Physical Realization of Quantum Information Processing Systems ....... 65 1.15 Universal Computers; The Circuit Model of Computation ............ 68 1.16 Quantum Gates, Circuits, and Quantum Computers ............... 74 1.17 Universality of Quantum Gates; Solovay-Kitaev Theorem ............ 79 1.18 Quantum Computational Models and Quantum Algorithms . ........ 82 1.19 Deutsch, Deutsch-Jozsa, Bernstein-Vazirani, and Simon Oracles ........ 89 1.20 Quantum Phase Estimation ............................ 96 1.21 Walsh-Hadamard and Quantum Fourier Transforms ...............102 1.22 Quantum Parallelism and Reversible Computing .................107 1.23 Grover Search Algorithm ..............................111 1.24 Amplitude Amplification and Fixed-Point Quantum Search ...........123 1.25 Error Models and Quantum Algorithms ......................130 1.26 History Notes ....................................132 1.27 Summary ......................................136 1.28 Exercises and Problems ...............................137 2 Measurements and Quantum Information 142 2.1 Measurements and Physical Reality ........................144 2.2 Copenhagen Interpretation of Quantum Mechanics ...............147 2.3 Mixed States and the Density Operator ......................149 2.4 Purification of Mixed States ............................156 2.5 Born Rule ......................................158 2.6 Measurement Operators ..............................159 2.7 Projective Measurements ..............................161 2.8 Positive Operator Valued Measures (POVM) ...................164 2.9 Neumark Theorem .................................167 2.10 Gleason Theorem ..................................168 2.11 Mixed Ensembles and their Time Evolution ...................172 2.12 Bipartite Systems; Schmidt Decomposition ....................174 4 2.13 Measurements of Bipartite Systems ........................176 2.14 Operator-Sum (Kraus) Representation ......................182 2.15 Entanglement; Monogamy of Entanglement . ...................185 2.16 Einstein-Podolski-Rosen (EPR) Thought Experiment ..............189 2.17 Hidden Variables ..................................193 2.18 Bell and CHSH Inequalities ............................199 2.19 Violation of Bell Inequality .............................202 2.20 Entanglement and Hidden Variables ........................206 2.21 Quantum and Classical Correlations ........................208 2.22 Measurements and Quantum Circuits .......................210 2.23 Measurements and Ancilla Qubits .........................214 2.24 Measurements and Distinguishability of Quantum States ............217 2.25 Measurements and an Axiomatic Quantum Theory ...............221 2.26 History Notes ....................................223 2.27 Summary and Further Readings ..........................225 2.28 Exercises and Problems ...............................228 3 Classical and Quantum Information Theory 230 3.1 The Physical Support of Information .......................233 3.2 Entropy; Thermodynamic Entropy ........................236 3.3 Shannon Entropy ..................................241 3.4 Shannon Source Coding ..............................251 3.5 Mutual Information; Relative Entropy ......................255 3.6 Fano Inequality; Data Processing Inequality ...................259 3.7 Classical Information Transmission through Discrete Channels .........261 3.8 Trace Distance and Fidelity ............................267 3.9 von Neumann Entropy ...............................269 3.10 Joint, Conditional, and Relative von Neumann Entropy .............274 3.11 Trace Distance and Fidelity of Mixed Quantum States .............275 3.12 Accessible Information in a Quantum Measurement; Holevo Bound ......282 3.13 No Broadcasting Theorem for General Mixed States ...............292 3.14 Schumacher Compression ..............................295 3.15 Quantum Channels .................................297 3.16 Quantum Erasure ..................................301 3.17 Classical Information Capacity of Noiseless Quantum Channels .........306 3.18 Entropy Exchange, Entanglement Fidelity, and Coherent Information. .....312 3.19 Quantum Fano and Data Processing Inequalities .................318 3.20 Reversible Extraction of Classical Information from Quantum Information . 322 3.21 Noisy Quantum Channels . ..........................324 3.22 Holevo-Schumacher-Westmoreland Noisy Quantum Channel Encoding Theorem 329 3.23 Capacity of Noisy Quantum Channels .......................334 3.24 Entanglement-Assisted Capacity of Quantum Channels .............338 3.25 Additivity and Quantum Channel Capacity . ................342 3.26 Applications of Information Theory ........................345 3.27 History Notes ....................................347 5 3.28 Summary and Further Readings ..........................348 3.29 Exercises and Problems ...............................351 4 Classical Error Correcting Codes 355 4.1 Informal Introduction to Error Detection and Error Correction .........357 4.2 Block Codes. Decoding Policies ..........................359 4.3 Error Correcting and Detecting Capabilities of a Block Code . .......363 4.4 Algebraic Structures and Coding Theory .....................366 4.5 Linear Codes ....................................375 4.6 Syndrome and Standard Array Decoding of Linear Codes ............383 4.7 Hamming, Singleton, Gilbert-Varshamov, and Plotkin Bounds .........387 4.8 Hamming Codes ...................................392 4.9 Proper Ordering, and the Fast Walsh-Hadamard Transform ...........394 4.10 Reed-Muller Codes .................................400 4.11 Cyclic Codes ....................................405 4.12 Encoding and Decoding Cyclic Codes .......................410 4.13 The Minimum Distance of a Cyclic Code; BCH Bound .............421 4.14 Burst Errors. Interleaving .............................424 4.15 Reed-Solomon Codes ................................427 4.16 Convolutional Codes ................................438 4.17 Product Codes . .................................445 4.18 Serially Concatenated Codes and Decoding Complexity .............446 4.19 Parallel Concatenated Codes - Turbo Codes ...................449 4.20 History Notes ....................................453 4.21 Summary and Further Readings ..........................454 4.22 Exercises and Problems ...............................457 5 Quantum Error Correcting Codes 461 5.1 Quantum Error Correction .............................463 5.2 A Necessary Condition for the Existence of a Quantum Code ..........468 5.3 Quantum Hamming Bound .............................469 5.4 Scale-up and Slow-down ..............................470 5.5 A Repetitive Quantum Code for a Single Bit-flip Error .............471 5.6 A Repetitive Quantum Code for a Single Phase-flip Error ............478 5.7 The Nine Qubit Error Correcting Code of Shor .................483 5.8 The Seven Qubit Error Correcting Code of Steane ................485 5.9 An Inequality for Representations in Different Bases ...............490 5.10 Calderbank-Shor-Steane (CSS) Codes .......................494 5.11 The Pauli Group ..................................500 5.12 Stabilizer Codes ...................................503 5.13 Stabilizers for Perfect Quantum Codes ......................512 5.14 Quantum Restoration Circuits ...........................515 5.15 Quantum Codes over GF (pk) ...........................518 5.16 Quantum Reed-Solomon Codes ..........................521 5.17 Concatenated Quantum Codes ...........................527 6 5.18 Quantum Convolutional and Quantum Tail-Biting Codes ............528 5.19 Correction of Time-Correlated Quantum Errors .................538 5.20 Quantum Error Correcting Codes as Subsystems .................541 5.21 Bacon-Shor Code ..................................544 5.22 Operator Quantum Error Correction .......................549 5.23 Stabilizers for Operator Quantum Error Correction ...............553 5.24 Correction of Systematic Errors Based on Fixed-Point Quantum Search ....555 5.25 Reliable Quantum Gates and Quantum Error Correction ............557 5.26 History Notes ....................................560 5.27 Summary and Further Readings ..........................560 5.28 Exercises and Problems ...............................562 6 Physical Realization of Quantum Information Processing Systems 565 6.1 Requirements for Physical Implementations of Quantum Computers ......567 6.2 Cold Ion Traps ...................................573 6.3 First Experimental Demonstration of a Quantum Logic Gate ..........583 6.4 Trapped Ions in Thermal Motion .........................588 6.5 Entanglement
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