Quantum Information Science

  1. Core Concepts of Quantum Information
    1. The Quantum Bit (Qubit)
      1. Definition and Properties
        1. Two-Level Quantum Systems
          1. Qubit as Unit of Quantum Information
            1. Comparison with Classical Bits
            2. Physical Realizations
              1. Spin-1/2 Systems
                1. Polarization of Photons
                  1. Energy Levels in Atoms
                    1. Superconducting Circuits
                    2. Mathematical Representation
                      1. State Vectors in Two-Dimensional Hilbert Space
                        1. Computational Basis States
                          1. General Qubit States
                          2. Superposition
                            1. Linear Combinations of Basis States
                              1. Superposition Principle
                                1. Physical Interpretation
                                  1. Measurement of Superposition States
                                  2. Qubit Visualization
                                    1. The Bloch Sphere
                                      1. Spherical Coordinates
                                        1. Geometric Representation
                                          1. Bloch Vector
                                          2. Pure States on the Bloch Sphere
                                            1. Mixed States and the Bloch Ball
                                              1. Geometric Operations
                                            2. Multiple Qubit Systems
                                              1. Two-Qubit Systems
                                                1. Tensor Product Structure
                                                  1. Four-Dimensional Hilbert Space
                                                    1. Computational Basis
                                                      1. General Two-Qubit States
                                                      2. Multi-Qubit Generalizations
                                                        1. n-Qubit Systems
                                                          1. Exponential Growth of Hilbert Space
                                                            1. Computational Complexity
                                                            2. Basis States and Representations
                                                              1. Computational Basis
                                                                1. Bell Basis
                                                                  1. Other Useful Bases
                                                                    1. Change of Basis Transformations
                                                                    2. Separable and Entangled States
                                                                      1. Separable States
                                                                        1. Product States
                                                                          1. Local Operations
                                                                          2. Entangled States
                                                                            1. Definition and Characterization
                                                                              1. Examples of Entangled States
                                                                                1. Entanglement Detection
                                                                            2. Quantum Entanglement
                                                                              1. Fundamental Concepts
                                                                                1. Definition of Entanglement
                                                                                  1. Nonlocal Correlations
                                                                                    1. Monogamy of Entanglement
                                                                                      1. Entanglement vs Classical Correlations
                                                                                      2. Bell States
                                                                                        1. Four Maximally Entangled States
                                                                                          1. Construction and Properties
                                                                                            1. Bell State Measurements
                                                                                              1. Applications in Protocols
                                                                                              2. Historical and Philosophical Context
                                                                                                1. EPR Paradox
                                                                                                  1. Einstein-Podolsky-Rosen Argument
                                                                                                    1. Local Realism
                                                                                                      1. Hidden Variable Theories
                                                                                                      2. Bell's Theorem
                                                                                                        1. Bell Inequalities
                                                                                                          1. CHSH Inequality
                                                                                                            1. Experimental Violations
                                                                                                              1. Loopholes and Tests
                                                                                                            2. Quantifying Entanglement
                                                                                                              1. Entanglement Measures
                                                                                                                1. Entanglement Entropy
                                                                                                                  1. Concurrence
                                                                                                                    1. Negativity
                                                                                                                      1. Entanglement of Formation
                                                                                                                      2. Entanglement Monotones
                                                                                                                        1. Operational Measures
                                                                                                                        2. Multipartite Entanglement
                                                                                                                          1. Three-Qubit Systems
                                                                                                                            1. GHZ States
                                                                                                                              1. W States
                                                                                                                                1. Classification of Multipartite Entanglement
                                                                                                                              2. Quantum States and Representations
                                                                                                                                1. Pure States
                                                                                                                                  1. State Vector Representation
                                                                                                                                    1. Normalization and Phases
                                                                                                                                      1. Equivalence Classes
                                                                                                                                      2. Mixed States
                                                                                                                                        1. Statistical Mixtures
                                                                                                                                          1. Density Matrix Formalism
                                                                                                                                            1. Definition and Properties
                                                                                                                                              1. Trace and Positivity
                                                                                                                                                1. Hermiticity
                                                                                                                                                2. Purity and Mixedness
                                                                                                                                                  1. Purity Measure
                                                                                                                                                    1. Maximally Mixed States
                                                                                                                                                      1. Relationship to Entropy
                                                                                                                                                    2. Partial Trace and Reduced States
                                                                                                                                                      1. Partial Trace Operation
                                                                                                                                                        1. Reduced Density Matrices
                                                                                                                                                          1. Subsystem States
                                                                                                                                                            1. Entanglement and Reduced States
                                                                                                                                                            2. State Tomography
                                                                                                                                                              1. Quantum State Reconstruction
                                                                                                                                                                1. Measurement Strategies
                                                                                                                                                                  1. Tomographic Completeness
                                                                                                                                                                2. Quantum Operations and Dynamics
                                                                                                                                                                  1. Unitary Evolution
                                                                                                                                                                    1. Reversible Quantum Dynamics
                                                                                                                                                                      1. Conservation Laws
                                                                                                                                                                        1. Schrödinger Picture vs Heisenberg Picture
                                                                                                                                                                        2. Quantum Gates
                                                                                                                                                                          1. Single-Qubit Gates
                                                                                                                                                                            1. Pauli Gates
                                                                                                                                                                              1. X Gate (Bit Flip)
                                                                                                                                                                                1. Y Gate
                                                                                                                                                                                  1. Z Gate (Phase Flip)
                                                                                                                                                                                  2. Hadamard Gate
                                                                                                                                                                                    1. Superposition Creation
                                                                                                                                                                                      1. Basis Transformation
                                                                                                                                                                                      2. Phase Gates
                                                                                                                                                                                        1. S Gate
                                                                                                                                                                                          1. T Gate
                                                                                                                                                                                            1. General Phase Gates
                                                                                                                                                                                            2. Rotation Gates
                                                                                                                                                                                              1. RX Gate
                                                                                                                                                                                                1. RY Gate
                                                                                                                                                                                                  1. RZ Gate
                                                                                                                                                                                                    1. Arbitrary Single-Qubit Rotations
                                                                                                                                                                                                  2. Two-Qubit Gates
                                                                                                                                                                                                    1. Controlled-NOT (CNOT) Gate
                                                                                                                                                                                                      1. Definition and Properties
                                                                                                                                                                                                        1. Entangling Capability
                                                                                                                                                                                                        2. Controlled-Z (CZ) Gate
                                                                                                                                                                                                          1. Controlled Phase Gates
                                                                                                                                                                                                            1. SWAP Gate
                                                                                                                                                                                                              1. iSWAP Gate
                                                                                                                                                                                                              2. Multi-Qubit Gates
                                                                                                                                                                                                                1. Toffoli Gate (CCNOT)
                                                                                                                                                                                                                  1. Fredkin Gate (CSWAP)
                                                                                                                                                                                                                    1. Generalized Controlled Gates
                                                                                                                                                                                                                  2. Universal Quantum Computation
                                                                                                                                                                                                                    1. Universal Gate Sets
                                                                                                                                                                                                                      1. Criteria for Universality
                                                                                                                                                                                                                        1. Solovay-Kitaev Theorem
                                                                                                                                                                                                                        2. Specific Universal Sets
                                                                                                                                                                                                                          1. Clifford + T Gates
                                                                                                                                                                                                                            1. Hadamard + Toffoli Gates
                                                                                                                                                                                                                              1. Continuous Gate Sets
                                                                                                                                                                                                                            2. Quantum Circuits
                                                                                                                                                                                                                              1. Circuit Model of Computation
                                                                                                                                                                                                                                1. Circuit Depth and Width
                                                                                                                                                                                                                                  1. Parallel and Sequential Operations
                                                                                                                                                                                                                                    1. Circuit Optimization
                                                                                                                                                                                                                                  2. Quantum Measurement
                                                                                                                                                                                                                                    1. Projective Measurements
                                                                                                                                                                                                                                      1. von Neumann Measurements
                                                                                                                                                                                                                                        1. Measurement Operators
                                                                                                                                                                                                                                          1. Orthogonal Projectors
                                                                                                                                                                                                                                            1. Measurement Outcomes and Probabilities
                                                                                                                                                                                                                                            2. Generalized Measurements
                                                                                                                                                                                                                                              1. POVM (Positive Operator-Valued Measure)
                                                                                                                                                                                                                                                1. Definition and Properties
                                                                                                                                                                                                                                                  1. Physical Realizations
                                                                                                                                                                                                                                                    1. Relationship to Projective Measurements
                                                                                                                                                                                                                                                    2. Measurement Strategies
                                                                                                                                                                                                                                                      1. Optimal Measurements
                                                                                                                                                                                                                                                        1. State Discrimination
                                                                                                                                                                                                                                                          1. Parameter Estimation
                                                                                                                                                                                                                                                          2. Fundamental Theorems
                                                                                                                                                                                                                                                            1. No-Cloning Theorem
                                                                                                                                                                                                                                                              1. Statement and Proof
                                                                                                                                                                                                                                                                1. Implications for Information Processing
                                                                                                                                                                                                                                                                  1. Quantum vs Classical Information
                                                                                                                                                                                                                                                                  2. No-Deleting Theorem
                                                                                                                                                                                                                                                                    1. No-Broadcasting Theorem
                                                                                                                                                                                                                                                                      1. No-Hiding Theorem
                                                                                                                                                                                                                                                                        1. Information Conservation
                                                                                                                                                                                                                                                                          1. Relationship to Unitarity

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                                                                                                                                                                                                                                                                    1. Foundations of Quantum Information Science

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                                                                                                                                                                                                                                                                    3. Quantum Computing