Quantum Information Science

Quantum Information Science (QIS) is an interdisciplinary field that merges principles from quantum mechanics with computer science and information theory to explore how information can be acquired, processed, and transmitted using quantum systems. It investigates the fundamental capabilities and limitations of information processing by harnessing uniquely quantum phenomena like superposition and entanglement, where a quantum bit (qubit) can exist in multiple states at once or be intrinsically linked to another, regardless of distance. While quantum computing is its most prominent application, QIS also encompasses the development of ultra-secure quantum communication networks, advanced quantum sensing for precision measurement, and the theoretical foundations that underpin these revolutionary technologies.

1. Foundations of Quantum Information Science
   1. Mathematical Preliminaries
      1. Linear Algebra for Quantum Mechanics
         1. Vector Spaces
            1. Definition and Properties
            2. Finite and Infinite Dimensional Spaces
            3. Subspaces
            4. Linear Independence
            5. Spanning Sets
         2. Hilbert Spaces
            1. Definition and Properties
            2. Completeness
            3. Separability
            4. Examples in Quantum Mechanics
         3. Basis and Dimension
            1. Orthogonal Bases
            2. Orthonormal Bases
            3. Change of Basis
            4. Dimension Theorems
         4. Inner Products
            1. Definition and Properties
            2. Complex Inner Products
            3. Cauchy-Schwarz Inequality
            4. Parallelogram Law
         5. Norms
            1. Definition and Properties
            2. Induced Norms
            3. Equivalence of Norms
            4. Completeness and Banach Spaces
         6. Orthogonality and Orthonormalization
            1. Orthogonal Vectors
            2. Gram-Schmidt Process
            3. Orthogonal Complements
            4. Projection Theorem
         7. Linear Operators
            1. Definition and Properties
            2. Bounded and Unbounded Operators
            3. Operator Norms
            4. Composition of Operators
         8. Matrix Representations
            1. Matrix of a Linear Operator
            2. Change of Basis for Matrices
            3. Rank and Nullity
            4. Matrix Decompositions
         9. Eigenvalues and Eigenvectors
            1. Characteristic Polynomial
            2. Algebraic and Geometric Multiplicity
            3. Diagonalization
            4. Eigenspaces
         10. Spectral Theory
             1. Spectral Decomposition
             2. Spectral Theorem for Normal Operators
             3. Functional Calculus
             4. Spectral Measures
         11. Special Classes of Operators
             1. Unitary Operators
                1. Definition and Properties
                2. Preservation of Inner Products
                3. Unitary Groups
             2. Hermitian Operators
                1. Self-Adjoint Properties
                2. Real Eigenvalues
                3. Spectral Theorem
             3. Normal Operators
                1. Commuting with Adjoint
                2. Spectral Properties
             4. Projectors
                1. Orthogonal Projections
                2. Properties of Projectors
                3. Projection-Valued Measures
         12. Tensor Products
             1. Definition and Properties
             2. Universal Property
             3. Tensor Product of Vector Spaces
             4. Tensor Product of Operators
             5. Tensor Product of States
             6. Kronecker Products
      2. Probability Theory
         1. Probability Spaces
            1. Sample Spaces
            2. Sigma-Algebras
            3. Probability Measures
            4. Kolmogorov Axioms
         2. Events and Operations
            1. Event Algebra
            2. Union and Intersection
            3. Complement and Difference
            4. Limits of Events
         3. Conditional Probability
            1. Definition and Properties
            2. Law of Total Probability
            3. Independence of Events
            4. Conditional Independence
         4. Bayes' Theorem
            1. Statement and Proof
            2. Bayesian Inference
         5. Random Variables
            1. Definition and Types
            2. Discrete Random Variables
            3. Continuous Random Variables
            4. Distribution Functions
         6. Probability Distributions
            1. Discrete Distributions
            2. Continuous Distributions
            3. Joint Distributions
            4. Marginal Distributions
         7. Expectation and Moments
            1. Expected Value
            2. Properties of Expectation
            3. Variance and Standard Deviation
            4. Higher Order Moments
         8. Joint and Conditional Distributions
            1. Joint Probability Mass Functions
            2. Joint Probability Density Functions
            3. Conditional Distributions
            4. Independence and Correlation
         9. Limit Theorems
            1. Law of Large Numbers
            2. Central Limit Theorem
            3. Convergence Concepts
   2. Principles of Classical Information Theory
      1. The Classical Bit
         1. Binary Representation
         2. Bit Operations
         3. Boolean Algebra
         4. Information Content
      2. Information Measures
         1. Shannon Entropy
            1. Definition and Interpretation
            2. Properties of Entropy
            3. Maximum Entropy Principle
         2. Joint and Conditional Entropy
            1. Chain Rule for Entropy
            2. Conditional Entropy Properties
         3. Mutual Information
            1. Definition and Properties
            2. Information Gain
            3. Data Processing Inequality
         4. Relative Entropy
            1. Kullback-Leibler Divergence
            2. Properties and Applications
      3. Classical Logic Gates
         1. Basic Gates
            1. NOT Gate
            2. AND Gate
            3. OR Gate
         2. Derived Gates
            1. NAND Gate
            2. NOR Gate
            3. XOR Gate
            4. XNOR Gate
         3. Universal Gate Sets
            1. NAND Universality
            2. NOR Universality
            3. Functional Completeness
      4. Classical Circuits
         1. Combinational Circuits
         2. Sequential Circuits
         3. Circuit Complexity
         4. Reversible Computation
      5. Classical Communication
         1. Channel Models
            1. Discrete Memoryless Channels
            2. Binary Symmetric Channel
            3. Gaussian Channels
         2. Channel Capacity
            1. Shannon's Channel Capacity Theorem
            2. Capacity Calculation Methods
            3. Rate-Distortion Theory
         3. Source Coding
            1. Huffman Coding
            2. Arithmetic Coding
            3. Lempel-Ziv Coding
         4. Channel Coding
            1. Error Detection and Correction
            2. Noisy Channel Coding Theorem
            3. Coding Bounds
      6. Error Correction Codes
         1. Linear Codes
            1. Generator Matrices
            2. Parity Check Matrices
            3. Syndrome Decoding
         2. Specific Codes
            1. Repetition Codes
            2. Hamming Codes
            3. Reed-Solomon Codes
            4. Convolutional Codes
         3. Performance Metrics
            1. Error Detection vs Correction
            2. Minimum Distance
            3. Code Rate and Efficiency
   3. Fundamentals of Quantum Mechanics
      1. Historical Development
         1. Planck's Quantum Hypothesis
         2. Bohr Model
         3. Wave-Particle Duality
         4. Development of Modern Quantum Theory
      2. The Postulates of Quantum Mechanics
         1. State Space Postulate
            1. Hilbert Space Formulation
            2. Pure and Mixed States
         2. Observable Postulate
            1. Hermitian Operators as Observables
            2. Spectral Decomposition
         3. Measurement Postulate
            1. Born Rule
            2. Measurement Outcomes
            3. State Collapse
         4. Evolution Postulate
            1. Unitary Evolution
            2. Schrödinger Equation
         5. Composite Systems Postulate
            1. Tensor Product Structure
            2. Entanglement
      3. Quantum States
         1. Wavefunctions and State Vectors
            1. Position and Momentum Representations
            2. Normalization
            3. Physical Interpretation
         2. Dirac Notation
            1. Bra-Ket Notation
            2. Inner Products
            3. Outer Products
            4. Completeness Relations
      4. Time Evolution
         1. Schrödinger Equation
            1. Time-Dependent Form
            2. Time-Independent Form
            3. Solutions and Eigenstates
         2. Unitary Time Evolution
            1. Time Evolution Operator
            2. Conservation Laws
            3. Symmetries and Noether's Theorem
      5. Quantum Operators
         1. Properties of Quantum Operators
            1. Linearity
            2. Hermiticity
            3. Unitarity
            4. Commutativity
         2. Commutation Relations
            1. Canonical Commutation Relations
            2. Angular Momentum Commutators
            3. Uncertainty Relations
         3. Operator Functions
            1. Exponential of Operators
            2. Operator Equations
      6. Quantum Measurement Theory
         1. Measurement Process
            1. Measurement Postulates
            2. Measurement Operators
            3. POVM Formalism
         2. Measurement Outcomes
            1. Probability Calculations
            2. Expectation Values
            3. Measurement Statistics
         3. Quantum Measurement Problem
            1. Measurement and Decoherence
            2. Interpretations of Quantum Mechanics
      7. Uncertainty Principles
         1. Heisenberg Uncertainty Principle
            1. Position-Momentum Uncertainty
            2. Energy-Time Uncertainty
            3. General Uncertainty Relations
         2. Measurement Disturbance
            1. Information-Disturbance Trade-offs
            2. Measurement Back-action