🔐Quantum Cryptography Unit 1 – Introduction to Quantum Mechanics

Key Concepts and Principles

• Quantum mechanics describes the behavior of matter and energy at the atomic and subatomic scales
• Fundamental concepts include quantization of energy, wave-particle duality, and the uncertainty principle
• Quantum systems exist in superposition of multiple states until measured, causing the wavefunction to collapse
• Entanglement occurs when two or more particles become correlated, even across vast distances
• Quantum cryptography leverages principles of quantum mechanics to ensure secure communication
  • Utilizes the no-cloning theorem and the ability to detect eavesdropping attempts
• Quantum computing harnesses quantum phenomena to perform certain computations exponentially faster than classical computers

Mathematical Foundations

• Complex numbers are essential in quantum mechanics, with the imaginary unit $i$ satisfying $i^2 = -1$
• Hilbert spaces provide a mathematical framework for describing quantum states
  • Vectors in Hilbert space represent physical states, while inner products define probability amplitudes
• Linear algebra is crucial, with quantum states represented as vectors and operators as matrices
• Eigenvalues and eigenvectors of operators correspond to measurable quantities and their possible values
• Schrödinger equation describes the time evolution of a quantum system: $i\hbar\frac{\partial}{\partial t}\Psi(x,t) = \hat{H}\Psi(x,t)$
  • $\Psi(x,t)$ is the wavefunction, $\hbar$ is the reduced Planck's constant, and $\hat{H}$ is the Hamiltonian operator
• Dirac notation (bra-ket notation) provides a convenient way to represent quantum states and operators

Quantum States and Superposition

• A quantum state is a complete description of a quantum system, represented by a vector in a Hilbert space
• Superposition allows a quantum system to exist in a linear combination of multiple states simultaneously
  • Mathematically, a superposition state is a weighted sum of basis states: $|\Psi\rangle = \sum_i c_i |i\rangle$
• Quantum bits (qubits) are the fundamental units of quantum information, existing in a superposition of $|0\rangle$ and $|1\rangle$ states
• Coherence refers to the ability of a quantum system to maintain superposition and exhibit interference
• Decoherence occurs when a quantum system interacts with its environment, causing loss of coherence and superposition
• Quantum state tomography is a process of reconstructing the quantum state from multiple measurements

Measurement and Observables

• Measurement in quantum mechanics is probabilistic and causes the wavefunction to collapse to an eigenstate of the measured observable
• Observables are physical quantities that can be measured, represented by Hermitian operators in the Hilbert space
  • Examples include position, momentum, energy, and spin
• Born rule states that the probability of measuring a particular eigenvalue is given by the square of the absolute value of the corresponding probability amplitude
• Projective measurements are described by a set of orthogonal projection operators that sum to the identity operator
• Expectation value of an observable is the average value obtained from repeated measurements on identically prepared systems
• Compatible observables have commuting operators and can be measured simultaneously with arbitrary precision

Quantum Entanglement

• Entanglement is a quantum phenomenon where two or more particles become correlated in such a way that their individual states cannot be described independently
• Entangled particles exhibit strong correlations that cannot be explained by classical physics
  • Einstein referred to this as "spooky action at a distance"
• Bell states are maximally entangled two-qubit states, such as the singlet state $|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)$
• Entanglement is a key resource in quantum cryptography protocols like quantum key distribution (QKD)
  • Ensures security by detecting eavesdropping attempts that disturb the entangled states
• Quantum teleportation utilizes entanglement to transmit quantum information without physically sending the quantum state
• Entanglement swapping allows establishing entanglement between particles that have never interacted directly

Wave-Particle Duality

• Wave-particle duality states that quantum entities exhibit both wave-like and particle-like properties
• Double-slit experiment demonstrates the wave nature of particles, showing interference patterns
  • Particles passing through double slits interfere with themselves, behaving like waves
• Photons, electrons, and other quantum particles can be described as wavepackets with a certain wavelength and frequency
• Matter waves, proposed by de Broglie, associate a wavelength to particles: $\lambda = \frac{h}{p}$, where $h$ is Planck's constant and $p$ is the particle's momentum
• Complementarity principle states that wave and particle aspects are complementary and cannot be observed simultaneously
• Quantum eraser experiments demonstrate the ability to "erase" which-path information and restore interference

Uncertainty Principle

• Heisenberg's uncertainty principle sets a fundamental limit on the precision with which certain pairs of physical properties can be determined simultaneously
  • Canonically conjugate variables, such as position and momentum, are subject to the uncertainty relation: $\Delta x \Delta p \geq \frac{\hbar}{2}$
• The more precisely one property is measured, the less precisely the other can be known
• Uncertainty principle arises from the wave nature of quantum mechanics and the non-commutative property of certain operators
• Implies that there is an inherent uncertainty in the values of certain observables, even in principle
• Quantum cryptography protocols exploit the uncertainty principle to detect eavesdropping attempts
  • Measuring one property (e.g., polarization) disturbs the conjugate property, revealing the presence of an eavesdropper
• Energy-time uncertainty relation, $\Delta E \Delta t \geq \frac{\hbar}{2}$, limits the precision of energy measurements in short time intervals

Applications in Cryptography

• Quantum key distribution (QKD) protocols, such as BB84, use quantum states to securely distribute cryptographic keys
  • Relies on the no-cloning theorem and the ability to detect eavesdropping through disturbances in the quantum states
• Quantum random number generation (QRNG) produces true random numbers by measuring quantum processes
  • Ensures high-quality randomness for cryptographic purposes
• Quantum digital signatures (QDS) use quantum states to provide secure authentication and non-repudiation
• Quantum-resistant cryptography aims to develop classical cryptographic algorithms that are secure against attacks by quantum computers
  • Examples include lattice-based cryptography and code-based cryptography
• Quantum-secured communication networks, such as quantum key distribution networks, are being developed for secure information transmission
• Blind quantum computing allows delegating computations to a quantum server while preserving the privacy of the input, output, and computation