Quantum Mechanics cheat sheet

This reference document provides a concise, mathematically rigorous summary of foundational concepts and theorems in quantum information theory. It establishes essential mathematical tools, including classical probability, Dirac distributions, Fourier transforms, and matrix theory. The text formally defines quantum states through density matrices, Schmidt decomposition, and purification, alongside fundamental linear algebra operations such as the partial trace. A significant portion is dedicated to measurement theory, detailing both von Neumann projective measurements and generalized measurements, including the proof that generalized measurements can be represented as von Neumann measurements on a larger Hilbert space. The document systematically outlines critical no-go theorems and paradoxes that rule out classical hidden-variable theories, explicitly covering the Bell/CHSH inequality, the Mermin-GHZ paradox, and the Kochen-Specker paradox. It explores the relationship between physical information and thermodynamics via von Neumann entropy, the fundamental postulate of statistical mechanics, and Landauer's erasure principle. Finally, it provides step-by-step algebraic breakdowns of foundational quantum information protocols, analyzing the delayed-choice quantum eraser, approximate quantum cloning, the universal-NOT gate, and the constraints on programmable quantum machines. This cheat sheet serves as a rapid, comprehensive reference for researchers and advanced students in quantum information science.