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Quantum Foundations

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Module 1 — hilbert spaces

1.1 complex vector spaces 1.2 inner products and norms 1.3 hilbert space completeness 1.4 orthonormal bases and dirac notation 1.5 linear operators and adjoints 1.6 spectral decomposition

Module 2 — postulates

2.1 state postulate 2.2 observable postulate 2.3 born rule 2.4 unitary time evolution 2.5 composite systems tensor products 2.6 symmetries and conservation laws

Module 3 — measurement theory

3.1 projective measurements 3.2 povms 3.3 naimark dilation 3.4 quantum non demolition 3.5 weak measurement 3.6 measurement back action

Module 4 — entanglement

4.1 tensor product states 4.2 separable vs entangled states 4.3 schmidt decomposition 4.4 bell states and chsh 4.5 entanglement measures 4.6 monogamy of entanglement

Module 5 — decoherence

5.1 density matrices 5.2 reduced density operator partial trace 5.3 system environment coupling 5.4 decoherence mechanisms 5.5 pointer basis and einselection 5.6 decoherence timescales

Module 6 — interpretations

6.1 the measurement problem 6.2 copenhagen interpretation 6.3 many worlds interpretation 6.4 bohmian mechanics 6.5 consistent histories and qbism 6.6 empirical equivalence and constraints

Module 7 — quantum information

7.1 qubits and quantum registers 7.2 no cloning theorem 7.3 quantum channels and cptp maps 7.4 von neumann entropy 7.5 holevo bound 7.6 teleportation and superdense coding

Module 8 — open systems

8.1 open vs closed systems 8.2 lindblad master equation 8.3 markovian vs non markovian 8.4 quantum trajectories 8.5 decoherence free subspaces 8.6 error correction foundations

Module 9 — quantum thermodynamics

9.1 thermal states and gibbs 9.2 quantum work and heat 9.3 fluctuation theorems 9.4 landauer principle 9.5 quantum heat engines 9.6 resource theories

Module 10 — foundations and open problems

10.1 bell theorem and nonlocality 10.2 kochen specker and contextuality 10.3 pbr theorem 10.4 quantum to classical transition 10.5 measurement problem technical 10.6 quantum gravity frontier