After successful completion of the course, students are able to understand and apply the fundamental concepts in quantum information science. Specifically, the students will acquire the necessary theoretical skills for
(1) formally describing quantum information systems,
(2) identifying, characterizing, and quantifying entanglement,
(3) understanding paradigmatic protocols for quantum communication
1 Basic formalism of quantum information
1.1 Pure and Mixed Quantum States: Hilbert space, pure states, review of Dirac notation, qubits, linear operators, Hermitian operators, unitary operators, projectors, expectation values, trace, mixedness/linear entropy, time evolution, Bloch decomposition, single-qubit examples.
1.2 Composite Systems: Tensor products of vectors and operators, expectation values, partial trace, reduced states, generalized Bloch decomposition
1.3 Entropy of Quantum States: Shannon & von Neumann entropy, properties, entropy of bipartite systems, subadditivity, Araki-Lieb inequality, concavity, relative entropy
1.4 Schmidt Decomposition & Purification: Schmidt decomposition theorem and proof, purification of mixed quantum states
1.5 Hilbert Space Geometry: Overlap of quantum states, Uhlmann fidelity, Uhlmann theorem, Bures distance, trace distance, relative entropy revisited
2 Entanglement & Correlations
2.1 Entanglement of Pure and Mixed States: Separability of pure states, entropy of entanglement, Example: Bell states, separability of mixed states, classical correlations vs. entanglement, mutual information
2.2 Entanglement and Nonlocality: EPR Paradox, Bell inequalities, CHSH inequalities, entanglement vs. non-locality, Tsirelson´s bound
2.3 Separability Criteria: Peres-Horodecki criterion, realignment criterion, Example: Werner states, CHSH criterion
2.4 Geometry of Entanglement: Convex structure of state space, entanglement witnesses. Example: geometry of Bell-diagonal states
2.5 Entanglement Quantification: LOCC, Nielsen majorization, entanglement monotones vs. entanglement measures, negativity, entanglement of formation, concurrence, monogamy of entanglement
