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 Elements of Quantum Information Processing
2.1 Non-Locality and Bell Inequalities: EPR Paradox, Bell inequalities, CHSH inequalities, entanglement vs. non-locality, Tsirelson´s bound
2.2 Contextuality: Gleason’s theorem, Kochen-Specker Theorem, Peres’ square, Mermin’s pentagram
2.3 Quantum Teleportation and Quantum Cryptography: teleportation protocol, entanglement swapping and dense coding, BB84 protocol and Ekert 91 protocol
2.4 Entanglement: Separability of pure states, entropy of entanglement, Example: Bell states, separability of mixed states, classical correlations vs. entanglement, mutual information, PPT criterion, entanglement witnesses
2.5 Quantum Channels, Quantum Operations, & Generalized Measurements: CPTP maps, Kraus operators, Stinespring dilation, church of the larger Hilbert space, POVMs, projective measurements, observables, distinguishing quantum states, Neumark dilations.
2.6 No-Cloning & Distinguishing Quantum States: Proof of no-cloning theorem, approximate cloning, no broadcasting, no deleting, distinguishing non-orthogonal quantum states