3 Elements of Quantum Information Processing
3.1 Generalized Measurements: POVMs, projective measurements, observables, distinguishing quantum states, Neumark dilations.
3.2 Quantum Teleportation: teleportation protocol, entanglement swapping and dense coding
3.3 Quantum Key Distribution: BB84 protocol and Ekert 91 protocol
3.4 Quantum Channels and Quantum Operations: CPTP maps, Kraus operators, Stinespring dilation, church of the larger Hilbert space
3.5 No-Cloning: Proof of no-cloning theorem, approximate cloning, no broadcasting, no deleting
3.6 Contextuality: Gleason’s theorem, Kochen-Specker Theorem, Peres’ square, Mermin’s pentagram
4 Elements of Quantum Computation
4.1 Quantum Gates and Quantum Circuits: reversible computation, quantum-extended Church-Turing thesis, circuit diagrams, Hadamard, Phase gate, CNOT, example: half-adder circuit, decompositions of single-qubit gates
4.2 Multi-Qubit Gates: controlled operations for two qubits, operations conditioned on the state of multiple qubits, useful identities, decomposition into single-qubit gates and CNOTS
4.3 Universal Quantum Computation: notions of universality for discrete/continuous sets of operations, Gray code
4.4 Quantum Algorithms: Deutsch-Josza algorithm, Grover’s algorithm, Shor‘s algorithm
4.5 Quantum Error Correction: classical vs. quantum error correction, repetition code, code words, logical states, syndromes, stabilizer formalism, topological codes, Kitaev’s toric code, transversal implementations, fault tolerance, Eastin-Knill theorem
4.6 Graph States & Measurement-Based Quantum Computation: graphs and graph states, gate teleportation, measurement-based implementation of single and two-qubit gates
5 Selected Advanced Topics in Quantum-Information Processing
5.1 Quantum Metrology: local and Bayesian estimation paradigms, quantum parameter estimation, unbiased estimators, (quantum) Cramér-Rao bound, (quantum) Fisher information, symmetric logarithmic derivative,
5.2 Quantum Information Processing with Continuous-Variable Systems: continuous variables, quadrature operators, symplectic form, s-parameterized quasi-probability distributions, Wigner function, Glauber-Sudarshan P function, Husimi Q function, Gaussian and non-Gaussian states, covariance matrix, entropies and separability of Gaussian states, Gaussian operations
5.3 High-Dimensional Entanglement: Schmidt number, mutually unbiased bases, fidelity witnesses, certification of high-dimensional entanglement
5.4 Multipartite Entanglement: multipartite quantum systems, LOCC classes, GHZ and W states, k-separability, k-producibility, entanglement depth, genuine multipartite entanglement, activation of genuine multipartite entanglement, Schmidt-rank vectors