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

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

Lecture with active participation of students.

Oral

This lecture series is a continuation of the previous course Quantum Information Theory I. As such, familiarity with the contents of the previous course is helpful but not strictly required. In particular, Chapters 3 and 4 are self-contained and can be followed with only basic knowledge of quantum mechanics.