Nach positiver Absolvierung der Lehrveranstaltung sind Studierende in der Lage fundamentale Konzepte der Quanteninformationstheorie zu verstehen und anzuwenden. Konkret werden die notwendigen Fähigkeiten vermittelt um:

(1) formal Quanteninformationssysteme formal zu beschreiben
(2) Verschränkung zu identifizieren, charakterisieren und in Quantentechnologien anzuwenden
(3) Paradigmatische Protokolle der Quantenkommunikation, Quantencomputation zu meistern
(4) simple Schaltungen auf einem echten Quantencomputer zu implementieren

1. Basic formalism of quantum information

1. Quantum States & Operations Hilbert space, pure states, review of Dirac notation, qubits, linear operators, Hermitian operators, unitary operators, projectors, expectation values.

2. Mixed Quantum States Properties, trace, expectation values, mixedness/linear entropy, time evolution.

3. Bloch decomposition, single-qubit examples.

4. Composite Systems Tensor products of vectors and operators, expectation values, partial trace, reduced states, generalized Bloch decomposition

5. Entropy of Quantum States Shannon & von Neumann entropy, properties, entropy of bipartite systems, subadditivity, Araki-Lieb inequality, concavity, relative entropy

6. Schmidt Decomposition & Purification Schmidt decomposition theorem and proof, purification of mixed quantum states

7. Hilbert Space Geometry Overlap of quantum states, Uhlmann fidelity, Uhlmann theorem, Bures distance, trace distance, relative entropy revisited

1. Entanglement & Correlations

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. Separability Criteria Peres-Horodecki criterion, realignment criterion, Example: Werner states, CHSH criterion

3. Geometry of Entanglement Convex structure of state space, entanglement witnesses. Example: geometry of Bell-diagonal states

4. Entanglement Quantification LOCC, Nielsen majorization, entanglement monotones vs. entanglement measures, negativity, entanglement of formation, concurrence, monogamy of entanglement

1. Elements of Quantum Information Processing

1. Generalized Measurements POVMs, projective measurements, observables, distinguishing quantum states, Neumark dilations.

2. Quantum Teleportation protocol, entanglement swapping and dense coding

3. Quantum Key Distribution Ekert 91 protocol and BB84 protocol

4. No-Cloning Proof of no-cloning theorem, approximate cloning, no broadcasting, no deleting.

1. Elements of Quantum Computation

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

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

3. Universal Quantum Computation Notions of universality for discrete/continuous sets of operations, Gray code

4. Quantum Algorithms Deutsch-Josza algorithm, Grover’s algorithm, comments on Shor‘s algorithm

Quantum Error Correction Why error correction? Repetition code, stabilizer codes

Partizipative Vorlesung.

Mündliche Prüfung am Ende der Vorlesung