Quantum Information Theory

Scuola Normale Superiore
LEVEL
Master
TYPE
Course
MODES
-
LANGUAGE
-
ECTS
60
PERIOD
07/01/2022 to 30/03/2022

Course Description

1. Intro, Mathematical Instruments (a) Intro to Q-INFO. (b) from Bit to Qubit. (c) Basics of Linear Algebra. 2. States: (a) Density operators and Ensembles (b) The Schmidt Decomposition (c) Reduced density operators and Purification (existence, how different purifications are connected) (d) The Bloch Sphere. 3. Measurements (a) POVMs: properties and representations (b) Naimark Theorem (b) Impossibility of discriminating among non orthogonal states (d) Helstrom Theorem (c) Quantum State Tomography. 4. Distances for quantum states: (a) Kolmogorov distance for probability distributions (b) Properties of TD and its operational meaning (c) Fidelity and its connection with the TD (d) Uhlmann Theorem. 5. Open quantum system dynamics (a) Basic Properties (b) Kraus and Stinespring representations (c) CP vs P (d) Contractivity, non-invertibility (e) Local, LOCC and separable maps (f) Qubit channels. 6. Entanglement Theory (a) Pure state entanglement (b) Mixed state entanglement (c) Locality and Realism (d) EPR paradox and the hidden variable hypothesis, Bell inequality (CHSH),Tsirelson bound, PR-Boxes (e) The GHZ state and multipartite entanglement (f) Separability criteria (g) Partial transpose e Reduction criteria (PPT states), Majorization criterion,Entanglement Witnesses (h) Entanglement Measures. 7. Possible and Impossible Machines (a) Possible Machines: Quantum Teleportation, Superdense Coding, Entanglement Swapping (b) Impossible Machines: Q-Bell telephone, Q-copier (No cloning theorem), Classical Teleportation, Joint-Measurement Machine. 8. Quantum Computation (a) Classical model of computation: The Turing Machine, Universal and Probabilistici TMs, Complexity classes, The Church-Turing Thesis, The Gate Array Model, Universal Gate sets,Reversible vs Non-reversible gates (b) The Landauer Principle and the second principle of thermodynamics (c) Quantum Gates: One-qubit gates, Generalized Euler decomposition, Universal sets and Approximate universal sets for one qubit,Two-qubit gates: C-NOT, C-U from C-NOT, U(N) from C-NOT gates, Gottesman-Knill theorem (d) Quantum Supremacy (e) Quantum Parallelism. 9. Quantum Algorithms (a) Deutch-Jozsa Algorithm (b) Berenstein-Vazirani Algorithm (c) Simon Algorithm (d) Quantum Fourier Transfrom (e) Period Finding Algorithm (f) Shor Algorithm (g) Grover Algorithm.10. Quantum Error Correction (a) Intro (b) 3 qubit code for Bit-Flip errors and Phase-Flip errors (c) 9 qubit code for all 1-qubit errors (c) Knill-Laflamme theorem. 11. Quantum Cryptography (a) Private key vs. public key algorithms, One-time Pad, Shor algorithm vs RSA (b) Quantum Key distribution protocols: BB84 protocol, B92 protocol, Ekert protocol.

Subject area

Digital communications IA electronics
Physics

Field area

Digital
Industry and Space

Educational-info

Competences

Introductory course on quantum information theory

Prerequisites

at least a Bachelor

Duration

45h

ECTS

60

Validation mode

Oral Examination

Maximum number of students

30

Organizer

Partner

Scuola Normale Superiore

Faculty

Scuola Normale Superiore

Department

Classe di Scienze

Contact or registration links