Examination procedure
<p>The final assessment will be an oral exam.</p>
Examination procedure notes
<p>The oral exam consists of a discussion of approximately 40 minutes, during which the candidate is asked to solve problems related to the material covered in the course.</p><p><br></p>
Prerequisites
Basic knowledge of Quantum Mechanics, including:
- Formalism of states and operators in Hilbert spaces
- Unitary evolution and quantum measurements (observables, expectation values)
- Pure and mixed states
- Superposition principle and uncertainty principle
- Fundamental examples (spin-½ system, harmonic oscillator, infinite potential well)
Fundamentals of Linear Algebra, including:
- Complex vector spaces and orthonormal bases
- Inner product, norms, and linear operators
- Eigenvalues, eigenvectors, and diagonalization
- Hermitian and unitary operators
- Trace, determinant, and density matrices
Syllabus
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
(c) Impossibility of discriminating among non orthogonal states
(d) Helstrom Theorem
(e) 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
(d) Knill-Laflamme theorem
(e) Stabilizer formalism
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
Bibliographical references
- M. A. Nielsen and I. L. Chuang, Quantum Computation And Quantum Information (Cambridge Un. Press 2000)
- J. Preskill, Lecture notes on Quantum Information and Computation. http://www.theory.caltech.edu/people/preskill/ph229/notes/book.ps
- B. Schumacher and M. Westmoreland, Quantum Processes Systems, and Information (Cambridge Un. Press 2010)