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Quantum Information Theory

Periodo di svolgimento

da Giovedì, 7 Gennaio 2021 a Martedì, 30 Marzo 2021
Ore del corso: 45
Ore dei docenti responsabili: 45

Modalità d'esame

  • Prova orale

Programma

 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

(d) 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

Obiettivi formativi

Corso base sui fondamenti di quantum information 

Riferimenti bibliografici

  • M. A. Nielsen and I. L. Chuang, Quantum Computation And Quantum Information (Cambridge Un. Press 2000)

  • J. Preskill, Lecture notes on Quantum Information and Computa- tion

    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)