Quantum Information Theory
Prerequisiti
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
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
(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
Obiettivi formativi
Develop a Solid Mathematical Foundation
Equip students with the essential linear algebra and probability tools required to formalize quantum information concepts, including Hilbert spaces, operators, and state representations.
Understand Quantum States and Measurements
Enable students to describe, analyze, and manipulate quantum states (pure and mixed), understand measurement theory (POVMs, projective measurements), and apply key theorems like Naimark and Helstrom.
Explore Quantum State Structure and Dynamics
Teach students to characterize quantum systems using reduced states, purifications, and Bloch sphere representations, and understand the principles governing open system dynamics (Kraus operators, CP maps).
Quantify and Classify Quantum Resources
Provide students with tools to distinguish and quantify quantum states using distances (trace distance, fidelity) and explore the operational meanings of these quantities.
Gain Deep Understanding of Quantum Entanglement
Introduce students to entanglement theory, from bipartite to multipartite systems, covering entanglement measures, separability criteria (PPT, reduction), and fundamental tests of non-classicality like Bell inequalities.
Analyze Feasibility of Quantum Operations
Train students to differentiate between physically realizable and forbidden operations in quantum mechanics (e.g., teleportation vs. no-cloning), thereby reinforcing the foundational constraints of the theory.
Master Quantum Computation Principles
Familiarize students with models of quantum computation, complexity theory, and physical constraints (Landauer’s principle), along with the architecture of quantum gates and circuits.
Apply and Evaluate Quantum Algorithms
Enable students to understand, implement, and analyze the performance of key quantum algorithms such as Shor’s, Grover’s, and Simon’s, highlighting the contrast with classical counterparts.
Understand Quantum Error Correction
Introduce students to the principles of fault-tolerant quantum computation and quantum error correction codes, including stabilizer formalism and the Knill-Laflamme conditions.
Explore Quantum Cryptography and Security
Provide students with an understanding of quantum cryptographic protocols and their relation to classical security concepts, emphasizing how quantum mechanics enhances or limits secure communication.
Encourage Critical Thinking and Research Orientation
Develop students’ ability to critically assess the assumptions and implications of quantum information protocols and to connect foundational principles with advanced research topics.
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 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)