Universität Wien FIND

260046 VO Blind men and an elephant: the marginal problem in quantum information theory & statistical physics (2018W)

2.50 ECTS (2.00 SWS), SPL 26 - Physik

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Details

Language: English

Examination dates

Lecturers

Classes (iCal) - next class is marked with N

Thursday 04.10. 17:30 - 19:00 Seminarraum, Zi. 3354A, Boltzmanngasse 5, 3. Stk., 1090 Wien (Kickoff Class)
Thursday 11.10. 17:30 - 19:00 Seminarraum, Zi. 3354A, Boltzmanngasse 5, 3. Stk., 1090 Wien
Thursday 18.10. 17:30 - 19:00 Seminarraum, Zi. 3354A, Boltzmanngasse 5, 3. Stk., 1090 Wien
Thursday 25.10. 17:30 - 19:00 Seminarraum, Zi. 3354A, Boltzmanngasse 5, 3. Stk., 1090 Wien
Thursday 08.11. 17:30 - 19:00 Seminarraum, Zi. 3354A, Boltzmanngasse 5, 3. Stk., 1090 Wien
Thursday 15.11. 17:30 - 19:00 Seminarraum, Zi. 3354A, Boltzmanngasse 5, 3. Stk., 1090 Wien
Thursday 22.11. 17:30 - 19:00 Seminarraum, Zi. 3354A, Boltzmanngasse 5, 3. Stk., 1090 Wien
Thursday 29.11. 17:30 - 19:00 Seminarraum, Zi. 3354A, Boltzmanngasse 5, 3. Stk., 1090 Wien
Wednesday 05.12. 17:30 - 19:00 Lise-Meitner-Hörsaal, Boltzmanngasse 5, 1. Stk., 1090 Wien
Thursday 13.12. 17:30 - 19:00 Seminarraum, Zi. 3354A, Boltzmanngasse 5, 3. Stk., 1090 Wien
Thursday 10.01. 17:30 - 19:00 Seminarraum, Zi. 3354A, Boltzmanngasse 5, 3. Stk., 1090 Wien
Thursday 17.01. 17:30 - 19:00 Seminarraum, Zi. 3354A, Boltzmanngasse 5, 3. Stk., 1090 Wien
Thursday 24.01. 17:30 - 19:00 Seminarraum, Zi. 3354A, Boltzmanngasse 5, 3. Stk., 1090 Wien
Thursday 31.01. 17:30 - 19:00 Seminarraum, Zi. 3354A, Boltzmanngasse 5, 3. Stk., 1090 Wien

Information

Aims, contents and method of the course

Many tasks in quantum information and statistical physics can be understood as a marginal problem. That is, they require us to figure out whether a number of observable quantities are integrated into a more general structure to which we do not have full access. Examples of these tasks are certifying security in quantum key distribution, determining whether a quantum state is separable and computing the free energy of a many-body system. In this course, we will study different expressions of the marginal problem and introduce a number of modern mathematical tools to tackle it.

Course program

1. Presentation
What is the marginal problem? And why should I care?
2. The basics of convex optimization.
Convex sets. Extreme points and polytopes. Convex functions. Jensen’s inequality. The hyperplane separation theorem. Convex optimization theory. Lagrangians and duals. Strong duality and the Slater criterion. Linear and semidefinite programming. The barrier method.
3. Nonlocality and entanglement theory
The nonlocality problem and its computational complexity. Entangled states. Hardness of entanglement detection. The symmetric extensions criterion. The quantum de Finetti theorem.
4. Quantum nonlocality
The NPA hierarchy, definition and convergence.
5. The entropic approach
Shannon and von Neumann Entropies. Entropic inequalities. Classical and quantum nonlocality via entropic methods.
6. 1D classical systems
The marginal problem in 1D. Computing the free energy of 1D systems. The thermodynamics of infinite translation-invariant (TI) 1D systems.
7. 2D classical systems
The classical problem in 2D square lattices. Resolution for variables of low cardinality. Undecidability of the energy minimization problem in 2D translation-invariant systems. The shape of 2D TI marginals.
8. 1D quantum systems
Complexity of energy minimization in 1D. Matrix Product States: definition and approximation results. Exactly solvable models. Entropic inequalities for TI 1D quantum systems.
9. Entanglement and nonlocality in 1D systems
Efficient characterization of near-neighbor reduced density matrices of separable states. Efficient characterization of near-neighbor local correlations. The infinite TI limit.
10. Beyond Bell scenarios
The inflation technique: method and proof of convergence.
11. Large quantum networks
Introduction to connector theory.

Assessment and permitted materials

Final oral exam

Minimum requirements and assessment criteria

Reasonable competence in quantum theory and linear algebra.

Examination topics

Reading list

D. Gottesman and S. Irani, The Quantum and Classical Complexity of Translationally Invariant Tiling and Hamiltonian Problems, Theory of Computing, Volume 9, Article 2, pp. 31-116 (2013).
M. Nielsen and I. L. Chuang, Quantum computation and quantum information, Cambridge University Press.
S. Goldstein, T. Kuna, J. L. Lebowitz, and E. R. Speer, Translation Invariant Extensions of Finite Volume Measures, Journal of Statistical, Physics 166 (2017), no. 3, 765–782.
S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press.
N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014).
Z. Wang and M. Navascués, Two Dimensional Translation-Invariant Probability Distributions: Approximations, Characterizations and No-Go Theorems, arXiv:1703.05640.
M. Navascués, S. Pironio and A. Acín, A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations, New J. Phys. 10, 073013 (2008).
Elie Wolfe, Robert W. Spekkens, Tobias Fritz, The Inflation Technique for Causal Inference with Latent Variables, arXiv:1609.00672.
M. Nielsen, Complete notes on fermions and the Jordan-Wigner transform, http://michaelnielsen.org/blog/complete-notes-on-fermions-and-the-jordan-wigner-transform/.
R. Orus, A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States, Annals of Physics 349, 117-158 (2014).
Z. Wang, S. Singh and M. Navascués, Entanglement and Nonlocality in Infinite 1D Systems, Phys. Rev. Lett. 118, 230401 (2017).
M. Navascués, M. Owari and M. B. Plenio, The power of symmetric extensions for entanglement detection, Phys. Rev. A 80, 052306 (2009).

Association in the course directory

MF 6, MaG 17, MaG 18, UF MA PHYS 01a, UF MA PHYS 01b

Last modified: Th 05.09.2019 12:27