As the title of the School suggests, the lectures are at an advanced level, appropriate for graduate students, postdocs and faculty working in related areas.
Mini-courses
Victor V. Albert |
Quantum Theory of Molecular Orientations |
| (NIST & UMD College Park, EU) (via Zoom) | |
Frédéric Holweck |
The Geometry of Quantum Contextuality |
|
(Laboratoire ICB, University of Technology of Belfort-Montbéliard, France) |
|
Andrei Klimov |
Introduction to Continuous and Discrete Phase-space Representations |
|
(University of Guadalajara, Jalisco, Mexico) |
|
Marek Kus |
Symplectic and Algebraic Geometry of Quantum States |
|
(Center for Theoretical Physics, Polish Academy of Sciences, Poland) |
|
Plenary Talk
Karol Zyczkowski |
Geometry of Quantum States |
| (Jagiellonian University, Poland) | |
Program
| Time | Wednesday, April 29 | Thursday, April 30 |
|---|---|---|
| 10:00 - 10:55 | Marek Kus | Andrei Klimov |
| 11:00 - 11:55 | Andrei Klimov | Victor Albert (Zoom) |
| 11:55 - 12:10 | Coffee Break | |
| 12:10 - 13:05 | Victor Albert (Zoom) | Frédéric Holweck |
| 13:10 - 14:05 | Frédéric Holweck | Marek Kus |
| 14:05 - 16:30 | Lunch | |
| 16:30 - 17:30 | Karol Zyczkowski | |
Mini-course contents
Quantum Theory of Molecular Orientations
We formulate a quantum phase space for rotational and nuclear-spin states of rigid molecules. For each nuclear spin isomer, we re-derive the isomer's admissible angular momentum states from molecular geometry and nuclear-spin data, introduce its angular position states using quantization theory, and develop a generalized Fourier transform converting between the two. We classify molecules into three types -- asymmetric, rotationally symmetric, and perrotationally symmetric -- with the last type having no macroscopic analogue due to nuclear-spin statistics constraints. We discuss two general features in perrotationally symmetric state spaces that are Hamiltonian- independent and induced solely by symmetry and spin statistics. First, we quantify when and how an isomer's state space is completely rotation-spin entangled, meaning that it does not admit any separable states. Second, we identify isomers whose position states house an internal pseudo-spin or "fiber" degree of freedom, and the fiber's Berry phase or matrix after adiabatic changes in position yields naturally robust operations, akin to braiding anyonic quasiparticles or realizing fault-tolerant quantum gates. We outline how the fiber can be used as a quantum error-correcting code and discuss scenarios where these features can be experimentally probed.
The Geometry of Quantum Contextuality
Quantum contextuality is one of the most fundamental and non-classical features of quantum theory. It captures the impossibility of explaining quantum measurement outcomes using non-contextual hidden-variable models — that is, models in which the value assigned to an observable is independent of which other compatible observables are measured alongside it. This No-go result was proven in 1967 by Kochen and Specker and has since been tested experimentally many times. Beyond its foundational significance, contextuality is now recognized as a resource for quantum advantage in quantum computation and quantum communication protocols. Much like entanglement, it represents a striking and counterintuitive departure from classical intuition. This mini-course explores the geometric structure underlying operator-based proofs of quantum contextuality. In particular, we will see how finite geometry provides a powerful and elegant framework for understanding contextual configurations of Pauli observables. We will also discuss how these configurations can be tested experimentally using online quantum computing platforms.
Lecture 1: The Geometry of Contextuality
In this first lecture, I will introduce a geometric description of the two- and three-qubit Pauli groups. I'll show how these operator sets naturally organize into finite geometric structures and I will then present the operator-based proofs of the Kochen–Specker Theorem due to Peres and Mermin (e.g., the Peres–Mermin square), and show how these configurations of operators fit naturally into the geometric framework developed earlier.
Lecture 2: The “Rio Negro” Inequality
Named after the Rio Negro river in South America, the Rio Negro inequality (introduced by Adán Cabello) provides an experimentally testable inequality derived from contextual operator configurations. These configurations consist in linear combination of multi- Pauli observables. We will derive the inequality, explain its conceptual meaning, and show how it can be implemented on an online quantum computer to experimentally demonstrate contextuality. This mini-course is intended for Master’s and PhD students with an interest in quantum physics and quantum information. The only prerequisite is familiarity with Pauli matrices and basic quantum mechanics.
Introduction to Continuous and Discrete Phase-space Representations
List of topics to be covered:
1) Idea of quantization with symmetries. Classical manifold and coherent states.
2) General structure of the Stratonovich-Weyl map.
3) Examples: HW, SU(2), SU(1,1), SU(N).
4) Generalized Pauli group. Discrete coherent states Discrete symplectic operations.
5) Discrete phase-space structure.
6) Discrete quasidistributions.
7) A few words about multipartite systems and the stabilizers in phase-space.
Symplectic and Algebraic Geometry of Quantum States
Since quantum information tasks are usually achieved by manipulating spin and similar systems, or, in general, systems with a finite number of energy levels, classification problems are usually treated in frames of linear algebra. Here, I propose to shift the attention to a geometric description. Treating consistently quantum states as points of a projective space rather than as vectors in a Hilbert space, it is possible to apply powerful methods of differential, symplectic, and algebraic geometry to attack the problem of equivalence of states with respect to the strength of correlations, or, in other words, to classify them from this point of view. Such classifications are interpreted as an identification of states with “the same correlation properties”, i.e., ones that can be used for the same information purposes, or, from yet another point of view, states that can be mutually transformed one into another by specific, experimentally accessible operations. It is clear that the latter characterization answers the fundamental question “what can be transformed into what via available means?”. Exactly such an interpretation, i.e., in terms of mutual transformability, can be clearly formulated in terms of actions of specific groups on the space of states and is the starting point for the proposed methods.
Venue
The school will take place at the Instituto de Ciencias Nucleares, located on the main campus of the Universidad Nacional Autónoma de México (UNAM) in Mexico City. Address: Circuito Exterior s/n, Ciudad Universitaria, 04510 Ciudad de México.
Some places of local interest (ICN, Radisson Paraiso, Plaza San Jacinto, Plaza Jardin Hidalgo) as well as various metro/metrobus stations, are marked by pins in the clickable map below
Organizers
Eduardo Serrano-Ensástiga (University of Liège, Belgium)
Chryssomalis Chryssomalakos (ICN UNAM, Mexico)
John Martin (University of Liège, Belgium)