Geometry and Entanglement of Multipartite Quantum States
This course introduces basic concepts in quantum information theory and the structure of multipartite quantum systems. Much of the content can be viewed as a study of the geometry of quantum states. Topics include quantum states, von Neumann and relative entropy, distances between quantum states, quantum channels, strong subadditivity and quantum Markov states; bipartite entanglement measures; local unitary equivalence, finite-depth circuits, and quantum cellular automata; stabilizer states, Clifford gates, and absolutely maximally entangled states.
While background material and applications to many-body physics will be kept to a minimum, the course will lay the foundations for several deeper topics in quantum many-body physics. Toward the end of the course, we will give brief introductions to multipartite quantum-state chirality and the entanglement bootstrap, closely related to the lecturer’s research interests.
The course is intended mainly for graduate students and postdoctoral researchers, although advanced undergraduates are also welcome. Emphasis will be placed on conceptual understanding, examples, open questions, and connections to current research. Problem sets will be distributed, ranging from standard exercises to open-ended and research-oriented questions. The course will be taught in English.
Course format. The course meets on Wednesdays from 13:30 to 16:55, corresponding to four BIMSA class-hours. The main lecture occupies the first three class-hours, 13:30–16:05. The final class-hour, 16:10–16:55, is reserved primarily for questions, discussion, examples, and further exploration of the lecture material. Students who need to catch an earlier shuttle from BIMSA are welcome to leave after the main lecture; essential new material will normally not be introduced during the final Q&A session.
While background material and applications to many-body physics will be kept to a minimum, the course will lay the foundations for several deeper topics in quantum many-body physics. Toward the end of the course, we will give brief introductions to multipartite quantum-state chirality and the entanglement bootstrap, closely related to the lecturer’s research interests.
The course is intended mainly for graduate students and postdoctoral researchers, although advanced undergraduates are also welcome. Emphasis will be placed on conceptual understanding, examples, open questions, and connections to current research. Problem sets will be distributed, ranging from standard exercises to open-ended and research-oriented questions. The course will be taught in English.
Course format. The course meets on Wednesdays from 13:30 to 16:55, corresponding to four BIMSA class-hours. The main lecture occupies the first three class-hours, 13:30–16:05. The final class-hour, 16:10–16:55, is reserved primarily for questions, discussion, examples, and further exploration of the lecture material. Students who need to catch an earlier shuttle from BIMSA are welcome to leave after the main lecture; essential new material will normally not be introduced during the final Q&A session.
Lecturer
Date
16th September ~ 9th December, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday | 13:30 - 16:55 | A14-101 | ZOOM A | 388 528 9728 | BIMSA |
Syllabus
Weeks 1–2: Quantum states, entropy, and quantum Markov states
Qubits and qudits; quantum states in tensor-product Hilbert spaces; pure and mixed states; von Neumann entropy and relative entropy; distances between quantum states and distinguishability; quantum channels; conditional mutual information; strong subadditivity; quantum Markov states and their structure.
Weeks 3–4: Bipartite entanglement measures
Bipartite mixed states; product and separable states; local operations and classical communication (LOCC); entanglement monotones, including squashed entanglement and entanglement negativity.
Weeks 5–6: Local unitaries, quantum circuits, and complexity
We introduce a hierarchy of unitary operations according to their complexity:
-on-site local unitaries and equivalence of states under such transformations;
-finite-depth local unitary circuits;
-quantum cellular automata;
-unitary transformations of higher complexity than the above classes.
Weeks 7–8: Stabilizer states, Clifford gates, and absolutely maximally entangled states
-Pauli algebra, stabilizer formalism, and the Clifford group and Clifford gates;
-absolutely maximally entangled (AME) states and perfect tensors;
-examples and constructions of AME states;
-nonexistence of AME states for four qubits.
Weeks 9–10: Multipartite chirality
-definition and examples of chiral quantum states;
-chirality of four-partite pure states and the modular commutator;
-chirality in nature — a colloquium-style discussion;
-chirality of bipartite mixed states;
-chiral separable states;
-chiral states related to reflection positivity.
Weeks 11–12: Introduction to the entanglement bootstrap
Entanglement-based constraints in many-body systems; the gapped entanglement bootstrap, from area-law entanglement to anyon theory; the gapless entanglement bootstrap beyond the area law; and a sketch of open problems, conjectures, and possible directions for further development.
Audience
Graduate
, Postdoc
, Researcher
, Advanced Undergraduate
Video Public
Yes
Notes Public
Yes
Language
English