Explore the cutting-edge intersection of optimal transport theory and quantum mechanics through this comprehensive workshop series from the Institute for Pure & Applied Mathematics. Delve into the non-commutative counterpart of Monge-Kantorovich optimal transport theory, examining how classical transport concepts translate to quantum systems through density operators and Hermitian matrices. Investigate the foundational work of Connes, Voiculescu, Carlen, and Maas in developing gradient flow theories for quantum settings, while exploring the rich mathematical structures that emerge from statistical manifolds and Riemannian geometry in non-commutative spaces. Examine key questions surrounding unifying principles in quantum transport, including the role of functional inequalities, curvature in non-commutative flows, and stochastic models for quantum evolution. Learn from leading experts as they present research on Weyl symbols of Gaussian semigroups, quantum algorithms through mathematical analysis, vector-valued optimal transport formulations, and viscosity solutions in non-commutative variables. Discover applications ranging from quantum Markov semigroups and Ricci curvature bounds to free probability theory, Lindblad evolution equations, and continuous quantum measurements. Gain insights into the Wasserstein distance in quantum mechanics, completely positive maps, matrix-valued optimal transport problems, and the integrability of Hamilton-Jacobi-Bellman equations in quantum contexts. This workshop addresses fundamental questions about natural time constants in quantum processes, the physical significance of various mathematical structures, and the potential for these frameworks to predict new quantum properties.

Syllabus

Ralph Sabbagh - On the Weyl symbols of Gaussian semigroups - IPAM at UCLA
Di Fang - How Mathematical Analysis can help better understand quantum algorithms - IPAM at UCLA
Katy Craig - Vector Valued Optimal Transport: From Dynamic to Kantorovich Formulations
Wilfrid Gangbo - Viscosity solutions in non-commutative variables - IPAM at UCLA
Marius Junge - Why Ricci curvature (almost) failed us in noncommutative dynamics - IPAM at UCLA
Peixue Wu - From non-commutative optimal transport to limitations of quantum simulations
Haonan Zhang - On the Ricci curvature lower bounds for quantum Markov semigroups - IPAM at UCLA
Franca Hoffmann - Covariance-Modulated Optimal Transport Geometry - IPAM at UCLA
Dimitri Shlyakhtenko - Transport in free probability theory - IPAM at UCLA
Eric Carlen - Approach to steady state for Lindblad evolutions equations in statistical mechanics
Francois Golse - Wasserstein Distance and the Observability Problem in Quantum Mechanics
Roy Araiza - Lipschitz Cost of Quantum Channels on von Neumann Algebras - IPAM at UCLA
David Andrew Jekel - Relating entropy and Wasserstein distance in free probability - IPAM at UCLA
Melchior Wirth - Completely positive maps and the V-transform - IPAM at UCLA
Andrew Jordan - Continuous quantum measurements: The most likely path with pre- and post-selection
Dmitry Vorotnikov - Matrix-valued problems reminiscent of optimal transport and applications to PDEs
Rocco Duvenhage - Quadratic Wasserstein distance between quantum dynamical systems - IPAM at UCLA
Dániel Virosztek - Optimal transport by quantum channels: non-quadratic problems, metric properties
Jean-Claude Zambrini - Integrability of Hamilton-Jacobi-Bellman equation - IPAM at UCLA
Karol Zyczkowski - Transforms of classical & quantum states & discrete dynamics in quantum measures