Robert Zeier (Technische Universität München): Controllability and symmetries of coupled quantum systems
→
Europe/Budapest
Tanacsterem
Tanacsterem
Description
We study finite-dimensional quantum systems and their symmetries using representation-theoretic methods in the context of Lie algebras and Lie groups. The convenience of this approch is exemplified in two particular cases. First, we ask what one can do with a given coupled quantum system whose time evolution is to a certain degree externally controllable. We present a technique based on analyzing symmetries which decides if a controlled quantum system can simulate a given effective Hamiltonian. Moreover, our technique can compare the respective computational power of two controlled quantum systems. We emphasize that our approach improves on the conventional approach of computing Lie-closures of generators and harnesses the symmetries of the quantum system in order to reduce Lie-algebraic computations to effective linear-algebra ones. On a mathematical level, this question is related to the study of how tensor products of representations decompose when restricted from a compact Lie algebra to one of its subalgebras. Second, coupled quantum systems exhibit an intricate structure not explained by properties of their subsystems alone. We study the plethora of emerging symmetries that are invariant under local operations, i.e., operations which do not lead to interactions between the subsystems. We enumerate these symmetries by computing Hilbert series and also explore connections to so-called Kronecker coefficients. We particularly focus on the case of three-qubit mixed states. Our approach provides a new foundation for better understanding non-local quantum states. In general, our point of view highlights the utility of algebraic and representation-theoretic tools for analyzing quantum systems.
