Kharkiv Quantum Seminar

Both in condensed matter physics, e.g. Eigenstate Thermalization Hypothesis, and in the physics of black holes a natural problem is how thermal a pure quantum state can appear to be. In black hole physics this is the essence of Hawking's Information Paradox, and connected to the (enormous) entropy increase of a black hole compared to the matter that could have given rise to a black hole. One can also turn the problem around and assume a pure quantum state with some specified partial thermal properties, such as the reduced states of individual modes, and try to estimate how thermal (or not thermal) are other properties. I will discuss these issues in the framework of Gaussian pure states and typical properties in ensembles of random states.

I will show that there is a natural set of constrained random symplectic transformations which give the required marginals, and for which one can estimate some multi-marginal properties, these being mode-mode correlations and entanglement (von Neumann entropy) of small subsets. Applied to black holes (very large phase space, specific choices of mode marginals etc) they are as thermal as can be. The more interesting case of entanglement of reasonably large subsets (say, comprising about half of the modes) is currently open, and I will describe the resulting matching problem (currently not solved).

The talk is based on joint work with Mario Kieburg, Lucas Hackl, available as  Quantum Science and Technology 10:045068 (2025), and on earlier work also with Pawel Horodecki and others.