**Speaker**: Ben Grossmann (ELEC)

**Time and place**: Tu, 17/09/2019: 10.00 - 11:00, ELEC seminar room

**Absract**: For this talk, we will discuss the topic of my PhD thesis, which consisted of two projects. The first project concerns quantum information theory and the linear preservers of "maximally entangled states". In this context, "maximally entangled states" are a set of (rank-1, positive semidefinite) matrices, which in quantum information theory correspond to the highest-information system configurations. The goal of the project is to answer the following: which linear maps (with matrix inputs and matrix outputs) produce a maximally entangled output for any maximally entangled input? In the quantum context, this corresponds to classifying the "quantum channels" that perfectly preserve information (in a certain sense). The second project concerns the "fractional minimal rank", a new tool related to the problem of finding low-rank completions for "partial matrices" (matrices with some known entries and some unknown entries). The lowest rank that can be achieved by filling in the unknowns in a partial matrix (i.e. "completing" the matrix) is called the "minimal rank" of that partial matrix. We define and consider the "fractional minimal rank", a lower bound for the minimal rank that is found by filling in the unknowns with square matrices rather than with numbers.

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