Every time the release button of a digital camera is pressed, several megabytes of raw data are recorded. But the size of a typical jpeg output file is only 10% of that. What a waste! Can't we design a process which records only the relevant 10% of the data to begin with?
Compressed sensing is a young theory that achieves this trick for certain signals. There has been a fruitful exchange of ideas between this field and quantum physics: Mathematical methods from quantum information have found many applications in classical compressed data acquisition tasks. Conversely, compressed sensing ideas have advanced the theory of quantum state estimation. I will introduce the basics of the theory and outline where we stand with regards to quantum tomography applications. I will mention very recent results in uncertainty quantification, as well as applications of the diamond norm and the Clifford group in compressed sensing.