  Process tensor matrix product operators (PT-MPOs) enable numerically exact
simulations for an unprecedentedly broad range of open quantum systems. By
representing environment influences in MPO form, they can be efficiently
compressed using established algorithms. The dimensions of inner bonds of the
compressed PT-MPO may be viewed as an indicator of the complexity of the
environment. Here, we show that the inner bonds themselves, not only their
dimensions, have a concrete physical meaning: They represent the subspace of
the full environment Liouville space which hosts environment excitations that
may influence the subsequent open quantum systems dynamics the most. This
connection can be expressed in terms of lossy linear transformations, whose
pseudoinverses facilitate the extraction of environment observables. We
demonstrate this by extracting the environment spin of a central spin problem,
the current through a quantum system coupled to two leads, the number of
photons emitted from quantum emitters into a structured environment, and the
distribution of the total absorbed energy in a driven non-Markovian quantum
system into system, environment, and interaction energy terms. Numerical tests
further indicate that different PT-MPO algorithms compress environments to
similar subspaces. Thus, the physical interpretation of inner bonds of PT-MPOs
both provides a conceptional understanding and it enables new practical
applications.
