Characterizing eigenstate thermalization via measures in the Fock space of operators

A common question of interest in quantum statistical mechanics, especially in the context of eigenstate thermalization and many-body localization, is, ``how well can two operators or states be distinguished using measurements on just a few sites at a time?" In this work,we define two easily computable measures --  he ``n-weight" and the  ``n-distinguishability“ that quantify the difference between two operators when given access to strictly n-site measurements. We then define a construction that allows viewing a general operator or many-body state as a single particle wavefunction on a complicated graph. Interestingly, the two measures acquire intuitive, geometric interpretations on this graph. The utility of the mapping to the graph will be demonstrated in the context of eigenstate thermalization, and related ongoing work on distinguishing between various quantum statistical phases will be mentioned.