# "Entropy" of a set of correlators in a quantum system

Please forgive the ill-posedness of this question; I am hoping someone can help me formulate what I am asking more clearly.

Consider the ground state of a one-dimensional quantum spin chain on $$N$$ sites. There are a finite number (exponential in $$N$$) of operators in the theory, and therefore one can compute a finite number of correlation functions in e.g. the ground state.

With access to the numerical value of every single correlator, one can presumably reconstruct the ground state density matrix. However, many correlators may be equivalent either by symmetry or by entanglement between spins in the ground state, especially in a critical system.

My question is whether there is a notion of "entropy" or "information" to characterize sets of correlation functions with "maximum entropy", i.e. sets that contain most of the information about the ground state while only including a subset of the $$\sim e^N$$ operators of the theory.

The entropy and subsystem entropy of the state is well studied. For example, one can find subsystems of quantum systems which contain "most" of the information about the entire system (i.e. Page's theorem says once you get to half the system, you're pretty much good). Perhaps a similar notion can be defined for sets of operators, especially in a situation with an operator-state correspondence in the thermodynamic limit (i.e. a critical spin chain).

Does there exist some notion of "entropy" for sets of operators (and hence correlation functions)i.e. a subadditive function which is maximized for the identity operator and zero for all $$\sim e^N$$ operators of the spin chain?

• PRX 10, 011020 (2020) might be of some interest. The paper doesn't address the notion of entropy you're after, but it does discuss the redundancy of information in the correlation functions. Commented Oct 21, 2023 at 16:05