The trade-off between diagonal and off-diagonal elements in the eigenstate thermalization hypothesis

Abstract: To bypass using local observables as intermediate quantities in proving the eigenstate thermalization hypothesis (ETH), we have introduced an observable-independent measure of distinguishability. In this paper, we establish the connection between this measure and several other ETH measures in a more natural way. We first demonstrate a universal trade-off relation between the diagonal and off-diagonal elements of the measure. We then extend this discussion to eigenstate typicality and the average observable. This trade-off relationship reveals that the exponential growth of off-diagonal elements directly suppresses their own values, as well as indirectly suppressing the diagonal elements. This provides a new perspective on the physical mechanisms underlying ETH. Finally, through numerical calculations on a one-dimensional Ising spin chain, we explore various trade-off relationships and examine strong and weak ETH.