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

In order to bypass the local measurement as an intermediate quantity in the proof of the eigenstate thermalization hypothesis (ETH), we have proposed a measurement-independent distinguishability measure. In this paper, we establish the connection between this measure and several other ETH measures in a more unified way. We first prove a universal trade-off relation for the diagonal and off-diagonal elements of the measure. Then we extend it to the discussion of eigenstate typicality and the average observable. In the trade-off relationship, the exponential increase of off-diagonal elements will directly suppress the value of off-diagonal elements, and the diagonal elements will also be suppressed indirectly. This relationship gives another perspective to understand the physical mechanism behind ETH. Finally, by numerically calculating one-dimensional Ising spin chain, we examine several trade-off relationships and strong and weak ETH.