Full characterization of modular values for finite-dimensional systems

•Deriving the relations between the modular values and weak values for n-dimensional systems.•Showing a linear relation for two-dimensional Hilbert space case.•Enable one to obtain weak value via modular value, which easier to measure.•Apply to the cases such as EPR, Hardy, and Cheshire Cat paradoxes.

Kedem and Vaidman obtained a relationship between the spin-operator modular value and its weak value for specific coupling strengths [14]. Here we give a general expression for the modular value in the n  -dimensional Hilbert space using the weak values up to (n−1)(n−1)th order of an arbitrary observable for any coupling strength, assuming non-degenerated eigenvalues. For two-dimensional case, it shows a linear relationship between the weak value and the modular value. We also relate the modular value of the sum of observables to the weak value of their product.