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.