Full characterization of modular values for two-dimensional systems

TL;DR

This paper derives a general formula linking weak and modular values in two-dimensional quantum systems, revealing the failure of sum rules and illustrating paradoxes related to nonlocality.

Contribution

It provides a comprehensive expression connecting weak and modular values for any coupling strength in 2D systems, expanding their theoretical understanding.

Findings

Derived a general relation between weak and modular values.

Showed the failure of sum rule for modular values.

Presented examples involving quantum paradoxes.

Abstract

Vaidman pointed out the importance of modular values, and related the modular value of a Pauli spin operator to its weak value for specific coupling strengths [Phys. Rev. Lett. 105, 230401 (2010)]. It would be useful if this relationship is generalized since a modular value, which assumes a finite strength of the measurement interaction, is sometimes more practical than a weak value, which assumes an infinitesimally small interaction. In this paper, we give a general expression that relates the weak value and the modular value of an arbitrary observable in the 2-dimensional Hilbert space for an arbitrary coupling strength. Using this expression, we show the "failure of sum rule" for modular values, which has a resemblance to the "failure of product rule" for weak values. We give examples of "failure of sum rule" for some interesting cases, i.e., paradoxes based on nonlocality, which include EPR paradox, Hardy's paradox, and Cheshire cat experiment.