[1602.01594] The two different aspects of the modular value

[ Authors ]
Le Bin Ho, Nobuyuki Imoto
[ Abstract ]
We show that the modular value is expressed by the average of the dynamic phase factors using the complex conditional probabilities. In relation to this expression, the chain rule of the conditional probabilities is also derived, which relates the initial-final-state modular value with the initial- transitional-state modular values and the transitional-final-state weak values. Then, we express the modular value with the relative change and the total phase shift (the geometric phases and the intrinsic phase), which is another aspect of the modular value. The modulus of modular value is found to play a significant role in the relative change of the quantum system under the dynamic evolution, and also, the argument of modular value is found to have connection with the total phase shift. By analyzing these two relations, we can apply to calibrate the value of coupling constant, which was used in modular value and the geometric phase, which has many implications. In contrast, one can fully determine the modular value (modulus and argument) by examining the relative change and the corresponding total phase shift.