We study the off-diagonal current fluctuation in a macroscopic quantum system measured in an ideal manner that is as close as possible to the classical ideal measurement. We show rigorously that not only extended but also localized states contribute to the off-diagonal current fluctuation. This result contrasts with the fact that only the extended states affect the off-diagonal (Hall) conductivity and apparently contradicts the naive expectation from the fluctuation-dissipation theorem that might directly connect these two quantities. More specifically, we study the off-diagonal current fluctuation in a disordered two-dimensional electron system in a strong magnetic field at low temperatures. The fluctuation is almost unchanged from that of the pure system reflecting the property mentioned above, being approximately proportional to the Landau level filling factor with high accuracy. Our finding paves the way to estimate the filling factor and the electron density from the off-diagonal current fluctuation observed in macroscopic systems.
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