Can quantum fluctuations be consistently monitored?

Abstract

Recent works on the decoherent histories formalism suggested that macroscopic quantities (extensive sums of local observables) in quantum many-body systems can be consistently monitored: The existence of past measurements does not alter future outcome distribution. Here, we show that fluctuations of macroscopic quantities cannot be consistently monitored in general, in contrast to their intensive mean value. Exceptions include fluctuations at infinite temperature, at critical points, and in semiclassical systems. We analytically quantify non-consistency in terms of susceptibility, and obtain related results on entropy growth under noisy unitary.

Figures (2)

• Figure 1: Joint and marginal outcome distributions from two-time ($t = 0,1$) monitoring of (rescaled) total magnetization in the quantum Ising model \ref{['eq:ising']} ($L = 16$) initialized in the ground state. The $t=0$ marginal histogram is plotted without filling color together with the $t = 1$ marginal in the top panels. We simulated $8000$ measurements for each data set. (a) Away from criticality ($J = 2/3 < 1$), the Gaussian fluctuations scaling as $\sim L^{1/2}$ cannot be consistently monitored: The measurement at $t = 0$ alters the outcome distribution at $t = 1$. See also Fig. \ref{['fig:diffs']}-(a). $\gamma_0 = \gamma_1 = 1$. (b) At quantum criticality ($J = 1$), the large critical fluctuations rescaled accordingly $q := \sum_j Z_j / L^{15/16}$ can be consistently monitored, $\gamma_0 = \gamma_1 = 2$.
• Figure 2: The difference between outcome variances $\delta := \left< \tilde{x}_t^2 \right> - \left< \tilde{x}_0^2 \right>$, in a two-time monitoring setup with the non-critical Ising model ($J = 2/3$), same as in Fig. \ref{['fig:hists']}-(a), except that we vary the monitoring strength $\gamma_0 = \gamma_{1} = \gamma$ and system size. (a) Data with ground state initial condition are compared to the analytical prediction $\delta \propto \gamma^4$ (dashed line). (b) The absolute difference $|\delta|$ averaged over $32$ Haar random initial conditions (per data point) tend to $0$ as $L$ increases.