Upper Bounds on Fluctuation Growths of Observables in Open Quantum Systems

TL;DR

This work establishes generator-independent upper bounds on the rate of fluctuation growth $d\sigma_A/dt$ for observables in open quantum systems, extending prior closed-system results. It derives the open-system evolution operator via both Taylor and Dyson formalisms, introducing a Hermitian pseudo-Hamiltonian $\Omega(t)$ that evolves the eigenvectors of the reduced density matrix, and relates $\frac{d}{dt}\langle A\rangle$ to $\langle \partial_t A\rangle$ and commutator dynamics with $\Omega$. Two bounds are obtained: (i) with knowledge of the pseudo-Hamiltonian, and (ii) without any specific evolution generator; both yield a bound of the form $\dot{\sigma}_A^2 \le 2\left( \langle(\partial_t A)^2\rangle + \frac{(\operatorname{tr}(\dot{\rho}\Delta A^2))^2}{4\sigma_A^2} \right)$ (generator-independent in Sec.4). The results are validated on a qubit with amplitude-damping noise, showing the inequalities hold for both time-independent and time-dependent observables, and for scenarios with or without explicit Hamiltonian dynamics; the analysis suggests that more detailed decomposition of dynamics yields looser bounds, while the bound remains a practical tool for controlling variance growth in noisy quantum devices.

Abstract

The upper bounds for the rate of fluctuation growth of an observable in both open and closed quantum systems have been studied actively recently. In our recent work we showed that the rate of fluctuation growth for an observable in a closed quantum system is upper bounded by the fluctuation of its corresponding velocity-like observable. That bound also indicated a tradeoff between the time derivatives of the mean and the standard deviation. In this paper we will look at open quantum systems in two cases. For the first case we find the generator of evolution for an open system employing both the Taylor expansion and the standard time-ordered evolution via the Dyson series, while in the second case we consider no specific information about the evolution of the system. We then find the rate of fluctuation growth in each case. Comparing the upper bounds for each case and considering the upper bound found for a closed system suggest that including more details by separating the contributions of the system and state dynamics seems to result in looser bounds for the rate of fluctuation growth.

Figures (1)

• Figure 1: A graphical representation illustrating the temporal progression of the inequality $\dot{\sigma}_{A}^{2}\leq\sigma_{\dot{A}}^{2}$ (or, alternatively, $\dot{\mu}_{A}^{2}+\dot{\sigma}_{A}^{2}\leq v_{A}^{2}$ with $\dot{\mu}_{A}^{2}\overset{\text{def}}{=}\left\langle \dot{A}\right\rangle ^{2}$ and $v_{A}^{2}\overset{\text{def}}{=}\left\langle \dot{A}^{2}\right\rangle$) in the $\left( \mu_{A}\text{, }\sigma_{A}\right)$-plane where $\mu_{A}\overset{\text{def}}{=}\left\langle A\right\rangle$ and $A\overset {\text{def}}{=}\sigma_{z}$. In our example, we have $\mu_{A}\left( t\right) =1-2e^{-\Gamma t}$, $\sigma_{A}\left( t\right) =2e^{-\frac{\Gamma}{2}t}\sqrt{1-e^{-\Gamma t}}$, and $v_{A}\left( t\right) =2\Gamma e^{-\frac{\Gamma}{2}t}$. The initial state of the system is assumed to be $\rho\left( 0\right) =\left\vert 1\right\rangle \left\langle 1\right\vert$. From our analysis, it happens that the inequality is violated (satisfied) for $\gamma\left( t\right) <1/4$ ($\gamma\left( t\right) \geq1/4$ ) or, in terms of time, for $t<(1/\Gamma)\ln(4/3)$ ($t\geq(1/\Gamma)\ln(4/3)$). Clearly, $\gamma\left( t\right)$ and $\Gamma$ are the damping probability and the decay rate, respectively. From a graphical standpoint, this violation is visualized in terms of a bold arrow of length $\left\langle \dot {A}\right\rangle ^{2}+\dot{\sigma}_{A}^{2}$ which is outside the circle of radius $v_{A}$. Moreover, when the inequality holds, the bold arrow is contained within the circle. Finally, observe that when $t$ approaches infinity, $\mu_{A}$ approaches one, $\sigma_{A}$ converges to zero, and the density operator of the system transitions to $\left\vert 0\right\rangle \left\langle 0\right\vert$.