From Quantum-Mechanical Acceleration Limits to Upper Bounds on Fluctuation Growth of Observables in Unitary Dynamics

TL;DR

The paper investigates fundamental speed limits on how rapidly fluctuations, quantified by the standard deviation, of quantum observables can grow under unitary dynamics, connecting to quantum speed limits and acceleration concepts. It provides two proofs of the key inequality $|d\sigma_A/dt|\le \sigma_{v_A}$, one revisiting Hamazaki's derivation and one using quantum-acceleration-based reasoning in projective Hilbert space, with $v_A=\dot A+\mathcal{L}^{\dagger}[A]$ and, for unitary dynamics, $v_A=\dot A+(i/\hbar)[H,A]$. Three illustrative examples—two-level qubits with tight and loose bounds and a harmonic oscillator in a finite Fock space—demonstrate when the bound is saturated and how fluctuation growth relates to signal quality in practice. The work lays a foundation for extending these uncertainty-based speed limits to open systems and for developing physically meaningful figures of merit in quantum control and metrology, with potential applications in nonequilibrium thermodynamics and quantum information processing.

Abstract

Quantum Speed Limits (QSLs) are fundamentally linked to the tenets of quantum mechanics, particularly the energy-time uncertainty principle. Notably, the Mandelstam-Tamm (MT) bound and the Margolus-Levitin (ML) bound are prominent examples of QSLs. Recently, the notion of a quantum acceleration limit has been proposed for any unitary time evolution of quantum systems governed by arbitrary nonstationary Hamiltonians. This limit articulates that the rate of change over time of the standard deviation of the Hamiltonian operator-representing the acceleration of quantum evolution within projective Hilbert space-is constrained by the standard deviation of the time-derivative of the Hamiltonian. In this paper, we extend our earlier findings to encompass any observable A within the framework of unitary quantum dynamics. This relationship signifies that the speed of the standard deviation of any observable is limited by the standard deviation of its associated velocity-like observable. Finally, for pedagogical purposes, we illustrate the relevance of our inequality by providing clear examples. We choose suitable observables related to the unitary dynamics of two-level quantum systems, as well as a harmonic oscillator within a finite-dimensional Fock space.

Figures (3)

• Figure 1: Numerical verification of the inequality $(\dot{\sigma}_{A})^{2}+\left( \dot{\mu}_{A}\right) ^{2}\leq\left\langle v_{A}^{2}\right\rangle$ with $\mu_{A}\overset {\text{def}}{=}\left\langle A\right\rangle$ for $\omega_{0}=\nu_{0}=1$ and $A\left( t\right) \overset{\text{def}}{=}a\left( t\right) \sigma _{x}+b\left( t\right) \sigma_{z}$, with $a\left( t\right) =b\left( t\right) =t$. Observe that $\left\langle v_{A}^{2}\right\rangle$ and $(\dot{\sigma}_{A})^{2}+\left( \dot{\mu}_{A}\right) ^{2}$ are represented by a dashed and a solid line, respectively. Formally, the evolution of the two-level quantum system on the Bloch sphere, starting from the initial state $\left\vert \psi\left( 0\right) \right\rangle =\left\vert +\right\rangle \overset{\text{def}}{=}(\left\vert 0\right\rangle +\left\vert 1\right\rangle )/\sqrt{2}$, can be viewed as specified by a time-dependent Hamiltonian of the form H$\left( t\right) \overset{\text{def}}{=}-\vec{\mu}\cdot \vec{B}\left( t\right)$, where $\vec{B}\left( t\right)$ is the time-varying magnetic field, $\vec{\mu}\overset{\text{def}}{=}-\mu _{\mathrm{B}}\vec{\sigma}$ is the magnetic moment of the electron, and $\mu_{\mathrm{B}}\overset{\text{def}}{=}e\hslash/(2m_{e})\simeq+9.27\times 10^{-24}$$\left[ \mathrm{MKSA}\right]$ is the Bohr magneton.
• Figure 2: On the left side, we plot the SNR$\left( A\right) \overset{\text{def}}{=}\left\langle A\right\rangle ^{2}/\mathrm{var}(A)$ versus time $t$ for the first (thin solid line) and the second (thick solid line) examples, respectively. On the right side, instead, we display the temporal behavior of the expectation value of the square of the velocity observable, $\left\langle v_{A}^{2}\right\rangle \overset{\text{def}}{=}\left\langle \left( dA/dt\right) ^{2}\right\rangle \geq\left( \dot{\mu}_{A}\right) ^{2}+\left( \dot{\sigma}_{A}\right) ^{2}$, for the first (thin solid line) and the second (thick solid line) examples, respectively. In all plots, we assume $\omega_{0}=\nu_{0}=1$ and $a\left( t\right) =b\left( t\right) =t$. Recall that the observables $A(t)$ being measured in the first and second examples are given by $a\left( t\right) \sigma_{x}$ and $a\left( t\right) \sigma_{x}+b\left( t\right) \sigma_{z}$, respectively. Finally, we point out that the displayed behaviors suggest that to a lower SNR$\left( A\right)$ there corresponds a higher $\left\langle v_{A}^{2}\right\rangle$.
• Figure 3: Numerical verification of the inequality $(\dot{\mu}_{A})^{2}+\left( \dot{\sigma}_{A}\right) ^{2}\leq\left\langle v_{A}^{2}\right\rangle$ using a squeezed coherent state and a time-dependent operator $\hat{A}\left( t\right) \overset{\text{def}}{=}\cos\left[ \theta\left( t\right) \right] \hat{x}+\sin\left[ \theta\left( t\right) \right] \hat{p}$, where $\theta\left( t\right) \overset{\text{def}}{=}\cos\left( t\right)$. In a), we plot the Wigner function $W\left( x\text{, }p\right)$ of the squeezed state used for the initial conditions in the position-momentum space. In addition, we note in a) that the operator $\hat{A}(t)$ forms an angle $\theta\left( t\right)$ with the position axis. In the example, the squeezed coherent state is characterized by $\alpha\overset{\text{def}}{=}2+i$, and $z\overset {\text{def}}{=}0.5+0.5i$. In b), we visualize the inequality $(\dot{\mu}_{A})^{2}+\left( \dot{\sigma}_{A}\right) ^{2}\leq\left\langle v_{A}^{2}\right\rangle$ as a function of $\theta$ in a polar plot. The black solid line represents $(\dot{\mu}_{A})^{2}+\left( \dot{\sigma}_{A}\right) ^{2}$ and is always bounded by the dashed line that describes $\left\langle v_{A}^{2}\right\rangle$. Thus, the inequality is constantly preserved. Finally, we assume $\omega=\hslash=m=1$ in all our numerical calculations.