Kolmogorov-Arnold-Moser Stability for Conserved Quantities in Finite-Dimensional Quantum Systems

We show that for any finite-dimensional quantum systems the conserved quantities can be characterized by their robustness to small perturbations: for fragile symmetries, small perturbations can lead to large deviations over long times, while for robust symmetries, their expectation values remain close to their initial values for all times. This is in analogy with the celebrated Kolmogorov-Arnold-Moser theorem in classical mechanics. To prove this result, we introduce a resummation of a perturbation series, which generalizes the Hamiltonian of the quantum Zeno dynamics.

Figure 1

Dynamics of the expectation of a randomly picked observable $M$ in a Heisenberg chain with $N=4$ as a function of time. We show its decomposition into nonconserved, robust, and fragile parts [Eqs. (11)–(13)]. The initial state and the perturbation $V$ are chosen randomly with strength $ϵ\parallel V\parallel =0.02\parallel H\parallel$. Shown is one realization.

Figure 2

Same setup and realization as in Fig. 1, but now showing the dynamics of the expectation of $Q_2=H$ (robust) and $Q_1={\sum }_{n=1}^N{\sigma }_{n,z}$ (fragile).

Figure 3

The free and perturbed evolutions for a qubit of (a) the robust monotone $f_{{\mathcal{M}}_{\text{robust}}}(\rho )=|⟨0|\rho |1⟩{|}^2$, with ${\mathcal{M}}_{\text{robust}}=-(i/\sqrt{2})[{\sigma }_z,·]$ and $\lambda =1$, and of (b) the fragile monotone $f_{{\mathcal{M}}_{\text{fragile}}}(\rho )$, with ${\mathcal{M}}_{\text{fragile}}=-(i/\sqrt{2})[{\sigma }_z,·]+|0⟩⟨0|·|0⟩⟨0|$ and $\lambda =1$, where $\mathcal{L}=-(i/2)\omega [{\sigma }_z,·]-\frac{1}{2}\kappa (1-{\sigma }_z·{\sigma }_z)$ and $ϵ\mathcal{V}=-(i/2)ϵg[{\sigma }_x,·]$. The perturbation creates coherence between the eigenstates of ${\sigma }_z$, $|0⟩$ and $|1⟩$. The initial state of the qubit is given by the coherence vector $(r_x,r_y,r_z)=(0.2,0,0.8)$, and the parameters are set at $g=\kappa$ and $ϵ=0.1$ ($\omega$ is irrelevant).