Symmetry Properties of Quantum Dynamical Entropy

TL;DR

This work advances the understanding of quantum dynamical entropy by deriving rigorous inequalities for AFL entropy under symmetry in finite-dimensional systems, encompassing Abelian, anticommuting, and non-Abelian cases. It shows that when measurements respect the symmetry, the cumulative AFL entropy saturates at lower values dictated by the sector structure, while symmetry-agnostic measurements yield the usual higher bounds; these insights are supported by numerical studies of perturbed quantum cat maps. The analytical results cover general unitary dynamics, including tensor-product and commutant partitions, and reveal how symmetry-induced decompositions constrain information production under repeated measurements. The findings highlight the critical role of symmetry in quantum dynamics with measurements and provide a adaptable framework for diagnosing quantum chaos across diverse models and probes, with potential connections to CNT entropy and holographic perspectives.

Abstract

As quantum analogs of the classical Kolmogorov-Sinai entropy, quantum dynamical entropies have emerged as important tools to characterize complex quantum dynamics. In particular, Alicki-Fannes-Lindblad (AFL) entropy, which quantifies the information production rate of a coherent quantum system subjected to repeated measurement, has received considerable attention as a potential diagnostic for quantum chaos. Despite this interest, the precise behavior of quantum dynamical entropy in the presence of symmetry remains largely unexplored. In this work, we establish rigorous inequalities of the AFL entropy for arbitrary unitary dynamics (single-particle and many-body) in the presence of various types of symmetry. Our theorems encompass three cases: Abelian symmetry, an anticommuting unitary, and non-Abelian symmetries. In particular, we show that, while the cumulative AFL entropy generally saturates to the dimensional bound at late times for chaotic dynamics, this saturation value is distinctively lower when the measurements respect the symmetries. We motivate our main results with numerical simulations of the perturbed quantum cat maps. Our findings highlight the crucial role of symmetry in quantum dynamics under measurements, and our framework is readily adaptable for investigating symmetry's influence across diverse probes of quantum chaos.

Figures (11)

• Figure 1: (a) Overview of the general setup. The system undergoes repeated steps of measurement and discrete (Floquet) time evolution. Without post-selection, this generates entanglement with the auxiliary measurement/environment Hilbert space. The entanglement cut is represented by the dashed line. The AFL entropy, defined in Section \ref{['sect:dynamical-entropy']}, is the maximal growth rate of this entanglement entropy over all possible measurement choices. The cumulative AFL entropy, defined in Section \ref{['sect:dynamical-entropy']} as a more suitable measure for finite-dimension systems, is the entanglement entropy under a particular set of measurements. The same construction is used in the computation of quantum RP resonances garcia-mata-2005-SpectralApproachyoshimura-2025-TheoryIrreversibility, spatiotemporal entanglement cotler-2018-SuperdensityOperatorsodonovan-2025-DiagnosingChaos, and butterfly-flutter fidelity dowling-2024-OperationalMetric. Post-selecting on the measurement results yields a projected ensemble, as used in MIPT Li-2018-ZenoSkinner-2019-mipt, deep thermalization ippoliti-2022-DeepTherm, and post-selected dynamical entropy slomczynski-2017-QuantumDynamical. This is discussed in more detail in Appendix \ref{['sect:projected-ensembles']}. (b) A sketch of the typical behavior of cumulative AFL entropy as a function of time. For chaotic dynamics, the initial growth is linear until nearing the dimensional capacity for entanglement and saturating. In the presence of symmetry, the growth and saturation are lowered, provided the measurements are compatible with the symmetry. Our analytical results in Sections \ref{['sect:abelian']}, \ref{['sect:tensorproduct']}, and \ref{['sect:nonabelian']} (along with numerical results in Section \ref{['sect:catmapresults']}) show that, the lowered value depends on how the Hilbert space decomposes under different types of symmetries. In contrast, our numerical results suggest the cumulative AFL entropy is not sensitive to the symmetries if the measurements do not respect them. Note that the analytical results apply to both single-particle and many-body unitary dynamics.
• Figure 2: Tensor network representations of $\ket{\Psi}$, $\Tilde{\rho}[\mathcal{X}]$, and $\sigma[\mathcal{X}]$. These are the pure state on $ESP$ and the reduced states on $E$ and $SP$ respectively. The top line of $X$ is the Kraus index, which is the environment Hilbert space $\mathcal{H}_E$. See cotler-2018-SuperdensityOperators for further diagrams and bridgeman-2017-HandwavingInterpretivebiamonte-2020-LecturesQuantum for details on this notation more generally.
• Figure 3: Tensor network representation of the pure state $\ket{\Psi}$ after $t$ time steps of measurement $X$ and unitary evolution $U$.
• Figure 4: Cumulative AFL entropy of quantum cat map with Abelian symmetries. (a)$R$ symmetry. The plot shows the cumulative AFL entropy with an $R$-symmetric partition. $R$ is a symmetry of the dynamics for dimensions 115 and 120 (dashed lines) only. The horizontal dash-dotted lines are the dimensional bounds for the largest appropriate dimension plotted: $2 \log N$ for no symmetry $(N=119)$, and the lowered bound $2 \log M + \log s$ for $R$ symmetry $(N=120)$. (b)$W$ symmetry. The cat maps for all dimensions plotted are symmetric under $W$. The plot shows the cumulative AFL entropy with random partitions (solid lines) and $W$-symmetric partitions (dashed lines). The horizontal dash-dotted lines show the dimensional bounds for the largest appropriate dimension plotted $(N=120)$: $2\log N$ for random partition, and the lowered bound of $2\log(N/2) + \log 2$ for $W$-symmetric partition.
• Figure 5: Cumulative AFL entropy of quantum cat map with an anticommuting unitary. The figure compares random partitions (solid lines) and two tensor product partitions: measurement of pseudospin-z (dashed lines), and no pseudospin measurement (dotted lines). The respective bounds of $2\log N$, $2\log(N/2) + \log 2$, and $2\log(N/2)$ for $N=118$ are shown by the horizontal dash-dotted lines.

Theorems & Definitions (7)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• proof