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arXiv:2603.29900

Dynamics of entanglement entropy for a locally monitored lattice gauge theory

Nisa Ara, Arpan Bhattacharyya, Nilachal Chakrabarti, Neha Nirbhan, Indrakshi Raychowdhury

quant-phSubmitted Mar 31, 2026• Announced Apr 1, 2026• 0 citations• Revised Jul 29, 2026 (v2)
v1 · Mar 31, 2026v2 · Jul 29, 2026
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Abstract

The $1+1$ dimensional $Z_2$ gauge theory is the simplest model that allows for quantum computation or quantum simulation to probe the fundamental aspects of a gauge theory coupled with dynamical fermions. To reliably benchmark such a system, it is crucial to understand the non-unitary quantum dynamics arising from the underlying non-Hermitian evolution and to model the effects of quantum measurements. In this work, we study post-selected filtering dynamics of physical observables for a $\mathbb {Z} _2$ gauge theory. Tensor network calculations are performed to dynamically probe entanglement entropy at larger lattice sizes. We report that projective measurement of local and diagonal observables (electric and mass energy densities) in the computational basis demonstrates the absence of any measurement-induced phase transition like phenomenon, as indicated by the system-size independence of the late-time saturation value of the bipartite entanglement entropy.

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The 1+11+11+1 dimensional Z2Z_2Z2​ gauge theory is the simplest model that allows for quantum computation or quantum simulation to probe the fundamental aspects of a gauge theory coupled with dynamical fermions. To reliably benchmark such a system, it is crucial to understand the non-unitary quantum dynamics arising from the underlying non-Hermitian evolution and to model the effects of quantum measurements. In this work, we study post-selected filtering dynamics of physical observables for a Z2\mathbb {Z} _2Z2​ gauge theory. Tensor network calculations are performed to dynamically probe entanglement entropy at larger lattice sizes. We report that projective measurement of local and diagonal observables (electric and mass energy densities) in the computational basis demonstrates the absence of any measurement-induced phase transition like phenomenon, as indicated by the system-size independence of the late-time saturation value of the bipartite entanglement entropy.
The 1+11+11+1 dimensional Z2Z_2Z2​ gauge theory is the simplest model that allows for quantum computation or quantum simulation to probe the fundamental aspects of a gauge theory coupled with dynamical fermions. To reliably benchmark such a system, it is crucial to understand the non-unitary quantum dynamics arising from the underlying non-Hermitian evolution and to model the effects of quantum measurements. This work focuses on monitoring ultra-local physical observables for a Z2\mathbb Z_2Z2​ gauge theory. Tensor network calculations are performed to dynamically probe entanglement entropy at larger lattice sizes. In this work, we report that continuously monitoring local and diagonal observables (electric and mass energy densities) in the computational basis demonstrates the absence of any measurement-induced phase transition, as indicated by the system-size independence of the late-time saturation value of the bipartite entanglement entropy.
Effects of monitoring on entanglement dynamics for 1+11+11+1D Z2\mathbb Z_2Z2​ lattice gauge theory
The (1+1)(1+1)(1+1)-dimensional Z2\mathbb Z_2Z2​ gauge theory is the simplest model that allows for quantum simulation to probe the fundamental aspects of a gauge theory coupled with dynamical fermions. To reliably benchmark such a system, it is crucial to understand the non-unitary quantum dynamics arising from effective non-Hermitian evolution and post-selected monitoring protocols. This work focuses on the post-selected non-Hermitian filtering dynamics of a Z2\mathbb Z_2Z2​ gauge theory, where the non-Hermitian terms are associated with local and non-local gauge-invariant operators naturally present in the theory. We interpret the resulting dynamics as post-selected filtering, where different operator sectors are coupled to loss channels with different rates. This gives a unified framework for both the local electric flux and particle-number terms and the non-local mesonic hopping term. Tensor network calculations are performed to probe the effect of the filtering for larger lattice sizes (up to 256-site systems). Using Matrix Product State calculations, the dynamics of entanglement entropy are studied as a function of the filtering rate and the coupling constant. We find that, under both local and non-local filtering, the late-time saturation value of the bipartite entanglement entropy remains independent of system size, providing no evidence of a measurement-induced phase transition-like phenomenon in the post-selected dynamics across the range of filtering strengths, evolution times, and system sizes considered here.
Entanglement and non-stabilizerness (magic) quantify two distinct departures of quantum systems from classical description: the former measures non-local correlations, while the latter measures the deviation from stabilizer states that can be efficiently simulated classically. Understanding magic in physically relevant quantum field theories is essential for identifying where quantum advantage may be realized in the early fault-tolerant quantum computing era. We calculate the gauge-invariant entanglement entropy and stabilizer Rényi entropy of the ground state of the (1+1)-dimensional SU(2) lattice gauge theory formulated in a dressed-site basis that enforces Gauss's law exactly. Using tensor networks, we obtain results for system sizes up to L=100L=100L=100 (300 qubits). We find a crossover denoted by g⋆g_{\star}g⋆​ where the ground state passes from a more magic-rich regime into a regime with less magic; this is also tracked by the sharpest change of both the entanglement entropy and lattice particle density. Our large-scale study of non-stabilizerness and entanglement entropy in a non-Abelian lattice gauge theory with matter provides new insight into the interplay of magic and entanglement in gauge theories, both of which are relevant for classical and early fault-tolerant quantum simulations.
Quantum-state complexity diagnostics provide valuable insight into many-body dynamics, information scrambling, and quantum computation. Here, we investigate the real-time dynamics of quantum complexity in 1+11+11+1-dimensional Abelian U(1) and non-Abelian SU(2) lattice gauge theories (LGTs), focusing on the disorder-free localization (DFL) regime. Using stabilizer Rényi entropy, participation Rényi entropy, and fermionic non-Gaussianity as measures of complexity, we observe, for both theories, two main behaviors as a function of the gauge coupling: at intermediate values, a power-law relaxation towards saturation, consistent with observations in many-body localization, and, at sufficiently large values, an ultraslow double-logarithmic growth, which we substantiate with a configuration-space bound verified by exact counting. Our results not only provide deeper insight into the dynamics of DFL but also highlight the role of gauge invariance in constraining quantum resources and are relevant to recent quantum simulations of LGTs.
The description of the entanglement dynamics of monitored noninteracting fermions, including the existence of measurement-induced phase transitions (MIPTs), is a challenging problem with conflicting results in the literature. The mapping of the problem onto a non-linear sigma model (NLSM) indicates that relatively large lattice sizes are required to determine the nature of the entanglement entropy (EE) in the thermodynamics limit. Here we address this problem numerically for monitored noninteracting fermions with U(1)U(1)U(1) symmetry. The use of graphics processing unit (GPU) techniques, even with outdated hardware, makes it possible to reach much larger lattice sizes (L=16384L = 16384L=16384 and 160×160160\times160160×160 in one (1d) and two (2d) dimensions respectively) than in previous studies which enables us to characterize quantitatively the entanglement dynamics. In 1d, we show that in order to confirm the absence of a MIPT, for both projective and homodyne measurements, predicted by the NLSM it is necessary to reach L∼10000L \sim 10000L∼10000. In 2d, also as predicted by the NLSM, we observe for both protocols a MIPT at finite monitoring rate characterized by a scale invariant mutual information. The critical monitoring strength depends on the protocol while the critical exponent ν≈1.3ν\approx 1.3ν≈1.3 governing the approach to the MIPT is similar in both cases. These features are not correctly predicted by the NLSM. Our results paves the way for a fully quantitative description of the entanglement dynamics of monitoring quantum systems.
Random quantum circuits provide a framework for probing universal dynamical phenomena in chaotic quantum many-body systems, by circumventing the microscopic details of Hamiltonian evolution. Bipartite entanglement measures can be used to characterize the dynamical phases that emerge in these systems. Typically, under random unitary evolution, bipartite entanglement entropies exhibit a linear growth, and attain steady-state values that scale extensively with the system volume. However, when local projective measurements are introduced, a suppression of steady-state bipartite entropies occurs, leading to a sub-extensive scaling with the system volume. This class of entanglement scaling transitions, from an extensive to a sub-extensive scaling regime, has sparked interest in a broader class of measurement-induced phase transitions, with a potential to further our understanding of fault-tolerance in quantum computing, and in realizing synthetic quantum phases on digital quantum simulators. In this work, we characterize entanglement scaling transitions in one-dimensional random quantum circuits composed of q-level systems, with the local Hilbert space dimension q ≥ 2. This predominantly numerical study employs generalizations of the stabilizer formalism and the Clifford group to higher-dimensional Hilbert spaces to simulate the measurement-induced transitions. We find that the nature of the entanglement scaling transition remains invariant under changes in the local Hilbert space dimension. However, the critical measurement rate inducing the transition increases with q, asymptotically approaching 1/2 in the limit q → ∞. To further explore this classical limit, we develop a minimal dynamical model that captures entanglement structures under unitary evolution. We analyze the entanglement structures in thermal stabilizer states using this model, and propose an approach for incorporating measurements within this framework.