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arXiv:2603.14292

Mixed-State Entanglement in a Minimal Model of Quantum Chaos

Tanay Pathak

quant-phcond-mat.stat-mechhep-thSubmitted Mar 15, 2026• Announced Mar 17, 2026• 0 citations
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Abstract

Understanding the dynamics of quantum correlations in many-body systems is a central problem in non-equilibrium quantum physics. We study the spread of mixed-state entanglement in a minimal model of quantum chaos, the kicked field Ising model. By combining the replica trick with the space-time duality of the model, we determine the exact spectrum of the partially transposed reduced density matrix. The resulting flat spectrum leads to exact relations between entanglement negativity, odd entropy and Rényi mutual information at early times. Numerical results further demonstrate that for equal tri-partitions and at late times, all entanglement measures saturate to the Haar-random values. In contrast, for unequal tri-partitions Rényi mutual information and negativity vanish at late times, implying that the corresponding reduced density matrix is factorizable. Extensive numerical simulations also show that the relation remains quantitatively valid for generic initial states, leading us to conjecture it for all initial states and all times.

Comments
5+5pages, 3 figures

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Understanding the dynamics of quantum correlations in many-body systems is a central problem in non-equilibrium quantum physics. We study the spread of mixed-state entanglement in a minimal model of quantum chaos, the kicked field Ising model. By combining the replica trick with the space-time duality of the model, we determine the exact spectrum of the partially transposed reduced density matrix. The resulting flat spectrum leads to exact relations between entanglement negativity, odd entropy and Rényi mutual information at early times. Numerical results further demonstrate that for equal tri-partitions and at late times, all entanglement measures saturate to the Haar-random values. In contrast, for unequal tri-partitions Rényi mutual information and negativity vanish at late times, implying that the corresponding reduced density matrix is factorizable. Extensive numerical simulations also show that the relation remains quantitatively valid for generic initial states, leading us to conjecture it for all initial states and all times.
We study the growth of entanglement between two adjacent regions in a tripartite, one-dimensional many-body system after a quantum quench. Combining a replica trick with a space-time duality transformation, we derive an exact, universal relation between the entanglement negativity and Rényi-1/2 mutual information that holds at times shorter than the sizes of all subsystems. Our proof is directly applicable to any local quantum circuit, i.e., any lattice system in discrete time characterized by local interactions, irrespective of the nature of its dynamics. Our derivation indicates that such a relation can be directly extended to any system where information spreads with a finite maximal velocity.
We investigate the dynamics of entanglement in a generalized version of the kicked Ising chain, extending the model from the standard qubit case (local dimension q=2q=2q=2) to higher local dimensions (q>2q > 2q>2). We identify the existence of ''dual-unitary'' points where the model's space-time duality allows for exact analytical solutions. Our analysis reveals that while a few unique dual-unitary points exist analytically for systems with local dimensions q=3q=3q=3 and q=4q=4q=4, such points do not exist for q≥5q \ge 5q≥5 due to the lack of a unique kicking strength that satisfies the required matrix element conditions. Utilizing the transfer matrix method and a replica trick specifically adapted for higher dimensions, we derive exact expressions for the growth of entanglement entropy in the q=3q=3q=3 (kicked Potts-type) model starting from a class of solvable initial states. Our results demonstrate that at the dual-unitary point, both Rényi and von Neumann entanglement entropies grow linearly with time until reaching a maximum value determined by the subsystem size.
In a non-equilibrium many-body system, the quantum information dynamics between non-complementary regions is a crucial feature to understand the local relaxation towards statistical ensembles. Unfortunately, its characterization is a formidable task, as non-complementary parts are generally in a mixed state. We show that for integrable systems, this quantum information dynamics can be quantitatively understood within the quasiparticle picture for the entanglement spreading. Precisely, we provide an exact prediction for the time evolution of the logarithmic negativity after a quench. In the space-time scaling limit of long times and large subsystems, the negativity becomes proportional to the Rényi mutual information with Rényi index α = 1 / 2 . We provide robust numerical evidence for the validity of our results for free-fermion and free-boson models, but our framework applies to any interacting integrable system.
Information scrambling is a hallmark of quantum chaos and thermalization in isolated quantum many-body systems. We investigate scrambling dynamics in the all-to-all interacting spin Sachdev-Ye-Kitaev (SYK)-qqq model using both pure- and mixed-state entanglement measures. We show that von-Neumann and Rényi entropies exhibit rapid growth followed by saturation near Haar-random values, signaling efficient scrambling. The scrambling rate reveals a nontrivial dependence on the interaction order, system size, and Hamiltonian scaling. We further employ mixed-state entanglement as a powerful probe of information scrambling. We numerically find a universal relation between the Rényi-1/2 mutual information and entanglement negativity for minimal interaction order in the early growth regime. Furthermore, entanglement negativity displays a Page-curve-like behavior under unequal subsystem partitioning, characterized by the birth, spread, and eventual death of quantum correlations. Our results provide a generic description of information scrambling using entanglement dynamics in all-to-all interacting spin systems with multi-body interactions.
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In this work, we explore the effects of a quantum quench on the entanglement measures of a two-body coupled oscillator system having quartic interaction. We use the invariant operator method, under a perturbative framework, for computing the ground state of this system. We give the analytical expressions for the total and reduced density matrix of the system having non-Gaussian, quartic interaction terms. Using this reduced density matrix, we show the analytical calculation of two entanglement measures viz., Von Neumann entanglement entropy using replica trick and Renyi entanglement entropy. Further, we give a numerical estimate of these entanglement measures with respect to the dimensionless parameter $(t/\ensuremath{\delta}t)$ and show its behavior in the three regimes, i.e., late time behavior, around the quench point and the early time behavior. We comment on the variation of these entanglement measures for different orders of coupling strength. The variation of Renyi entropy of different orders has also been discussed.