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arXiv:2607.08615

Operational meaning of Markov gap in tripartite entanglement of quantum dynamics

Zongsheng Zhou, Riqiang Zhang, Yu-Xiang Zhang

quant-phcond-mat.stat-mechSubmitted Jul 9, 2026• Announced Jul 10, 2026• 0 citations
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Abstract

We investigate how irreducible multipartite entanglement, a long-range correlation by nature, can emerge from short-range dynamics far from equilibrium. Focusing on the Markov gap as a probe of irreducible tripartite entanglement (IrTE) in free-fermion chains, we uncover qualitatively distinct dynamical behaviors: the Markov gap grows either quasi-linearly or in staircase-like jumps depending on the initial state. We also propose attainable upper and lower bounds for the onset time of IrTE based on the Lieb-Robinson bound. Strikingly, the Markov gap saturates to a volume-law value on a timescale $t\sim\! L^2$, much slower than the ballistic spreading of bipartite correlations. To understand what information about the wavefunctions is revealed by the Markov gap calculation, we introduce the concept of essential tripartite fermion (ETF) and an associated tripartite null matrix. The value of Markov gap closely tracks the number of small singular values of this tripartite null matrix, yielding a transparent, operational physical interpretation of the measure. We further demonstrate that several dynamical signatures persist in the interacting XXZ chain.

Comments
8 pages, 4 figures, 9 page supplemental material

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We investigate how irreducible multipartite entanglement, a long-range correlation by nature, can emerge from short-range dynamics far from equilibrium. Focusing on the Markov gap as a probe of irreducible tripartite entanglement (IrTE) in free-fermion chains, we uncover qualitatively distinct dynamical behaviors: the Markov gap grows either quasi-linearly or in staircase-like jumps depending on the initial state. We also propose attainable upper and lower bounds for the onset time of IrTE based on the Lieb-Robinson bound. Strikingly, the Markov gap saturates to a volume-law value on a timescale t∼ ⁣L2t\sim\! L^2t∼L2, much slower than the ballistic spreading of bipartite correlations. To understand what information about the wavefunctions is revealed by the Markov gap calculation, we introduce the concept of essential tripartite fermion (ETF) and an associated tripartite null matrix. The value of Markov gap closely tracks the number of small singular values of this tripartite null matrix, yielding a transparent, operational physical interpretation of the measure. We further demonstrate that several dynamical signatures persist in the interacting XXZ chain.
We investigate the dynamics of multipartite entanglement after quenches from initial states which generate multiplets of quasiparticle excitations beyond the usual pair structure. We focus on the dynamics of tripartite entanglement through the lens of the Markov gap -- a computable quantity that signals irreducible tripartite entanglement when positive. In the XX spin chain, we show that the Markov gap is positive at intermediate times, implying the presence of tripartite entanglement. After a time delay, the Markov gap increases and then decays at longer times, thus exhibiting an entanglement barrier. We argue that those qualitative features are consistent with an interpretation of the spreading of tripartite entanglement by triplets of tripartite-entangled quasiparticles.
A quantum quench is the simplest protocol to investigate nonequilibrium many-body quantum dynamics. Previous studies on the entanglement properties of quenched quantum many-body systems mainly focus on the growth of entanglement entropy. Several rigorous results and phenomenological guiding principles have been established, such as the no-faster-than-linear entanglement growth generated by generic local Hamiltonians and the peculiar logarithmic growth for many-body localized systems. However, little is known about the dynamical behavior of the full entanglement spectrum, which is a refined character closely related to the topological nature of the wave function. Here, we establish a rigorous and universal result for the entanglement spectra of 1D SPT systems evolving out of equilibrium. Our result is derived both for free-fermion SPT systems and interacting ones. For free-fermion systems with AZ symmetries, we prove that the single-particle entanglement gap after quenches obeys essentially the same Lieb-Robinson bound as that on the equal-time correlation, provided that there is no dynamical symmetry breaking. As a notable byproduct, we obtain a new type of Lieb-Robinson velocity which is related to the band dispersion with a complex wave number and reaches the minimum as the maximal (relative) group velocity. Within the framework of tensor networks, i.e., for SPT MPSs evolved by symmetric and trivial MPUs, we also identify a Lieb-Robinson bound on the many-body entanglement gap for general quenched interacting SPT systems. This result suggests high potential of tensor-network approaches for exploring rigorous results on long-time quantum dynamics. Influence of partial symmetry breaking, effects of disorder, and the relaxation property in the long-time limit are also discussed. Our work establishes a paradigm for exploring rigorous results of SPT systems out of equilibrium.
Entanglement equilibration need not mark the end of quantum-information dynamics in quantum thermalization. We show that the complete Schmidt spectra of a fixed tripartition can become stationary while its multipartite correlation structure, distinguished by local-unitary invariants, continues to evolve. This separation occurs exactly within a finite window in self-dual kicked-Ising dynamics and asymptotically in Haar-random circuits. We further demonstrate an independently tunable timescale for multipartite dynamics using quantum-error-correcting codes. When coupled to a locally scrambling bath, ordinary entanglement approaches equilibrium through rapid local scrambling, while weak logical--bath coupling drives a slower reorganization of multipartite correlations. These results reveal a deeper distinction than the ordering of equilibration times: bipartite entanglement and multipartite correlation structure can probe distinct physical channels of quantum-information dynamics, rather than merely different stages of the same relaxation process.
In this paper, we investigate Quantum Phase Transitions (QPTs) in the long-range XXZ spin chain using two information-theoretic probes: The half-chain entanglement entropy and a witness of genuine multipartite nonlocality. Both quantities effectively identify the phase boundary, but exhibit distinct scaling behaviors: the entanglement entropy grows linearly with system size N, while the multipartite nonlocality increases exponentially, with [Formula: see text]. By independently tuning the power-law decay exponents of the transverse ([Formula: see text]) and longitudinal ([Formula: see text]) couplings, we uncover contrasting trends in the critical point [Formula: see text]: Increasing [Formula: see text] shifts it to larger values, while increasing [Formula: see text] shifts it to smaller ones. These monotonic and saturating dependencies are captured by an empirical exponential model, offering a compact and physically transparent description of the critical behavior. Our results highlight the crucial role of interaction range in driving criticality, with implications for engineered quantum systems and long-range entanglement control.
We uncover a "junction law" for genuine multipartite entanglement, suggesting that in gapped local systems multipartite entanglement is controlled and effectively localized near junctions where subsystem boundaries meet. Using the Rényi-2 genuine multi-entropy GM2(q)\mathrm{GM}^{(\mathtt{q})}_2GM2(q)​ as a diagnostic of genuine q\mathtt{q}q-partite entanglement, we establish this behavior in (2+1)(2+1)(2+1)-dimensional gapped free-fermion lattices with correlation length ξξξ. For partitions with a single junction, GM2(q)\mathrm{GM}^{(\mathtt{q})}_2GM2(q)​ exhibits a universal scaling crossover in L/ξL/ξL/ξ, growing for L≪ξL\llξL≪ξ and saturating to a ξξξ-dependent constant for L≫ξL\ggξL≫ξ, up to O(e−L/ξ)\mathcal{O}(e^{-L/ξ})O(e−L/ξ) corrections. In sharp contrast, for partitions without a junction, GM2(q)\mathrm{GM}^{(\mathtt{q})}_2GM2(q)​ is exponentially suppressed in L/ξL/ξL/ξ and drops below numerical resolution once L≫ξL\ggξL≫ξ. We observe the same pattern for q=3\mathtt{q}=3q=3 (tripartite) and q=4\mathtt{q}=4q=4 (quadripartite) cases, and further corroborate this localization by translating the junction at fixed system size. We also provide a geometric explanation of the junction law in holography. Altogether, these results show that in this gapped free-fermion setting genuine multipartite entanglement is localized within a correlation-length neighborhood of junctions.
Many-body entanglement in condensed matter systems can be diagnosed from equilibrium response functions through the use of entanglement witnesses and operator-specific quantum bounds. Here, we investigate the applicability of this approach for detecting entangled states in quantum systems driven out of equilibrium. We use a multipartite entanglement witness, the quantum Fisher information, to study the dynamics of a paradigmatic fermion chain undergoing a time-dependent change of the Coulomb interaction. Our results show that the quantum Fisher information is able to witness distinct signatures of multipartite entanglement both near and far from equilibrium that are robust against decoherence. We discuss implications of these findings for probing entanglement in light-driven quantum materials with time-resolved optical and x-ray scattering methods.