An imperfect many-body revival does not by itself certify the entanglement of the returning state. We give a finite-record protocol for graph-edge localizable entanglement along scar wavepackets of a graph-dressed PXP chain. The target cluster state has nonzero energy variance, so neither an exact target eigenstate nor a dark-state embedding is assumed. Fresh binary probes of the undeformed Hamiltonians Pauli terms have a fully separable explanation, even within the dressed blockade sector. Adding local graph-stabilizer probes makes established entanglement witnesses accessible with simultaneous confidence bounds. For a specified square-root instrument, an imposed worst-case disturbance budget fixes the strength that minimizes the equal-allocation Hoeffding sufficient sampling cost. A local commutator bound accounts for finite-duration ancilla pulses while the Hamiltonian remains active. We test the protocol on chains through twenty spins, under perturbed dynamics, and with independent implementations. At the first twenty-spin return, synthetic weak records certify all nineteen graph edges. Generator, two-color, bounded-weight, and full-group benchmarks separate this localizable resource from genuine multipartite certification. The protocol measures recoverable entanglement in a known scar wavepacket, with explicit calibration and limits on its physical and statistical interpretation.
An imperfect many-body revival does not by itself certify the entanglement of the returning state. We give a finite-record protocol for graph-edge localizable entanglement along scar wavepackets of a graph-dressed PXP chain. The target cluster state has nonzero energy variance, so neither an exact target eigenstate nor a dark-state embedding is assumed. Fresh binary probes of the undeformed Hamiltonians Pauli terms have a fully separable explanation, even within the dressed blockade sector. Adding local graph-stabilizer probes makes established entanglement witnesses accessible with simultaneous confidence bounds. For a specified square-root instrument, an imposed worst-case disturbance budget fixes the strength that minimizes the equal-allocation Hoeffding sufficient sampling cost. A local commutator bound accounts for finite-duration ancilla pulses while the Hamiltonian remains active. We test the protocol on chains through twenty spins, under perturbed dynamics, and with independent implementations. At the first twenty-spin return, synthetic weak records certify all nineteen graph edges. Generator, two-color, bounded-weight, and full-group benchmarks separate this localizable resource from genuine multipartite certification. The protocol measures recoverable entanglement in a known scar wavepacket, with explicit calibration and limits on its physical and statistical interpretation.
Characterizing the Hamiltonian that a quantum processor actually implements is central to calibrating and validating current quantum hardware. Many devices, however, operate with generators that are time dependent by design. Here we develop a rigorous and experimentally friendly protocol for learning time-dependent many-body Hamiltonians from continuous weak measurement records. The key observation is that interaction sparsity reduces the global reconstruction to a set of local inverse problems, whose number is controlled by the interaction connectivity rather than by the system size. Pure separable probe states suffice to drive these inversions, and a graph-coloring construction embeds them into a small number of global product-state preparations. We derive explicit reconstruction-error bounds and a sample-complexity theorem that cleanly separates the finite-sampling statistical noise from the deterministic bias of the iterative state update, and we validate the protocol on time-dependent spin chains with up to n=8 qubits. Beyond these results, our analysis provides a rigorous foundation for time-dependent Hamiltonian learning from continuous monitoring in many-body systems, establishing a framework that extends naturally to many platforms and probe ensembles.
Identifying a low-entanglement eigenstate inside a many-body spectrum and preserving it during measurement are distinct tasks. We construct an explicit local ring Hamiltonian with an exact cluster-state eigenvector and study continuous monitoring of its stabilizer defects. For arbitrary mixed inputs, the conditional cluster fidelity is the initial target weight divided by the no-observed-click probability. A positive defect-operator gap gives finite-time bounds that hold for noncommuting Hamiltonian dynamics, nonnormal effective generators and imperfect detection. At fixed total monitoring rate, the guaranteed exponent falls inversely with system size; high conditional fidelity does not remove the preparation cost set by the initial overlap. Exact diagonalization up to eleven qubits gives finite-size evidence for a cluster-state outlier in a chaotic spectral background. Independent matrix and trajectory calculations verify the dynamics and a conservative coherent-error bound. This construction specializes established scar embedding and nondemolition verification frameworks, with explicit measurement assumptions, finite-time guarantees and resource limitations.
Evidence Paper I of the Existence Equation series. Quantum many-body scars — persistent, non-thermalizing revivals observed in constrained quantum systems — are conventionally explained by analyzing the spectral properties of the PXP Hamiltonian. This paper demonstrates that the PXP Hamiltonian is not a modeling choice but a geometric consequence of a single constraint: nearest-neighbor exclusion on a one-dimensional chain. Input: 3 items. Nothing else. #InputDescription 11D chainL sites, each with two states (|0⟩ and |1⟩) 2ConstraintNo two adjacent sites simultaneously in |1⟩ 3Flip ruleAt each site, flip if and only if constraint is satisfied No Hamiltonian written. No projection operators. No Pauli matrices. No quantum-mechanical formula invoked. The constraint decides. Output: The PXP Hamiltonian — derived, not assumed. LHilbert dim D‖T_constraint − H_PXP‖Revival T/T_∞First peak F₁Thermal 1/D 8550.000.780.7961.8 × 10⁻² 101440.000.860.7686.9 × 10⁻³ 123770.000.980.7432.7 × 10⁻³ 149870.000.980.7191.0 × 10⁻³ 162,5840.000.980.6953.9 × 10⁻⁴ 186,7650.000.980.6731.5 × 10⁻⁴ 2017,7110.000.980.6545.6 × 10⁻⁵ 2246,3680.000.980.6362.2 × 10⁻⁵ 24121,3930.000.990.6188.2 × 10⁻⁶ The constraint-built transition matrix and the textbook PXP Hamiltonian are the same object — to machine precision, at every system size tested, up to D = 121,393. Why this matters: The conventional direction of inference is: Hamiltonian → spectrum → structure. This paper reverses it: constraint → structure → dynamics → symmetry. The constraint is the cause; the Hamiltonian is the consequence. Within the Existence Equation framework, scars have a structural explanation. The nonlinear term α|Ψ|²Ψ enforces hard-core exclusion in the strong-coupling limit, condensing the field into a binary manifold with nearest-neighbor exclusion. The smoothing operator ∇²Ψ cannot act — spatial spreading is blocked. The deviation field has only one remaining option: it must reorganize temporally, circulating along closed trajectories in phase space. A quantum many-body scar is the temporal trace of deviation forced to close on itself. In the standard view, scars are anomalous because they violate thermalization. In the constraint view, scars are expected because spatial smoothing is blocked and temporal reorganization is the only remaining option. The absence of a scar — thermalization despite complete spatial obstruction — would be the anomaly requiring explanation. Spectral evidence: At L = 14 (D = 987), only 22 of 987 eigenstates carry significant Néel overlap. These scar states form a tower distributed symmetrically across the spectrum. Their participation entropy lies systematically below the thermal bulk (Page value), confirming that they are localized near the closed temporal trajectory — not spread across the full Hilbert space. Revival convergence: The revival period converges toward T_∞ ≈ 4.72/Ω (the thermodynamic limit from Turner et al., 2018), reaching 98.6% at L = 24. First revival peaks exceed the thermal expectation 1/D by three to five orders of magnitude. The system does not thermalize. It returns. The structural principle: Constraint → structure → symmetry, never the reverse. This principle extends across the full Evidence Program: the same mechanism that produces scars here (temporal phase closure) produces FQHE topological degeneracy (EP II), helical chirality (EP III), non-commutative algebra (EP IV), dark matter (EP V), the Tsirelson bound (EP VI), and Josephson physics (EP VII) — each as a structural consequence of the same equation. Evidence Program — Complete Summary: EPED mechanismWhat emergesKey result I α|Ψ|²Ψ strong-coupling → binary occupation; ∇²Ψ → nearest-neighbor exclusion PXP Hamiltonian (not assumed, derived) ‖T − H_PXP‖ = 0 II Same exclusion on 2D torus; spatial phase closure → winding number sectors Topological degeneracy (without Coulomb interaction); anyon braiding q-fold at ν = p/q; 2π/q phase (0.0% error) III ∇²Ψ + (1/c²)Ψ̈: simultaneous spatial and temporal phase closure Helical scars with spontaneous chirality |ΔF| = 1.78 × 10⁻¹⁵ IV α ≠ 0 → amplitude–phase coupling; commutative operators projected onto constrained subspace Non-commutative algebra with integer eigenvalues j_max = ⌊L/2⌋ (exact) V ½A²|∇Φ|² (phase energy) persists where |∇A|² (amplitude) has decayed Topological phase persistence as structural origin of dark matter ⟨v_θ·r⟩ = 2.0000 ± 0.0003 VI Axiom 1.1 (discreteness) → ±1 projection of continuous phase; 2 axes × √2 diagonal Tsirelson bound S = 2√2 without Hilbert space Exact (analytic + numerical) VII (1/c²)Ψ̈ → 0 (NR limit): ED reduces to GP; ε = k²/(4λ) controls validity boundary Josephson effect (DC + AC); GP as low-energy ED; f_Φ̇ force channel (no GP counterpart) DC: R² = 0.998; 4 falsifiable predictions at ε ~ 0.1 All code and figures: https://github.com/Galileo-leo/existence-equation Part of the Existence Equation Evidence Paper series (EP I–VII). Parent work: The Existence Equation: The Grammar of Persistence (doi:10.5281/zenodo.19364254)
We study the task of localizing multipartite entanglement in pure quantum states onto a subsystem by measuring the remaining systems. To this end, we fix a multipartite entanglement measure and consider two quantities: the multipartite entanglement of assistance (MEA), defined as the entanglement measure averaged over the post-measurement states and maximized over arbitrary measurements; and the localizable multipartite entanglement (LME), defined in the same way but restricted to only local single-system measurements. Both quantities generalize previously considered bipartite entanglement localization measures. In our work we choose the n-tangle, the genuine multipartite entanglement concurrence and the concentratable entanglement (CE) as the underlying seed measure, and discuss the resulting MEA and LME quantities. First, we prove easily computable upper and lower bounds on MEA and LME and establish Lipschitz-continuity for the n-tangle and CE-based LME and MEA. Using these bounds we investigate the typical behavior of entanglement localization by deriving concentration inequalities for the MEA evaluated on Haar-random states and performing numerical studies for small tractable system sizes. We then turn our attention to protocols that transform graph states. We give a simple criterion based on a matrix equation to decided whether states with a specified n-tangle value can be obtained from a given graph state, providing no-go theorems for a broad class of such graph state transformations beyond the usual “local Clifford plus local Pauli measurement” framework. This analysis is generalized to weighted graph states, which provide a realistic error model in current experiments preparing graph state. Our entanglement localization framework certifies the near-optimality of recently discussed local-measurement protocols to transform uniformly weighted line graph states into GHZ states, even when considering arbitrary entangled measurements. Finally, we demonstrate how our MEA and LME quantities can be used to detect critical phenomena such as phase transitions in transversal field Ising models. Since entanglement localization is operationally relevant throughout quantum networking and measurement-based quantum computation, our framework of results based on the MEA and LME has the potential for broad applications in these fields.
Quantum many-body scars enable persistent non-ergodic dynamics in otherwise thermalizing systems, yet their stabilization typically relies on fine-tuned initial states or engineered Hamiltonian perturbations. Here we show that lattice geometry alone can serve as a powerful and experimentally accessible control knob for inducing and enhancing scarring. By transforming a one-dimensional chain into a quasi-one-dimensional triangle-decorated lattice, we find that the fully polarized state -- normally thermalizing in the PXP model -- exhibits pronounced fidelity revivals, slow entanglement growth, and strong overlap with a tower of weakly entangled eigenstates. We trace this behavior to a geometry-induced restructuring of the constrained Hilbert space, whereby the adjacency graph decomposes into hypercube subgraphs that enforce coherent population transfer and stabilize an emergent approximate su(2) algebra. We propose a direct implementation in programmable arrays of tweezer-trapped Rydberg atoms, where the triangle-decorated geometry can be realized using spatial light modulators and the resulting scarring dynamics probed via time-resolved measurements of excitation density. Our results establish lattice connectivity as a design principle for engineering non-ergodic dynamics in constrained quantum systems.
Certifying motional entanglement generated by weak forces between levitated nanoparticles requires distinguishing entangled from separable states using finite, noisy measurement records. We show that the number of measurements required to certify Gaussian entanglement can vary among equally entangled states, and quantify this variation. We derive an attainable lower bound showing that, near the separability threshold, the required number of measurements diverges inversely with the square of the distance from this threshold. We translate the required number of measurements into an experimental certification time, comparing two protocols under comparable experimental conditions: release--recapture, in which particles evolve freely before measurement, and continuous monitoring, in which the steady state is measured continuously. We find that release--recapture enables entanglement certification at interaction strengths several orders of magnitude weaker. Moreover, minimizing certification time can favor states with less entanglement, revealing a trade-off between entanglement generation and its certification. These results highlight the role of certification in the design and optimization of ongoing efforts to probe weak-force-mediated entanglement between levitated nanoparticles.