From Local Nonclassicality to Entanglement: A Convexity Law for Single-Excitation Dynamics

TL;DR

The paper introduces a resource-centric, convexity-based law governing the dynamical interconversion of local non-classicality and entanglement under excitation-preserving XY dynamics. By exploiting the convexity of Wigner negativity and an exact mapping to beam-splitter dynamics, it proves a tight bound: the sum of local negativities satisfies $\mathcal{N}_{\text{tot}}(t) \le \mathcal{N}_1$, with saturation only at endpoints when the excitation is fully localized. Numerically, the authors validate the bound for two-qubit systems and extend the analysis to N-qubit PST chains, uncovering a dark-transport phenomenon where the resource moves into multi-body correlations while remaining invisible locally. In the native CV framework and across diverse non-Gaussian seeds, the redistribution dynamics exhibit universal behavior when normalized by the seed budget, underscoring a general principle of coherent resource flow. The results suggest practical hardware metrics for characterizing and benchmarking coherent dynamics, with potential experimental paths via CV tomography and resource-based performance benchmarks that quantify deviations from ideal budgeted trajectories.

Abstract

We prove a simple dynamical law for excitation-preserving interactions: the {sum of local Wigner negativities} is upper-bounded by a fixed budget set by the initially excited state. For the single-excitation sector of the XY model (and its beam-splitter analogue), this convexity bound equals the negativity of the seed state and is saturated only when the excitation is fully localized. At intermediate times the sum lies strictly below the bound due to phase-space overlap in local mixtures, quantitatively accounting for entanglement growth as a redistribution of a finite, budgeted resource into shared correlations. We establish the result analytically for two bodies and corroborate it numerically in engineered state-transfer chains, where it reveals a coherence-enabled dark transport: the resource becomes locally invisible while being stored in multi-body coherences. The predicted trajectory of the summed local negativity provides a practical hardware metric: deviations from the ideal, budgeted curve diagnose decoherence and control error.

Figures (5)

• Figure 1: Numerical simulation of resource dynamics and the convexity bound in the two-subsystem exchange model (CV embedding), initialized in $\ket{\psi(0)}=\ket{0}_A\ket{1}_B$. (a) The concurrence $\,\mathcal{C}(t)\,$ oscillates between 0 and 1. (b) Local Wigner negativities $\mathcal{N}_A(t)$ and $\mathcal{N}_B(t)$ are out of phase, evidencing redistribution. (c) The summed local negativity $\mathcal{N}_{\mathrm{tot}}(t)=\mathcal{N}_A(t)+\mathcal{N}_B(t)$ (left axis) remains below the convexity bound $\mathcal{N}_1$ (dashed), saturating it only when the excitation is fully localized on either subsystem (e.g., at times $t/T\in\{0, 0.5, 1.0, \dots\}$). The shaded band shows $\pm1\sigma$ around the mean of $\mathcal{N}_{\mathrm{tot}}(t)$. The right axis plots the tracking gap $\Delta(t)=\max\![0,\mathcal{N}_1-\mathcal{N}_{\mathrm{tot}}(t)]$, which quantifies the reduction due to overlap cancellation at intermediate times.
• Figure 2: Time-resolved phase-space visualization of resource conversion (CV embedding). Wigner functions for subsystem A (top row) and subsystem B (bottom row) are shown. The sequence illustrates the coherent transfer of non-classicality, initially localized on subsystem B ($t/T=0.00$), through a maximally entangled state where the non-classicality is shared and reduced by overlap cancellation ($t/T=0.25$), to its complete localization on subsystem A ($t/T=0.50$). This demonstrates the redistribution of a bounded resource whose sum follows a predictable trajectory below the convexity bound.
• Figure 3: Convexity-bounded redistribution and coherence-protected transport in a 4-qubit PST chain. (a) Resource dynamics summary. The total local negativity ($\sum_k \mathcal{N}_k$, solid blue) remains below the initial resource budget ($\mathcal{N}_1$, orange dashed), vanishing during mid-transport. This disappearance is explained by the maximum single-site probability ($\max_k p_k$, purple dotted) dropping into the negativity-suppressed region ($p < 0.5$, shaded grey). In contrast, the maximum 2-qubit block probability ($\max_b p_b^{(2)}$, green dash-dotted) remains high, sustaining the resource. (b) Heatmap of the underlying excitation probability ($p_k$) showing perfect state transfer from site 0 to site 3. (c) Heatmap of local negativity ($\mathcal{N}_k$) revealing a “dark transport corridor”: the resource is only visible at the chain ends, appearing to be lost during transit. (d) Heatmap of block-2 negativity ($\mathcal{N}_b^{(2)}$) resolves the puzzle, showing the resource propagating as a continuous, bright wave. The negativity is not lost but is protected in the correlations between qubits, becoming visible only when examining a sufficiently large, coherent block.
• Figure 4: Resource transfer and the convexity bound in a continuous-variable system. Top: Wigner function snapshots of mode B show the transfer of the non-classical Fock state $\ket{1}$ to the vacuum state $\ket{0}$. Bottom: Summed local Wigner negativity $\mathcal{N}^{\mathrm{CV}}_{\mathrm{tot}}(t)$ over one period. The numerically computed trajectory (blue line) lies below the convexity upper bound (gray dotted line), saturating it only at the swap points. The gap at intermediate times is due to physical overlap cancellation, while the small superimposed oscillations are a numerical artifact of Hilbert-space truncation.
• Figure 5: Universality of bounded–budget redistribution for distinct non-Gaussian seed states. (a) Absolute sum of local Wigner negativities, $\sum_i \mathcal{N}_i(t)$, for three seeds prepared on mode $B$: single photon $|1\rangle$ (blue), odd cat $\mathcal{C}^-_{\alpha=1.4}$ (orange), and squeezed $|1\rangle$ with $r=0.35$ (green). Thin dashed lines indicate each seed’s initial budget $\mathcal{N}_{\rm seed}$. Time is shown in units of the exchange period $T$. (b) The same trajectories normalized by their own $\mathcal{N}_{\rm seed}$ collapse onto a common curve, indicating that the redistribution dynamics are largely independent of the seed’s detailed structure.