Anti-Flatness and Non-Local Magic in Two-Particle Scattering Processes

TL;DR

The paper investigates how fundamental two-body interactions generate quantum complexity, quantified by non-local magic and anti-flatness, by mapping two-spin systems arising in low-energy nucleon-nucleon scattering and high-energy Møller scattering to two qubits. Using S-matrix analyses and helicity-amplitude formalism, it derives that the non-local linear magic power satisfies ${\cal M}_{\rm lin}^{(NL)}(\hat{S}) = 4\, {\cal F}_A(\hat{S})$, with explicit expressions such as ${\overline{\overline{{\cal M}_{\rm lin}^{(NL)}}}}(\hat{S}) = \frac{1}{48}\,[11+5\cos(4\Delta\delta)]\sin^2(2\Delta\delta)$ in NN, where $\Delta\delta=\delta_1-\delta_0$. The Clifford-averaged anti-flatness relates to the total magic via ${\langle {\cal F}_A (\Gamma|\psi\rangle) \rangle_{\mathcal{C}} = c(d,d_A)\, {\cal M}_{\rm lin}(|\psi\rangle)$ with $c(4,2)=\tfrac{1}{10}$ in the Møller context, implying experimental accessibility of anti-flatness from a single final-state particle. The study reveals that nuclear forces can disentangle initial entanglement and generate non-local magic, while quantum electrodynamics in Møller scattering with entangled inputs tends to preserve stabilizer structure and yield no non-local magic, underscoring qualitative differences in complexity generation between nuclear and electromagnetic interactions. These results advance understanding of quantum complexity in fundamental processes and inform strategies for simulating larger many-body systems.

Abstract

Non-local magic and anti-flatness provide a measure of the quantum complexity in the wavefunction of a physical system. Supported by entanglement, they cannot be removed by local unitary operations, thus providing basis-independent measures, and sufficiently large values underpin the need for quantum computers in order to perform precise simulations of the system at scale. Towards a better understanding of the quantum-complexity generation by fundamental interactions, the building blocks of many-body systems, we consider non-local magic and anti-flatness in two-particle scattering processes, specifically focusing on low-energy nucleon-nucleon scattering and high-energy Moller scattering. We find that the non-local magic induced in both interactions is four times the anti-flatness (which is found to be true for any two-qubit wavefunction), and verify the relation between the Clifford-averaged anti-flatness and total magic. For these processes, the anti-flatness is a more experimentally accessible quantity as it can be determined from one of the final-state particles, and does not require spin correlations. While the MOLLER experiment at the Thomas Jefferson National Accelerator Facility does not include final-state spin measurements, the results presented here may add motivation to consider their future inclusion.

Figures (7)

• Figure 1: The non-local magic power, $\overline{\overline{{\cal M}_{\rm lin}^{(NL)}}}(\hat{S})$ and the total magic powers $\overline{{\cal M}_{\rm lin}}(\hat{S})$ and $\overline{\overline{{\cal M}_{\rm lin}}}(\hat{S})$ for low-energy S-wave nucleon-nucleon scattering as a function of momentum in the laboratory. The difference in phase shifts are determined from the Nijmegen phase-shift analysis PhysRevC.49.2950NNonline. The entanglement power is also shown, as a dashed line.
• Figure 2: The linear magic, ${\cal M}_{\rm lin}(\ket{\chi_i})$, the non-local linear magic, ${\cal M}_{\rm lin}^{(NL)}(\ket{\chi_i})$ and the anti-flatness, ${\cal F}_A(\ket{\chi_i})$ (multiplied by a factor 4), for outgoing states $\ket{\chi_i} = \mathcal{N} \hat{\cal A} \ket{\psi_i}$, corresponding to the distinct groups of initial-state helicity wavefunctions $\ket{\psi_i}$ for high-energy Møller scattering. The upper panels from left to right show the results for tensor-product states from Group-1, 2, 3, while the lower panels show the results for Group-4, 5a, 5b. The green curves associated with the ${\cal M}_{\rm lin}$ are from Eq. (\ref{['eq:MollerM2ana']}), while the dashed blue curves associated with the ${\cal F}_A$ are from Eq. (\ref{['eq:MollerAF']}). The pink curves are from numerical minimization of ${\cal M}_{\rm lin}^{(NL)}(\ket{\chi_i})$, using Eq. (\ref{['eq:NLM']}).
• Figure 3: The Clifford-averaged anti-flatness, $\langle \mathcal{F}_A (\Gamma \ket{\chi_i}) \rangle_\mathcal{C}$, (green points) compared with $c(d,d_A) {\mathcal{M}}_{\rm lin}(\ket{\chi_i})$ (blue curve) in Møller scattering from initial states $|\psi_{5a}\rangle$ (left panel) and $|\psi_{5b}\rangle$ (right panel) in Eq. (\ref{['eq:Mollerinitialstates']}). The un-averaged values of anti-flatness are shown in Fig. \ref{['fig:MOLLHECOMPgr1to5b']}.
• Figure 4: The total linear magic, non-local linear magic (anti-flatness), and linear entanglement entropy in NN scattering and Møller scattering final states, averaged over all 24 entangled stabilizer states.
• Figure 5: Total linear magic (top), linear entanglement entropy (middle), and non-local linear magic (anti-flatness $\times 4$) in NN scattering. The left (right) panels show the results for outgoing states from initial unentangled (entangled) stabilizer states.