Local Minimum of Spin-Sector Magic at the CP-Conserving Point in Low-Energy Neutron-Proton Scattering

Abstract: We study Magic generation in elastic neutron-proton scattering within a leading low-energy spin-sector ansatz that retains the one-pion-exchange spin structures and treats each scattering direction as a conditional two-qubit spin map. We show that the direction-averaged Magic is locally minimized at the CP-conserving (CPC) point $\barθ=0$ at the Clifford point $f_{\rm CPC}=π/4$, and for the representative non-Clifford CPC backgrounds analyzed here. At $f_{\rm CPC}=π/4$, the CPC spin map reduces to SWAP up to a phase and therefore generates zero Magic from stabilizer inputs. We further evaluate the complete spin-sector Magic functional by averaging over all 60 two-qubit stabilizer inputs and over scattering directions, and find that the curvature at $\barθ=0$ is positive only within specific windows of the effective CPC phase $f_{\rm CPC}$. These results identify the CPC point as a local Magic minimum within the restricted low-energy spin sector considered here.

• The paper establishes that at the CP-conserving point, the averaged non-Clifford magic—quantified by the stabilizer Rényi entropy M2—is locally minimized.
• It employs a spin-effective Hamiltonian model and conditional two-qubit maps to simulate elastic np scattering and quantify quantum resource generation.
• Numerical results reveal a quadratic increase in magic due to CP-violating perturbations, underscoring robust signatures for effective field theory analyses.

Local Magic Minima at the CP-Conserving Point in Low-Energy Neutron-Proton Scattering

Introduction and Motivation

This work investigates the information-theoretic properties of neutron-proton ($np$) scattering at low energies, embedding CP-violation diagnostics within the quantum resource framework of nonstabilizerness (Magic). The central technical innovation is to model the two-nucleon spin dynamics as a conditional two-qubit map, analyze the Clifford/non-Clifford structure of these spin operations, and systematically quantify the generated Magic via the stabilizer Rényi entropy ($M_2$). The analysis is motivated by patterns seen in other sectors of the Standard Model, where extrema of entanglement or Magic have been found to align with physically distinguished or naturally fine-tuned parameters. Here, the focus is on the strong CP problem, specifically the physical vacuum angle $\bar\theta$, and whether the CP-conserving (CPC) point ($\bar\theta=0$) is structurally distinguished from the perspective of non-Clifford resource generation in scattering.

Spin Map Construction and Magic Quantification

The analysis begins by reducing the elastic $np$ scattering process to a spin-effective Hamiltonian motivated by leading-order chiral EFT. The retained CP-conserving term takes the form $f_{\rm CPC}\,\bm\sigma_1 \cdot \bm\sigma_2$, while the CP-violating part is linear in $\bar\theta$ via a spin-difference operator $\epsilon\,(\bm\sigma_1 - \bm\sigma_2)\cdot\hat n$, where $\hat n$ parameterizes the momentum transfer direction. The resulting two-qubit conditional spin unitary,

$U(\hat n) = e^{-i H_{\rm eff}(\hat n)},$

acts on arbitrary spin input states.

Magic is quantified operationally: each of the 60 pure two-qubit stabilizer states is evolved under $M_2$0 for a fixed $M_2$1, and the output Magic is computed using $M_2$2: $M_2$3 where $M_2$4 is the phase-free Pauli group for $M_2$5 qubits ($M_2$6 here) and $M_2$7.

Magic is then averaged over all stabilizer inputs and over $M_2$8 on the sphere, producing a basis-independent, direction-averaged diagnostic of non-Clifford resource generation for elastic $M_2$9 scattering in the spin sector.

Local Magic Minimum at the CP-Conserving Point

For the Clifford spin-exchange point $\bar\theta$0, corresponding to the SWAP gate up to global phase, the spin map is a Clifford rotation, so Magic vanishes identically regardless of input. Away from this point, nonzero Magic arises. The central question is: for small $\bar\theta$1, is there a local extremum of averaged Magic at the CPC point, and what is its character?

Figure 1: The complexity landscape at the CPC Clifford point in the leading spin-effective model. The solid purple line shows direction-averaged stabilizer Rényi entropy as a function of the scaled CPV coordinate. The red dot marks the local minimum at $\bar\theta$2.

Numerical computations confirm that at $\bar\theta$3, the averaged Magic is strictly minimized at $\bar\theta$4 (Figure 1). The CPV perturbation increases Magic quadratically in $\bar\theta$5 due to the symmetry $\bar\theta$6, so the CPC point is a strict stationary point and a local minimum: $\bar\theta$7 in the spin sector.

Robustness Under CPC Background Variation

One critical issue is whether this minimum depends on fine-tuning to the Clifford/SWAP point or is robust under generic CPC backgrounds ($\bar\theta$8 values). Additional simulations for various $\bar\theta$9 show the persistence of the Magic minimum at $\bar\theta=0$0 over a significant range of CPC couplings.

Figure 2: Robustness under variations of the effective CPC coupling. Direction-averaged Magic across several $\bar\theta=0$1 illustrates that the local minimum at $\bar\theta=0$2 persists for a range of CPC backgrounds, not just at the Clifford point.

Moving away from the Clifford point lifts the CPC baseline Magic, but for all sampled $\bar\theta=0$3 values, the local minimum remains at the CP-conserving point. This supports the view that the extremum is not artificial, but a structural feature of the spin-effective model in a physically relevant regime.

Explicit Spin-Sector Curvature Analysis

The local minimum is characterized by evaluating the curvature of the Magic functional at $\bar\theta=0$4 (i.e., $\bar\theta=0$5). To this end, a dense scan over the CPC coupling parameter $\bar\theta=0$6 is performed. The curvature,

$\bar\theta=0$7

is computed explicitly via finite differences on the complete two-qubit functional averaged over all stabilizer states and directions.

Figure 3: Finite spin-sector Magic curvature. Left: $\bar\theta=0$8 (curvature at $\bar\theta=0$9) as a function of $np$0. Rigorous sign changes demonstrate that the positivity of curvature—and thus the extremum—is windowed, not universal. Right: baseline Magic as a function of $np$1.

The scan in Figure 3 reveals that positive curvature—hence, a local minimum at $np$2—is observed in explicit windows of $np$3, which include the Clifford point and representative CPC backgrounds relevant to low-energy $np$4 scattering. Notably, sign reversals occur outside these windows, indicating that the information-theoretic extremum is not generic for all CPC spin-exchange phases but is sharply localized in the parameter space allowed by the physical $np$5-wave channel.

Implications and Future Directions

This study gives a concrete information-theoretic characterization of the CP-conserving point in low-energy $np$6 scattering: within a leading spin-effective treatment retaining OPE spin structures, the generation of Magic—i.e., the non-Clifford computational resource—is locally minimized at $np$7 over a physically relevant domain of CPC backgrounds. When interpreted via the mapping of $np$8 to the physical singlet-triplet $np$9-wave phase shift difference,

$f_{\rm CPC}\,\bm\sigma_1 \cdot \bm\sigma_2$0

this implies that the phase-shift trajectory for low-energy $f_{\rm CPC}\,\bm\sigma_1 \cdot \bm\sigma_2$1 scattering falls within the positive-curvature domain where the minimum is present.

Theoretically, this adds to the body of evidence that fundamental parameters (e.g., Standard Model couplings, now the strong CP angle) are associated with resource minima in the quantum computation sense. Practically, this work motivates further EFT studies to determine the precise physical location of the $f_{\rm CPC}\,\bm\sigma_1 \cdot \bm\sigma_2$2 trajectory and to generalize from the spin-effective proxy to the full, multi-channel $f_{\rm CPC}\,\bm\sigma_1 \cdot \bm\sigma_2$3-matrix in chiral EFT. In particular, the susceptibility of the Magic minimum to higher-order and non-spin-exchange interactions will be crucial for comprehensive resource-theoretic diagnostics of strong-interaction processes.

Conclusion

This paper establishes that within a spin-effective, OPE-retained description of low-energy $f_{\rm CPC}\,\bm\sigma_1 \cdot \bm\sigma_2$4 scattering, the CP-conserving vacuum angle $f_{\rm CPC}\,\bm\sigma_1 \cdot \bm\sigma_2$5 is locally distinguished as a minimum of non-Clifford Magic generation for a broad and physically motivated set of CPC spin-exchange backgrounds. The positive curvature at the minimum is shown both numerically and via explicit computation of the finite spin-sector functional. The result is robust within specified CPC phase windows corresponding to relevant experimental phase-shift ranges, but is not globally generic in parameter space. Extension of these findings to the complete chiral EFT scattering matrix will require further partial-wave and input-output quantum resource analyses across all spin and momentum channels.

This work provides both a new lens for the strong CP problem and concrete impetus for future investigation of quantum information-theoretic structure in nucleon-nucleon interactions.