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arxiv: 2508.01806 · v2 · submitted 2025-08-03 · 🪐 quant-ph · q-bio.QM

Quantum Optimal Control for Coherent Spin Dynamics of Radical Pairs via Pontryagin Maximum Principle

Ugur G. Abdulla , Jose H. Rodrigues , Jean-Jacques Slotine This is my paper

Pith reviewed 2026-05-19 00:59 UTC · model grok-4.3

classification 🪐 quant-ph q-bio.QM
keywords optimal controlradical pairsPontryagin Maximum Principlesinglet yieldspin dynamicsbang-bang controlelectromagnetic fieldsquantum biology
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The pith

Pontryagin Maximum Principle applied to filtered spin models identifies bang-bang electromagnetic fields maximizing singlet yield in radical pairs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to find the external electromagnetic field shape that drives radical pair spins toward maximum singlet yield in biochemical reactions. It models the dynamics with a Schrödinger system including Zeeman and hyperfine terms, then introduces a one-parameter family of control problems by coupling the system to the field via filtering equations. The authors prove Fréchet differentiability and apply the Pontryagin Maximum Principle in Hilbert space to establish the bang-bang nature of the optimal control, while developing an iterative method to compute it. Simulations using this method and gradient projection show convergence and that filtering changes the yield maxima by less than one percent versus non-filtered cases. A reader might care because the work points toward possible experiments confirming quantum coherence effects in biological magnetoreception.

Core claim

By coupling the Schrödinger system for radical pair spin dynamics to the control field through filtering equations, the Pontryagin Maximum Principle is proved in Hilbert space for a one-parameter family of problems, the bang-bang structure of the optimal electromagnetic control is established, and an iterative Pontryagin Maximum Principle method is developed whose numerical application demonstrates less than 1% change in the maxima of the singlet yield compared to non-filtering bang-bang controls.

What carries the argument

The iterative Pontryagin Maximum Principle (IPMP) method for computing bang-bang optimal electromagnetic field controls in the one-parameter family of filtered Schrödinger systems for radical pair spins.

Load-bearing premise

The spin Hamiltonians with Zeeman and hyperfine terms are assumed to accurately model the dynamics, and the one-parameter filtering coupling is taken as a suitable physical representation of the control.

What would settle it

Measuring the singlet yield in a radical pair experiment under the identified bang-bang optimal field and checking whether the maximum deviates by more than 1% from the value obtained with the filtered regular optimal field.

Figures

Figures reproduced from arXiv: 2508.01806 by Jean-Jacques Slotine, Jose H. Rodrigues, Ugur G. Abdulla.

Figure 1
Figure 1. Figure 1: FIG. 1: Approximated optimal control and magnetic field obtained fo [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: 1-proton case. Asymptotic of [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Approximated optimal control and magnetic field obtained fo [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: 2-proton. Asymptotic of [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Approximated optimal control and magnetic field obtained fo [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: 3-proton. Asymptotic of [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Approximated optimal control and magnetic field obtained fo [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: 4-proton. Asymptotics of the cost functional [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Approximated optimal control and magnetic field obtained fo [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: 5-proton. Asymptotics of the cost functional [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Approximated optimal control and magnetic field obtained [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: 6-proton. Asymptotics of the cost functional [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Approximated optimal control and magnetic field obtained [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: 7-proton. Asymptotics of the cost functional [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: 1-proton. Approximated optimal controls obtained for in [PITH_FULL_IMAGE:figures/full_fig_p028_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: 1-proton. Approximated optimal controls obtained for in [PITH_FULL_IMAGE:figures/full_fig_p028_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: 1-proton. Approximated bang-bang optimal control [PITH_FULL_IMAGE:figures/full_fig_p029_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: 1-proton. Approximated bang-bang optimal control [PITH_FULL_IMAGE:figures/full_fig_p029_18.png] view at source ↗
read the original abstract

This paper aims to devise the shape of the external electromagnetic field that drives the spin dynamics of radical pairs to a quantum coherent state through maximization of the triplet-born singlet yield in biochemical reactions. The model is a Schr\"{o}dinger system with spin Hamiltonians given by the sum of Zeeman interaction and hyperfine coupling interaction terms. We introduce a one-parameter family of optimal control problems by coupling the Schr\"{o}dinger system to a control field through filtering equations for the electromagnetic field. Fr\'echet differentiability and the Pontryagin Maximum Principle in Hilbert space are proved, and the bang-bang structure of the optimal control is established. A new iterative Pontryagin Maximum Principle (IPMP) method for the identification of the bang-bang optimal control is developed. Numerical simulations based on IPMP and the gradient projection method (GPM) in Hilbert spaces are pursued, and the convergence, stability, and the regularization effect are demonstrated. Comparative analysis of filtering with regular optimal electromagnetic field versus non-filtering with bang-bang optimal field ({\it Abdulla et al, Quantum Sci. Technol., {\bf9}, 4, 2024}) demonstrates that the change of the maxima of the singlet yield is less than 1\%. The results open a venue for a potential experimental work on magnetoreception as a manifestation of quantum biological phenomena.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript develops an optimal control approach for maximizing the singlet yield in radical-pair spin dynamics relevant to biochemical reactions. It models the system as a Schrödinger equation with Zeeman and hyperfine terms, introduces a one-parameter family of problems by coupling the dynamics to filtering equations for the control field, proves Fréchet differentiability and the Pontryagin Maximum Principle in Hilbert space, establishes the bang-bang structure of the optimal control, proposes an iterative PMP (IPMP) algorithm, and reports numerical comparisons (via IPMP and gradient projection) showing that the maxima of the singlet yield differ by less than 1% between the filtered regular optimal field and the non-filtered bang-bang control from prior work.

Significance. If the central results hold, the paper supplies a rigorous infinite-dimensional optimal-control framework for quantum-coherent spin dynamics in radical pairs, including proofs of differentiability and PMP together with a new IPMP method and numerical evidence of convergence and stability. These elements could support further theoretical and experimental exploration of magnetoreception as a quantum-biological phenomenon, provided the physical modeling assumptions are validated.

major comments (1)
  1. [Abstract and Numerical Simulations] Abstract and Numerical Simulations section: The headline comparative result states that the change in the maxima of the singlet yield is less than 1% between filtering with the regular optimal electromagnetic field and non-filtering with the bang-bang optimal field of Abdulla et al. (2024). This bound is reported for a specific choice of the filtering parameter in the one-parameter family; the manuscript does not present systematic variation of this parameter, nor recomputation of the yield surface under modest changes to the hyperfine coupling tensors, leaving open whether the <1% difference is robust or an artifact of the chosen parameter values.
minor comments (2)
  1. [Model Formulation] The definition and physical motivation of the one-parameter filtering equations could be stated more explicitly when first introduced, including how the parameter enters the coupling to the Schrödinger dynamics.
  2. [Numerical Results] Figure captions for the numerical convergence plots should include the specific values of the filtering parameter and hyperfine strengths used, to allow direct reproduction of the reported stability results.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the major comment below and indicate the revisions planned for the next version of the manuscript.

read point-by-point responses

  1. Referee: Abstract and Numerical Simulations section: The headline comparative result states that the change in the maxima of the singlet yield is less than 1% between filtering with the regular optimal electromagnetic field and non-filtering with the bang-bang optimal field of Abdulla et al. (2024). This bound is reported for a specific choice of the filtering parameter in the one-parameter family; the manuscript does not present systematic variation of this parameter, nor recomputation of the yield surface under modest changes to the hyperfine coupling tensors, leaving open whether the <1% difference is robust or an artifact of the chosen parameter values.

    Authors: We agree that the reported difference of less than 1% is shown for one representative value of the filtering parameter, chosen to illustrate the regularization effect while preserving close agreement with the non-filtered case. The one-parameter family is introduced precisely to permit such tuning, and the numerical results already demonstrate stability for the selected value. In the revised manuscript we will add a short sensitivity study in the Numerical Simulations section, presenting singlet-yield maxima for two or three nearby values of the filtering parameter and confirming that the difference remains below 2%. With respect to the hyperfine coupling tensors, these are taken from established literature values for the radical-pair model under study; recomputing the full yield surface for varied tensors would require a separate optimization campaign for each new parameter set. We will nevertheless insert a clarifying paragraph noting that the PMP derivation and the IPMP algorithm are independent of the specific tensor values, so that the relative performance between filtered and non-filtered controls is expected to be robust under modest tensor perturbations. revision: partial

Circularity Check

0 steps flagged

No significant circularity; core proofs and numerics are self-contained.

full rationale

The paper derives Fréchet differentiability of the cost functional, the Pontryagin Maximum Principle in Hilbert space, and the bang-bang structure of the optimal control directly from the coupled Schrödinger-filtering system and standard optimal control theory. It introduces and analyzes a new IPMP iterative method whose convergence and regularization properties are demonstrated via independent numerical experiments with both IPMP and GPM. The comparative singlet-yield statement references prior work only for the non-filtering baseline and does not serve as a load-bearing premise for the new theoretical results or the primary claims about control structure and stability. No step reduces by construction to a fitted input, self-definition, or unverified self-citation chain.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The paper relies on established quantum mechanical models for radical pairs and applies mathematical optimal control theory without introducing new physical entities.

free parameters (1)
  • filtering parameter
    The one-parameter family introduced for coupling the control field through filtering equations.
axioms (1)
  • domain assumption The dynamics of radical pair spins are governed by the Schrödinger equation with Zeeman and hyperfine interaction terms.
    Standard model for spin dynamics in radical pairs as stated in the abstract.

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