Observations about utilitarian coherence in the avian compassLuke D. Smith, Jean Deviers, and Daniel R. Kattniga)

Living Systems Institute and Department of PhysicsUniversity of Exeter, Stocker Road, Exeter EX4 4QD, United Kingdom

(Dated: 10 November 2021)

It is hypothesised that the avian compass relies on spin dynamics in a recombining radical pair. Quantumcoherence has been suggested as a resource to this process that nature may utilise to achieve increasedcompass sensitivity. To date, most theoretical studies have been conducted for toy radical pair models.Consequently, the true functional role of coherence in these natural systems has remained speculative. Here,we investigate large radical pair models with up to 21 nuclear spins, inspired by the putative magnetosensoryprotein cryptochrome. By varying relative radical orientations, we reveal correlations of several coherencemeasures with compass fidelity. Whilst electronic coherence is found to be an ineffective predictor of compasssensitivity, a robust correlation of compass sensitivity and a global coherence measure is established. Theresults demonstrate the importance of realistic models, and appropriate choice of coherence measure, inelucidating the quantum nature of the avian compass.

Quantum coherence and entanglement are central en-ablers of quantum technologies and quantum informationprocessing. The emerging field of quantum biology1–5

hypothesizes that these rudiments of quantumness couldlikewise empower biological processes of living systems.In particular, long-lived quantum coherence is an over-arching theme of many of the focus points of quantumbiology, such as the avian magnetoreceptor, and pho-tosynthetic energy transport6–10. However, our under-standing and quantification of coherence as an opera-tional resource11–13, and its role in the complex quantumsystems of life, is still limited.

The avian compass and several related magnetosensi-tive feats are hypothesised to rely on coherent spin dy-namics in a radical pair14 (RP), putatively formed inthe blue-light sensitive flavo-protein cryptochrome. Theradical pair comprises two unpaired electrons, the com-bined spin angular momentum of which are describedin terms of singlet and triplet states. Typically, onlythe singlet state can recombine to re-form the diamag-netic resting state, while both singlets and triplets cangive rise to spin-independent structural rearrangementsof the protein and thus signal to down-stream processes(see the scheme presented in Fig. 1)15. Magnetosensi-tivity emerges in these settings from a change in singletand triplet state populations and thus yields for differentmagnetic field orientations. This is driven by the coher-ent singlet-triplet interconversions which result from thehyperfine coupling with magnetic nuclei in the radicalsunder perturbation by the comparably weak Zeeman in-teraction of the electron magnetic moments with the geo-magnetic field (50µT). The nonstationary singlet-tripletcoherence is an essential requirement for this process. Inits absence (i.e., if singlet and triplet states are eigen-states of the Hamiltonian), no interconversion betweensinglet and triplet states occurs and the reaction yield willbe insensitive to the applied magnetic field. On the other

a)Electronic mail: d.r.kattnig@exeter.ac.uk

hand, electronic spin state decoherence, through interac-tion with the environment of nuclear spins, provides anatural companion to the radical pair spin dynamics andprerequisite for its sensitivity to magnetic fields16. In or-der to elicit sensitivity to a 50µT field the spin coherencemust persist for about a microsecond, i.e., the reciprocalof the electron Larmor precession frequency (1.4 MHz), orlonger. Longer-lived singlet-triplet coherences promise amore precise magnetic field measurement, which couldunderpin a compass of exquisite acuity17.

Current models of magnetoreception implicate the pro-tein cryptochrome as the host of the magnetosensitiveradical pair15. A photo-generated radical pair comprisinga flavin anion radical (FAD•−) and radical cation derivedfrom a surface-exposed tryptophan residue (W•+) is apopular model supported by in vitro studies18. Alterna-tive radical pair models have been put forward based onin vivo observations and modelling19–24 (e.g. the flavinsemiquinone/superoxide pair). Here, we focus on theformer and an additional model comprising a tyrosineradical partner instead of the tryptophan22. A coherentlifetime of the radical pair of 1–10µs has been deemedrealistic based on predictions of spin relaxation processesand animal behavioural experiments25–27.

Several works have suggested that coherence is a re-source for the sensitivity of the compass, insofar asa larger coherence corresponds to increased directionalmagnetic sensitivity. Cai and Plenio28 have analysedmany randomly chosen prototype radical pair systems(with 5–6 nuclear spins) and found that their “globalcoherence” of the electron and nuclear spin system is apredictor of compass sensitivity. Kominis has introduceda formal measure of coherence, which demonstrated thatsinglet-triplet coherence29 of radical pairs provides an op-erational advantage to magnetoreception in simple modelsystems involving one nuclear spin. In a previous study22,we have observed that the electron coherences in realisticmulti-nuclear radical pairs comprising up to 21 nuclearspins are long-lived compared to the electronic entan-glement. We concluded that the compass is coherence-

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FIG. 1. a Structure of an avian cryptochrome (PDB identifier: 6PU0, Columba livia), including the central electron transferchain comprising four tryptophans (W) labelled A (W395), B (W372), C (W318) and D (W369) and ending in the surfaceexposed tyrosine Y319. Photo-excitation of FAD in cryptochrome initiates consecutive electron transfer reactions of adjacentdonors/acceptor pairs (red arrows), producing sequential radical pairs of the form [FAD•− / W•+] and possibly [FAD•− /Y•] (see e.g. ref. (18)). The well separated radical pairs involving WC and WD have been implicated with magnetoreception.Alternative radical pair models have been discussed, e.g., in the context of dark-state reoxidation19–21. b Generic radical-pairreaction scheme. The radical pair is born in the singlet state (spin multiplicity indicated by superscript labels, total electronicspin: S = 0) which, via coherent interconversion, can interconvert to a triplet state (S = 1). Here, A• is assumed to beFAD and B• is a tryptophan (W) or tyrosine (Y) residue. The radical pair may form singlet and triplet products with rateconstants kS and kT , respectively. The singlet channel typically comprises radical pair recombination and spin-independentprotein structural rearrangements; the latter also contributes to the triplet channel. c Graphical representation of hyperfineinteractions in FAD•− (top), Y• (bottom left) and W•+ (bottom right). Here, in the direction given by the unit vector d, theplotted surfaces are drawn at distance ||Ad|| from the location of the nucleus in question, whereby 1 A corresponds to 17 MHz.Surfaces are coloured according to the sign of the projection, with blue and red corresponding to positive and negative signs,respectively. The molecules are shown in their respective standard orientation.

driven while entanglement does not confer an operationaladvantage of the compass. The latter point has also beenelaborated on in ref. (30) with comparable conclusions.Le and Olaya-Castro have recently investigated a three-spin system subject to collisional, symmetry-breakingenvironmental interactions31. The authors demonstratethat magnetosensitivity in this system requires coherenceand suggest that a small degree of coherence, regardlessof basis, is likely a quantum resource for biomolecularsystems.

The property of coherence as a facilitator of increasedcompass sensitivity was recently questioned by Jain etal., who suggest that the electronic coherence swiftly de-cays with the number of coupled nuclei. The authorsdemonstrate that the compass can provide high sensi-tivity despite operating in a parameter regime without“sustained” electron spin coherence32. This conclusion

was derived by comparing systems with up to 6 mag-netic nuclei coupled with identical, aligned and axial hy-perfine tensors (i.e., with identical transverse and domi-nant longitudinal components). While this system showssustained electron coherences for vanishing transverse hy-perfine components, the system with maximal sensitivitydid not sustain the coherences (with “sustained” con-strued in relation to the artificial reference system). Asof now, it is not clear if these findings translate to morerealistic conditions or models, but the conclusion is cer-tainly surprising, as it is in stark deviance to the above-mentioned studies.

Here, we investigate the utilitarian character ofcoherence for the compass based on models that arebiologically relevant. In particular, we focus on radicalpairs that have been implicated with cryptochrome-magnetoreception, namely flavin/tryptophan and

3

flavin/tyrosine radical pairs with up to 21 nuclear spins.Our study thus endeavours to address the character ofcoherence in the actual magnetoreceptor, as currentlyenvisaged, rather than the principle role as elaboratedin previous studies using strongly simplified/abstractmodels. Instead of varying the hyperfine interactionparameters, which are predetermined by the identityof the radical pair and only weakly dependent on themolecular environment33, we vary the relative orienta-tion of the two radicals. This is a parameter which couldhave been subject to evolutionary optimisation in theprotein cryptochrome and is more likely to reflect anypotential quantum advantage that is exercisable in thewell-defined biological system, if it exists. Specifically,considering more realistic modelling conditions, we anal-yse whether the relative orientations of optimal compassfidelity are correlated with the electronic coherence (asanticipated based on investigations22,28,29) or not (asadvocated in a recent study by Jain et al.32). Ourresults question the effectiveness of coherence, measuredby electronic coherence quantifiers, as the sole driver ofmagnetoreception and explanation of increased compasssensitivity. However, global coherence is realized as apersistent effectuator of compass fidelity, confirming thehypothesized quantum nature of the processes even forrealistic systems with relatively short radical lifetimes of1µs.

I. METHODS

Radical pair model. We consider radical pairs sub-ject to coherent evolution under the local Hamiltoni-ans of their constituent radicals and spin-selective re-combination proceeding with the same rate constant,kS = kT = k, for the singlet and triplet channel. Inter-radical interactions are neglected, as is spin relaxation.In this simplified, but commonly studied scenario17,22,23,the spin dynamics of the RP and its associated magneticfield effects (MFEs) can be described by the followingmaster equation for the spin density operator ρ(t):

dρ(t)

dt= −i

[H, ρ(t)

]− kS

2

{PS , ρ(t)

}− kT

2

{PT , ρ(t)

}= −i

[H, ρ(t)

]− kρ(t), (1)

where [] and {} denote the commutator and anti-commutator, respectively. Here, the Hamiltonian is ofthe form H = HA + HB with subscripts A and B refer-ring to the two radicals, and PS and PT = 1 − PS areprojection operators onto the singlet state and tripletmanifold, respectively. The chemical reactivity has beenincluded using the Haberkorn approach34, which givesrise to minimal singlet-triplet dephasing (alternative ap-proaches have been suggested, but for kS = kT the differ-ences of various master equation are lessened29,35). The

Hamiltonians Hi, i ∈ {A,B}, account for the Zeemaninteractions between the electron spins and the applied

magnetic field and the hyperfine interactions between theelectron and nuclear spins within radical i (~ = 1):

Hi =∑j

Si ·Ai,j · Ii,j + ~ωi · Si. (2)

Here, the Larmor precession angular frequency is given

by ~ωi = −γi ~B, with γi denoting the gyromagnetic ratio

of the electron in radical i and ~B the applied magneticfield. Ai,j is the hyperfine coupling tensor between the

jth nuclear spin and the ith electron spin; Ii,j and Siare the corresponding vector operators of nuclear andelectron spin angular momentum.

Equation (1), with H = HA+HB, is opportune insofaras it allows one to express the singlet yield of the recom-bination reaction in terms of the spin correlation tensors

(SCTs), T(i)α,β(t), with components22,36,37

T(i)α,β(t) =

1

ziTr[Si,αSi,β(t)]

=1

ziTr[Si,αe

iHitSi,βe−iHit], (3)

where zi denotes the dimension of the nuclear subspaceof the Hilbert space of radical i. Specifically, the singletyield of a radical pair born in the singlet configuration,ρ(0) = PS/(z1z2), is given by

YS = k

∫ ∞0

pS(t) exp (−kt) dt, (4)

where

pS(t) =1

zAzBTr[PSPS(t; k = 0)]

=1

4+∑α,β

T(A)α,β (t)T

(B)α,β (t). (5)

Here, PS(t; k = 0) is the singlet projection operatorin the Heisenberg picture evaluated for k = 0. AsT(i)(t) is defined for the individual radicals, i.e., calcula-ble within the individual Hilbert space of radical i, equa-tions (3) to (5) allow the treatment of radical pairs witha reasonably complex hyperfine coupling pattern involv-ing many coupled nuclei without common symmetry ofthe interactions, as expected for the radicals implicatedwith the magnetoreception processes. Furthermore, ifthe SCTs are available on a grid of magnetic field ori-entations (spanning a hemisphere), the SCTs of radicalsthat have been rotated (by rotation matrix R) can beefficiently reconstructed from the unrotated SCTs due tothe property22.

T(i)(t; ~B0, {RAi,jR−1}) = RT(i)(t;R−1 ~B0, {Ai,j})R−1

(6)

Here, T(i)(t; ~B0, {Ai,j}) denotes the spin correlation ten-

sor for a given magnetic field, ~B0, and set of hyperfine

4

tensor parameters, {Ai,j}, and T(i)(t;R−1 ~B0, {Ai,j})the spin correlation tensor of the radical with rotatedorientation. Equation (6) allows the efficient evaluationof the singlet yield of complex radical pairs as a func-tion of their mutual orientation in space, as has beendemonstrated in a previous study22. Here, we will adaptthis approach to evaluate the dependence of coherencemeasures and compass fidelity on relative radical pairorientation.

To this end, observe that for k = 0 the electronic spindensity operator σ(t) = Trn [ρ(t)] (with Trn denoting thetrace over all nuclear spins of both radicals) can be ex-pressed in terms of T(i)(t), i.e.

σ(t) =1

4+

∑α,β∈{x,y,z}

cαβ(t)SA,αSB,β , (7)

where

cαβ(t) = −4∑

γ∈{x,y,z}

T (A)γα (t)T

(B)γβ (t). (8)

Equations (6) and (7) thus provide a means to calculateelectronic coherence quantifiers of complex radical sys-tems subject to arbitrary relative reorientations of theconstituent radicals. We will relate these coherence mea-sures to widely used measures of compass fidelity, namelythe absolute (∆S), and relative anisotropy (ΓS), definedas

∆S = maxϑ,ϕ

YS(ϑ, ϕ)−minϑ,ϕ

YS(ϑ, ϕ), (9)

and

ΓS =∆S

Y S, (10)

where the polar angle ϑ and azimuthal angle ϕparametrize the direction of the applied magnetic field inthe protein frame, and the mean quantum yield is givenby

Y S =1

4π

∫ π

0

dϑ

∫ 2π

0

dϕ sin(ϑ)YS(ϑ, ϕ). (11)

Measures of coherence. Various measures to quan-tify coherence have been developed11–13. Several suchmeasures have previously been used in the context ofthe avian compass28,29,32,38,39. One canonical measure toquantify coherence is the relative entropy of coherence,which, in terms of the normalized density operator ρ anda chosen basis, {|n〉}dn=1, of the d-dimensional Hilbertspace, is defined as

Cr[ρ] = S[IC(ρ)]− S[ρ], (12)

where S[ρ] = −Tr[ρ log(ρ)] is the von Neumann en-tropy and IC denotes the dephasing operation IC(ρ) =∑n |n〉〈n|ρ|n〉〈n|, which maps any quantum state into an

incoherent state in the chosen basis.

Another measure, the l1-norm of coherence11 is definedas

Cl1 [ρ] =∑n 6=m

|〈n|ρ|m〉|. (13)

Kominis has suggested the alternative measure definedby

Cst[ρ] = S[PS ρPS + PT ρPT ]− S[ρ], (14)

which reports singlet-triplet coherence while being inde-pendent of basis and unaffected by the degree of tripletcoherence, which the author suggests was superior in as-sessing coherence in the radical pair-based compass29.An alternative l1 based measure of singlet-triplet coher-ence was used in ref. (40). The relative entropy of ρ

with respect to the maximally mixed state, S[ρ||1/d

]=

log d− S [ρ], has likewise been employed to quantify theamount of basis-independent coherence31.

The coherence quantifiers introduced above can be em-ployed to the entire density operator, ρ, as well as to theelectronic part, σ, and are either basis-independent ordefined with respect to a basis. In the present study,two clear choices include the singlet-triplet basis and theup-down basis of the electronic subspace, both of whichwill be used to evaluate Cr and Cl1 below. To distinguishthese choices, we will henceforth index the coherence la-bels with G (global, based on ρ), E (electronic, based onσ, and ST (singlet-triplet basis) and UD (up-down ba-sis). For example, CE,UDr corresponds to the electroniccoherence assessed via the relative entropy quantifier inthe up-down basis, which is the measure used by Jainet al.32. Note that, for all definitions used above, thenon-trace preserving density matrix characteristic of re-combining radical pair systems must be renormalized (byTr[ρ]; which here simply yields the k = 0 result). Asthe coherence measures Ci(t), i ∈ {r, l1, st}, are time-dependent we further introduce the coherence yield asthe time-averaged coherence measure weighted by the ex-ponential decay kinetics of the radical pair (cf. equation(4)):

Ci = k

∫ ∞0

Ci(t) exp(−kt) dt. (15)

For comparison to the anisotropy, we further definemeasures accounting for the variation of Ci with the mag-netic field direction as

µ[Ci] =Ci(ρmax) + Ci(ρmin)

2, (16)

and

∆[Ci] = Ci(ρmax)− Ci(ρmin), (17)

where ρmax and ρmin corresponds to the density matrixassociated with maximum and minimum singlet yieldover magnetic field orientations. Furthermore, a field-independent measure is introduced as [Ci]B=0, which isevaluated for B = 0.

5

Cai and Plenio have suggested an alternative, oper-ational global quantifier of coherence28 in the contextof chemical compass in terms of the field-independent(B = 0) singlet recombination yield due to the coher-ent part of the initial density operator GC(ρ(0)), withGC(ρ) = ρ − IC(ρ) evaluated in the eigenbasis of thehyperfine Hamiltonian:

[CGy ]B=0 =∣∣∣YS(ρ(0) = (z1z2)−1GC(PS);B = 0

)∣∣∣. (18)

In this paper we utilise both this measure and a gener-alised field-dependent version (with the magnetic field inthe extremal directions) to allow us to also evaluate µ[CGy ]

and ∆[CGy ].

II. RESULTS

Large spin system electronic coherence. Pre-vious studies on coherence in the avian compass typi-cally use systems with a modest number of nuclear spins(only one or 5–6) and often randomly assumed hyperfineinteractions28,29,32,38,41. Under the assumptions as laidout above, we have been able to overcome this limitationand evaluate fidelities and coherence measures for flavin-tryptophan and flavin-tyrosine radical pairs with up to21 nuclear spins and judiciously chosen hyperfine param-eters, as summarized in the Supplementary Information(SI). Our implementation relied on GPU computing re-alized using CUDA, which vastly outperformed the cor-responding calculations on CPUs. We have varied therelative orientation of the radicals, as parametrized byEuler angles, α, β and γ, defining the orientation of theflavin radical relative to the radical partner. Using anangular resolution of 3 degrees, we have probed 878,400relative orientations for each combination of radicals. Foreach radical orientation, we sampled 10242 orientationsof ϑ and ϕ (5121 unique directions) to evaluate the fi-delity measures ΓS and ∆S and several associated coher-ence measures as introduced above. In what follows, wefocus on the compass anisotropy ∆S and the coherencemeasures µ[Ci] and ∆[Ci], for both the ST and UD basis,and the l1- and relative entropy of coherence quantifiers.Alternative measures are discussed in the SI, wherebyqualitatively corresponding conclusions emerged. For allsimulations, a magnetic field strength of 50 µT (compa-rable to the geomagnetic field in Northern Europe) and aradical lifetime of k−1 = 1µs were assumed. The latter isin line with the order-of-magnitude lifetime of the mag-netosensitive radical pairs in cryptochromes as observedfor in vitro studies14,18 and the anticipated spin relax-ation times27. Results for larger magnetic field strengthmay be found in the SI; these show increased anisotropybut reduced correlation with coherence measures (Figs.S1 and S2).

For flavin-tryptophan systems the results are presentedin Fig. 2, which shows plots of the absolute anisotropyvs. various electronic coherence measures. The data re-

FIG. 2. Compass sensitivity, as captured by the anisotropy∆S , is plotted against electronic coherence measures µ[Ci] (a,c, e) and ∆[Ci] (b, d, f), for 878,400 relative orientationsof flavin-tryptophan radical pairs. Data has been colouredaccording to the relative orientation angle β, ranging from0◦ (blue) to 180◦ (yellow). Both up-down (UD) and singlet-triplet (ST) basis are considered, and a linear fit (red line)with associated Pearson correlation coefficient R is displayed.

veals diverse, i.e., not universal, and complex relationsof fidelity and coherence measures. Surprisingly, ananti-correlation with respect to µ[Ci] is observed that ispresent in both the UD and ST basis, but larger in theformer. Specifically, ∆S is found to anti-correlate withµ[CE,UDr ], which is based on the same coherence measureas used by Jain et al.32, giving rise to a remarkable cor-relation coefficient of R = −0.927. In contrast, a positive(but weaker) correlation is observed with respect to the∆[CE,STr ] measure. Similar results, shown in the SI, arefound for correlations involving the relative anisotropyΓS (Fig. S3).

Furthermore, the results show structured subsets of thedata that may present positive or negative correlations.In our data discussed below, and in comparable studiesin the literature, such structures in the correlation data

6

appear to be absent for a small number of nuclear spins.This suggests that for comprehensive considerations ofcoherence in biology, systems must be sufficiently andrealistically complex.

In the case of ∆[CE,STr ], the colouration of the plotaccording to the angle β demonstrates a relationshipto the bands in which small values of β correspond toa stronger positive correlation (see Fig. 2d). However,maximal anisotropy is realized for ∆[CE,STr ] ≈ 0, a con-dition which appears to likewise approximately hold forthe other coherence anisotropy measures used (see Fig. 2band f). The implication is that, for this system, maximalcompass sensitivity is not realized by minimizing elec-tronic coherence in one extremal direction while maxi-mizing it in the other, a strategy which, at least a priori,might have appeared auspicious in maximizing yield dif-ferences and thus compass fidelity.

Results for flavin-tyrosine radical pairs are shown inFig. 3, in which smaller correlations are found than forthe flavin-tryptophan model. An anti-correlation is stillobserved for µ[CE,STr ] and ∆S , and likewise the weakerpositive correlation persists between ∆[CE,STr ] and ∆S .As in the flavin-tryptophan models, subsets of the datapresent positive and negative correlation. However, thestructure of these subsets, and relationship to the angle βis different. Here, for mid-range values of β, which corre-spond to maximal compass sensitivity, the results suggesta marked anti-correlation in µ[CE,STr ] and a strong posi-tive correlation in ∆[CE,STr ]. Similar results, shown in theSI (Figs. S3 and S4), are found for correlations involvingthe relative anisotropy ΓS .

The results on large spin systems are unexpected inview of previous studies17,29,38,42 and thought provokingwith respect to Jain et al.’s suggestion of a compass op-erating in a regime of low coherence32. Here, our resultsbased on the same coherence measure as used Jain et al.(CE,UDr ), appear to not only corroborate this conclusion,but lead to the supposition that the lack of electronic co-herence is advantageous. This could imply an incoherent,relaxation-driven16,43 character of the avian compass.

Small spin system electronic and global coher-ence. We note that the coherence measures employedabove, including CE,UDr as used by Jain et al., have notpreviously been assessed with respect to their princi-pal predictive power of compass performance. We havethus decided to systematically explore the correlation ofcompass fidelity and the various coherence measures em-ployed. Similar to28, we elected to focus on systems with5 randomly chosen hyperfine interactions, whereby wepartly retained the dominant hyperfine interactions ofthe flavin N5 and N10 nuclei, as these are consideredessential to the processes. Both systems of reference-probe topology23,24 (with all hyperfine interactions con-fined to one radical) and systems with more symmetri-cally distributed hyperfine interactions were consideredseparately, as were two radical pair lifetimes, k−1 = 1µsand k−1 = 10µs. Table I summarizes the various sys-tems studied, which will be referred to via labels A to

FIG. 3. Compass sensitivity, as captured by the anisotropy∆S , is plotted against electronic coherence measures µ[Ci] (a,c, e) and ∆[Ci] (b, d, f), for 878,400 relative orientations offlavin-tyrosine radical pairs. Data has been coloured accord-ing to the relative orientation angle β, ranging from 0◦ (blue)to 180◦ (yellow). Both up-down (UD) and singlet-triplet (ST)basis are considered, and a linear fit (red line) with associatedPearson correlation coefficient R is displayed.

H. Further details on the randomly assigned hyperfineinteractions can be found in the SI. Fig. 4 provides agraphical representation of the correlations of fidelity andcoherence measures. A more comprehensive analysis ex-tending to alternative coherence measures is provided inthe SI (Figs. S5 and S6).

As was the case for many nuclear spins, we find thatthere is significant variation of correlations for differentsystems and measures, overall supporting the view thatelectronic coherence is not the most effective predictor ofcompass sensitivity. For systems A and B, which retainN5 and N10 in one radical and employ random hyperfinecouplings in the other, we find low correlation betweenanisotropy and electronic coherence. Specifically, for atleast 2400 random radical pairs considered, an increase in

7

anisotropy is rarely accompanied by an increase in elec-tronic coherence. However, as was the case in the studypresented by Cai and Plenio28, a stronger correlation isfound with global coherence measures, in particular insystem A. CGy corresponds to the coherent contributionto the singlet yield, and its broad success suggests that itis important that the global coherence quantifier relatesto operation. Further measures and correlation plots sup-porting these findings can be found in the SI (Figs. S5–S8).

Large spin system global coherence. The con-firmed applicability of [CGy ]B=0 motivated us to attempta reassessment of the coherence-fidelity correlation of thelarge spin systems based on this global coherence mea-sure. While global coherence measures are in general pro-hibitively expensive to evaluate for spin systems of therelevant size, [CGy ]B=0 can be approximated with reason-able effort from the analytical expression for the singletyield (eq. (S7) in the SI; implemented for graphics pro-cessing unit using CUDA). Using Monte Carlo samplingof the matrix elements of the singlet projection operatorin the eigenbasis of the combined Hilbert space of bothradicals assists in accelerating the calculation (see SI).Utilising this we have been able to evaluate [CGy ]B=0 of

FIG. 4. Correlation coefficients between anisotropy and co-herence measures are plotted for a range of measures, shownon the y-axis, and for 8 system types (shown in Table I forsystems A-H), each comprising 5 nuclear spins. Black linespartitioning columns group together similar systems. Thefirst column entry of a partition represents the system fork−1 = 1µs and the second column entry represents the samesystem with the longer lifetime of k−1 = 10µs. Likewise,the row partition separates the electronic measures from theglobal measure. Bar plots at the top and at the right siderepresent the fraction of the sum of absolute values of corre-lations within the plot, for a respective column or row. Moreextensive versions of this figure covering many more coherencemeasures is provided in the SI (Figs. S5 and S6).

TABLE I. Parameter choices for systems A-H. Hyperfine in-teractions are chosen as either N5, N10 and a set of 3 ran-dom hyperfine interactions {Ai,j}n=3, or 5 random hyperfineinteractions assigned to radical A• and B• as {Ai,j}n=2 and{Ai,j}n=3, or {Ai,j}n=5. Under the hyperfine interactions col-umn, radical A• is referred to as 1, and radical B• is referredto as 2 for clarity.

System Hyperfine interactions Lifetime k−1 (µs)A 1: N5, N10

2: {Ai,j}n=3

1

B 1: N5, N102: {Ai,j}n=3

10

C 1: {Ai,j}n=2

2: {Ai,j}n=3

1

D 1: {Ai,j}n=2

2: {Ai,j}n=3

10

E 1: N5, N10, {Ai,j}n=3

2: none1

F 1: N5, N10, {Ai,j}n=3

2: none10

G 1: {Ai,j}n=5

2: none1

H 1: {Ai,j}n=5

2: none10

the flavin-tryptophan and flavin-tyrosine systems for arepresentative number of relative orientations.

In Fig. 5 we plot compass anisotropy ∆S of these sys-tems as a function of global coherence [CGy ]B=0. For both

systems, a strong positive correlation of [CGy ]B=0 and ∆S

is found, thereby reinforcing the results for 5-6 nuclearspins, as studied previously28 and above. Specifically, wefind a remarkable correlation for flavin-tryptophan forwhich R = +0.975 based on 1784 randomly chosen rela-tive orientations, in stark contrast to the anti-correlationof R = −0.927 found with the electronic coherence mea-sure µ[CE,UDr ]. The results for flavin-tyrosine are morestructured, but likewise give rise to a strong overall cor-relation with R = +0.825. This result confirms the sup-position of Cai and Plenio, who proposed that globalsystem-environment coherence is crucial for the compasssensitivity, based on simpler model systems28. Alongsideour other results, this implies that the total system can-not be reduced to consideration of electronic subsystems.Nuclear coherences and nuclear-electronic coherences arean integral part of the processes.

III. DISCUSSION

The radical-pair compass, by its design, relies on thecoherent interconversion of singlet and triplet electronicstates and, thus, the presence of a certain degree ofsinglet-triplet coherence is essential14,16,31. However, wefind that the amount of electronic coherence does notgenerally correlate with increased compass performanceacross differently oriented systems. Instead, in both large

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FIG. 5. Compass anisotropy ∆S is plotted against the globalmeasure of coherence [CGy ]B=0 that represents the coherencecontribution, of the total state, to the singlet yield. a Correla-tion plot for the flavin-tryptophan model. b Correlation plotfor flavin-tyrosine model. Data has been coloured accordingto the radical relative orientation angle β, ranging from 0◦

(blue) to 180◦ (yellow), and a linear fit (red line) with asso-ciated Pearson correlation coefficient R is displayed.

spin systems considered, we observe a surprising anti-correlation, between selected mean electronic coherencemeasures and the reaction yield anisotropy. In contrast,global coherence measures are found to predict compassperformance more effectively than electronic measures.Overall, our results suggest that nature, rather than max-imising and prolonging the possible electronic coherence,must utilise an interplay of several factors on relevanttimescales to increase compass sensitivity. These includethe effect of the magnetic field on coherent dynamics, en-gineering of radical pair lifetimes, decay channels and thehyperfine-driven singlet-triplet dissipation16,26,27. Com-plex radical systems cannot be reduced to considera-tion of electronic subsystems. Nuclear coherences andnuclear-electronic coherences are an integral part of the

processes, which must not be neglected. On the contrary,for FAD/W the anti-correlated electronic coherence mea-sures suggest that global coherence is realized by sacri-ficing electronic coherence to boost the compass fidelity.The crucial relevance of the nuclear degrees of freedomhas also become apparent in a recent study exploring theeffects of accumulated nuclear polarisation for repeatedlyre-excited radical pairs as a means to boost anisotropicMFEs44. Conversely, incoherent operations on nuclearspins can have a significant effect on the singlet-tripletcoherence29.

Advancing our understanding of non-trivial quantumeffects in biology requires the precise characterisationof putative quantum feats as well as asserting whetherthis non-classicality enables robust function. In the con-text of the avian compass, resourceful non-classicality hasbeen implicated based on the correlation of compass fi-delity and coherence measures. However, previous stud-ies have focused on toy models, which did not live up tothe complexity expected for the actual system. There-fore, at best the principal feasibility could be deduced,while the physiological relevance remained opaque. Here,a robust correlation of compass fidelity and global co-herence measures is established for large spin systemsand realistic parameters, thereby corroborating the viewthat coherence—but not electronic coherence—is truly aquantum resource for magnetoreception.

Non-classicality in complex spin systems represen-tative of flavin-tryptophan has previously been pos-tulated based on the comparison with semi-classicalcalculations36,42,45. For long coherent lifetimes, an in-crease in sensitivity emerging as a sharp spike in the frac-tional yield of the singlet product17 is apparent in quan-tum simulations, but absent from the semi-classical de-scription, thereby demonstrating a clear-cut quantum ad-vantage. This “quantum needle”, however, only emergessignificantly for coherent lifetimes that are likely unre-alistically long, at least in view of the expected spin re-laxation rates in cryptochromes and direct animal be-havioural experiments using rf-magnetic fields to in-terfere with compass navigation (specifically, their fre-quency dependence)25–27. In this regard, the correlationof compass fidelity and the coherence measure [CGy ]B=0 asfound here is remarkable, because it suggests a manifestquantum advantage despite the comparably short life-time of 1µs, for which the semi-classical and fully quan-tum approaches concur.

In summary, our results demonstrate Cai and Plenio’sclaim28, that global coherence is a resource, in the com-plex system regime, whilst simultaneously supporting theclaim32 of Jain et al. that sensitivity can be obtainedwithout sustained electronic coherence for some systems.The connection between seemingly contrasting results,concerning electronic coherence as a resource, is foundwhen it is considered in the wider setting of several di-verse systems and complexities as presented in our study.Future studies could steadily increase the relevant com-plexity of models in a tractable manner and consider al-

9

ternative models of the radical reaction46,47 such as theflavin semiquinone/O•−2 radical pair, studied e.g. as afunction of the binding site of the latter. As O•−2 is de-void of hyperfine interactions, this system is invariantto the mutual orientation of the radicals and thus notamenable to the study design as chosen here. It should benoted that even the large systems studied here have been(necessarily) idealized. The asymmetry of the reactionshas been neglected, by setting k = kS = kT , and inter-radical interactions (such as electron-electron dipolar in-teractions) has been ignored, both of which have beenassociated with peculiar quantum feats46–51. Addition-ally, the open quantum system nature of the system couldbe analysed in further detail52,53, employing numericallyexact methods54–56 for greater accuracy and to iden-tify if non-Markovian dynamics57–59 are of importancein realistic natural and artificial systems. Lastly, mea-sures capturing nuclear and electronic subsystems28,29

with clear links to compass operation should be devel-oped to assist in the interpretation of experiments60, suchas those utilising quantum control61, alongside protocolsfor their implementation62. Other resources63–66 such asentanglement67 may also/still be of interest for systems.However, it has been shown that entanglement is unlikelyto be of direct relevance to avian magnetoreception22,30.We anticipate that the approach and observations of thispaper will aid these future endeavours through identifica-tion of the most effective use of coherence measures, andby paving a way to a more realistic assessment of theputatively utilitarian character of coherence in magne-toreception. Regarding this, we recommend that resultsshould always be rationalised with respect to the com-plexity of the system.

IV. ACKNOWLEDGMENTS

We gladly acknowledge the use of the University ofExeter High-Performance Computing facility. This workwas supported by the UK Defence Science and Tech-nology Laboratory (DSTLX-1000139168), the Office ofNaval Research (ONR award number N62909-21-1-2018)and the EPSRC (grant EP/V047175/1). The robin inFig. 1 is extracted from the artwork “Robin Vintage ArtPoster” released under Public Domain license by KarenArnold.

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