Therefore $R(M)=1+0(\delta^{2})$ uniformly over all configurations as desired.
∎

To prove the main proposition from Section 1.1, we just need to put our pieces
together.

###### Proof of Proposition 1.2.

Obviously we will take the domain $U(\delta)$ in the statement of Proposition
1.2 to be the domain $U(\delta)$ from Definition 3.35. Proposition 1.2
contains three statement that we need to check :

  1. 1.

Seen as a closed set of $\mathbb{C}$, $U(\delta)$ is at Hausdorff distance
$O(\delta)$ of $U$,

  2. 2.

$\sup_{v\in\partial U(\delta)}|\delta h^{{\sharp\delta}}(v)-h(v)|=0(\delta)$,

  3. 3.

There exists a map $\phi$ from $U_{ex}$ to $\mathbb{D}$ such that

$h^{\sharp\delta}-\mathbb{E}(h^{\sharp\delta})\to\frac{1}{2\pi\chi}h^{GFF}\circ\phi.$

Items (1) and (2) were already proved in Proposition 3.38. More precisely
Proposition 3.38 states that $U(\delta)=U^{\sharp\delta}$ away from the
boundary and in particular if we embed both using regular hexagons
$d_{\mathcal{H}}(U(\delta,U^{\sharp\delta}))=O(\delta)$. On the other hand by
construction $d_{\mathcal{H}}(U^{\sharp\delta},U)=O(\delta)$ which proves the
first point. The second point is explicitly stated in the proposition.

For item (3), we again obviously want to use the function $\phi$ defined in
Proposition 3.5. By Proposition 4.10, the random walk on the graphs
$\Gamma(\delta)$ satisfy a central limit theorem. They also clearly satisfy
the other assumptions for the main theorem of [4] (the main so called “uniform
crossing” estimate is stated in Lemma 2.13) so if we write
$h_{\Gamma}^{\sharp\delta}$ its winding field, we have

$h_{\Gamma}^{\sharp\delta}-\mathbb{E}(h_{\Gamma}^{\sharp\delta})\to\frac{1}{\chi}h^{GFF}$

where $h^{GFF}$ is a Dirichlet Gaussian free field in $\phi(U)$. The
convergence is in a strong moment sense, see [4] for details. Translating this
convergence in terms of dimer height function, it exactly means that for the
height function $h_{\tilde{U}}$

$h_{\tilde{U}}^{\sharp\delta}-\mathbb{E}(h_{\tilde{U}}^{\sharp\delta})\to\frac{1}{2\pi\chi}h^{GFF}\circ\phi.$

since the map from $\tilde{U}$ to $\Gamma$ is asymptotically given by $\phi$.
Now by Lemma 5.1, the dimers measures on $\tilde{U}(\delta)$ and on
$U_{\mathcal{H}}(\delta)$ are at most $O(\delta^{-2})$ away in total
variation. By Lemma 5.2, this measure is also at most $O(\delta^{2})$ away in
total variation from the measure on $U(\delta)$. Overall we get

$h^{\sharp\delta}-\mathbb{E}(h_{\Gamma}^{\sharp\delta})\to\frac{1}{2\pi\chi}h^{GFF}\circ\phi.$

We still need to replace $\mathbb{E}(h_{\Gamma}^{\sharp\delta})$ by
$\mathbb{E}(h^{\sharp\delta})$ and for this we will bound moments. By applying
Lemma 5.5 in [4] to two neighbouring points, together with Proposition 5.4
there to control the
$m^{\sharp\delta}(v_{i})-\mathbb{E}(h^{\sharp\delta}(v_{i}))$ term, we have
for all $\delta$ small enough, for all $v$

$\operatorname{Var}(h_{\tilde{U}}^{\sharp\delta}(v))\leq C(1+\log^{4}\delta)$

for some $C$. Since $h_{\mathcal{H}}$ is just a conditioned version of
$h_{\tilde{U}}^{\sharp\delta}$, we have

$\operatorname{Var}(h^{\sharp\delta}_{\mathcal{H}}(v))\leq\mathbb{E}[(h^{\sharp\delta}_{\mathcal{H}}(v)-\mathbb{E}h^{\sharp\delta}_{\tilde{U}}(v))^{2}]\leq(1+O(\delta^{2}))C(1+\log^{4}\delta).$

Similarly, since the Radon-Nycodym derivative of
$h_{\mathcal{H}}^{\sharp\delta}$ with respect to $h^{\sharp\delta}$ is of
order $1+0(\delta^{2})$, we also have

$\operatorname{Var}(h^{\sharp\delta}(v))\leq(1+O(\delta^{2}))C(1+\log^{4}\delta).$

Together with the fact that the total variation distance between
$h_{\tilde{U}}^{\sharp\delta}$ and $h^{\sharp\delta}$ is at most
$O(\delta^{2})$, this shows that
$\mathbb{E}(h_{\tilde{U}}^{\sharp\delta})-\mathbb{E}(h^{\sharp\delta})=O(\delta^{2}\log^{2}\delta)$
which concludes the proof. ∎

###### Proof of Proposition 1.3.

This was essentially already done in the proof of Proposition 1.2 just above.
Note that Lemma 5.5 in [4] also applies to higher moment than 2 so we have for
all $k$ and $v$

$\mathbb{E}[|h_{\tilde{U}}(v)-\mathbb{E}[h_{\tilde{U}}(v)]|^{k}]\leq
C_{k}(1+\log^{2k}(\delta).$

We already proved that
$\mathbb{E}[h_{\tilde{U}}(v)]-\mathbb{E}[h^{\sharp\delta}(v)]=O(\delta^{2}\log^{2}\delta)$
so we can conclude exactly as in the proof of Proposition 1.2. ∎

###### Proof of Proposition 1.4.

Note that for this statement, since there is no centring of the height, the
choice of reference flow is important. On the other hand, apart from that, the
comparison between $\mathbb{E}h^{\sharp\delta}_{\tilde{U}}$ and
$\mathbb{E}h^{\sharp\delta}$ is already done in the proof of Proposition 1.2.
Let us therefore use the standard reference flow of $1/3$ along every edge of
$\mathcal{H}$ to compute $h^{\sharp\delta}_{\tilde{U}}$ so that it is enough
to prove

$|\delta\mathbb{E}h^{\sharp\delta}_{\tilde{U}}-h^{\mathcal{C}}|\leq
C\delta(1+\log\frac{1}{\delta}).$

Let $h_{\Gamma}^{\sharp\delta}$ be the height computed using the flow
$M_{ref}$ so that $h_{\Gamma}^{\sharp\delta}$ matches directly the winding of
the spanning tree. By Proposition 4.12 in [4], we see that each exponential
scale contribute at most $O(1)$ to the expected winding of a branch. On the
other hand, the minimal size of a face on the graph $\Gamma(\delta)$ is of
order $\delta^{3}$ so

$|\mathbb{E}h^{\sharp\delta}_{\Gamma}|\leq C(1+\log\frac{1}{\delta}).$

On the other hand, the only difference between $h^{\sharp\delta}_{\gamma}$ and
$h^{\sharp\delta}_{\tilde{U}}$ comes from the change of reference flow and a
global scaling so

$h_{\tilde{U}}^{\sharp\delta}=h^{\sharp\delta}_{\Gamma}+h_{ref}.$

The control of $h_{ref}$ from Lemma 3.31 concludes the proof. ∎

##  Appendix A Continuity with respect to $h^{\mathcal{C}}$

In this section, we prove the continuity statements in Propositions 1.2, 1.3
and 1.4. The main difficult points are the continuity of $\phi$, and the
functions $M^{\pm,\delta}$ so the proof is analytical in nature. We will not
try to be optimise the topology we use on $h^{\mathcal{C}}$ instead trying to
make the argument as easy as possible even for reader not familiar with PDEs.
In particular, we will not discuss any estimate on the regularity of
$h^{\mathcal{C}}$ coming from the fact that it solves a (non-linear) elliptic
equation. We only note that to the best of our admittedly limited knowledge,
the fact that the PDE satisfied by limit shapes admits solutions with frozen
faces makes it hard to apply directly the standard theory of elliptic
operators.

We first recall standard results. The state $W^{k,p}$ for $k\in\mathbb{N}$ and
$p\in[1,\infty]$ is the space of distributions which are represented together
with all their partial derivatives by functions in $L^{p}$. It is a Banach
space for the norm

$\lVert f\rVert_{W^{k,p}}=\sum_{i+j\leq
k}\lVert\frac{\partial^{i+j}f}{\partial^{i}x\partial^{j}y}\rVert_{L^{p}}.$

For $p>2$, elements of $W^{1,p}$ are $1-2/p$ Hölder functions and satisfy
(Theorem A.6.1 in [2]).

$|f(x)-f(y)|\leq\frac{4p}{p-2}|x-y|^{1-p/2}\lVert\nabla f\rVert_{p}$

so in particular for $p>2$ and in a bounded domain, the $W^{k,p}$ norm is
stronger than the uniform norm on all derivatives up to order $k-1$. We will
denote as usual the space of $\alpha$-Hölder functions by
$\mathcal{C}^{\alpha}$ and by $\mathcal{C}^{k,\alpha}$ the space of functions
whose derivatives of order $k$ are $\alpha$-Hölder.

We will also need some results on the solution of classical PDEs.

######  Theorem A.1 (Theorem 4.7.2 in [2]).

If $h$ is a $\mathcal{C}^{\alpha}$ function on $\mathbb{C}$, then the equation

$\frac{\partial f}{\partial\bar{z}}=h$

admits an unique solution in $\mathcal{C}^{1,\alpha}$, up to a linear function
$a+bz$. It satisfies the estimate

$\lVert\frac{\partial f}{\partial
z}\rVert_{\mathcal{C}^{\alpha}}+\lVert\frac{\partial
f}{\partial\bar{z}}\rVert_{\mathcal{C}^{\alpha}}\leq\frac{5}{\alpha(1-\alpha)}\lVert
h\rVert_{\mathcal{C}^{\alpha}}.$

######  Theorem A.2 (Theorem 4.5.3 in [2]).

For any $p>1$, there exists $C>0$ such that for any $f\in
W^{1,p}(\mathbb{C})$,

$\lVert\frac{\partial f}{\partial z}\rVert_{L^{p}(\mathbb{C})}\leq
C\lVert\frac{\partial f}{\partial\bar{z}}\rVert_{L^{p}(\mathbb{C})}.$

Note that holomorphic functions cannot be in $L^{p}(\mathbb{C})$ so the fact
that $\frac{\partial f}{\partial\bar{z}}$ only determines a function $f$ up to
an holomorphic term does not contradict our theorem.

######  Theorem A.3 (Theorem 5.1.1 in [2]).

For every $\mu_{\mathrm{max}}\in(0,1)$, there exists $P>2$ such that the
following holds for all $p\in(2,P)$. If $\mu$ and $\phi$ are functions with
compact support and if $\lVert\mu\rVert_{\infty}\leq\mu_{\mathrm{max}}$ and
$\phi\in L^{p}$ then the equation

$\frac{\partial f}{\partial\bar{z}}=\mu\frac{\partial f}{\partial z}+\phi$

has a unique solution with $f\in W^{1,p}$ and $f(z)=O(1/z)$ at infinity.
Furthermore the solution operator $\phi\to f$ is bounded from $L^{p}$ to
$W^{1,p}$.

######  Theorem A.4 (Theorem 5.1.2, Theorem 5.3.2 and equation 5.10 in [2]).

For every $\mu_{\mathrm{max}}\in(0,1)$, there exists $P>2$ such that the
following holds for all $p\in(2,P)$. For all $\mu$ with compact support and
$\lVert\mu\rVert_{\infty}\leq\mu_{\mathrm{max}}$, there exists a unique
function $f\in W^{1,2}_{\mathrm{loc}}$ such that

$\displaystyle\frac{\partial f}{\partial\bar{z}}$
$\displaystyle=\mu\frac{\partial f}{\partial z}$ $\displaystyle f(z)$
$\displaystyle=z+O(1/z)\text{ at infinity}.$

This solution is called the principal solution of the equation $\frac{\partial
f}{\partial\bar{z}}=\mu\frac{\partial f}{\partial z}$. Furthermore $f$ is an
homeomorphism of $\mathbb{C}$ and $f(z)-z\in W^{1,p}(\mathbb{C})$.

######  Lemma A.5 (Lemma 5.3.1 in [2]).

For every $\mu_{\mathrm{max}}\in(0,1)$, there exists $P>2$ such that the
following holds for all pairs $p,s$ such that $2<p<ps<P$. For all
$\mu_{1},\mu_{2}$ satisfying $|\mu_{i}|\leq\mu_{0}1_{B(0,r)}$ for some $r$,
the principal solutions $f_{1},f_{2}$ for $\mu_{1}$ and $\mu_{2}$ satisfy

$\lVert\frac{\partial f_{1}}{\partial\bar{z}}-\frac{\partial
f_{2}}{\partial\bar{z}}\rVert_{L^{p}}\leq
C(p,s,\mu_{\mathrm{max}})r^{\frac{2}{ps}}\lVert\mu_{1}-\mu_{2}\rVert_{L^{ps/(s-1)}}.$

We can now turn to the continuity of the function $\phi$ with respect to
$h^{\mathcal{C}}$. We fix domains $U$ and $U_{ex}$, both inside $B(0,r)$. We
let $h_{1}$ and $h_{2}$ be two non-extremal dimer limit shapes, dropping the
superscript since we are always in the continuum in this section. We let
$\Phi_{1}$ and $\Phi_{2}$ be as in Definition 3.3 and we let

$\mu_{i}=\frac{\frac{d\Phi_{i}}{d\bar{z}}}{\frac{d\Phi_{i}}{dz}}=\frac{\Phi-e^{i\pi/3}}{\Phi-e^{-i\pi/3}}.$

Since the $h_{i}$ are non extremal we have $|\mu_{i}|\leq{\mathrm{max}}$ for
some $\mu_{\mathrm{max}}<1$ and actually note that the value of
$\mu_{\mathrm{max}}$ controls how close to extremal the gradients of the
$h_{i}$ are. We let $3\geq P>2$ be chosen as in Theorems A.3 and A.4 and to
ease notations we will let $C$ denote constants which can change line by line
which only depend on the parameters given explicitly as in Lemma A.5.

######  Lemma A.6.

If $h_{1}$ and $h_{2}$ are in $W^{35,3}(U_{ex})$ then one can extend to
$\mu_{i}$ to $\mathbb{C}$ so that $\mu_{i}\in W^{34,3}$ and the principal
solution $f_{i}$ are in $W^{34,p}(U_{ex})$ with $\frac{\partial
f_{i}}{\partial\bar{z}}\in W^{33,p}(\mathbb{C})$ for all $p<P$. Furthermore we
have

$\displaystyle\lVert\mu_{i}\rVert_{W^{34,3}(\mathbb{C})}$ $\displaystyle\leq
C(\mu_{\mathrm{max}})\lVert h_{i}\rVert_{W^{34,3}(U_{ex})}$
$\displaystyle\lVert\mu_{1}-\mu_{2}\rVert_{W^{34,3}(\mathbb{C})}$
$\displaystyle\leq C(\mu_{\mathrm{max}})\lVert
h_{1}-h_{2}\rVert_{W^{35,3}(U_{ex})}$ $\displaystyle\lVert
f_{1}-f_{2}\rVert_{W^{34,p}(\mathbb{C})}$ $\displaystyle\leq
C(\mu_{\mathrm{max}},r,p)\lVert h_{1}-h_{2}\rVert_{W^{35,3}(U_{ex})}$

###### Proof.

Note that the function $\nabla h_{i}\to\Phi_{i}$ is smooth over all non-
extremal gradients. In particular $\Phi_{i}\in W^{34,3}$ and we can bound its
norm in term of $h_{i}$. Similarly we have
$\mu_{i}=\frac{\Phi_{i}-e^{i\pi/3}}{\Phi-e^{-i\pi/3}}$ and since
$\operatorname{Im}(\Phi)>0$ we easily obtain the first two points.

For the last point, we first note that for any differential operator $D$,
$Df_{i}$ satisfies the equation

$\frac{\partial Df_{i}}{\partial\bar{z}}=\mu\frac{\partial Df_{i}}{\partial
z}+D\mu\frac{\partial f_{i}}{\partial z}.$

Note that this equation is of the type of Theorem A.3 since $D_{m}u$ has
compact support, $D\mu$ is continuous because $\mu\in W^{34,3}$ and
$\frac{\partial f_{i}-z}{\partial z}\in L^{p}$ by Theorem A.4. Therefore by
uniqueness in Theorem A.3, $Df_{i}-Dz\in W^{1,p}$ for all $p$ and since this
was true for all differential operator, $f_{i}-z\in W^{2,p}$. However note
that if $D^{\prime}$ is another differential operator, we have

$\frac{\partial D^{\prime}Df_{i}}{\partial\bar{z}}=\mu\frac{\partial
D^{\prime}Df_{i}}{\partial z}+D^{\prime}\mu\frac{\partial Df_{i}}{\partial
z}+D^{\prime}D\mu\frac{\partial f_{i}}{\partial z}+D\mu\frac{\partial
D^{\prime}f_{i}}{\partial z}.$

and we can reapply the argument above to get that $f_{i}-z\in W^{3,p}$.
Clearly we can keep iterating until we obtain $f_{i}-z\in W^{34,p}$.

Now we note that $f_{1}-f_{2}$ satisfies the equation For the last point, we
note that $f_{1}-f_{2}$ satisfies the equation

$\frac{\partial(f_{1}-f_{2})}{\partial\bar{z}}=\frac{\mu_{1}+\mu_{2}}{2}\frac{\partial(f_{1}-f_{2})}{\partial
z}+\frac{\mu_{1}-\mu_{2}}{2}\frac{\partial(f_{1}+f_{2})}{\partial z}$

with condition $f_{1}-f_{2}=O(1/z)$ at infinity. This is again of the type of
Theorem A.3. On the other hand, by Lemma A.5,

$\lVert\frac{\partial(f_{1}-f_{2})}{\partial\bar{z}}\rVert_{L^{p}}\leq
C(\mu_{\mathrm{max}},p,r)\lVert\mu_{1}-\mu_{2}\rVert_{L^{\infty}}\leq
C(\mu_{\mathrm{max}},p,r)\lVert\mu_{1}-\mu_{2}\rVert_{W^{1,3}},$

By Theorem A.2, the bound on $\frac{\partial(f_{1}-f_{2})}{\partial\bar{z}}$
extends to $\frac{\partial(f_{1}-f_{2})}{\partial z}$ and since the solution
operator in Theorem A.3 is continuous, we obtain

$\lVert f_{1}-f_{2}\rVert_{W^{1,p}}\leq
C(\mu_{\mathrm{max}},p,r)\lVert\mu_{1}-\mu_{2}\rVert_{W^{1,3}}.$

We can then iterate as above to bound the derivatives. ∎

In order to speak of the continuity of the choice of function $\phi$, it is
necessary to have a consistent choice of conformal map between $f_{i}(U_{ex})$
and $\mathbb{D}$. for this we fix $z\in U$ and we take the convention that
$f_{i}(z)$ is sent to $0$ with real positive derivative. We call $g_{i}$ the
conformal map from $\mathbb{D}$ to $f_{i}(U_{ex})$ so that
$\phi=g_{i}^{-1}\circ f_{i}$.

######  Lemma A.7.

With the above convention, the function $\phi$ depends continuously on the
function $h^{\mathcal{C}}$ when we consider the
$W^{34,p}_{\mathrm{loc}}(U_{ex})$ topology for $\phi$ and the
$W^{35,3}(U_{ex})$ topology on $h^{\mathcal{C}}$.

###### Proof.

First we note that $f(U_{ex})$ is a Jordan domain whose boundary depends
continuously on $h$ (say for the uniform topology). It is easy to see that
this implies by the Caratheodory kernel theorem that $g$ depends continuously
on $h$ for the uniform topology on compacts. Since the $g$ are conformal maps,
the convergence of the functions also implies the convergence of all their
derivatives.

Since $g$ is a conformal map, it has a non-zero derivative inside $\mathbb{D}$
and therefore its inverse must also depend continuously on $h^{\mathcal{C}}$.
We have already proved that $f$ depends continuously on $h^{\mathcal{C}}$ and
the composition of a function in $W^{34,p}$ with a $\mathcal{C}^{\infty}$
function cannot change the local regularity, therefore $\phi$ depends
continuously on $h^{\mathcal{C}}$ as desired. ∎

######  Lemma A.8.

The functions $M_{n}^{\pm}$ defined in the proof of Lemma 3.8 for $1\leq n\leq
30$ depend continuously on $h^{\mathcal{C}}$ when we consider the
$\mathcal{C}^{34-n}$ topology on $M_{n}^{\pm}$ and the $W^{35,3}$ topology on
$h^{\mathcal{C}}$. In particular for $\delta\leq 1$, the functions
$M^{+\delta}$ and $M^{-,\delta}$ can be bounded in $L^{\infty}$ norm by a
continuous function of $h^{\mathcal{C}}$.

###### Proof.

Recall that the functions $M_{n}^{+}$ are built iteratively as solution of the
equations

$\frac{\partial(M_{n}\circ\phi^{-1}}{\partial\bar{z}}=\frac{J_{n+1}}{\partial_{y}\phi(\partial_{1}+\Phi\partial_{2})\operatorname{Re}(\phi)}$

where $J_{n+1}$ is a polynomial expression involving $\Phi$, derivatives of
$\phi$ and $H$ and derivatives of $M_{k}$ for $k\leq n$. Furthermore it is
easy to see that all terms in $J_{n+1}$ involve at most derivatives up to
order $n+2$ even when we consider any $M_{k}$ term already as a derivative of
order $k$.

Since $\phi\in W^{34,p}$ for some $p>2$, $\phi$ is in $\mathcal{C}^{33}$ with
$\alpha$-Hölder derivatives of order $33$ for some $\alpha$. Similarly
$h^{\mathcal{C}}\in W^{35,3}$ and therefore $H$ is $\mathcal{C}^{34}$ with
Hölder derivatives. We see that $J_{2}$ is in $\mathcal{C}^{33}$ and applying
Theorem A.1 to all derivatives of $M_{1}\circ\phi^{-1}$ we see that $M_{1}$ is
in $\mathcal{C}^{33}$. It is also clear by composition that $M_{1}$ is a
continuous function of $h^{\mathcal{C}}$. We can then iterate the argument for
all $M_{n}$ to obtain the result. ∎

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# Observations about utilitarian coherence in the avian compass

Luke D. Smith  Jean Deviers  Daniel R. Kattnig  Living Systems Institute and
Department of Physics
University of Exeter, Stocker Road, Exeter EX4 4QD, United Kingdom
* Corresponding author<EMAIL_ADDRESS>
###### Abstract

Abstract. It is hypothesised that the avian compass relies on spin dynamics in
a recombining radical pair. Quantum coherence has been suggested as a resource
to this process that nature may utilise to achieve increased compass
sensitivity. To date, the true functional role of coherence in these natural
systems has remained speculative, lacking insights from sufficiently complex
models. Here, we investigate realistically large radical pair models with up
to 21 nuclear spins, inspired by the putative magnetosensory protein
cryptochrome. By varying relative radical orientations, we reveal correlations
of several coherence measures with compass fidelity. Whilst electronic
coherence is found to be an ineffective predictor of compass sensitivity, a
robust correlation of compass sensitivity and a global coherence measure is
established. The results demonstrate the importance of realistic models, and
appropriate choice of coherence measure, in elucidating the quantum nature of
the avian compass.

††preprint: APS/123-QED

## Introduction

Quantum coherence and entanglement are central enablers of quantum
technologies and quantum information processing. The emerging field of quantum
biology [1, 2, 3, 4, 5] hypothesizes that these rudiments of quantumness could
likewise empower biological processes of living systems. In particular, long-
lived quantum coherence is an overarching theme of many of the focus points of
quantum biology, such as the avian magnetoreceptor, and photosynthetic energy
transport [6, 7, 8, 9, 10]. However, our understanding and quantification of
coherence as an operational resource [11, 12, 13], and its role in the complex
quantum systems of life, is still limited.

The avian compass and several related magnetosensitive feats are hypothesised
to rely on coherent spin dynamics in a radical pair [14] (RP), putatively
formed in the blue-light sensitive flavo-protein cryptochrome. The radical
pair comprises two unpaired electrons, the combined spin angular momentum of
which are described in terms of singlet and triplet states. Typically, only
the singlet state can recombine to re-form the diamagnetic resting state,
while both singlets and triplets can give rise to spin-independent structural
rearrangements of the protein and thus signal to down-stream processes (see
the scheme presented in Fig. 1) [15]. Magnetosensitivity emerges in these
settings from a change in singlet and triplet state populations and thus
yields for different magnetic field orientations. This is driven by the
coherent singlet-triplet interconversions which result from the hyperfine
coupling with magnetic nuclei in the radicals under perturbation by the
comparably weak Zeeman interaction of the electron magnetic moments with the
geomagnetic field ($\approx 50\mskip 3.0mu\mathrm{\mu}$T). The nonstationary
singlet-triplet coherence is an essential requirement for this process. In its
absence (i.e., if singlet and triplet states are eigenstates of the
Hamiltonian), no interconversion between singlet and triplet states occurs and
the reaction yield will be insensitive to the applied magnetic field. On the
other hand, electronic spin state decoherence, through interaction with the
environment of nuclear spins, provides a natural companion to the radical pair
spin dynamics and prerequisite for its sensitivity to magnetic fields [16]. In
order to elicit sensitivity to a $50\mskip 3.0mu\mathrm{\mu}$T field the spin
coherence must persist for about a microsecond, i.e., the reciprocal of the
electron Larmor precession frequency ($\approx 1.4\mskip 3.0mu$MHz), or
longer. Longer-lived singlet-triplet coherences promise a more precise
magnetic field measurement, which could underpin a compass of exquisite acuity
[17]. However, a coherent lifetime of the radical pair of $1$–$10\mskip
3.0mu\mathrm{\mu}$s has been deemed realistic based on predictions of spin
relaxation processes and animal behavioural experiments [18, 19, 20].

Figure 1: a Structure of an avian cryptochrome (PDB identifier: 6PU0, Columba
livia), including the central electron transfer chain comprising four
tryptophans (W) labelled A (W395), B (W372), C (W318) and D (W369) and ending
in the surface exposed tyrosine Y319. Photo-excitation of FAD in cryptochrome
initiates consecutive electron transfer reactions of adjacent donors/acceptor
pairs (red arrows), producing sequential radical pairs of the form [FAD∙- /
W∙+] and possibly [FAD∙- / Y∙] (see e.g. ref. 21). 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 reoxidation [22, 23, 24]. b Generic radical-pair reaction scheme.
The radical pair is born in the singlet state (spin multiplicity indicated by
superscript labels, total electronic spin: $S=0$) which, via coherent
interconversion, can interconvert to a triplet state ($S=1$). Here, A∙ is
assumed to be FAD and B∙ is a tryptophan (W) or tyrosine (Y) residue. The
radical pair may form singlet and triplet products with rate constants $k_{S}$
and $k_{T}$, respectively. The singlet channel typically comprises radical
pair recombination and spin-independent protein structural rearrangements; the
latter also contributes to the triplet channel. c Graphical representation of
hyperfine interactions in FAD∙- (top), Y∙ (bottom left) and W∙+ (bottom
right). Here, in the direction given by the unit vector $\mathbf{d}$, the
plotted surfaces are drawn at distance $||\mathbf{Ad}||$ from the location of
the nucleus in question, whereby $1\mskip 3.0mu$Å corresponds to $17\mskip
3.0mu$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 orientations.

Current models of magnetoreception implicate the protein cryptochrome as the
host of the magnetosensitive radical pair [15]. A photo-generated radical pair
comprising a flavin anion radical (FAD${}^{\bullet-})$ and radical cation
derived from a surface-exposed tryptophan residue (W∙+) is a popular model
supported by in vitro studies [21]. Alternative radical pair models have been
put forward based on in vivo observations and modelling [25, 22, 26, 23, 24,
27] (e.g. the flavin semiquinone/superoxide pair). Here, we focus on the
former and an additional model comprising a tyrosine radical partner instead
of the tryptophan [25].

Several works have suggested that coherence is a resource for the sensitivity
of the compass, insofar as a larger coherence corresponds to increased
directional magnetic sensitivity. Cai and Plenio [28] have analysed many
randomly chosen prototype radical pair systems (with 5–6 nuclear spins) and
found that their “global coherence” of the electron and nuclear spin system is
a predictor of compass sensitivity. Kominis has introduced a formal measure of
singlet-triplet coherence [29] and demonstrated that singlet-triplet coherence
of radical pairs provides an operational advantage to magnetoreception in
simple model systems involving one nuclear spin. In a previous study [25], we
have observed that the electron coherences in realistic multi-nuclear radical
pairs comprising up to 21 nuclear spins are long-lived compared to the
electronic entanglement. We concluded that the compass is coherence-driven
while entanglement does not confer an operational advantage of the compass.
The latter point has also been elaborated on in ref. 30 with comparable
conclusions. Le and Olaya-Castro have recently investigated a three-spin
system subject to collisional, symmetry-breaking environmental interactions
[31]. The authors demonstrate that magnetosensitivity in this system requires
coherence and suggest that a small degree of coherence, regardless of basis,
is likely a quantum resource for biomolecular systems.

The property of coherence as a facilitator of increased compass sensitivity
was recently questioned by Jain et al., who suggest that the electronic
coherence swiftly decays with the number of coupled nuclei. The authors
demonstrate that the compass can provide high sensitivity despite operating in
a parameter regime without “sustained” electron spin coherence [32]. This
conclusion was derived by comparing systems with up to 6 magnetic nuclei
coupled with identical, aligned and axial hyperfine tensors (i.e., with
identical transverse and dominant longitudinal components). While this system
shows sustained electron coherences for vanishing transverse hyperfine
components, the system with maximal sensitivity did not sustain the coherences
(with “sustained” construed in relation to the artificial reference system).
As of now, it is not clear if the findings of previous coherence measure
studies translate to more realistic conditions or models, but the conclusion
of Jain et al. is certainly surprising, as it is in stark deviance to the
abovementioned studies.

Here, we investigate the utilitarian character of coherence for the compass
based on models that are biologically relevant. In particular, we focus on
radical pairs that have been implicated with cryptochrome-magnetoreception,
namely flavin/tryptophan and flavin/tyrosine radical pairs with up to 21
nuclear spins. Our study thus endeavours to address the character of coherence
in the actual magnetoreceptor, as currently envisaged, rather than the
principle role as elaborated in previous studies using strongly
simplified/abstract models. Instead of varying the hyperfine interaction
parameters, which are predetermined by the identity of the radical pair and
only weakly dependent on the molecular environment [33], we vary the relative
orientation of the two radicals. This is a parameter which could have been
subject to evolutionary optimisation in the protein cryptochrome and is more
likely to reflect any potential quantum advantage that is exercisable in the
well-defined biological system, if it exists. Specifically, considering more
realistic modelling conditions, we analyse whether the relative orientations
of optimal compass fidelity are correlated with the electronic coherence (as
anticipated based on investigations [28, 29, 25]) or not (as advocated in a
recent study by Jain et al. [32]). Our results question the effectiveness of
coherence, measured by electronic coherence quantifiers, as the sole driver of
magnetoreception and explanation of increased compass sensitivity. However,
global coherence is realized as a persistent effectuator of compass fidelity,
confirming the hypothesized quantum nature of the processes even for realistic
systems with relatively short radical lifetimes of $1\mskip
3.0mu\mathrm{\mu}$s.

## Results

Radical pair model. We consider radical pairs subject to coherent evolution
under the local Hamiltonians of their constituent radicals and spin-selective
recombination proceeding with the same rate constant, $k_{S}=k_{T}=k$, for the
singlet and triplet channel. Inter-radical interactions are neglected, as is
spin relaxation. In this simplified, but commonly studied scenario [17, 25,
26], the spin dynamics of the RP and its associated magnetic field effects
(MFEs) can be described by the following master equation for the spin density
operator $\hat{\rho}(t)$:

$\displaystyle\frac{{{\rm{d}}\hat{\rho}(t)}}{{{\rm{d}}t}}$
$\displaystyle=-i\left[{\hat{H},\hat{\rho}(t)}\right]-\frac{{{k_{S}}}}{2}\left\\{{{{\hat{P}}_{S}},\hat{\rho}(t)}\right\\}-\frac{{{k_{T}}}}{2}\left\\{{{{\hat{P}}_{T}},\hat{\rho}(t)}\right\\}$
$\displaystyle=-i\left[{\hat{H},\hat{\rho}(t)}\right]-k\hat{\rho}(t),$ (1)

where [] and {} denote the commutator and anti-commutator, respectively. Here,
the Hamiltonian is of the form $\hat{H}={\hat{H}_{\rm{A}}}+{\hat{H}_{\rm{B}}}$
with subscripts A and B referring to the two radicals, and $\hat{P}_{S}$ and
$\hat{P}_{T}=\hat{1}-\hat{P}_{S}$ are projection operators onto the singlet
state and triplet manifold, respectively. The chemical reactivity has been
included using the Haberkorn approach [34], which gives rise to minimal
singlet-triplet dephasing (alternative approaches have been suggested, but for
$k_{S}=k_{T}$ the differences of various master equation are lessened [29,
35]). The Hamiltonians $\hat{H}_{i}$, $i\in\\{A,B\\}$, account for the Zeeman
interactions between the electron spins and the applied magnetic field and the
hyperfine interactions between the electron and nuclear spins within radical
$i$ ($\hbar=1$):

$\displaystyle{\hat{H}_{\rm{i}}}=\sum\limits_{j}{{{{\bf{\hat{S}}}}_{i}}\cdot{{\bf{A}}_{i,j}}\cdot{{{\bf{\hat{I}}}}_{i,j}}}+{\vec{\omega}_{i}}\cdot{{\bf{\hat{S}}}_{i}}.$
(2)

Here, the Larmor precession angular frequency is given by
$\vec{\omega}_{i}=-\gamma_{i}\vec{B}$, with $\gamma_{i}$ denoting the
gyromagnetic ratio of the electron in radical $i$ and $\vec{B}$ the applied
magnetic field. $\mathbf{A}_{i,j}$ is the hyperfine coupling tensor between
the $j$th nuclear spin and the $i$th electron spin; $\hat{\mathbf{I}}_{i,j}$
and $\hat{\mathbf{S}}_{i}$ are the corresponding vector operators of nuclear
and electron spin angular momentum.

Here, we solve Eq. (1) for realistically complex spin systems based on an
approach employing spin correlation tensors (see Methods for details) [25, 36,
37], to yield the quantum yield of recombination via the singlet channel,
$Y_{S}(\vartheta,\varphi)$. Its sensitivity to changes in the applied magnetic
field direction determine the compass fidelity, which we here assess in terms
of established measures, namely the absolute anisotropy defined as

$\displaystyle\Delta_{S}=\max_{\vartheta,\varphi}Y_{S}(\vartheta,\varphi)-\min_{\vartheta,\varphi}Y_{S}(\vartheta,\varphi),$
(3)

where the polar angle $\vartheta$ and azimuthal angle $\varphi$ parametrize
the direction of the applied magnetic field in the protein frame. We will
relate this sensitivity measure to quantifiers of the quantum coherence
present in the radical pair during its magnetosensitive spin evolution.

Measures of coherence. Various measures to quantify coherence have been
developed [11, 13, 12]. Several such measures have previously been used in the
context of the avian compass [28, 29, 38, 39, 32]. One canonical measure to
quantify coherence is the relative entropy of coherence, which, in terms of
the normalized density operator $\hat{\rho}$ and a chosen basis,
$\\{|n\rangle\\}_{n=1}^{d}$, of the $d$-dimensional Hilbert space, is defined
as

$\displaystyle\mathcal{C}_{r}[\hat{\rho}]=S[\mathbb{I}\mathbb{C}(\hat{\rho})]-S[\hat{\rho}],$
(4)

where $S[\hat{\rho}]=-\mathrm{Tr}[\hat{\rho}\log(\hat{\rho})]$ is the von
Neumann entropy and $\mathbb{IC}$ denotes the dephasing operation
$\mathbb{IC}(\hat{\rho})=\sum_{n}|n\rangle\langle n|\hat{\rho}|n\rangle\langle
n|$, which maps any quantum state into an incoherent state in the chosen
basis.

Another measure, the $l_{1}$-norm of coherence [11] is defined as

$\displaystyle\mathcal{C}_{l_{1}}[\hat{\rho}]=\sum_{n\neq m}|\langle
n|\hat{\rho}|m\rangle|.$ (5)

Furthermore, we employ an additional measure suggested by Kominis defined by

$\displaystyle\mathcal{C}_{st}[\hat{\rho}]=S[\hat{P}_{S}\hat{\rho}\hat{P}_{S}+\hat{P}_{T}\hat{\rho}\hat{P}_{T}]-S[\hat{\rho}],$
(6)

which reports singlet-triplet coherence while being independent of basis and
unaffected by the degree of triplet coherence, which the author suggests was
superior in assessing coherence in the radical pair-based compass [29].
Similarly, an alternative $l_{1}$-based measure of singlet-triplet coherence
was used in ref. 40. The relative entropy of $\hat{\rho}$ with respect to the
maximally mixed state, $S\left[{\hat{\rho}||\hat{1}/d}\right]=\log
d-S\left[{\hat{\rho}}\right]$, has likewise been employed to quantify the
amount of basis-independent coherence [31]. Several alternative measures have
been discussed in the literature [40, 29, 31]. We provide a comprehensive
collection of results for these alternatives in the Supporting Information
(SI), but focus on the above-mentioned measures in the main text for
conciseness.

The coherence quantifiers introduced above can be employed to the entire
density operator, $\hat{\rho}$, as well as to the electronic part,
$\hat{\sigma}$, and are either basis-independent or defined with respect to a
basis. In the present study, two clear choices include the singlet-triplet
basis and the up-down basis of the electronic subspace, both of which will be
used to evaluate $\mathcal{C}_{r}$ and $\mathcal{C}_{l_{1}}$ below. To
distinguish these choices, we will henceforth index the coherence labels with
$\mathcal{G}$ (global, based on $\hat{\rho}$), $\mathcal{E}$ (electronic,
based on $\hat{\sigma}$), and ST (singlet-triplet basis) and UD (up-down
basis). For example, $\mathcal{C}_{r}^{\mathcal{E},U\\!D}$ corresponds to the
electronic coherence assessed via the relative entropy quantifier in the up-
down basis, which is the measure used by Jain et al. [32]. Note that, for all
definitions used above, the non-trace preserving density matrix characteristic
of recombining radical pair systems must be renormalized (by
$\mathrm{Tr}[\hat{\rho}]$; which here simply yields the $k=0$ result). As the
coherence measures $\mathcal{C}_{i}(t)$, $i\in\\{r,l_{1},st\\}$, are time-
dependent we further introduce the coherence yield as the time-averaged
coherence measure weighted by the exponential decay kinetics of the radical
pair:

$\displaystyle\bar{\mathcal{C}}_{i}=k\int_{0}^{\infty}\mathcal{C}_{i}(t)\exp(-kt)\,\mathrm{d}t.$
(7)

For comparison to the anisotropy, we further define measures accounting for
the variation of $\bar{\mathcal{C}}_{i}$ with the magnetic field direction as

$\displaystyle\mu[\bar{\mathcal{C}}_{i}]$
$\displaystyle=\frac{\bar{\mathcal{C}}_{i}(\hat{\rho}_{\max})+\bar{\mathcal{C}}_{i}(\hat{\rho}_{\min})}{2},$
(8)

and

$\displaystyle\Delta[\bar{\mathcal{C}}_{i}]$
$\displaystyle=\bar{\mathcal{C}}_{i}(\hat{\rho}_{\max})-\bar{\mathcal{C}}_{i}(\hat{\rho}_{\min}),$
(9)

where $\hat{\rho}_{\max}$ and $\hat{\rho}_{\min}$ corresponds to the density
matrix associated with maximum and minimum singlet yield over magnetic field
orientations. Furthermore, a field-independent measure is introduced as
$[\bar{\mathcal{C}}_{i}]_{B=0}$, which is evaluated for $B=0$.

Cai and Plenio have suggested an alternative, operational global quantifier of
coherence [28] in the context of a chemical compass in terms of the field-
independent ($B=0$) singlet recombination yield due to the coherent part of
the initial density operator $\mathbb{GC}(\hat{\rho}(0))$, with
$\mathbb{GC}(\hat{\rho})=\hat{\rho}-\mathbb{IC}(\hat{\rho})$ evaluated in the
eigenbasis of the hyperfine Hamiltonian:

$\displaystyle[\mathcal{C}_{y}^{\mathcal{G}}]_{B=0}=\Big{|}Y_{S}\Big{(}\hat{\rho}(0)=\mathrm{Tr}[\hat{P}_{S}]^{-1}\mathbb{GC}(\hat{P}_{S});B=0\Big{)}\Big{|}.$
(10)

In this paper we utilise both this measure and a generalised field-dependent
version (with the magnetic field in the extremal directions) to allow us to
also evaluate $\mu[\mathcal{C}_{y}^{\mathcal{G}}]$ and
$\Delta[\mathcal{C}_{y}^{\mathcal{G}}]$.

Large spin system electronic coherence. Previous studies on coherence in the
avian compass typically use systems with a modest number of nuclear spins
(only one or 5–6) and often randomly assumed hyperfine interactions [28, 29,
38, 41, 32]. Under the assumptions as laid out above, we have been able to
overcome this limitation and evaluate fidelities and coherence measures for
flavin-tryptophan and flavin-tyrosine radical pairs with up to 21 nuclear
spins and judiciously chosen hyperfine parameters, as summarized in the SI.
Our implementation relied on GPU computing realized using CUDA, which vastly
outperformed the corresponding calculations on CPUs. We have varied the
relative orientation of the radicals, as parametrized by Euler angles,
$\alpha$, $\beta$ and $\gamma$, defining the orientation of the flavin radical
relative to the radical partner. Using an angular resolution of 3 degrees, we
have probed 878,400 relative orientations for each combination of radicals.
For each radical orientation, we sampled 10242 orientations of $\vartheta$ and
$\varphi$ (5121 unique directions) to evaluate the fidelity measure
$\Delta_{S}$ and several associated coherence measures as introduced above. In
what follows, we focus on the compass anisotropy $\Delta_{S}$ and the
coherence measures $\mu[\bar{\mathcal{C}}_{i}]$ and
$\Delta[\bar{\mathcal{C}}_{i}]$, for both the ST and UD basis, and the
$l_{1}$\- and relative entropy of coherence quantifiers. Alternative measures
are discussed in the SI, whereby qualitatively corresponding conclusions
emerged. For all simulations, a magnetic field strength of 50 $\mu$T
(comparable to the geomagnetic field in Northern Europe) and a radical
lifetime of $k^{-1}=1\mskip 3.0mu\mu$s were assumed. The latter is in line
with the order-of-magnitude lifetime of the magnetosensitive radical pairs in
cryptochromes as observed for in vitro studies [14, 21] and the anticipated
spin relaxation times [20]. Results for larger magnetic field strength may be
found in the SI; these show increased anisotropy but reduced correlation with
coherence measures (Figs. S1 and S2).

Figure 2:  Compass sensitivity, as captured by the anisotropy $\Delta_{S}$, is
plotted against electronic coherence measures $\mu[\bar{\mathcal{C}}_{i}]$ (a,
c, e) and $\Delta[\bar{\mathcal{C}}_{i}]$ (b, d, f), for 878,400 relative
orientations of flavin-tryptophan radical pairs. The anisotropy $\Delta_{S}$
quantifies the maximal change in the recombination quantum yield as the
direction of the applied magnetic field is varied in the frame of the
cryptochrome protein, with larger values corresponding to greater sensitivity.
Here, we use the recombination weighted relative entropy of coherence
$\bar{\mathcal{C}}_{r}$ defined in Eq. (4), and the $l_{1}$-norm measure
$\bar{\mathcal{C}}_{l_{1}}$ defined in Eq. (5) evaluated in the Zeeman basis,
also known as the “up-down” (UD) basis, and the singlet-triplet (ST) basis.
Coherence is assessed by its mean value $\mu[\bar{\mathcal{C}}_{i}]$ as
defined in Eq. (8), and difference $\Delta[\bar{\mathcal{C}}_{i}]$ as defined
in Eq. (9), with respect to the magnetic field directions associated with
maximal and minimal recombination yield. Alternative measures are covered in
the Supporting Information. Data has been coloured according to the relative
orientation angle $\beta$, ranging from $0^{\circ}$ (blue) to $180^{\circ}$
(yellow) and a linear fit (red line) with associated Pearson correlation
coefficient $R$ is displayed.

For flavin-tryptophan systems the results are presented in Fig. 2, which shows
plots of the absolute anisotropy vs. various electronic coherence measures.
The absolute anisotropy $\Delta_{S}$ measures the maximal spread of the
quantum yield of radical pair recombination as the direction of the
geomagnetic field is varied relative to the sensory protein. Thus,
$\Delta_{S}$ indicates the contrast of the sensor via a relative difference in
reaction products with respect to magnetic field orientation and is widely
accepted as a measure of compass sensitivity. In this way, Fig. 2 correlates
this measure of compass sensitivity with measures quantifying the electronic
coherence present during the lifetime of the radical pair. The data reveals
diverse, i.e., not universal, and complex relations of fidelity and coherence
measures. Surprisingly, an anti-correlation with respect to
$\mu[\bar{\mathcal{C}}_{i}]$ is observed that is present in both the UD and ST
basis, but larger in the former. Specifically, $\Delta_{S}$ is found to anti-
correlate with $\mu[\bar{\mathcal{C}}_{r}^{\mathcal{E},U\\!D}]$, which is
based on the same coherence measure as used by Jain et al. [32], giving rise
to a remarkable correlation coefficient of $R=-0.927$. In contrast, a positive
(but weaker) correlation is observed with respect to the
$\Delta[\bar{\mathcal{C}}_{r}^{\mathcal{E},UD}]$ measure. Similar results,
shown in the SI, are found for correlations involving the relative anisotropy
$\Gamma_{S}$ (Fig. S3), which is a measure of compass sensitivity $\Delta_{S}$
relative to the mean recombination yield.

Furthermore, the results show structured subsets of the data that may present
positive or negative correlations. In our data discussed below, and in
comparable studies in the literature, such structures in the correlation data
appear to be absent for a small number of nuclear spins. This suggests that
for comprehensive considerations of coherence in biology, systems must be
sufficiently and realistically complex.

In the case of $\Delta[\bar{\mathcal{C}}_{r}^{\mathcal{E},ST}]$, the
colouration of the plot according to the angle $\beta$ demonstrates a
relationship to the bands in which small values of $\beta$ correspond to a
stronger positive correlation (see Fig. 2d). However, maximal anisotropy is
realized for $\Delta[\bar{\mathcal{C}}_{r}^{\mathcal{E},ST}]\approx 0$, a
condition which appears to likewise approximately hold for the other coherence
anisotropy measures used (see Fig. 2b and f). The implication is that, for
this system, maximal compass sensitivity is not realized by minimizing
electronic coherence in one extremal direction while maximizing it in the
other, a strategy which, at least a priori, might have appeared auspicious in
maximizing yield differences and thus compass fidelity.

Results for flavin-tyrosine radical pairs are shown in Fig. 3, in which
smaller correlations are found than for the flavin-tryptophan model. An anti-
correlation is still observed for
$\mu[\bar{\mathcal{C}}_{r}^{\mathcal{E},ST}]$ and $\Delta_{S}$, and likewise
the weaker positive correlation persists between
$\Delta[\bar{\mathcal{C}}_{r}^{\mathcal{E},ST}]$ and $\Delta_{S}$. As in the
flavin-tryptophan models, subsets of the data present positive and negative
correlation. However, the structure of these subsets, and relationship to the
angle $\beta$ is different. Here, for mid-range values of $\beta$, which
correspond to maximal compass sensitivity, the results suggest a marked anti-
correlation in $\mu[\bar{\mathcal{C}}_{r}^{\mathcal{E},ST}]$ and a strong
positive correlation in $\Delta[\bar{\mathcal{C}}_{r}^{\mathcal{E},ST}]$.
Similar results, shown in the SI (Figs. S3 and S4), are found for correlations
involving the relative anisotropy $\Gamma_{S}$.

The results on large spin systems are unexpected in view of previous studies
[17, 29, 38, 42] and thought provoking with respect to Jain et al.’s
suggestion of a compass operating in a regime of low coherence [32]. Here, our
results based on the same coherence measure as used Jain et al.
($\bar{\mathcal{C}}_{r}^{\mathcal{E},U\\!D}$), appear to not only corroborate
this conclusion, but lead to the supposition that the lack of electronic
coherence is advantageous. This could imply an incoherent, relaxation-driven
[16, 43] character of the avian compass.

Small spin system electronic and global coherence. We note that the coherence
measures employed above, including $\mathcal{C}_{r}^{\mathcal{E},U\\!D}$ as
used by Jain et al., have not previously been assessed with respect to their
principal predictive power of compass performance. We have thus systematically
explored the correlation of compass fidelity and the various coherence
measures employed, both for systems of reference-probe topology [26, 27] (with
all hyperfine interactions confined to one radical) and systems with more
symmetrically distributed hyperfine interactions, and for radical pair
lifetimes of $k^{-1}=1\mskip 3.0mu\mu$s and $k^{-1}=10\mskip 3.0mu\mu$s.
Details of this exploration are provided in the SI (Figs. S5–S8). In brief, we
again find that there is significant variation of correlations for different
systems and measures, overall supporting the view that electronic coherence is
not the most effective predictor of compass sensitivity. In particular systems
that combine the hyperfine interactions of N5 and N10 in flavin with random
hyperfine couplings in the other radical exhibit low correlation between
anisotropy and electronic coherence. However, as was the case in the study
presented by Cai and Plenio [28], a stronger correlation is found with global
coherence measures, in particular $\mathcal{C}^{\mathcal{G}}_{y}$. This
measure corresponds to the coherent contribution to the singlet yield, and its
broad success suggests that it is important that the global coherence
quantifier relates to operation. Further measures and correlation plots
supporting these findings can be found in the SI (Figs. S5–S8).

Figure 3:  Compass sensitivity, as captured by the anisotropy $\Delta_{S}$, is
plotted against electronic coherence measures $\mu[\bar{\mathcal{C}}_{i}]$ (a,
c, e) and $\Delta[\bar{\mathcal{C}}_{i}]$ (b, d, f), for 878,400 relative
orientations of flavin-tyrosine radical pairs. Here, we use equivalent
measures to Fig. 2, where additional details are provided; alternative
measures are detailed in the SI. Data has been coloured according to the
relative orientation angle $\beta$, ranging from $0^{\circ}$ (blue) to
$180^{\circ}$ (yellow). Both up-down (UD) and singlet-triplet (ST) bases are
considered, and a linear fit (red line) with associated Pearson correlation
coefficient $R$ is displayed.

Large spin system global coherence. The confirmed applicability of
$[\mathcal{C}_{y}^{\mathcal{G}}]_{B=0}$ motivated us to attempt a reassessment
of the coherence-fidelity correlation of the large spin systems based on this
global coherence measure. While global coherence measures are in general
prohibitively expensive to evaluate for spin systems of the relevant size,
$[\mathcal{C}_{y}^{\mathcal{G}}]_{B=0}$ can be approximated with reasonable
effort from the analytical expression for the singlet yield (eq. (S14) in the
SI; implemented for graphics processing unit using CUDA). Using Monte Carlo
sampling of the matrix elements of the singlet projection operator in the
eigenbasis of the combined Hilbert space of both radicals assists in
accelerating the calculation (see SI). Utilising this we have been able to
evaluate $[\mathcal{C}_{y}^{\mathcal{G}}]_{B=0}$ of the flavin-tryptophan and
flavin-tyrosine systems for a representative number of relative orientations.

Figure 4: Compass anisotropy $\Delta_{S}$ as a measure of compass sensitivity
is plotted against the global measure of coherence
$[\mathcal{C}_{y}^{\mathcal{G}}]_{B=0}$, an operational measure that
represents the coherence contribution of the total nuclear and electronic
state to the singlet yield evaluated in the absence of a magnetic field. a
Correlation plot for the flavin-tryptophan model. b Correlation plot for
flavin-tyrosine model. Data has been coloured according to the radical
relative orientation angle $\beta$, ranging from $0^{\circ}$ (blue) to
$180^{\circ}$ (yellow), and a linear fit (red line) with associated Pearson
correlation coefficient $R$ is displayed.

In Fig. 4 we plot compass anisotropy $\Delta_{S}$ of these systems as a
function of global coherence $[\mathcal{C}_{y}^{\mathcal{G}}]_{B=0}$. For both
systems, a strong positive correlation of
$[\mathcal{C}_{y}^{\mathcal{G}}]_{B=0}$ and $\Delta_{S}$ is found, thereby
reinforcing the results for 5-6 nuclear spins, as studied previously [28] and
above. Specifically, we find a remarkable correlation for flavin-tryptophan
for which $R=+0.975$ based on 1784 randomly chosen relative orientations, in
stark contrast to the anti-correlation of $R=-0.927$ found with the electronic
coherence measure $\mu[\bar{\mathcal{C}}_{r}^{\mathcal{E},U\\!D}]$. The
results for flavin-tyrosine are more structured, but likewise give rise to a
strong overall correlation with $R=+0.825$. This result confirms the
supposition of Cai and Plenio, who proposed that global system-environment
coherence is crucial for the compass sensitivity, based on simpler model
systems [28]. Alongside our other results, this implies that the total system
cannot be reduced to consideration of electronic subsystems. Nuclear
coherences and nuclear-electronic coherences are an integral part of the
processes.

## Discussion

The radical-pair compass, by its design, relies on the coherent
interconversion of singlet and triplet electronic states and, thus, the
presence of a certain degree of singlet-triplet coherence is essential [14,
31, 16]. However, we find that the amount of electronic coherence does not
generally correlate with increased compass performance across differently
oriented systems. Instead, in both large spin systems considered, we observe a
surprising anti-correlation, between selected mean electronic coherence
measures and the reaction yield anisotropy. In contrast, global coherence
measures are found to predict compass performance more effectively than
electronic measures.

Quantum biology and coherence. Overall, our results suggest that nature,
rather than maximising and prolonging the possible electronic coherence, must
utilise an interplay of several factors on relevant timescales to increase
compass sensitivity. These include the effect of the magnetic field on
coherent dynamics, engineering of radical pair lifetimes, decay channels and
the hyperfine-driven singlet-triplet dissipation [20, 19, 16]. Complex radical
systems cannot be reduced to consideration of electronic subsystems. Nuclear
coherences and nuclear-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 measures suggest that global coherence is
realized by sacrificing electronic coherence to boost the compass fidelity.
The crucial relevance of the nuclear degrees of freedom has also become
apparent in a recent study exploring the effects of accumulated nuclear
polarisation for repeatedly re-excited radical pairs as a means to boost
anisotropic MFEs [44]. Conversely, incoherent operations on nuclear spins can
have a signiﬁcant effect on the singlet-triplet coherence [29].

Quantum advantage at short lifetimes. Non-classicality in complex spin systems
representative of flavin-tryptophan has previously been postulated based on
the comparison with semi-classical calculations [42, 36, 45]. For long
coherent lifetimes, an increase in sensitivity emerging as a sharp spike in
the fractional yield of the singlet product [17] is apparent in quantum
simulations, but absent from the semi-classical description, thereby
demonstrating a clear-cut quantum advantage. This “quantum needle”, however,
only emerges significantly for coherent lifetimes that are likely
unrealistically long, at least in view of the expected spin relaxation rates
in cryptochromes and direct animal behavioural experiments using rf-magnetic
fields to interfere with compass navigation (specifically, their frequency
dependence) [18, 19, 20]. In this regard, the correlation of compass fidelity
and the coherence measure $[\mathcal{C}_{y}^{\mathcal{G}}]_{B=0}$ as found
here is remarkable, because it suggests a manifest quantum advantage despite
the comparably short lifetime of $1\mskip 3.0mu\mu$s, for which the semi-
classical and fully quantum approaches concur.

Realistic model complexity. Advancing our understanding of non-trivial quantum
effects in biology requires the precise characterisation of putative quantum
feats as well as asserting whether this non-classicality enables robust
function. In the context of the avian compass, resourceful non-classicality
has been implicated based on the correlation of compass fidelity and coherence
measures. However, previous studies have focused on toy models, which did not
live up to the complexity expected for the actual system. Therefore, at best
the principal feasibility could be deduced, while the physiological relevance
remained opaque. Here, a robust correlation of compass fidelity and global
coherence measures is established for large spin systems and realistic
parameters, thereby corroborating the view that coherence—but not electronic
coherence—is truly a quantum resource for magnetoreception.

In summary, our results demonstrate Cai and Plenio’s claim [28], that global
coherence is a resource, in the complex system regime, whilst simultaneously
supporting the claim [32] of Jain et al. that sensitivity can be obtained
without sustained electronic coherence for some systems. The connection
between seemingly contrasting results, concerning electronic coherence as a
resource, is found when it is considered in the wider setting of several
diverse systems and complexities as presented in our study. Future studies
could steadily increase the relevant complexity of models in a tractable
manner and consider alternative models of the radical reaction [46, 47] such
as the flavin semiquinone/O${}_{2}^{\bullet-}$ radical pair, studied e.g. as a
function of the binding site of the latter. As O${}_{2}^{\bullet-}$ is devoid
of hyperfine interactions, this system is invariant to the mutual orientation
of the radicals and thus not amenable to the study design as chosen here. It
should be noted that even the large systems studied here have been
(necessarily) idealized. The asymmetry of the reactions has been neglected, by
setting $k=k_{S}=k_{T}$, and inter-radical interactions (such as electron-
electron dipolar interactions) has been ignored, both of which have been
associated with peculiar quantum feats [48, 49, 47, 46, 50, 51]. Additionally,
the open quantum system nature of the system could be analysed in further
detail [52, 53], employing numerically exact methods [54, 55, 56] for greater
accuracy and to identify if non-Markovian dynamics [57, 58, 59] are of
importance in realistic natural and artificial systems. We anticipate that the
approach and observations of this paper will aid these future endeavours
through identification of the most effective use of coherence measures, and by
paving a way to a more realistic assessment of the putatively utilitarian
character of coherence in magnetoreception. Regarding this, we recommend that
results should always be rationalised with respect to the complexity of the
system.

## Methods

Numerical implementation. We have considered radical pair systems with up to
21 nuclear spins. The ability to simulate such large spin systems relies on
the separability of the problem in the absence of inter-radical interactions,
which allows us to express the electronic density operator in terms of spin
correlation tensors (SCTs), defined for each individual radical [25, 36]. Thus
the problem of treating 21 nuclear spins for flavin/tryptophan reduces to
independently propagating spin systems comprising 12 nuclear spins for flavin,
and 9 nuclear spins for tryptophan. To assess the effect of the mutual
orientation of the two radicals in space we additionally rely on an
implementation that rotates the SCTs rather than recalculating them [25]. The
SCTs have been calculated on graphic processing units (GPU) using purpose-
designed CUDA kernels, which vastly outperforms similar CPU computations.
Taken together this provides a means to calculate electronic coherence
quantifiers and singlet recombination yields of complex radical systems
subject to arbitrary relative reorientations of the constituent radicals. We
have also devised a GPU implementation to calculate recombination yields
directly, which we used alongside Monte Carlo sampling of the expectation
values of the singlet projection operator in the eigenbasis of the Hamiltonian
to calculate the global coherence measure
$[\mathcal{C}_{y}^{\mathcal{G}}]_{B=0}$.

Additional details on the implementation, including the CUDA-kernels devised,
are provided in the Supporting Information.

## Data availability

The datasets generated and analyzed during the current study have been made
available via Open Research Exeter (ORE), the University of Exeter’s online
repository: https://doi.org/10.24378/exe.3923.

## Acknowledgments

We gladly acknowledge the use of the University of Exeter High-Performance
Computing facility. This work was supported by the UK Defence Science and
Technology Laboratory (DSTLX-1000139168), the Office of Naval Research (ONR
award number N62909-21-1-2018) and the EPSRC (grant EP/V047175/1). The robin
in Fig. 1 is extracted from the artwork “Robin Vintage Art Poster” released
under Public Domain license by Karen Arnold.

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