Quantum relative entropy as a quantifier of singlet-triplet coherence in the radical-pair mechanism of biological magnetic sensing

Radical-pair reactions pertinent to biological magnetic field sensing have been shown to be an ideal system for demonstrating the paradigm of quantum biology, the exploration of quantum coherene effects in complex biological systems. We here provide yet another fundamental connection between this biochemical spin system and quantum information science, related to the coherent spin motion driven by the magnetic interactions within these molecules. We introduce and explore a formal measure quantifying singlet-triplet coherence of radical-pairs using the concept of quantum relative entropy. The ability to quantify singlet-triplet coherence opens up a number of possibilities in the study of magnetic sensing with radical-pairs. We first use the explicit quantification of singlet-triplet coherence to affirmatively address the major premise of quantum biology, namely that quantum coherence provides an operational advantage to magnetoreception. Secondly, we use the concept of incoherent operations, which underlies the introduction of our singlet-triplet coherence measure, to show that incoherent manipulations of nuclear spins can have a dire effect on singlet-triplet coherence when the radical-pair exhibits electronic-nuclear entanglement. Finally, we study the role of magnetic interactions within the radical-pair in promoting quantum coherence, in particular unraveling a subtle effect related to exchange interactions.

Indeed, it was shown that to understand the fundamental quantum dynamics of radical-pair reactions one needs to introduce quantum meaurement theory, quantum coherence quantifiers and quantum trajectories elucidating the physics at the single molecule level [12][13][14][15]. These works also led to a new master equation describing the time evolution of the radical-pair spin dynamics [6], qualitatively and quantitatively departing from the theory attributed to Haberkorn [16], which has been traditionally used in spin chemical calculations. Moreover, quantum information concepts like violation of entropy bounds were taken advantage of to further demonstrate [17] the inadequacy of the long-standing theoretical foundations of spin chemistry [18] in a general way unaffected by the precise knowledge, or lack thereof, of molecular parameters. Most recently, quantum metrology methods were introduced to study the fundamental limits to quantum sensing of magnetic fields using radical-pair reactions and treating them as biochemical quantum magnetometers [19].
Following these developments, there have been several other approaches exploring radical-pair quantum dynamics [20][21][22], essentially concurring with the basic aforementioned findings, namely that a new fundamental theory based on quantum measurements is required to understand these spin-dependent biochemical reactions, that tools of modern quantum metrology are indeed useful to address their dynamics, and that in gen-eral, radical-pairs are an ideal system demonstrating the paradigm of quantum biology.
Yet apart from any quantitative or qualitative differences in accounting for radical-pair quantum dynamics in the various approaches being explored, it is broadly accepted that radical-pairs do exhibit quantum coherence, in particular singlet-triplet coherence defined by the spin space of the two electronic spins of the pair. The role of quantum coherence in magnetoreception has to some extent been addressed [23], but it is far from being conclusively understood. In particular, the role of singlet-triplet coherence has not been explored, since a quantitative and physically intuitive measure of singlet-triplet coherence has not been formally defined, apart from some empirical approaches [14]. There have also been discussions on whether a semiclassical treatment could replace coherent spin dynamics in radical-pair magnetoreception [24], but again, such discussions are phenomenological and of limited predictive power unless a concrete singlet-triplet coherence measure is established. This way it will be straightforward to find the classical limit of the relevant dynamics by gradually eliminating singlet-triplet coherence, while being able to exaclty quantify its presence.
Interestingly, during the last few years quantum coherence measures have been formally investigated in quantum information science in the context of resource theories [25][26][27][28][29][30][31]. In this work we introduce and formally analyze quantum relative entropy as a singlet-triplet coherence measure. We rigorously prove the properties of the introduced quantifier, and offer some intuitive examples to explain its workings.
Since the theoretical formulation of the singlet-triplet coherence quantifier involves the definition of the socalled incoherent operations, we show, as a byproduct of their definition, that a class of incoherent operations originate from operating on just the nuclear spins of the radical-pair. Having such operations out our disposal, we demonstrate that incoherently operating on just the nuclear spins can have dire consequences for the electronic singlet-triplet coherence when there is electron-nuclear entanglement. This simple and intuitively understood result opens up a number of studies on the effect of nuclear spin dynamics on the radical-pair spin dynamics.
As the main application of our singlet-triplet quantifier, we then explore the role of singlet-triplet coherence in magnetoreception. We investigate the correlation of singlet-triplet coherence with the figure of merit for the operation of radical-pair reactions as a compass. Having an explicit quantifier of singlet-triplet coherence, we can controllably suppress it while always quantifying it, and thus study the compass figure of merit as we transition from a highly coherent to highly incoherent regime.
As another application, we study the role of specific magnetic interactions in promoting such coherence, in particular the exchange interaction. We find a subtle effect alluding to optimal values of exchange couplings promoting the quantum advantage of singlet-triplet coherence.
The structure of the paper is the following. In the following subsection we motivate the need to introduce a measure of singlet-triplet cohererence in simple and intuitive terms. In Sec. II we recapitulate our previous attemps at defining such a measure, which, being empirical, had several shortcomings. In Sec. III we present the definition of a singlet-triplet coherence quantifier based on relative entropy and fully analzyze its properties at a formal level. In Sec. IV we study magnetoreception using the firmly established singlet-triplet coherence measure, and argue quantitatively in support of the main premise of quantum biology, namely that quantum coherence indeed provides an operational advantage to biological magnetic sensing with radical-pairs. We conclude with a summary and an outlook in Sec. V.

A. Motivation
When discussing radical-pair states, we refer to a Hilbert space comprising of two electron spins, one for each radical, and any number of nuclear spins residing in both radicals. For the simplest possible approach, one could even consider a fictitious radical-pair without nuclear spins. In such a case, the singlet-triplet (S-T) basis states of the two-electron system are denoted by |s = (|↑↓ − |↓↑ )/ √ 2 for the singlet, and |t j for the triplets (j = 0, ±1), where |t 1 = |↑↑ , |t 0 = (|↑↓ + |↓↑ )/ √ 2 and |t −1 = |↓↓ . To motivate this work, we can ask a few questions, the answers to which are anything but obvious: (a) Which state is "more" singlet-triplet coherent, |ψ 1 = (|s + |t 0 )/ √ 2, or |ψ 2 = (|s + |t 1 + |t 0 + |t −1 )/2 ? Clearly, both states are pure and normalized, and both involve a superposition of the singlet and some of the triplet states, but which is "more coherent" ?
Finally, regarding the magnetic interactions that directly drive or indirectly affect coherent singlet-triplet oscillations in the radical-pair state, which are at the heart of spin dynamics of this system, one might also ask: (d) which are the interactions promoting singlet-triplet coherence and why ?
The lack of a straightforward and broadly understood answer to the aforementioned questions (a)-(d), and several others one could ask along these lines, demonstrates that a formal, as well as physically intuitive decription of singlet-triplet coherence in radical pairs is indeed required, and this is exactly the purpose of this work.

II. DEFINITIONS AND PREVIOUS WORK
Radical-pair reaction dynamics are depicted in Fig.  1. A charge transfer following the photoexcitation of a donor-acceptor dyad DA (not shown) leads to the radicalpair (also called charge-separated state) D •+ A •− , where the two dots represent the two unpaired electron spins of the two radicals. The initial state of the two unpaired electrons of the radical-pair is usually a singlet, denoted by S D •+ A •− . In theory, any other initial state is equally permissible, but in practice the precursor neutral molecule is in a singlet state, preserved by the photoexcitation process.
Both D and A contain a number of magnetic nuclei (their initial state is usually fully mixed) which hyperfinecouple to the respective unpaired electron of D and A. The resulting magnetic Hamiltonian H involves all such hyperfine couplings, and extra terms accounting for the electronic Zeeman interaction with an applied magnetic field, exchange and dipolar interactions etc.
The initial electron singlet state (and for that matter the triplet-state) is not an eigenstate of H, hence the initial formation of S D •+ A •− is followed by S-T mixing, a coherent oscillation of the spin state of the electrons (and concomitantly the nuclear spins), designated by S D •+ A •− T D •+ A •− . This coherent spin motion has a finite lifetime since charge recombination, i.e. charge transfer from A back to D, terminates the reaction and leads to the formation of the neutral reaction products. There are two kinds of neutral products, singlet and triplet. Purely singlet (triplet) radical pairs would produce singlet (triplet) neutral products at a rate k S (k T ). Both rates are in principle known or measurable parameters (in general different) of the specific molecular system under consideration.
The electronic spin basis states have been presented in Sec. I A, they are the well-known four states consisting of the singlet and triplet states |s , |t −1 , |t 0 and |t 1 . When considering radical-pairs with nuclear spins, the tensor product structure will be explicitly given. For example, for the case of a single nuclear spin-1/2, a singlet electonic state and a spin-up nucleus will be denoted as |s ⊗ |⇑ .

A. Singlet and triplet projectors and basis states
The singlet and triplet projection operators are Q S = (1/4 − s 1 · s 2 ) ⊗ 1 and Q T = (3/4 + s 1 · s 2 ) ⊗ 1, with 1 being the unit operator in the nuclear spin space (the dimension of the unit matrix should be evident from the context), and s 1 and s 2 the electron spins of the two radicals. The projectors Q S and Q T are orthogonal and complete, i.e. Q S Q T = Q T Q S = 0 and Q S + Q T = 1 (for example in the last relation 1 refers to the total Hilbert space electrons+nuclei).
Using the completeness property we can multiply any given radical-pair density matrix ρ from left and right with 1 = Q S + Q T and write where ρ SS = Q S ρQ S , ρ ST = Q S ρQ T , ρ T S = Q T ρQ S and ρ T T = Q T ρQ T . We will make frequent use of the identity (1) in the following. We can already identify ρ SS + ρ T T as the S-T "incoherent part" of ρ, and ρ ST + ρ T S the S-T "coherent part" of ρ, to be formally defined and quantified later.
The density matrix, the projectors and any other operator relevant to a particular radical-pair having M nuclei with nuclear spins I 1 , I 2 , ..., I M have dimension d = 4d nuc , where 4 is the multiplicity of the two electronspin space, and d nuc = (2I 1 + 1)(2I 2 + 1)...(2I M + 1) is the dimension of the nuclear spin space.

B. Reaction super-operators
We first introduced the quantification of S-T coherence in radical-pairs in [13]. We will briefly reiterate the motivation for this introduction. The special property of radical-pair reaction dynamics not found in most open quantum systems usually considered in the quantum information literature is that the time evolution of their quantum state is not preserving the trace of the radicalpair density matrix ρ.
At any given time interval dt, the number of singlet and triplet neutral reaction products is given by dn S = k S dtTr{ρQ S } and dn T = k T dtTr{ρQ T }, respectively. This is an incoherent and irreversible process, i.e. it is evident that any S-T coherence of the precursor radical pairs, of number dn S + dn T , leading to the dn S and dn T singlet and triplet products, is irreversibly lost. The question, however, is how to update the density matrix of the surviving radical-pairs, taking into account not only the unitary evolution driven by H, but also the statedependent recombination of radical-pairs and the reduction of their trace. In other words, if at time t the radicalpair system is described by ρ t , what is ρ t+dt , given that The answer to this question, we claim, depends on "how much" S-T coherent are the radical-pairs described by ρ t , hence the need for an S-T coherence quantifier. The reader is referred to review [6] for further details.
However, this work is decoupled from this discussion on the exact form of the reaction super-operators, because we choose k S = k T = k, a choice that simplifies the reaction dynamics considerably.

C. First measure of S-T coherence is an l2-norm
Our first S-T coherence quantifier [13] was introduced in analogy with the coherence for a light field pertaining to a double-slit interferometer. Specifically, we defined [13] The definition (2) exactly aimed at quantifying the strength of the coherent part ρ ST + ρ T S . In particular, the form of the nominator in Eq. (2) was chosen because ρ ST and ρ T S are traceless due to the orthogonality of Q S and Q T , but their product has a nonzero trace. For example, taking as ρ = |ψ ψ|, with ψ = α |s + β |t 0 and any coefficients α and β, it follows from (2) that p coh = 1, i.e. any coherent superposition of |s and |t 0 is maximally coherent. Considering, as a second example, a partially coherent state like ρ = |α| 2 |s s| + |β| 2 |t 0 t 0 | + c |s t 0 | + c * |t 0 s|, with |c| 2 < |αβ| 2 , it would follow that p coh = |c| 2 /|αβ| 2 < 1. One problem of this definition is that any coherent superposition e.g. α |s + β |t 0 , whatever the coefficients α and β, is mapped into a maximum (equal to 1) coherence measure, whereas it would make intuitive sense that the more asymmetrical the superposition (e.g. the closer to one is the singlet probability) the smaller the S-T coherence should be. Yet another problem is that p coh defined in Eq. (2) is not permissible based on the formal requirements set forth at [25], because it is a so-called l 2norm, i.e. p coh scales with the square of the off-diagonal elements of ρ.

D. Second measure of S-T coherence is an l1-norm
In [14] we introduced the l 1 -norm, so-called [25] because it scales linearly with the off-diagonal elemets of the density matrix.

Shortcomings of previous definition
Our second attempt using Eq. (3) was motivated by the l 1 -norm introduced in [25], where the authors considered a general density matrix ρ and defined i,j,i =j |ρ ij | as the l 1 -norm coherence measure. This is the sum of the absolute value of all off-diagonal elements of ρ. The expression j=0,± |α s β j | derived from Eq. (3) in the case of a pure radical-pair state aims at realizing exactly such a sum of "off-diagonal" elements, now "off-diagonal" relating to the block-diagonal decomposition of ρ in the singlet and triplet subspace (see [6] for examples).
However, this expression has two shortcomings. First, the normalizing factor of 4/3 is incorrect, because the maximum value of the sum j=0,± |α s β j | is not 3/4 but √ 3/2, and occurs for |α s | = 1/ √ 2 and |β j | = 1/ √ 6. Second, the sum j=0,± |α s β j | is biased towards triplet states, i.e. it produces a higher coherence measure when more triplet states enter the superposition, for a given triplet character of the state. For example the superpo- j=−1 |t j and 1 √ 2 (|s +|t 0 ) both have an exectation value of Q S equal to 1/2, i.e. it is for both equally uncertain whether they are found in the singlet or triplet subspace upon a measurement of Q S , yet the former is maximally coherent whereas the latter is not, according to Eq. (3). However, we would intuitively expect that maximum S-T coherence would be attributed to the state having maximum quantum uncertainty in a measurement of Q S or Q T . Furthermore, the definition (3) is not readily amenable to a formal analysis of its properties.
All the aforementioned shortcomings of the previous defintions in Eq. (2) and Eq. (3) are alleviated by the quantifier defined in the following based on the quantum relative entropy.

III. FORMAL DEFINITION OF SINGLET-TRIPLET COHERENCE IN RADICAL-PAIRS BASED ON THE QUANTUM RELATIVE ENTROPY
We will now develop of formal theory of singlettriplet coherence in radical-pairs using a central concept of quantum information, the quantum relative entropy, adapted to our case. The resource theory of quantum cohence is built [25] on the notion of (i) incoherent states and (ii) incoherent operations. In [25] the authors consider a d-dimensional Hilbert space spanned by the basis states |i , with i = 1, 2, ...d, and define as incoherent states all density matrices of the form where the non-negative weights δ i sum up to unity, In our case we consider a d-dimensional Hilbert space of the radical-pair under consideration, which is spanned by d states, but we do not care about a "global" coherence, but only about coherence between the singlet and triplet subspaces, hence the following definition.

A. Definition of S-T incoherent states
Singlet-triplet incoherent radical-pairs are those for which the radical-pair density matrix ρ has the property that ρ = Q S ρQ S + Q T ρQ T ≡ ρ SS + ρ TT , or equivalently, those for which the density matrix has the property that Q S ρQ T + Q T ρQ S ≡ ρ ST + ρ TS = 0. This definition is straightforward, since as noted in Sec. II A with regard to the identity (1), any radical-pair density matrix can be written as ρ = ρ SS + ρ TT + ρ ST + ρ TS due to the completeness of the projectors Q S and Q T . Let all incoherent states (for a particular radical-pair Hilbert space) define the set I.

B. Definition of S-T incoherent operations
The authors in [25] consider a set of Kraus operators K n , with n K † n K n = 1. These are called incoherent if for all n it is K n IK † n ⊂ I. In our case, the two projectors Q S and Q T qualify as a set of incoherent operations, since Q S Q S + Q T Q T = Q S + Q T = 1, and for any ρ ⊂ I, i.e. ρ = ρ SS + ρ TT it is Q S ρQ S = ρ SS ⊂ I and Q T ρQ T = ρ TT ⊂ I. The pair Q S and Q T are incoherent operators of central significance for discussing S-T coherence, but by no means are they the only ones. For example, any operators acting only on the nuclear spins are S-T incoherent operators. Consider K n = 1 ⊗ k n , where now 1 is the unit matrix in the electronic spin subspace (4-dimensional), and k n are tracepreserving Kraus operators acting in the nuclear spin subspace and satisfying n k † n k n = 1 (here 1 refers to the nuclear spin subspace). Take a density matrix ρ ⊂ I.
It can be readily seen that Q S and Q T commute with K n , therefore That is, we have shown that for any ρ ⊂ I it will be K n ρK † n ⊂ I, thus K n are incoherent operators.

C. Definition of S-T coherence quantifier based on relative entropy
For any radical-pair density matrix ρ we define the singlet-triplet coherence quantifier as where S r is the von-Neuman entropy of the density matrix r. Since radical-pair reactions are non-trace preserving, the trace of the radical-pair state ρ is in general 0 ≤ Tr{ρ} ≤ 1. For the definition (4) to work, we first need to normalize the radical-pair state ρ with Tr{ρ} (see Appendix of [17] for a relevant discussion). In the following we will always imply that whenever we calculate C ρ we do so for radical-pair density matrices that have been appropriately normalized to have unit trace. At first sight, the quantum relative entropy is not present in the definition (4). However, equation (4) readily follows by first defining, along the lines of [25], where now S ρ||δ ≡ Tr{ρ log ρ}−Tr{ρ log δ} is the quantum relative entropy of the radical-pair density matrices ρ and δ. Indeed, by denotinĝ and for δ ⊂ I it is [25] S ρ||δ = S ρ||δ + S ρ − S ρ . However, since the quantum relative entropy is always positive or zero, S ρ||δ ≥ 0, it is seen that min δ⊂I S ρ||δ = 0, the minimum obviously taking place for δ =ρ, since it is known that for any density matrix r it is S r||r = 0. Hence C ρ = S ρ||ρ = S ρ − S ρ .

D. Properties of C ρ
We here present the basic properties of the definition (4).

Value of C ρ for incoherent states
For all ρ ⊂ I it is C ρ = 0.
Proof This follows trivially from the definition (4), since if r ⊂ I it is r = Q S rQ S + Q T rQ T , hence C r = S r − S r = 0. According to [25], all incorerent states r ⊂ I should have C r = 0, so the measure (4) satisfies this basic criterion for an acceptable coherence quantifier.

Minimum value of C ρ
For any ρ it is C ρ ≥ 0.
Proof The proof follows from the fact that under a nonselective (or blind) measurement the entropy does not decrease. This can be found in pp. 75 of [32] and pp.92 of [33]. In our case, defining a measurement by the Kraus operators Q S and Q T , which satisfy , the non-selective post-measurement state is given by Q S ρQ S + Q T ρQ T . Indeed, the measurement results are q s = 1 or q s = 0, taking place with probabilities p S = Tr{ρQ S } and p T = Tr{ρQ T }, with the selective post-measurement states being ρ S = Q S ρQ S /p S and ρ T = Q T ρQ T /p T , respectively.
The non-selective post-measurement state is p S ρ S + p T ρ T which indeed equals Q S ρQ S + Q T ρQ T . Since, by the theorem found in [32,33] it is S Q S ρQ S + Q T ρQ T ≥ S ρ , it follows that indeed C ρ ≥ 0.

Maximum value of C ρ
For any ρ it is C ρ ≤ 1.
Proof First, the quantum relative entropy is jointly convex in both of its arguments [34], i.e.
for any λ ∈ [0, 1]. Applying this property for ρ 1 = ρ 2 = ρ, σ 1 = ρ S , σ 2 = ρ T , λ = p S , and given thatρ = p S ρ S + p T ρ T , it follows that Now, the interpretation of the quantum relative entropy S ρ||σ is [35] the extent to which one can distinguish two different states ρ and σ, in particular by a series of quantum measurements and their resulting statistics. Let us first consider S ρ||ρ S . This reflects the extent to which by doing some measurement on ρ we can use the measurement statistics to distinguish ρ from ρ S . We can choose as measurement the measurement of Q S , the result of which can be either 0 or 1. Clearly if the state we were measuring was ρ S = Q S ρQ S /Tr{ρQ S }, we would only obtain 1 for every measurement performed in N identically prepared systems. But if the state of each of those identical copies is ρ, measuring Q S we will obtain 1 only some of the times. The probability to obtain 1 in all N such measurements will clearly be p N S . This is the probability that ρ S would "pass" our test and we would confuse the actual state ρ with ρ S . But it is known [35] that for an optimal measurement using N identical copies of our system, the probability that we will mistakenly confuse ρ for ρ S is 2 −N S ρ||ρS . Our choice of measurement is not necessarily optimal, hence Finally, using the inequality (8) we get where is the Shannon entropy of the pair of probabilities {p S , p T }. Since this Shannon entropy has maximum 1 (when the logarithms are calculated with base 2), the maximum occuring for p S = p T = 1/2, we finally show that the maximum of C ρ is 1.
Proof The most general pure state of a radical-pair can be written as where |χ s and |χ j are nuclear spin states "living" in a nuclear spin space of dimension d nuc (defined in Sec. II A), dependent on the number and spin of nuclear spins of the particular radical-pair. Setting ρ = |ψ ψ|, in order to calculate C ρ , we need to calculate the entropies S ρ and S ρ . The former is zero since ρ is a pure state. To calculate the latter, we writê where for brevity we denoted by |φ T = 1 j=−1 β j |t j ⊗ |χ j the triplet-subspace component of the most general pure state |ψ . The matrixρ clearly has a block-diagonal form, one block being the singlet and the other the triplet subspace. To calculate S ρ we need to find the eigenvalues ofρ. They are easily obtained by finding the eigenvectors and corresponding eigenvalues ofρ by construction.
For example, the state |s ⊗ |χ s is an eigenvector of the singlet block diagonal ofρ with eigenvalue |α s | 2 = Tr{ρQ S } = p S . Remaining in this singlet subspace block-diagonal, we can span the nuclear spin space with d nuc orthogonal basis states, one being |χ s itself. Hence the other d nuc − 1 eigenvalues of the singlet block-diagonal ofρ are zero. Similarly, the unnormalized state |φ T is an eigenstate of |φ T φ T | with eigenvalue φ T |φ T = |β −1 | 2 + |β 0 | 2 + |β 1 | 2 = Tr{ρQ T } = p T . We can clearly span this triplet subspace with 3d nuc orthogonal basis states, one of which is |φ T itself. Hence the other eigenvalues of |φ T φ T | are zero. Thus, the stateρ has two nonzero eigenvalues, p S = Tr{ρQ S } and p T = Tr{ρQ T } = 1 − p S , hence the coherence measure of the most general pure radical-pair state (10) is exactly equal to H[p S , p T ].

Connection with quantum uncertainty
For the general pure state |ψ of Equation (10) it can be easily seen that the quantum uncertainty of Q S , given by ∆q s Evidently, ∆q s is maximized for |α s | = 1/ √ 2, i.e. the maximally coherent pure states |ψ maxC also have maximum uncertainty in their singlet (or equivalently triplet) character. This is intuitively satisfactory, since thinking at the level of a simple qubit, we intuitively relate the maximum coherence state (|0 + |1 )/ √ 2 with the fact that this state is maximally uncertain regarding a measurement in the computational basis {|0 , |1 }.

Additional comments
Finally, all other conditions of an acceptable measure of coherence defined in [25], like (i) monotonicity under incoherent completely positive and trace preserving maps, (ii) monotonicity under selective measurements on average and (iii) convexity, are automatically satisfied as has been shown in for the relative entropy measure defined therein.

Fictitious radical-pair with no nuclear spins
We will first consider the simplest system, a fictitious radical-pair with no nuclear spins, hence a fourdimensional Hilbert space spanned by |s and |t j , with j = 0, ±1. This example can be treated analytically and illustrate the properties of singlet-triplet coherence in a transparent way. In fact, high magnetic fields, at which the magnetic Hamiltonian is dominated by the electronic Zeeman terms, readily lead to this approximation and in such cases singlet-triplet mixing is driven by a difference in the g-factor for the electronic spins in the two radicals [36]. Explicitly, consider a Hamiltonian H = ω 1 s 1z + ω 2 s 2z , where ω 1 and ω 2 are the Larmor frequencies of the electrons in the two radicals, taken to be different. If the initial state is |ψ 0 = |s it is easily seen that |ψ t = cos Ωt 2 |s − sin Ωt 2 |t 0 , where Ω = ω 1 − ω 2 . Thus the singlet and triplet probabilities are p S = cos 2 Ωt 2 and p T = sin 2 Ωt 2 , respectively. The state |ψ t is pure, hence it has zero entropy, S(ρ t ) = 0, where we set ρ t = |ψ t ψ t |. The incoherent stateρ t = Q S ρQ S + Q T ρQ T isρ t = p S |s s| + p T |t 0 t 0 |. It readily follows that the eigenvalues ofρ t are p S and p T , hence S ρ t = −p S log p S − p T log p T . Thus the coherence measure for |ψ t is C |ψ t ψ t | = S ρ t −S ρ t = −p S log p S − p T log p T . In Fig. 2a we plot the time evolution of the singlet probability p S = Tr{ρ t Q S } and the S-T coherence measure C ρ t . Evidently, C |ψ t ψ t | is zero when |ψ t is a pure singlet or a pure triplet, and reaches its maximum value of 1 in between the maxima of p S , i.e. at those times where we have the most uncertain coherent superposition of |s and |t 0 , of the form 1 √ 2 (|s ± |t 0 ). We can now introduce a singlet-triplet dephasing through the operation ρ → ρ − K d dt(Q S ρQ T + Q T ρQ S ), i.e. by removing from ρ its coherent part ρ ST + ρ TS at a rate K d (we will elaborate more about this in the following section). In the presence of such an S-T dephasing mechanism, a pure initial state necessarily evolves into a mixed state, which in general satisfies the master . This is easy to solve analytically in the considered example, since the problem is essentially reduced to a two dimensional system spanned by |s and |t 0 . For K d < 2|Ω| the off-diagonal density matrix elements are oscillatory and decay exponentially at a rate K d /2. An analytic expression can be obtained for the time-evolved density matrix ρ t , but it is a bit cumbersome. For K d 2|Ω| an excellent approximation is ρ t = 1 2 (1 + e −K d t/2 cos Ωt) |s s| + The singlet probability is now Tr{ρ t Q S } = 1 4 (1 + e −K d t/2 cos Ωt) 2 . The eigenvalues of ρ t andρ t are e 1 = 1 2 (1 − e −K d t/2 ), e 2 = 1 2 (1 + e −K d t/2 ) andê 1 = 1 2 (1 − e −K d t/2 cos Ωt),ê 2 = 1 2 (1 + e −K d t/2 cos Ωt), respectively. Thus we can readily calculate the entropies S ρ t and S ρ t , and from them C ρ t = S ρ t − S ρ t = −ê 1 logê 1 −ê 2 logê 2 + e 1 log e 1 + e 2 log e 2 . These analytic results for Tr{ρ t Q S } and C ρ t are now shown in Fig. 2b.

Incoherent operations on nuclear spins
The phenomenological wealth of coherence phenomena increases dramatically by considering realistic radicalpairs involving one ore more nuclear spins. In particular, plots similar to Fig. 2 can also be produced by introducing just one nuclear spin and calculating the evolution of S-T coherence in various scenarios. However, here we will explicitly mention as a second example a more subtle effect having to do with the fact that the pair {Q S , Q T } are not the only incoherent operations, as mentioned in Sec. III B. As shown in the previous example, singlet-triplet dephasing and concomittant decrease in the S-T measure C can be readily induced by performing a measurement of the "singlet character" of the radical-pair density matrix. This measurement is described by the Kraus operators K 1 = Q S and K 2 = Q T and leads to the previously mentioned master equation including the S-T dephasing term proportional to K d . But with the presence of nuclear spins, there is yet another mechanism, irrelevant to the electronic spins, by which S-T dephasing can be produced. This can happen when there is electronic-nuclear entanglement.

IV. SINGLET-TRIPLET COHERENCE AS A RESOURCE FOR MAGNETORECEPTION
Having defined a measure for S-T coherence, we will now explore whether S-T coherence is a resource for biological magnetic sensing, in particular for radical-pair magnetoreception. That is, we will focus on the compass aspect of the radical-pair mechanism, and attempt to figure out the role S-T coherence plays in the workings of precise heading estimation. As we will show, this is not an innocuous question, and care should be exercised in claiming whether or not S-T coherence provides an operational advantage to the compass.

A. Some introductory comments
To be explicit, it is known that an anisotropic hyperfine coupling can render the radical-pair reaction yields dependent on the angle φ of the external magnetic field with respect to a molecule-fixed coordinate frame. As is usually the case, let us consider a single-nuclear-spin radical pair, with Hamiltonian H = s 1 ·A·I+ω cos φ(s 1x +s 2x )+ω sin φ(s 1y +s 2y ), (12) where A is the hyperfine tensor coupling the electron spin s 1 of one radical with the single nuclear spin I of that radical, and ω the external magnetic field lying on the x − y plane and producing the Zeeman terms of the two elecronics spins. Both A and ω are in units of frequency, and as usual we omit the nuclear Zeeman term. We will now calculate the singlet reaction yield as a function of the angle φ, always starting at t = 0 with a singlet state for the electrons and a fully mixed nuclear spin state, ρ 0 = Q S /Tr{Q S }. As mentioned in Sec. II B, we assume that k S = k T = k, in which case the quantum dynamics of the radical-pair reaction are simplified, and the differences between our master equation and Haberkorn's are less exacerbated. In particular, when k S = k T = k it is dρ/dt = e −kt R, with dR/dt = −i[H, R] − k(Q S RQ T + Q T RQ S ) according to our theory, whereas dR/dt = −i[H, R] according to Haberkorn's, i.e. we have an additional dephasing term −k(Q S RQ T + Q T RQ S ) inherent in our description of the dynamics. In the following we will anyhow use a significantly stronger dephasing term of the form −K d (Q S RQ T + Q T RQ S ), with K d k, hence it really does not matter for this discussion which of the two master equations we use [37].
In any case, we will now calculate the singlet reaction yield, Y S (φ) = ∞ 0 dtkTr{ρ t Q S }, and plot it is a function of φ in order to define the figure of merit for the compass function of the reaction. We explicitly include the time dependence of the density matrix to remind the reader that ρ t is the density matrix at time t, evolved by the master equation starting from ρ 0 , and the yield Y S depends on φ since the Hamiltonian H, affecting the evolution of ρ t , depends on φ.
An example of such a φ-dependence of Y S is shown in Fig. 3. What we define as figure of merit is the maximum slope of the function Y S (φ), because it is this slope that determines the useful compass "signal", i.e. the change of the reaction yield resulting from a change of heading φ around φ 0 . In the following we will quantify this figure of merit with the quantiy δY S = max φ0 |Y S (φ 0 + )−Y S (φ 0 − )|.

B. Correlation of δYS and C
We will now explore the connection between singlettriplet coherence as quantified by C and the figure of merit of the compass, δY S , as defined before. In particular, since C depends on the time-dependent density matrix ρ t , we define a mean value of C ρ t along the whole reaction as C = ∞ 0 dtke −kt C ρ t . We remind the reader that in order to calculate C ρ t using the definition (4), we always have to first normalize ρ t by Tr{ρ t } (since this trace changes with time due to the reaction), and then calculate the entropies in (4).
We use the Hamiltonian of Eq. (12), and for completeness we add an exchange term of the form −Js 1 · s 2 . We use a diagonal hyperfine tensor, randomizing all three diagonal elements A jj , with j = x, y, x. We also randomize the exchange coupling J. For each set of parameters A xx , A yy , A zz and J we calculate δY S and C. Additionally, we calculate the mean values along the reaction of the singlet and triplet expectation values, Q S t = Tr{ρ t Q S } and Q T t = Tr{ρ t Q T }, and call themq S = Now, the density matrix evolution is ρ t = e −kt R, where the density matrix R satisfies the master equation dR/dt = −i[H, R] − K d (Q S RQ T + Q T RQ S ). We repeat the aforementioned calculations for four different values of the dephasing rate K d , in particular for K d = 0, K d = k, K d = 5k and K d = 10k. As mentioned previously, we vary K d appreciably in order to explore the effect of suppressing S-T coherence.
The main results of this simulation, making a clear case that S-T coherence is indeed a resource for magnetoreception, are shown in Fig. 4. We will make a number of observations, and then provide their interpretation in the following subsection.
(1) In Fig. 4a we show for each value of the dephasing rate K d the distribution of 5000 pairs of δY S andC. We first note that for increasing K d , the distribution moves to smaller δY S and smallerC. This is more evident in Fig.  4b, where we plot the mean value of these two quantities over the sample of 5000 points, for each value of K d .
(2) Irrespective of K d , there seems to be an appreciable correlation coefficient (around 0.3) between δY S and C, as shown in Fig. 4c. Moreover, the correlation between the sample means δY S and C , presented in Fig. 4b, is much larger, and has the value 0.95.
(3) Finally, in Figs. 4(d1)-4(d4) we plot the distribution of the average values along the reaction of the singlet and triplet character of the radical-pair state, quantified bȳ q S andq T defined above. Also shown are the mean values over the sample of 5000 points of these distributions.

C. Interpretation
We will now interpret the aforementioned observations. The first question to ask is, does singlet-triplet coherence provide a quantum advantage to the operation of the compass? The answer should clearly be affirmative, because of three facts: (a) Due to the correlation between δY S andC at a specific value of K d , large values of S-T coherence C are on average connected with large figures of merit δY S for the compass. (b) Strong S-T dephasing produced by increasing K d leads to small values ofC and small values of δY S . (c) At the same time the average singlet and triplet populations, as seen in Figs. 4(d1)-4(d4), are not affected by the increasing K d . In other words, we have a process by which an initially singlet radical-pair state is coherently transformed into a triplet, and back and forth, but it is the underlying coherence and not the population exchange that seems to be directly connected with the figure of merit δY S . It should be clear that we were able to arrive at these conclusions because we have an explicit quantifier of S-T coherence.
To be precise, however, we should limit the affirmative answer to this statement: what the previous findings demonstrate is that singlet-triplet coherence allows for a quantum advantage not present in conditions of singlettriplet incoherence. Whether such advantage is actually realized in nature, or in other words, whether the actual molecular parameters of the naturally occuring compass are such that the compass operating point is among those exhibiting large S-T coherence is a different question. However, this is not of fundamental interest for quantum biology. In contrast, what is of interest is what is in principle possible with such biochemical spin-dependent reactions. If it is found that they naturally work in a regime of large S-T coherence it would be quite an exciting finding, but even if this is not the case, knowing what is in principle possible would allow for the design of an artificial compass (or magnetometer in general) taking advantage of quantum coherence effects.
Put differently, since the correlation coefficient of δY S with C is at the level of 0.3 (see Fig. 4c), the mean of δY S over the 5000 points is seen to drop with the decreasing mean of C (see Fig. 4b), but not excessively, i.e. by about 60% from K d = 0 up to K d = 10k. In contrast, considering the subset of highest δY S for each K d , the drop with decreasing C is much more significant (order of magnitude). However, as previously mentioned, it is unknown if the natural compass has evolved exploring those molecular parameters placing it at the high-δY S and high-C regime (the upper part of the stripes in Fig.  4a). These points will be further elaborated upon in the following subsection regarding the role of exchange interaction. (12) including an exchange term. Each point results from a random set of parameters 0 ≤ Axx, Ayy, Azz ≤ 10k, −10 ≤ J/k ≤ 10 and ω/k = 1. The dephasing rate K d was given four different values, K d = 0, K d = k, K d = 5k and K d = 10k, with 5000 points for each value of K d . Initial state was always a singlet with mixed nuclear spin, ρ0 = QS/Tr{QS}. (b) Mean value of δYS versus mean value of C over all 5000 points for each value of K d . (c) Correlation coeffecient between δYS and C for each value of K d . (d1)-(d4) Distribution of average value along the reaction,qS andqT, of the singlet and triplet character of the radical-pair state, respectively, for each value of K d . Also shown is the mean of each distribution (black lines). The distributions are seen to become more narrow at high K d . This is because the dephasing operation is equivalent to a quantum measurement with Kraus operators QS and QT, the narrowing reflecting the measurement-induced localization of the radical-pair's state.

D. The role of the exchange interaction
The exchange interaction is known to play a subtle role in magnetoreception [38,39]. To explore the role of exchange interactions, we included them in the Hamiltonian, with an exchange coupling in the interval −10k ≤ J ≤ 10k. Now we split this interval into 5 subintervals of width ∆J = 4k, and study the correlation of the figure of merit δY S with the S-T coherenceC. The result is plotted in Fig. 5a for the case K d = k, i.e the case where the dynamics are described by our master equation. In the same figure (right y-axis) we also plot the mean vaule, δY S , of δY S for those same subintervals of J.
The behavior of δY S with J is known since the work of [38], i.e. it is already known that large values of J supress the figure of merit ∆Y S of the compass, as is evident in Fig. 5a. We here observe two additional, counterintuitive effects. Namely, (i) for small values of J the correlation between singlet-triplet coherenceC and the figure of merit δY S is significantly suprressed, to recover for large values of |J|. (ii) Moreover, there is also an asymmetry in Corr{δY S ,C} between J > 0 and J < 0.
The interpretation of both of these observations is rather challenging, and we will at the moment only offer an educated guess. It is conceivable that they originate from the positions of level crossings in the eigenvalues of the magnetic Hamiltonian, in particular from the complex interplay of energy shifts between the singlet and triplet states produced by the exchange interaction on the one hand, and on the other hand Zeeman shifts of the triplet states.
However, there is a subtle conjecture to be made, resting on two points. Firstly, it is again noted that the absolute value of the figure of merit δY S is small for large |J| (right y-axis of Fig. 5). Based on the reasoning in [38], it is expected that |J| is large. In conjunction with the behavior of Corr{∆Y S ,C}, seen in the left y-axis of Fig. 5, it follows that if indeed |J| is large, then the correlation between δY S andC will be large, albeit at low absolute values of δY S .
Secondly, we concluded before that singlet-triplet coherence is indeed a resource, because increasing the dephasing rate K d led to a smaller S-T coherence and at the same time a smaller figure of merit. But one could argue that the (natural) compass operates at a particular K d , i.e. it is perhaps hard to imagine that the compass evolved "searching" for the optimal K d . On the other hand, it is conceivable that the compass evolved searching for the optimum distance between the radicals (which sets the value of J), and the optimum nuclear spin arrangement (setting the optimum hyperfine couplings). In that case, and assuming that the inter-radical distance is such that J is large, we can still make the case that S-T coherence is a resource because of the large correlation observed between δY S andC at large |J|.
That leaves one more possibility, the case where |J| is small. In Fig. 4b we plot the correlation between δY S andC for the interval |J|/k < 2, and for all four values of K d . As mentioned before, this correlation is small for K d = 1, and also for K d = 0. But the correlation recovers for larger valus of K d . Based on this, we anticipate that there is another regime of the reaction dynamics, which we have not addressed in this work, and where the correlation could be significant even for small |J|. Namely, we here considered equal recombination rates k S = k T = k, in order to simplify reaction dynamics and decouple from the ongoing discussion on the form of the reaction super-operators. However, the regime k T k S is also interesting, and based on Fig. 4b it is conceivable that in this regime there is a large correlation between figure of merit and S-T coherence for all values of the exchange coupling J. The study of this possibility will be undertaken elsewhere.

V. CONCLUSIONS
Quantum coherence is a fundamental resource for modern quantum technology. Its formal quantification has been established in recent years, in fact along the lines used to quantify entanglement, which came before. One quantifier of quantum coherence is the quantum relative entropy between the density matrix describing the quantum state of the system under consideration and its diagonal version.
We have here adapted this quantifier to radical-pairs, defining singlet-triplet coherence as the relative entropy between the radical-pair density matrix and its blockdiagonal version in the singlet-triplet subspaces. We have then established the properties of this singlet-triplet coherence quantifier at a formal level. Having an explicit quantifier of singlet-triplet coherence, one can study the fundamental properties of biological magnetic sensing in various regimes, for example in regimes where the relevant spin states are highly coherent or highly incoherent.
By doing so, we have shown that singlet-triplet coherence is indeed a quantum resource for magnetoreception, since (i) it is highly correlated with the figure of merit of the radical-pair compass, (ii) both singlet-triplet coherence and the figure of merit decrease significantly in the presence of singlet-triplet dephasing, and (iii) the singlet/triplet populations remain on average unaffected by such dephasing. Thus, due to the observations (i)-(iii) we conclude that it is not the singlet-triplet population exchange but the underlying singlet-triplet coherence that promotes precise heading of the radical-pair compass. Finally we explored the subtle role played by exchange interactions in promoting the correlation between singlettriplet coherence and the figure of merit of the chemical compass. Along the same lines one could explore the role of other interactions entering the Hamiltonian, in particular when more nuclear spins are included.
Last but not least, while defining the incoherent operations needed in the formulation of singlet-triplet coherence quantification, we gave an example where incoherent operations on nuclear spins only can have a significant effect on the singlet-triplet coherence, which is of electronic nature. This can happen when nuclear spins and electrons in the radical-pair are entangled, which is in general the case. This observation opens up a promis-ing direction of studying the effects of nuclear spin dynamics, e.g. the interaction with the environment of the radical-pair's nuclear spins, and their consequences on radical-pair spin dynamics.