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![(a) Magnetic sensitivity of the singlet reaction yield, B = |dY S /dB|, and (b) uncertainty δB in the estimation of the magnetic field, normalized by the absolute quantum limit δB F. The blue dashed line reproduces the result of the reaction control scheme of [45], while the red solid line is the result of this work. Our reaction control scheme approaches δB F within a factor of 2. (c) The minimum of the red solid line in (b) is plotted as a function of the exchange coupling J. For J = 0.65A = 229k, we obtain δB min = 2δB F. But nearby values of J are induced by molecular vibrations; hence averaging trace (c) leads to the realistic uncertainty 2.2 times away from δB F .](profile/Kyriacos-Vitalis/publication/311066540/figure/fig1/AS:476904860983297@1490714387359/a-Magnetic-sensitivity-of-the-singlet-reaction-yield-B-dY-S-dB-and-b.png)
(a) Magnetic sensitivity of the singlet reaction yield, B = |dY S /dB|, and (b) uncertainty δB in the estimation of the magnetic field, normalized by the absolute quantum limit δB F. The blue dashed line reproduces the result of the reaction control scheme of [45], while the red solid line is the result of this work. Our reaction control scheme approaches δB F within a factor of 2. (c) The minimum of the red solid line in (b) is plotted as a function of the exchange coupling J. For J = 0.65A = 229k, we obtain δB min = 2δB F. But nearby values of J are induced by molecular vibrations; hence averaging trace (c) leads to the realistic uncertainty 2.2 times away from δB F .
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Radical-ion pairs and their reactions have triggered the study of quantum effects in biological systems. This is because they exhibit a number of effects best understood within quantum information science, and at the same time are central in understanding the avian magnetic compass and the spin transport dynamics in photosynthetic reaction centers....
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Citations
... Qualitatively, the closer toh the numerical value of the ER is for any given sensor, the more "quantum" can the sensor be characterized. This offers an alternative perspective in the study of quantum biology of magnetoreception [87][88][89][90][91][92][93][94][95][96]. This discussion is not about a choice between "black" or "white," since "quantumness" can now be quantified, and spans a continuum. ...
... But in any case, with N 0 cry radical-pairs collectively producing the magnetic sensing response, the sensor volume scales as N 0 cry , meaning that the magnetic sensitivity δB scales as 1/ N 0 cry . This is the standard statistical gain over many independent repetitions of a measurement [129], usually referred to as standard quantum limit (see also Ref. [95]). ...
A large number of magnetic sensors, like superconducting quantum interference devices, optical pumping, and nitrogen vacancy magnetometers, were shown to satisfy the energy resolution limit. This limit states that the magnetic sensitivity of the sensor, when translated into a product of energy with time, is bounded below by Planck's constant, ℏ . This bound implies a fundamental limitation as to what can be achieved in magnetic sensing. Here we explore biological magnetometers, in particular three magnetoreception mechanisms thought to underly animals' geomagnetic field sensing: the radical-pair, the magnetite, and the MagR mechanism. We address the question of how close these mechanisms approach the energy resolution limit. At the quantitative level, the utility of the energy resolution limit is that it informs the workings of magnetic sensing in model-independent ways and thus can provide subtle consistency checks for theoretical models and estimated or measured parameter values, particularly needed in complex biological systems. At the qualitative level, the closer the energy resolution is to ℏ , the more “quantum” is the sensor. This offers an alternative route towards understanding the quantum biology of magnetoreception. It also quantifies the room for improvement, illuminating what nature has achieved, and stimulating the engineering of biomimetic sensors exceeding nature's magnetic sensing performance.
Published by the American Physical Society 2025
... Qualitatively, the closer toh the numerical value of the ER is for any given sensor, the more "quantum" can the sensor be characterized. This offers an alternative perspective in the study of quantum biology of magnetoreception [87][88][89][90][91][92][93][94][95][96]. This discussion is not about a choice between "black" or "white", since "quantumness" can now be quantified, and spans a continuum. ...
... Qualitatively, the closer to ℏ the numerical value of the ER is for any given sensor, the more "quantum" can the sensor be characterized. This offers an alternative perspective in the study of quantum biology of magnetoreception [87][88][89][90][91][92][93][94][95][96]. This discussion is not about a choice between "black" or "white", since "quantumness" can now be quantified, and spans a continuum. ...
A large number of magnetic sensors, like superconducting quantum interference devices, optical pumping and nitrogen vacancy magnetometers, were shown to satisfy the energy resolution limit. This limit states that the magnetic sensitivity of the sensor, when translated into a product of energy with time, is bounded below by Planck’s constant, ħ. This bound implies a fundamental limitation as to what can be achieved in magnetic sensing. Here we explore biological magnetometers, in particular three magnetoreception mechanisms thought to underly animals’ geomagnetic field sensing: the radical-pair, the magnetite and the MagR mechanism. We address the question of how close these mechanisms approach the energy resolution limit. At the quantitative level, the utility of the energy resolution limit is that it informs the workings of magnetic sensing in model-independent ways, and thus can provide subtle consistency checks for theoretical models and estimated or measured parameter values, particularly needed in complex biological systems. At the qualitative level, the closer the energy resolution is to ħ, the more “quantum” is the sensor. This offers an alternative route towards understanding the quantum biology of magnetoreception. It also quantifies the room for improvement, illuminating what Nature has achieved, and stimulating the engineering of biomimetic sensors exceeding Nature’s magnetic sensing performance.
... Information theoretic approaches have proved fruitful in investigating features of radical-pairs such as the fundamental limits of sensitivity under dim light relevant for nocturnal migratory species [52], as well as providing insight into other quantum phenomena such as quantum coherence [27,[53][54][55][56][57][58][59], where we have previously demonstrated the importance of treatments comprising many nuclear spins inherent to biological systems [60,61]. The QFI has previously been utilised to interrogate the fundamental quantum limits of magnetic field sensitivity of radical-pairs in model studies of sensing the magnitude [62] and direction of the field [63]. Whilst these studies have provided foundational insight into the precision capabilities of a radical-pair compass, they are restricted to a toy model comprised of a single nuclear spin and neglected inter-radical interactions and asymmetric reaction kinetics. ...
... Traditionally, the change of this flux with the direction of the magnetic field is considered as decisive in determining the compass sensitivity, whereby the maximal change of the singlet recombination quantum yield, Φ S = F r /k 0 c, is a widely applied fidelity measure. Though some investigations have also considered the case of access to time-resolved measurements of sensitivity [62], we choose to employ the steady-state picture as it more closely corresponds to the physiological conditions of magnetoreception under constant light excitation and allows a formal rationalisation of the form of the steady-state, to which the quantum parameter estimation approaches are applied. By calculating the average singlet yield over all directions of the magnetic field B(ϕ, θ) via ...
... In appendix B we further demonstrate that ∆ 2 θ saturates the Cramér-Rao bound set by F θ with measurement elements Π n ∈ {P S ,P T }, whereP S and P T = 1 −P S are the singlet and triplet projection operators, respectively. Note that [62] has suggested a Poisson binomial distribution rather than a binomial distribution from an analysis of the time-dependence of the reaction yield. However, this result appears to imply that time-dependent recombination statistics are accessible, which we do not assume is possible here. ...
Quantum sensing enables the ultimate precision attainable in parameter estimation. Circumstantial evidence suggests that certain organisms, most notably migratory songbirds, also harness quantum-enhanced magnetic field sensing via a radical-pair-based chemical compass for the precise detection of the weak geomagnetic field. However, what underpins the acuity of such a compass operating in a noisy biological setting, at physiological temperatures, remains an open question. Here, we address the fundamental limits of inferring geomagnetic field directions from radical-pair spin dynamics. Specifically, we compare the compass precision, as derived from the directional dependence of the radical-pair recombination yield, to the ultimate precision potentially realisable by a quantum measurement on the spin system under steady-state conditions. To this end, we probe the quantum Fisher information and associated Cramér-Rao bound in spin models of realistic complexity, accounting for complex inter-radical interactions, a multitude of hyperfine couplings, and asymmetric recombination kinetics, as characteristic for the magnetosensory protein cryptochrome. We compare several models implicated in cryptochrome magnetoreception and unveil their optimality through the precision of measurements ostensibly accessible to nature. Overall, the comparison provides insight into processes honed by nature to realise optimality whilst constrained to operating with mere reaction yields. Generally, the inference of compass orientation from recombination yields approaches optimality in the limits of complexity, yet levels off short of the theoretical optimal precision bounds by up to one or two orders of magnitude, thus underscoring the potential for improving on design principles inherent to natural systems.
... Today, OQC has developed into a mature discipline that is central to modern quantum technologies associated with the second quantum revolution [5], with numerous applications across quantum information processing and quantum metrology [1]. Following on from an earlier proposal for controlling the dynamics of radical-pair reactions through modulating the exchange interaction via an optically switchable bridging group [6,7], a recent suggestion extends coherent control to radical-pair reactions mediated through radio-frequency (rf) magnetic fields [8], thereby putting a renewed spotlight on the quantum control of * d.r.kattnig@exeter.ac.uk spin-chemical reactions. In the present work, we go on to demonstrate the implementation of a gradient-ascent pulse engineering-(GRAPE) inspired approach, made computationally feasible via novel algorithmic modifications, toward realizing open-loop control of singlet recombination yields for radical-pair dynamics in weak magnetic fields. ...
... This could translate to further applications [56,[61][62][63] in both quantum metrology and quantum information processing. In particular, potential engineering principles derived from gleaning deeper physical insights gained from artificially driving radical-pair reactions could conceivably inspire a new generation of quantum enhanced sensing technologies for magnetometry boasting heightened sensitivity and robustness against noise [7,61,[63][64][65][66][67]. But to realize this goal, and future applications of optimal quantum control in a potentially biological context, efficient approaches to control design are vital. ...
We realize arbitrary-wave-form-based control of spin-selective recombination reactions of radical pairs in the low-magnetic-field regime. To this end, we extend the gradient-ascent pulse engineering (GRAPE) paradigm to allow for optimizing reaction yields. This overcomes drawbacks of previously suggested time-local optimization approaches for the reaction control of radical pairs, which were limited to high biasing fields. We demonstrate how efficient time-global optimization of the recombination yields can be realized by gradient-based methods augmented by time blocking, sparse sampling of the yield, and evaluation of the central single time-step propagators and their Fréchet derivatives using iterated Trotter-Suzuki splittings. Results are shown for both a toy model, previously used to demonstrate coherent control of radical-pair reactions in the simpler high-field scenario and, furthermore, for a realistic exciplex-forming donor-acceptor system comprising 16 nuclear spins. This raises prospects for the spin control of actual radical-pair systems in ambient magnetic fields, by suppressing or boosting radical reaction yields using purpose-specific radio-frequency wave forms, paving the way for reaction-yield-dependent quantum magnetometry and potentially applications of quantum control to biochemical radical-pair reactions. We demonstrate the latter aspect for two radical pairs implicated in quantum biology.
Published by the American Physical Society 2024
... Information theoretic approaches have proved fruitful in investigating features of radical-pairs such as the fundamental limits of sensitivity under dim light relevant for nocturnal migratory species [51], as well as providing insight into other quantum phenomena such as quantum coherence [27,[52][53][54][55][56][57][58], where we have previously demonstrated the importance of treatments comprising many nuclear spins inherent to biological systems [59,60]. The QFI has previously been utilised to interrogate the fundamental quantum limits of magnetic field sensitivity of radical-pairs in model studies of sensing the magnitude [61] and direction of the field [62]. Whilst these studies have provided foundational insight into the precision capabilities of a radical-pair compass, they are restricted to a toy model comprised of a single nuclear spin and neglected inter-radical interactions and asymmetric reaction kinetics. ...
... Traditionally, the change of this flux with the direction of the magnetic field is considered as decisive in determining the compass sensitivity, whereby the maximal change of the singlet recombination quantum yield, Φ S = F r /k 0 c, is a widely applied fidelity measure. Though some works have also considered the case of access to time-resolved measurements of sensitivity [61], we choose to employ the steady-state picture as it more closely corresponds to the physiological conditions of magnetoreception under constant light excitation and allows a formal rationalisation of the form of the steady-state, to which the quantum parameter estimation approaches are applied. ...
... In Appendix B we further demonstrate that ∆ 2 θ saturates the Cramér-Rao bound set by F θ with measurement elements Π n ∈ {P S ,P T }, whereP S andP T = 1 1 1 −P S are the singlet and triplet projection operators, respectively. Note that [61] has suggested a Poisson binomial distribution rather than a binomial distribution from an analysis of the time-dependence of the reaction yield. ...
Quantum sensing enables the ultimate precision attainable in parameter estimation. Circumstantial evidence suggests that certain organisms, most notably migratory songbirds, also harness quantum-enhanced magnetic field sensing via a radical-pair-based chemical compass for the precise detection of the weak geomagnetic field. However, what underpins the acuity of such a compass operating in a noisy biological setting, at physiological temperatures, remains an open question. Here, we address the fundamental limits of inferring geomagnetic field directions from radical-pair spin dynamics. Specifically, we compare the compass precision , as derived from the directional dependence of the radical-pair recombination yield, to the ultimate precision potentially realisable by a quantum measurement on the spin system under steady-state conditions. To this end, we probe the quantum Fisher information and associated Cramér-Rao bound in spin models of realistic complexity, accounting for complex inter-radical interactions, a multitude of hyperfine couplings, and asymmetric recombination kinetics, as characteristic for the magnetosensory protein cryptochrome. We compare several models implicated in cryptochrome magnetoreception and unveil their optimality through the precision of measurements ostensibly accessible to nature. Overall, the comparison provides insight into processes honed by nature to realise optimality whilst constrained to operating with mere reaction yields. Generally, the inference of compass orientation from recombination yields approaches optimality in the limits of complexity, yet plateaus short of the theoretical optimal precision bounds by up to one or two orders of magnitude, thus underscoring the potential for improving on design principles inherent to natural systems.
... Quantum biology promises to push the quantum-toclassical transition barrier [1], and to some extent include into the quantum world the more complex setting of biological systems. Spin is an archetypal quantum system, often weakly coupled to decohering environments [2], hence retrospectively it is not surprising that spinchemical effects in biomolecular reactions [3,4] have become a major paradigm for quantum biology [5][6][7][8][9][10][11][12][13]. Excitation transfer in photosynthesis [14][15][16][17][18][19][20][21] and quantum spectroscopic effects in olfaction [22][23][24] have also been studied in the context of quantum biology, with more possibilities being contemplated [25,26]. ...
... We will now consider radical-pair reactions, which realize biological magnetic sensing [4]. Quantum uncertainties in the reaction yields were investigated [12] in the context of precision limits of radical-pair magnetometers, but with no relevance to the underlying coherence. We will here unravel the connection of such uncertainties with the coherent dynamics of radical-pairs. ...
A fundamental concept of quantum physics, the Wigner Yanase information, is here used as a measure of quantum coherence in spin-dependent radical-pair reactions pertaining to biological magnetic sensing. This measure is connected to the uncertainty of the reaction yields, and further, to the statistics of a cellular receptor-ligand system used to biochemically convey magnetic-field changes. Measurable physiological quantities, such as the number of receptors and fluctuations in ligand concentration, are shown to reflect the introduced Wigner-Yanase measure of singlet-triplet coherence. We arrive at a quantum-biological uncertainty relation, connecting the product of a biological resource and a biological figure of merit with the Wigner-Yanase coherence. Our approach can serve a general search for quantum-coherent effects within cellular environments.
... On the other hand, quantum theory has been applied to spin chemistry problems such as the magnetoreception phenomenon found in birds. [15][16][17][18][19][20][21][22][23][24][25][26][27][28]. Since electrons and qubits are both spin-1/2 systems, we aim to utilize quantum computers to simulate spin chemistry quantum dynamics that may be out of reach for classical digital computers. ...
Quantum dynamics of the radical pair mechanism is a major driving force in quantum biology, materials science, and spin chemistry. The rich quantum physical underpinnings of the mechanism are determined by a coherent oscillation (quantum beats) between the singlet and triplet spin states and their interactions with the environment, which is challenging to experimentally explore and computationally simulate. In this work, we take advantage of quantum computers to simulate the Hamiltonian evolution and thermal relaxation of two radical pair systems undergoing the quantum-beat phenomena. We study radical pair systems with nontrivial hyperfine coupling interactions, namely, 9,10-octalin+/p-terphenyl-d14 and 2,3-dimethylbutane/p-terphenyl-d14 that have one and two groups of magnetically equivalent nuclei, respectively. Thermal relaxation dynamics in these systems are simulated using three methods: Kraus channel representations, noise models on Qiskit Aer and the inherent qubit noise present on the near-term quantum hardware. By leveraging the inherent qubit noise, we are able to simulate noisy quantum beats in the two radical pairs better than with any classical approximation or quantum simulator. While classical simulations of paramagnetic relaxation grow errors and uncertainties as a function of time, near-term quantum computers can match the experimental data throughout its time evolution, showcasing their unique suitability and future promise in simulating open quantum systems in chemistry.
... Moreover, quantum information concepts such as the violation of entropy bounds were used 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, a quantum metrology approach addressed the fundamental limits to quantum sensing of magnetic fields using radical-pair reactions by treating them as biochemical quantum magnetometers [19]. ...
Radical-pair reactions pertinent to biological magnetic field sensing are an ideal system for demonstrating the paradigm of quantum biology, the exploration of quantum coherence effects in complex biological systems. We here provide yet another fundamental connection between this biochemical spin system and quantum information science. We introduce and explore a formal measure quantifying the 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. Second, we use the concept of incoherent operations 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 unravel subtle effects related to exchange interactions and their role in promoting quantum coherence.
... Moreover, quantum information concepts like violation of entropy bounds were used 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, a quantum metrology approach addressed the fundamental limits to quantum sensing of magnetic fields using radical-pair reactions by treating them as biochemical quantum magnetometers [19]. ...
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.
... Part of the recent quest to explore the possibilities biological systems offer for novel quantum technology realizations [11][12][13] has been the question of how well biological photodetectors, in particular rhodopsin molecules in retinal rod cells [14], compare with modern photodetectors [15]. Historically, the role of photon statistics in vision was addressed in the 1930's [16] and elucidated with human behavioral experiments in the 1940's [17][18][19], single cell responses have been recorded since the 1970's [20][21][22][23][24], while it was only recently that the quantum physics of human vision [25] was addressed with the modern experimental [26][27][28][29][30] and theoretical [31][32][33][34] tools of quantum optics. ...
It is known that the eye's scotopic photodetectors, rhodopsin molecules and their associated phototransduction mechanism leading to light perception, are efficient single photon counters. We here use the photon counting principles of human rod vision to propose a secure quantum biometric identification based on the quantum-statistical properties of retinal photon detection. The photon path along the human eye until its detection by rod cells is modeled as a filter having a specific transmission coefficient. Precisely determining its value from the photodetection statistics registered by the conscious observer is a quantum parameter estimation problem that leads to quantum secure identification method. The probabilities for a false positive and a false negative identification of this biometric technique can readily approach and , respectively. The security of the biometric method can be further quantified by the physics of quantum measurements. An impostor must be able to perform quantum thermometry and quantum magnetometry with energy resolution better than , in order to foil the device by non-invasively monitoring the biometric activity of a user.
