Quantum relative entropy shows singlet-triplet coherence is a resource in the radical-pair mechanism of biological magnetic sensing

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Quantum relative entropy shows singlet-triplet coherence is a resource in the radical-pair mechanism of biological magnetic sensing

I. K. Kominis

Phys. Rev. Research 2, 023206 – Published 21 May 2020

Abstract

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.

Received 13 January 2020
Accepted 1 May 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.023206

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum effects in biological systems Quantum sensing Resource theories

Quantum Information, Science & TechnologyPhysics of Living Systems

Authors & Affiliations

I. K. Kominis

Department of Physics, University of Crete, 71003 Heraklion, Greece and Institute for Theoretical and Computational Physics, University of Crete, 70013 Heraklion, Greece

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Vol. 2, Iss. 2 — May - July 2020

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Figure 1
Radical-pair reaction dynamics. A charge transfer following the photoexcitation (not shown here) of a donor-acceptor dyad DA produces a singlet state radical-pair ${}^{S}D^{• +} A^{• -}$ , which is coherently converted to the triplet radical pair, ${}^{T}D^{• +} A^{• -}$ , due to intramolecule magnetic interactions embodied in the spin Hamiltonian $H$ . Simultaneously, spin-selective charge recombination leads to singlet and triplet neutral products with respective rates $k_{S}$ and $k_{T}$ .
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Figure 2
Singlet probability $Tr {ρ_{t} Q_{S}}$ (black solid line), and coherence measure $C [[ρ_{t}]]$ (red dashed line), for a two-electron system evolving by $H = ω_{1} s_{1 z} + ω_{2} s_{2 z}$ , with the initial state at $t = 0$ being $| s 〉$ . The S-T Rabi frequency is $Ω = ω_{1} - ω_{2}$ . (a) Unitary evolution driven only by $H$ . S-T coherence is zero whenever the state is a pure singlet ( $Tr {ρ_{t} Q_{S}} = 1$ ) or a pure triplet ( $Tr {ρ_{t} Q_{S}} = 0$ ), and maximum (equal to 1) when the state is $(| s 〉 - i | t_{0} 〉) / \sqrt{2}$ . (b) Nonunitary evolution caused by S-T dephasing at a rate $K_{d} = Ω / 5$ .
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Figure 3
Example of the angular dependence of the singlet reaction yield, for the Hamiltonian of Eq. (10), having a diagonal hyperfine tensor with $A_{y y} = A_{z z} = 0, A_{x x} / k = 10$ , and $ω / k = 1$ . The figure of merit of the compass is the value $δ Y_{S}$ , which quantifies the highest slope of $Y_{S}$ vs $ϕ$ .
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Figure 4
(a) Correlation map of the compass' figure of merit $δ Y_{S}$ vs the S-T coherence averaged along the reaction $\bar{C}$ , as calculated from the master equation $ρ = e^{- k t} R$ , with $d R / d t = - i [H, R] - K_{d} (R_{ST} + R_{TS})$ , where $H$ is the Hamiltonian of Eq. (10) including an exchange term. Each point results from a random set of parameters $0 \leq A_{x x}, A_{y y}, A_{z z} \leq 10 k, - 10 \leq J / k \leq 10$ , and $ω / k = 1$ . The dephasing rate $K_{d}$ was given the values $K_{d} = 0, K_{d} = k, K_{d} = 5 k$ , and $K_{d} = 10 k$ , with 5000 points for each. Initial state was always a singlet with mixed nuclear spin, $ρ_{0} = Q_{S} / Tr {Q_{S}}$ . (b) Mean value of $δ Y_{S}$ vs mean value of $\bar{C}$ over all 5000 points for each value of $K_{d}$ . (c) Correlation coefficient between $δ Y_{S}$ and $\bar{C}$ for each value of $K_{d}$ . (d1)–(d4) Distribution of $Y_{S}$ and $Y_{T}$ 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 $Q_{S}$ and $Q_{T}$ , the narrowing reflecting the measurement-induced localization of the radical pair's state.
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Figure 5
(a) Correlation between the compass figure of merit $δ Y_{S}$ and the singlet-triplet coherence measure $\bar{C}$ (left $y$ axis, black disks) and the mean value of $δ Y_{S}$ (right $y$ axis, red squares), for various subintervals of the exchange coupling, shown in the gray boxes (in units of the reaction rate $k$ ). Here, $K_{d} / k = 1$ . (b) Same correlation, but for the particular subinterval $| J | < 2$ and for all four values of $K_{d}$ .
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