Quantum information processing in the radical-pair mechanism: Haberkorn's theory violates the Ozawa entropy bound

Radical-ion-pair reactions, central for understanding the avian magnetic compass and spin transport in photosynthetic reaction centers, were recently shown to be a fruitful paradigm of the new synthesis of quantum information science with biological processes. We show here that the master equation so far constituting the theoretical foundation of spin chemistry violates fundamental bounds for the entropy of quantum systems, in particular the Ozawa bound. In contrast, a recently developed theory based on quantum measurements, quantum coherence measures, and quantum retrodiction, thus exemplifying the paradigm of quantum biology, satisfies the Ozawa bound as well as the Lanford-Robinson bound on information extraction. By considering Groenewold's information, the quantum information extracted during the reaction, we reproduce the known and unravel other magnetic-field effects not conveyed by reaction yields.

Figure 1

The simulations were performed with an isotropic Hamiltonian $\mathcal{H}=A\mathbf{I}·{\mathbf{s}}_D$ and asymmetric recombination rates $k_S=A/100$ and $k_T=A/5$. (a)–(c) The initial state was $|S\rangle \otimes |\Uparrow \rangle$ and the spin-randomizing term had a rate $\gamma =A/2000$. (d)–(f) The initial state was the fully mixed triplet $Q_T/\mathrm{Tr}\left\{Q_T\right\}$ and there was no spin-randomizing term ($\gamma =0$). The right $y$ axis in (c) depicts the time dependence of $q_S=\mathrm{Tr}\left\{\rho Q_S\right\}/\mathrm{Tr}\left\{\rho \right\}$ and demonstrates that the Lanford-Robinson bound is saturated, as expected, at those times when the radical-pair state is close to an eigenstate of the measurement operator $Q_S$.

Figure 2

For the same Hamiltonian and recombination rates as in Fig. 1, a pure singlet initial state $|S\rangle \otimes |\Uparrow \rangle$, and various spin-randomization rates $\gamma$, we plot the initial and final entropies resulting from (a)–(c) Haberkorn's and (d)–(f) Kominis's master equation. The first row ($\gamma =A/2000$) reiterates Figs. 1 and 1. By increasing $\gamma$ it is seen that (b) Haberkorn's violation first becomes milder, while for still higher $\gamma$ (c) the violation turns into a satisfaction of the bound. This behavior demonstrates that for a large spin relaxation rate $\gamma$, singlet-triplet coherence quantified by $p_{\mathrm{coh}}$ swiftly decays to zero and HME becomes a good approximation describing a mixture of radical pairs that indeed are incoherent.

Figure 3

(a) Singlet reaction yield dependence (left $y$ axis) on the magnetic field using (1), depicting low- and high-field effects. (b) Extracted (Groenewold) information ${\mathcal{I}}_G\left(B\right)$, calculated from (5), exhibits both low- and high-field effects, but also reveals a different field effect at $B=A$. This is evidenced by measuring the coherence ${\rho }_{35}$, the relevant yield shown on the right $y$ axis of (a). The calculation was performed with $\mathcal{H}=A\mathbf{I}·{\mathbf{s}}_D+B(s_{Dz}+s_{Az}), k_S=k_T=A/20$, initial state $|S\rangle \otimes |\Uparrow \rangle$, and no spin-randomizing term ($\gamma =0$).