Identifying the quantum correlations in light-harvesting complexes
Kamil Bradler, Mark M. Wilde, Sai Vinjanampathy, Dmitry B. Uskov
IIdentifying the quantum correlations in light-harvesting complexes
Kamil Br´adler and Mark M. Wilde
School of Computer Science, McGill University, Montreal, Quebec, Canada H3A 2A7
Sai Vinjanampathy
Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana, USA 70803
Dmitry B. Uskov
Department of Physics and Engineering Physics,Tulane University, New Orleans, Louisiana, USA 70118 (Dated: August 23, 2018)One of the major efforts in the quantum biological program is to subject biological systems tostandard tests or measures of quantumness. These tests and measures should elucidate if non-trivialquantum effects may be present in biological systems. Two such measures of quantum correlationsare the quantum discord and the relative entropy of entanglement. Here, we show that the relativeentropy of entanglement admits a simple analytic form when dynamics and accessible degrees offreedom are restricted to a zero- and single-excitation subspace. We also simulate and calculate theamount of quantum discord that is present in the Fenna-Matthews-Olson protein complex duringthe transfer of an excitation from a chlorosome antenna to a reaction center. We find that thesingle-excitation quantum discord and relative entropy of entanglement are equal for all of ournumerical simulations, but a proof of their general equality for this setting evades us for now. Also,some of our simulations demonstrate that the relative entropy of entanglement without the single-excitation restriction is much lower than the quantum discord. The first picosecond of dynamics isthe relevant timescale for the transfer of the excitation, according to some sources in the literature.Our simulation results indicate that quantum correlations contribute a significant fraction of thetotal correlation during this first picosecond in many cases, at both cryogenic and physiologicaltemperature.
PACS numbers: 03.67.Mn, 03.65.Yz, 82.39.JnKeywords: light-harvesting complexes, quantum biology, quantum discord, relative entropy of entanglement,quantum mutual information
I. INTRODUCTION
Quantum biology aims to understand if, how, and why biological systems exploit quantum-mechanical effects fortheir functionality or for an evolutionary advantage [1–4]. Exemplary biological systems that may exploit quantumeffects vary from photosynthetic light-harvesting complexes [5], to the avian compass for bird navigation [6], to theolfactory system [7]. Ongoing theoretical research indicates that light-harvesting complexes exploit an environment-assisted quantum-walk like effect to enhance energy transport [8–11], the avian compass exploits a radical ion-pairmechanism for increased sensitivity of the earth’s geomagnetic field [12–14], and the olfactory system may exploitphonon-assisted tunneling for enhanced detection of smell [7].Light-harvesting complexes seem particularly suitable as biological systems to harness quantum-mechanical effects.Their lengthscales and energyscales are on the order where we would expect quantum-mechanical laws to apply [15],but what remains less clear is if they can still harness quantum effects such as entanglement even at physiologicaltemperature. A recent numerical study addresses this question by showing that light-harvesting complexes coulddemonstrate stronger-than-classical temporal correlations, even at physiological temperature [16]. However, it remainsan open task to devise and conduct a realistic experimental protocol that demonstrates an irrevocable test of non-classical temporal correlations for light-harvesting complexes.The aim of the study in Ref. [16], as well as that in Refs. [17, 18], is to address one of the important “quantumbiological questions”:
Does the biological system exhibit “quantumness” according to a standard test or measure?
Theoretical machinery from quantum information science [19], developed specifically for understanding quantumcomputational and communication devices, should be of immense utility in answering this question. Several articleshave already begun exploiting such tools. Ref. [16] exploits the Leggett-Garg test of non-classicality [20] to suggestthat light-harvesting complexes might exhibit stronger-than-classical temporal correlations, while Refs. [17, 18] utilizestandard entanglement-based measures of quantum correlations [21–23] to suggest that they might exhibit stronger-than-classical spatial correlations. Such standard measures of quantum behavior are more convincing than, say, aclaim that wavelike motion in population elements of a density matrix is a signature of quantumness [24].The studies in Refs. [17, 18] both suggest that it might be possible to observe spatial quantum correlations inTypeset by REVTEX a r X i v : . [ qu a n t - ph ] J u l the Fenna-Matthews-Olson (FMO) light-harvesting protein complex, but the quantum correlation measures exploitedthere, such as concurrence [25], the measure based on the global relative entropy of entanglement [23], and logarithmicnegativity [26], might not capture all of the spatial quantum correlations that are present in a given quantum system.For example, consider a bipartite quantum system in the state:12 | (cid:105) (cid:104) | A ⊗ | + (cid:105) (cid:104) + | B + 12 |−(cid:105) (cid:104)−| A ⊗ | (cid:105) (cid:104) | B , (1)where |±(cid:105) ≡ ( | (cid:105) ± | (cid:105) ) / √ A and Bob possesses the system B . The state in (1) is aseparable state [27], meaning that Alice and Bob can prepare it by means of local quantum operations and classicalcommunication. Yet, it is fundamentally non-classical because the states | (cid:105) and |−(cid:105) on Alice’s local system or | (cid:105) and | + (cid:105) on Bob’s local system are indistinguishable from one another (they are non-orthogonal). It could be possiblefor two bacteriochlorophyll sites of a light-harvesting complex to admit a state of the above class or a mixture of suchstates, but the quantum correlation measures mentioned above all vanish for such a separable state, despite its statusas a fundamentally non-classical state. Thus, these measures might not capture the full “quantumness” that mightbe present in a light-harvesting complex.A different measure of quantum correlations, known as the quantum discord [28, 29], captures all correlations in aquantum state that are non-classical. The discord is a measure of the total correlations present in a shared quantumstate, reduced by the classical correlations obtainable when one party performs a local measurement. For example, thestate in (1) registers a non-vanishing quantum discord and therefore possesses non-classical correlations according tothis measure, even though it does not violate a Bell inequality [30] due to it being a separable state. Other examplesdemonstrate that the quantum discord in a quantum system can remain positive even if the entanglement vanishesafter a finite time [31]. States that register non-vanishing discord can lead to exponential speedups [32, 33] in aquantum computational model known as the “one-clean qubit model” [34], where the only requirement for a speedupis access to a single qubit in a pure state. Although it is unlikely that a light-harvesting complex could be performingexotic quantum computational speedups of the aforementioned nature or in the standard way [35], one cannot rule outthe possibility that a light-harvesting complex exhibiting non-vanishing discord could be exploiting “quantumness”of this form for enhanced energy transport.In this paper, we systematically study the presence of quantum correlations in the FMO light-harvesting complex,both at cryogenic and physiological temperature. Our first contribution is a simple formula for computing the relativeentropy of entanglement [36] when a light-harvesting complex evolves according to the open quantum system modelwell studied in the quantum biological literature [9, 11]. This formula assumes that dynamics are restricted to a zero-and single-excitation subspace, and it reduces the a priori computationally intensive optimization task for computingthe single-excitation relative entropy of entanglement to a simple calculation with quantum entropies. These resultsfor the single-excitation relative entropy of entanglement generalize those of Sarovar et al . in Ref. [17] for the globalrelative entropy of entanglement. We then calculate the quantum discord for several phenomenologically motivated“bipartite cuts” of the FMO protein complex and find that it is equivalent to the single-excitation relative entropy ofentanglement in all cases presented here. These results demonstrate that quantum correlations contribute a significantfraction of the total correlation during the highly relevant first picosecond of dynamics in many cases (some sourcesin the literature [10, 11] indicate that the average time it takes for an excitation to trap to the reaction center is onepicosecond after it arrives from the chlorosome antenna). In other cases, these quantities are equivalent and contributea non-negligible fraction of the total correlation during the first picosecond. This contribution indicates that non-classical spatial correlations may be playing a role in the efficient transfer of an excitation from the chlorosome antennato the reaction center. Our final contribution is to study the relative entropy of entanglement without its optimizationrestricted to the single-excitation subspace, and we find that it can be significantly less than the restricted relativeentropy of entanglement while only using just a small fraction of doubly-excited states in the optimization.We structure this paper as follows. We first briefly review both the dynamical model of the FMO protein complex,the definition of the quantum discord, the definition of the relative entropy of entanglement, and we then discusshow to compute these quantities in the FMO protein complex model. Our analytic result states that the relativeentropy of entanglement, when restricted to the zero- and single-excitation subspace, is equal to a simple formulathat is a difference of entropies. Appendix A provides a proof of this theorem. Section IV presents the results of ournumerical simulations under various initial configurations and temperatures, and Section V discusses the unrestrictedoptimization of the relative entropy of entanglement. We then conclude with observations and open questions forfuture research. II. REVIEWA. FMO Complex Dynamics
The FMO protein complex is the crucial, light-harvesting component of the green sulfur bacteria prosthecochlorisaestuarii , that develop in dimly-lit, anoxic environments such as stratified lakes or sulfur springs [37]. It is a trimer,consisting of three identical subunits. We study one unit of the trimer, consisting of seven bacteriochlorophyll sitesthat act as a “molecular wire,” transferring energetic excitations from a photon-receiving antenna to a reaction center.A photon impinges on the antenna, producing an electronic excitation, dubbed an exciton, that then proceeds to theunit with seven sites. While traversing the seven sites, the exciton can either recombine, corresponding to an energeticloss, or it can trap to the reaction center for energy storage.We characterize the quantum state of the exciton as a density operator in the site basis: ρ ≡ (cid:88) m,n ∈{ G, ,..., ,S } ρ m,n | m (cid:105) (cid:104) n | , where the state | m (cid:105) indicates that the exciton is present at site m . The sites can be any of the seven sites in theprotein complex m ∈ { , . . . , } , a ground state | G (cid:105) that represents the loss or recombination of the exciton, or a sinkstate | S (cid:105) that implies that the exciton has trapped to the reaction center. We adopt the following “qubit” conventionin this work (as adopted in previous works [17, 38, 39]), by assigning site states to tensor-product states: | G (cid:105) ≡ | g (cid:105) | g (cid:105) | g (cid:105) | g (cid:105) | g (cid:105) | g (cid:105) | g (cid:105) | g (cid:105) S , | (cid:105) ≡ | e (cid:105) | g (cid:105) | g (cid:105) | g (cid:105) | g (cid:105) | g (cid:105) | g (cid:105) | g (cid:105) S , ... | (cid:105) ≡ | g (cid:105) | g (cid:105) | g (cid:105) | g (cid:105) | g (cid:105) | g (cid:105) | e (cid:105) | g (cid:105) S , | S (cid:105) ≡ | g (cid:105) | g (cid:105) | g (cid:105) | g (cid:105) | g (cid:105) | g (cid:105) | g (cid:105) | e (cid:105) S , where g indicates the absence of an excitation and e indicates the presence of an excitation at a particular site. Theexcitation number is a conserved quantity in the absence of light-matter interaction events [40], and this observationrestricts the protein dynamics to a zero- and single-excitation subspace. Thus, the above nine states are the onlystates that we consider for the exciton while it traverses the seven sites.Observe that “tracing over any site” in this qubit representation has the effect of placing the population term forthat site into the population of the ground state. For example, suppose that we trace over all sites except for the firstone. The resulting density matrix has the form:( ρ GG + ρ + . . . + ρ + ρ SS ) | G (cid:105) (cid:104) G | + ρ | (cid:105) (cid:104) | + ρ G | (cid:105) (cid:104) G | + ρ G | G (cid:105) (cid:104) | . Such manipulations are important for computing correlation measures for any bipartite cut of the sites in the FMOcomplex.Evolution of the density matrix occurs according to a combination of both coherent and incoherent dynamics. Atight-binding Hamiltonian of the following form governs coherent evolution across the seven site states | (cid:105) , . . . , | (cid:105) : H ≡ (cid:88) m E m | m (cid:105) (cid:104) m | + (cid:88) n We first recall various informational measures of a quantum state before briefly reviewing the motivation for thequantum discord as a measure of quantum correlations. Suppose that two parties Alice and Bob share a quantumstate ρ AB . The von Neumann entropy of this state is as follows: H ( AB ) ρ ≡ − tr (cid:8) ρ AB log ρ AB (cid:9) , where the logarithm is base two. The entropy H ( AB ) ρ measures the uncertainty about the total quantum state ρ AB in units of bits, whenever the logarithm is base two [45]. Similarly, we can compute the marginal entropies H ( A ) and H ( B )—these entropies are with respect to the respective reduced states ρ A and ρ B , obtained by a respective partialtrace over Bob or Alice’s share of the state ρ AB .The quantum mutual information I ( A ; B ) is a measure of the total correlations, both classical and quantum, sharedbetween two parties, where I ( A ; B ) ρ ≡ H ( A ) ρ + H ( B ) ρ − H ( AB ) ρ . (3)The quantum mutual information admits a natural operational interpretation as the amount of noise necessary todestroy all correlations present in a bipartite state [46]. Its other operational interpretations are as the maximumamount of secret information that one party can send to another if they use a shared state as the basis for a one-timepad cryptosystem [47], the entanglement-assisted classical capacity of a quantum channel [48–50], and more recentlythe amount of private quantum information that a sender can transmit to a receiver while being eavesdropped on bya uniformly accelerating third party [51]. All of these operational interpretations justify the notion of the quantummutual information measuring correlations between two parties in units of bits.Suppose now that Alice would like to extract classical information from the quantum state ρ AB . She performs avon Neumann measurement {| x (cid:105) (cid:104) x |} on her share of the state, where the states {| x (cid:105)} form an orthonormal basis. IfAlice obtains classical result x from the measurement, the resulting conditional quantum state of the whole system is | x (cid:105) (cid:104) x | X ⊗ ρ Bx , where ρ Bx ≡ p ( x ) Tr X (cid:110) ( | x (cid:105) (cid:104) x | X ⊗ I B ) ρ AB ( | x (cid:105) (cid:104) x | X ⊗ I B ) (cid:111) , (4) p ( x ) ≡ Tr (cid:110) | x (cid:105) (cid:104) x | X ρ A (cid:111) , (5)and we now label Alice’s system with X because it is classical. If we then take the expectation over all of Alice’soutcomes, the description of the system is a classical-quantum state: σ XB ≡ (cid:88) x p ( x ) | x (cid:105) (cid:104) x | X ⊗ ρ Bx . (6)We might think that a natural measure of the classical correlations present in a bipartite state is the amount ofcorrelations in the resulting classical-quantum state: I c ( A ; B ) ρ ≡ max { Λ A } I ( X ; B ) σ , (7)The above expression features a maximization over all of the von Neumann measurements Λ A that Alice could perform.These measurements result in a state of the form σ in (6), and the quantum mutual information I ( X ; B ) σ is withrespect to such a state. In fact, Henderson and Vedral proposed such a measure [28], and Devetak and Winter laterjustified this measure by providing an operational interpretation of it as the maximum amount of common randomness(perfect classical correlations) that two parties can extract from a bipartite quantum state [52].The two correlation measures in (3) and (7), the first a measure of total correlations and the second a measureof classical correlations, suggests that a measure of the quantum correlations should be the difference of these twoquantities. The quantum discord D ( A ; B ) ρ is such a measure [29, 53], defined as the difference of total and classicalcorrelations: D ( A ; B ) ρ ≡ I ( A ; B ) ρ − I c ( A ; B ) ρ . (8)The discord is always non-negative, by the quantum data processing inequality of quantum information theory [54],and it is generally not symmetric with respect to the parties A and B : ∃ ρ : D ( A ; B ) ρ (cid:54) = D ( B ; A ) ρ . It captures all of the quantum correlations, including entanglement, but also the quantumness in separable states of theform in (1). Zurek has suggested a physical interpretation of the discord as the difference in efficiency between quantumand classical Maxwell’s demons [55], but it still lacks a clear operational interpretation in the sense of Refs. [46–52].Nevertheless, we still employ the discord as a measure of the quantum correlations in the FMO complex.We can rewrite the quantum discord as the following expression: D ( A ; B ) ρ = I ( B (cid:105) A ) ρ + min { Π x } (cid:88) x p ( x ) H (cid:0) ρ Bx (cid:1) , (9)by performing straightforward manipulations of the entropies in (8). In the above, I ( A (cid:105) B ) is the coherent informa-tion [54], equal to the negative of a conditional entropy: I ( B (cid:105) A ) ρ = − H ( B | A ) ρ = H ( A ) ρ − H ( AB ) ρ , and the probabilities p ( x ) and conditional density operators are as they appear respectively in (4) and (5). C. Relative Entropy of Entanglement The relative entropy of entanglement is an entanglement measure from quantum information theory [36], and webriefly review its definition. The relative entropy D (cid:0) ρ AB || ω AB (cid:1) of two bipartite states ρ AB and ω AB is as follows [54]: D (cid:0) ρ AB || ω AB (cid:1) ≡ Tr (cid:8) ρ AB log ρ AB (cid:9) − Tr (cid:8) ρ AB log ω AB (cid:9) . (10)This measure, in some sense, quantifies the “distance” between two bipartite states, but it is not a distance measurein the strict mathematical sense because it fails to be symmetric. Though, this intuition is useful for constructingan entanglement measure. We might naturally expect a good measure of entanglement to be the distance of a givenbipartite state to the closest separable state σ AB ≡ (cid:80) i p ( i ) σ Ai ⊗ σ Bi . The relative entropy of entanglement R (cid:0) ρ AB (cid:1) is such a measure, defined as the minimization of the relative entropy over all separable states: R (cid:0) ρ AB (cid:1) ≡ min σ ∈S D (cid:0) ρ AB || σ AB (cid:1) , where S is the class of separable states. This measure satisfies the properties of an entanglement measure and appearsextensively in the quantum information theory literature [36]. We mention that Ref. [17] employed this entanglementmeasure for determining quantum correlations in the FMO complex, but they considered the global relative entropyof entanglement rather than a bipartite version of it as we consider in this work. III. FORMULA FOR THE SINGLE-EXCITATION RELATIVE ENTROPY OF ENTANGLEMENT The restriction of dynamics to the zero- and single-excitation subspace allows for significant simplifications tothe theory of energetic transfer in the FMO complex. We can also apply this restriction to the relative entropy ofentanglement, by restricting the optimization over separable states to the zero- and single-excitation subspace. Let R e (cid:0) ρ AB (cid:1) denote the relative entropy of entanglement with this restriction applied to its optimization. Theorem 1below states that R e (cid:0) ρ AB (cid:1) is equal to a simple formula that is a difference of entropies. Thus, the theorem significantlysimplifies the computation of this quantity. Theorem 1 Consider a density operator ρ AB restricted to the zero- and single-excitation subspace. Let ∆ (cid:0) ρ AB (cid:1) ≡ α | G (cid:105) (cid:104) G | + ρ Ae ⊗ | G (cid:105)(cid:104) G | B + | G (cid:105)(cid:104) G | A ⊗ ρ Be , where α is the population of the ground state, ρ Ae is the projection of Alice’s part of ρ AB into the single-excitationsubspace, and ρ Be is defined in a similar way. Suppose that the dynamics for this density operator never induce anycoherences between the zero- and single-excitation subspaces (as in the dynamics in (2)). Then the single-excitationrelative entropy of entanglement R e ( ρ ) is equal to the difference of the entropy of ∆ (cid:0) ρ AB (cid:1) and ρ AB : R e (cid:0) ρ AB (cid:1) = H (cid:0) ∆ (cid:0) ρ AB (cid:1)(cid:1) − H (cid:0) ρ AB (cid:1) . The full proof appears in Appendix A. It exploits standard properties of the von Neumann entropy and a pertur-bative argument. IV. SIMULATION RESULTS We conducted several simulations at both cryogenic temperature (77 ◦ K) and physiological temperature (300 ◦ K) andfor the initial state being a pure state at site one, six, and the mixture of the previous two. These simulations calculateboth the quantum mutual information, quantum discord, and single-excitation relative entropy of entanglement withrespect to several “bipartite cuts” of the sites in the FMO complex. Throughout this section, we refer to quantumdiscord, but all our simulations indicated that the single-excitation relative entropy of entanglement is equal to thequantum discord. These results provide strong evidence that the quantum discord is equal to the formula in Theorem 1for these cases, but a general proof of this conjecture eludes us for now. Our different simulation cases are as follows:1. We first considered the cut where system A consists of site three and system B consists of sites one and six. Wepicked this cut because the initial state of the complex is at site one, six, or the mixture, and the objective ofthe FMO “molecular wire” is to transfer the excitation from these initial sites to site three. If spatial quantumcorrelations play a role in the transfer of the excitation, one would expect a state with this bipartite cut toregister a non-negligible amount of quantum discord.2. The next bipartite cut that we considered is with the A system consisting of sites one and two and the B systemconsisting of site three. We picked such a cut because recent analysis of the FMO Hamiltonian [41] suggeststhat a superposition state of sites one and two gives an efficient energetic pathway for the excitation to transferto site three. The energy of the superposed state is closer to the energy of site three than it is to the energy ofeither site one or site two. We would again expect such a cut would register a non-negligible amount of quantumdiscord if quantum correlations play a role in the transfer of the excitation.3. Finally, after conducting the above simulations, we conducted simulations with respect to the cut where system A consists of site three and system B consists of all other sites. The quantum mutual information for this caseshould be larger than for the case of the other cuts, by the quantum data processing inequality.Figure 1 plots the results of the first simulation for cryogenic temperature and for each initial state. The figureindicates that the quantum discord contributes a significant fraction of the total correlation for short timescales (lessthan one picosecond). This contribution of the quantum discord to total correlation is significant, considering thatthe average transfer time of the excitation to site three occurs around the order of one picosecond in our model. Forlonger timescales (greater than one picosecond), the total correlation between A (sites one and six) and B (site three)increases to its maximum at around 2-4 ps, and the quantum discord no longer contributes any significant amount tothe total correlation. Even though the total correlation rises so much higher for longer timescales than it is for shortertimescales, the increase and peak are not relevant for excitation transfer, i.e., “they arrive too late” given that this (b) −2 −1 (a) −2 −1 (c) −2 −1 FIG. 1: (Color online) Mutual information I ( A ; B ) and quantum discord D ( A ; B ) with system A as site three and system B as sites one and six. The figure plots these quantities at cryogenic temperature (77 ◦ K) as a function of time when the initialstate is (a) a pure state at the first site, (b) a pure state at the sixth state, (c) an equal mixture of the two previous states. Ineach of the above plots, the quantum discord is a significant fraction of the total correlations during the first picosecond. (b) −2 −3 Time (ps) MutualinformationDiscord (a) −2 −1 (c) −2 FIG. 2: (Color online) Mutual information I ( A ; B ) and quantum discord D ( A ; B ) with system A as site three and system B assites one and six. The figure plots these quantities at physiological temperature (300 ◦ K) as a function of time when the initialstate is (a) a pure state at the first site, (b) a pure state at the sixth state, (c) an equal mixture of the two previous states. Ineach of the above figures, the quantum discord is a non-negligible fraction of the total correlation during the first picosecond. (a) −2 −1 (b) −2 −3 Time (ps) MutualinformationDiscord (c) −2 −1 FIG. 3: (Color online) Mutual information I ( A ; B ) and quantum discord D ( A ; B ) with system A as site three and system B as sites one and two. The figure plots these quantities at cryogenic temperature (77 ◦ K) as a function of time when the initialstate is (a) a pure state at the first site, (b) a pure state at the sixth state, (c) an equal mixture of the two previous states.In each of the above cases, the quantum discord is a significant fraction of the total correlation during the first picosecond. Infact, for the second case, the quantum discord contributes nearly all of the total correlation during the first picosecond. (b) −2 −1 −3 Time (ps) MutualinformationDiscord (a) −2 −1 (c) −2 −1 FIG. 4: (Color online) Mutual information I ( A ; B ) and quantum discord D ( A ; B ) with system A as site three and system B assites one and two. The figure plots these quantities at physiological temperature (300 ◦ K) as a function of time when the initialstate is (a) a pure state at the first site, (b) a pure state at the sixth state, (c) an equal mixture of the two previous states. Ineach of the above cases, the quantum discord contributes a fraction of the total correlation during the first picosecond. In thesecond case, it again contributes nearly all of the total correlation during the first picosecond. transfer occurs on the order of 1 ps. The total correlation then washes away as time increases beyond 10 ps, and itshould nearly vanish for times around 1 ns because this time is the average recombination time of the exciton, andno correlations should persist after a recombination.Figure 2 plots the results of the first simulation for physiological temperature. The difference between cryogenicand physiological temperature is a qualitative shrinking by a factor of two, but the fraction of quantum discord thatcontributes to total correlation for short timescales is lower than it is for cryogenic temperatures. The local dephasingat each site acts to destroy both total and quantum correlation, but it appears to have a more harmful effect onquantum correlation. Despite the low amount of quantum discord registered at physiological temperature, it couldstill be that a light-harvesting complex is harnessing this small amount of quantumness (that is a significant fractionof total correlation) on this short timescale to enhance excitation transfer.Figures 3 and 4 plot the results of the second simulation for both respective temperatures and for each initialstate. These results are qualitatively similar to the previous bipartite cut, but the most striking difference is that thequantum discord contributes all of the correlation (it is equal to the quantum mutual information) for short timescaleswhen the initial state is at site six for both cryogenic and physiological temperatures (see Figures 3(b) and 4(b)).Figures 5 and 6 plot the results of our final simulation, where the bipartite cut has system B as site three andsystem A as all other sites. The results for quantum discord are again qualitatively similar to previous results, butthe quantum mutual information is significantly higher than for the previous cases (this is expected because thequantum data processing inequality states that correlations can only decrease when discarding subsystems). Thequantum discord again contributes a significant fraction of the total correlation for short timescales and contributesonly a small fraction for later timescales. Again, physiological temperature decoherence mitigates the presence ofquantumness. V. COMPARISON OF THE QUANTUM DISCORD WITH THE STANDARD RELATIVE ENTROPYOF ENTANGLEMENT In this section, we focus on computing the relative entropy of entanglement in (10) without the assumption that theset of separable states σ AB involved in the minimization of D ( ρ AB || σ AB ) is limited to the single-excitation subspace.This implies that σ AB may include multiply-excited states (for example, states for which both systems A and B carryan exciton).A question may arise as to why such multiply-excited states are relevant for the optimization of the relative entropyof entanglement. After all, the Hamiltonian of the system commutes with the operator corresponding to the totalnumber of excitations in the system, and the dissipation operators in (2) only decrease the number of excitationsby dumping them to the sink or to the reaction center. There are two different ways to answer this question. Thefirst one is simple yet somewhat formal—Vedral established the relative entropy of entanglement in the context ofquantum information theory as a measure of entanglement for a general set of states with no subspace limitationsimposed. Therefore, even though the form of specific Hamiltonian and dissipation terms in (2) leads to the absence of (b) −2 −1 (a) −2 −1 Time (ps) Mutual informationDiscord (c) −2 −1 FIG. 5: (Color online) Mutual information I ( A ; B ) and quantum discord D ( A ; B ) with system A as site three and system B as all other sites (1-2, 4-7). The figure plots these quantities at cryogenic temperature (77 ◦ K) as a function of time when theinitial state is (a) a pure state at the first site, (b) a pure state at the sixth state, (c) an equal mixture of the two previousstates. The quantum discord contributes a significant fraction of the total correlation during the first picosecond. (b) −2 −1 (a) −2 −1 (c) −2 −1 FIG. 6: (Color online) Mutual information I ( A ; B ) and quantum discord D ( A ; B ) with system A as site three and system B as all other sites (1-2, 4-7). The figure plots these quantities at physiological temperature (300 ◦ K) as a function of time whenthe initial state is (a) a pure state at the first site, (b) a pure state at the sixth state, (c) an equal mixture of the two previousstates. The quantum discord does not contribute a large fraction of the total correlation here. multiply-excited states in the density operator, there is no particular reason to consider that multiply-excited statesand the mode of preparation of a state are irrelevant in this setting. From this perspective, entanglement is determinedsolely by the state rather than by the history and mode of state evolution. The relative entropy of entanglement is ameasure of the “distance” of a state to the closest separable state and excluding multiply-excited states from the setof separable states introduces a distortion to the original concept. Actually, the measure of such a distortion may berather large, as we found out performing optimization of D ( ρ AB || σ AB ) in the full space (see Figure 7(a)).Our second answer is more physically motivated. In general, while analyzing the exciton dynamics in a lightharvesting system, one assumes that the intensity of light, which determines the photon flux, is quite small. Then itfollows that the probability of simultaneous creation of two excitations is negligible. By performing the optimizationof the D ( ρ AB || σ AB ) in the full space, which includes multiply-excited states that can have the form | E (cid:105) A | E (cid:105) B , andcomputing the amount of multiply-excited state population in the closest separable state σ AB , we can set up a limitfor the light intensity when the single-excitation assumption will become meaningless even in the context of excitation-preserving dynamics. In accordance with Caratheodory’s theorem, we generated the generic separable state σ AB as asum of 2 = 64 arbitrary pure separable states (cid:80) n =1 c n | ψ n (cid:105) (cid:104) ψ n | . Here, we limited our computation to a simple examplewhen site A = 1 , B = 3, such that | ψ n (cid:105) = ( α o | g (cid:105) | g (cid:105) + α | e (cid:105) | g (cid:105) + α | g (cid:105) | e (cid:105) + α | e (cid:105) | e (cid:105) ) ⊗ ( β | g (cid:105) + β | e (cid:105) ) . (a) (b) FIG. 7: (a) This figure simulates the dynamics of the FMO complex at 77 ◦ K and calculates both the single-excitation relativeentropy of entanglement and the full relative entropy of entanglement for A = 1 , B = 3. (b) The ratio of the numberof doubly-excited states to the number of singly-excited states in the optimal separable state for the relative entropy ofentanglement is rather small. The optimization is performed by the gradient method in the space of coefficients c n , α i , and β j . Figure 7(a) showsresults of the computation for T = 77 ◦ K. We note that relative entropy of entanglement for the single-excitation-subspace case has a similar shape to the full-space relative entropy of entanglement, but it is approximately fivetimes smaller. Figure 7(b) demonstrates that the population of doubly-excited states in the optimal separable state isnegligibly small—it is three to five orders of magnitude smaller than the population of singly-excited states. We findit remarkable that such a negligible admixture of doubly-excited states changes the relative entropy of entanglementby more than a factor of five. VI. CONCLUSIONS We presented results quantifying the quantum correlations present in a biological system at cryogenic and physi-ological temperature. Theorem 1 proves that the single-excitation relative entropy of entanglement admits a simpleform. We then simulated the dynamics of the FMO complex and calculated the quantum discord and the quantummutual information for various phenomenologically motivated bipartite cuts of the sites in the FMO protein complex.It is surprising that the quantum discord and the single-excitation relative entropy of entanglement are equivalentfor these simulations, but we have not been able to find a general proof that the quantum discord of this system isequivalent to the formula in Theorem 1. The results of our simulations indicate that quantum correlations contributea significant fraction of the total correlation during the first picosecond of dynamics in many cases. Our last contribu-tion was to study the relative entropy of entanglement with an unrestricted optimization, and we found that a smallfraction of doubly-excited states contribute significantly to reducing the relative entropy of entanglement.Open questions remain for this line of research. Our last contribution above suggests an intriguing open questionof whether optimizing the relative entropy of entanglement over a single-excitation subspace, as done here and inRef. [17], is operationally justified. We should also compute the correlation of the quantum discord with the measureof transfer time in Ref. [9] of the excitation from the antenna to the reaction center. This correlation might indicatehow relevant “quantumness” is for the transfer of the excitation. Finally, it might be interesting to determine howefficiency or transfer time is affected by differing amounts of quantum discord, if there can be efficient energy transferwithout quantum discord, and how discord compares with entanglement in the general case.1 Acknowledgments The authors acknowledge useful discussions with Mohan Sarovar and thank the anonymous referees for useful com-ments. K. B. acknowledges support from the Office of Naval Research under grant No. N000140811249. M. M. W. ac-knowledges support from the MDEIE (Qu´ebec) PSR-SIIRI international collaboration grant. D. B. U. acknowledgessupport from the National Science Foundation under grant No. PHY-0545390. Appendix A We first establish some notation before proving Theorem 1. Let our zero- and single-excitation space be spannedby the following states: | G (cid:105) AB ≡ | g (cid:105) | g (cid:105) · · · | g (cid:105) n , | (cid:105) AB ≡ | e (cid:105) | g (cid:105) · · · | g (cid:105) n , ... | n (cid:105) AB ≡ | g (cid:105) | g (cid:105) · · · | e (cid:105) n . The number of states is thus n + 1. We introduce a bipartite splitting of the above states so that the correspondingsubsystems A and B are spanned by {| G (cid:105) A , | (cid:105) A , . . . , | n a (cid:105) A } and {| G (cid:105) B , | (cid:105) B , . . . , | n b (cid:105) B } , respectively, using the sameconvention as above. Note that n a + n b + 1 = n , and the original basis written in terms of the split bases reads {| G (cid:105) A | G (cid:105) B , | k (cid:105) A | G (cid:105) B , | G (cid:105) A | j (cid:105) B } , where k = 1 , . . . , n a and j = 1 , . . . , n b . Let Π Ae and Π Be denote the projectors onto the excited subspaces of theindividual subsystems of A and B : Π Ae ≡ n a (cid:88) k =1 | k (cid:105)(cid:104) k | A , Π Be ≡ n b (cid:88) j =1 | j (cid:105)(cid:104) j | B . (A1)The projector onto the full bipartite single-excitation subspace is as follows:Π ABe ≡ | G (cid:105)(cid:104) G | A ⊗ Π Be + Π Ae ⊗ | G (cid:105)(cid:104) G | B . (A2)Let Π ABg ≡ | G (cid:105)(cid:104) G | AB . Then Π ABg + Π ABe is a projector onto the zero- and single-excitation subspace. We have that (cid:104) G | AB ρ AB | k (cid:105) AB = 0 because the dynamics in (2) do not induce any correlations between the ground state and thesingle-excitation subspace of the density matrix ρ AB . The following chain of inequalities then gives another way towrite the density matrix ρ AB that is more useful to us: ρ AB = (Π ABg + Π ABe ) ρ AB (Π ABg + Π ABe )= α Π ABg + Π ABe ρ AB Π ABg + Π ABg ρ AB Π ABe + Π ABe ρ AB Π ABe = α Π ABg + Π ABe ρ AB Π ABe = α Π ABg + (cid:0) | G (cid:105)(cid:104) G | A ⊗ Π Be + Π Ae ⊗ | G (cid:105)(cid:104) G | B (cid:1) ρ AB (cid:0) | G (cid:105)(cid:104) G | A ⊗ Π Be + Π Ae ⊗ | G (cid:105)(cid:104) G | B (cid:1) = α Π g + (cid:0) | G (cid:105)(cid:104) G | A ⊗ Π Be (cid:1) ρ AB (cid:0) | G (cid:105)(cid:104) G | A ⊗ Π Be (cid:1) + (cid:0) | G (cid:105)(cid:104) G | A ⊗ Π Be (cid:1) ρ AB (cid:0) Π Ae ⊗ | G (cid:105)(cid:104) G | B (cid:1) + (cid:0) Π Ae ⊗ | G (cid:105)(cid:104) G | B (cid:1) ρ AB (cid:0) | G (cid:105)(cid:104) G | A ⊗ Π Be (cid:1) + (cid:0) Π Ae ⊗ | G (cid:105)(cid:104) G | B (cid:1) ρ AB (cid:0) Π Ae ⊗ | G (cid:105)(cid:104) G | B (cid:1) = α Π g + ρ Ae ⊗ | G (cid:105)(cid:104) G | B + | G (cid:105)(cid:104) G | A ⊗ ρ Be + τ AB . (A3)The first equality follows because ρ AB lives in the zero- and single-excitation subspace. The second equality followsby expanding. The third equality follows because the density operator ρ AB does not have any correlations betweenthe ground state | G (cid:105) and any excited state. The fourth equality follows from the relation in (A2). The fifth equalityfollows by expanding, and the last follows from the following definitions: ρ Ae ≡ (cid:0) Π Ae ⊗ | G (cid:105)(cid:104) G | B (cid:1) ρ AB (cid:0) Π Ae ⊗ | G (cid:105)(cid:104) G | B (cid:1) , (A4) ρ Be ≡ (cid:0) | G (cid:105)(cid:104) G | A ⊗ Π Be (cid:1) ρ AB (cid:0) | G (cid:105)(cid:104) G | A ⊗ Π Be (cid:1) , (A5) τ AB ≡ (cid:0) Π Ae ⊗ | G (cid:105)(cid:104) G | B (cid:1) ρ AB (cid:0) | G (cid:105)(cid:104) G | A ⊗ Π Be (cid:1) (A6)+ (cid:0) | G (cid:105)(cid:104) G | A ⊗ Π Be (cid:1) ρ AB (cid:0) Π Ae ⊗ | G (cid:105)(cid:104) G | B (cid:1) . ρ Ae is the projection of ρ AB into Alice’s single-excitation subspace, ρ Be is similarly defined, and τ AB is theblock-off-diagonal part of ρ AB living in the single-excitation subspace. 1. Proof that R e (cid:0) ρ AB (cid:1) = H (cid:0) ∆ (cid:0) ρ AB (cid:1)(cid:1) − H (cid:0) ρ AB (cid:1) Proof. Our first candidate to maximize the relative entropy of entanglement is a state of the following form σ = β Π g + σ Ae ⊗ | G (cid:105)(cid:104) G | B + | G (cid:105)(cid:104) G | A ⊗ σ Be , (A7)because it is in the most general form of a single-excitation separable state with no correlation between the groundstate and the other states. The operators σ Ae and σ Be are some positive operators that live in the single-excitationsubspaces with respective projectors Π Ae and Π Be . So for now, we restrict the minimization in the relative entropy ofentanglement to states of the above form, leading to the following minimization:˜ R e (cid:0) ρ AB (cid:1) ≡ min σ Tr (cid:8) ρ AB log ρ AB (cid:9) − Tr (cid:8) ρ AB log σ (cid:9) . (A8)We first determine a spectral decomposition of σ Ae and σ Be as follows: σ Ae = n a (cid:88) k =1 a k | φ k (cid:105)(cid:104) φ k | A , σ Be = n b (cid:88) k =1 b k | φ k (cid:105)(cid:104) φ k | B . (A9)Let ω A ≡ log (cid:0) σ Ae (cid:1) and ω B ≡ log (cid:0) σ Be (cid:1) . The following equalities then hold τ AB log σ = (cid:2)(cid:0) Π Ae ⊗ | G (cid:105)(cid:104) G | B (cid:1) ρ AB (cid:0) | G (cid:105)(cid:104) G | A ⊗ Π Be (cid:1) + (cid:0) | G (cid:105)(cid:104) G | A ⊗ Π Be (cid:1) ρ AB (cid:0) Π Ae ⊗ | G (cid:105)(cid:104) G | B (cid:1)(cid:3) log (cid:0) α Π g + σ Ae ⊗ | G (cid:105)(cid:104) G | B + | G (cid:105)(cid:104) G | A ⊗ σ Be (cid:1) = (cid:0) Π Ae ⊗ | G (cid:105)(cid:104) G | B (cid:1) ρ AB (cid:0) | G (cid:105)(cid:104) G | A ⊗ ω B (cid:1) + (cid:0) | G (cid:105)(cid:104) G | A ⊗ Π Be (cid:1) ρ AB (cid:0) ω A ⊗ | G (cid:105)(cid:104) G | B (cid:1) The first equality follows by definition, and the second follows by considering the overlapping support of the differentsubspaces. We then find thatTr (cid:8) τ AB log σ (cid:9) = Tr (cid:8)(cid:0) Π Ae ⊗ | G (cid:105)(cid:104) G | B (cid:1) ρ AB (cid:0) | G (cid:105)(cid:104) G | A ⊗ ω B (cid:1) + (cid:0) | G (cid:105)(cid:104) G | A ⊗ Π Be (cid:1) ρ AB (cid:0) ω A ⊗ | G (cid:105)(cid:104) G | B (cid:1)(cid:9) = Tr (cid:8)(cid:0) | G (cid:105)(cid:104) G | A ⊗ ω B (cid:1) (cid:0) Π Ae ⊗ | G (cid:105)(cid:104) G | B (cid:1) ρ AB (cid:9) + Tr (cid:8)(cid:0) ω A ⊗ | G (cid:105)(cid:104) G | B (cid:1) (cid:0) | G (cid:105)(cid:104) G | A ⊗ Π Be (cid:1) ρ AB (cid:9) = 0 . The second equality follows from cyclicity of the trace, and the last equality follows because the zero- and single-excitation subspaces are orthogonal. Therefore,Tr (cid:8) ρ AB log σ (cid:9) = Tr (cid:8) ( ρ AB − τ AB ) log σ (cid:9) . (A10)Observe that ρ AB − τ AB is a state of the form σ with β = α , σ Ae = ρ Ae , and σ Be = ρ Be . Consider thatTr (cid:8) ( ρ AB − τ AB ) log( ρ AB − τ AB ) (cid:9) ≥ Tr (cid:8) ( ρ AB − τ AB ) log σ (cid:9) , because the relative entropy D (cid:0) ( ρ AB − τ AB ) || σ (cid:1) ≥ R e (cid:0) ρ AB (cid:1) in (A8) attains its globalminimum at ρ − τ AB . Thus, ˜ R e (cid:0) ρ AB (cid:1) = H (cid:0) ∆ (cid:0) ρ AB (cid:1)(cid:1) − H (cid:0) ρ AB (cid:1) . The expression in (A7) is not the most general single-excitation separable state because it does not include coherenceelements between the ground state and the single-excitation subspace. That is, matrix elements of the following formcould be nonzero: | G (cid:105) A | G (cid:105) B (cid:104) φ k | A (cid:104) G | B + H.C. (A11) | G (cid:105) A | G (cid:105) B (cid:104) G | A (cid:104) φ k | B + H.C. (A12)3We now show that it is not necessary to consider states of this more general form by appealing to a perturbationtheory argument [56]. Consider a solution ω ≡ ρ − τ AB of (A8). Let us take a small shift of this solution: ω → ω + (cid:15) (cid:0) | G (cid:105) A | G (cid:105) B (cid:104) φ k | A (cid:104) G | B + H.C. (cid:1) . (A13)Consider that ω | φ k (cid:105) A | G (cid:105) B = a k | φ k (cid:105) A | G (cid:105) B , (A14)and ω | G (cid:105) A | G (cid:105) B = α | φ k (cid:105) A | G (cid:105) B . (A15)Since ω is Hermitian, we can apply perturbation theory to compute the new states and energies. Brillouin-Wignerperturbation theory states that if such a Schr¨odinger-like equation above is perturbed via ω → ω + (cid:15)dω. (A16)where dω = | G (cid:105) A | G (cid:105) B (cid:104) φ k | A (cid:104) G | B + H.C., then the new states are given by | φ k (cid:105) A | G (cid:105) B → | φ k (cid:105) A | G (cid:105) B + (cid:15) (cid:88) | φ m (cid:105)(cid:54) = | φ k (cid:105) A | G (cid:105) B | φ m (cid:105) a k − E m (cid:104) φ m | dω | φ k (cid:105) A | G (cid:105) B + O ( (cid:15) ) . Here E m refers to the eigenvalue of the state | φ m (cid:105) . Substituting for dω , we obtain | φ k (cid:105) A | G (cid:105) B → | φ k (cid:105) A | G (cid:105) B + (cid:15)a k − α | G (cid:105) A G (cid:105) B + O ( (cid:15) ) . Similarly, applying BWPT to the second Schr¨odinger like equation above yields | G (cid:105) A | G (cid:105) B → | G (cid:105) A | G (cid:105) B + (cid:15) (cid:88) | φ m (cid:105)(cid:54) = | G (cid:105) A | G (cid:105) B | φ m (cid:105) α − E m (cid:104) φ m | dω | G (cid:105) A | G (cid:105) B + O ( (cid:15) ) . Again, substituting for dω gives | G (cid:105) A | G (cid:105) B → | G (cid:105) A | G (cid:105) B + (cid:15)α − a k | φ k (cid:105) A G (cid:105) B + O ( (cid:15) ) . Within BWPT, the eigenvalues are calculated as a k → a k + (cid:15) (cid:104) φ k | A (cid:104) G | B dω | φ k (cid:105) A | G (cid:105) B + (cid:88) | φ m (cid:105)(cid:54) = | k (cid:105) A | G (cid:105) B (cid:15) a k − E m |(cid:104) φ m | dω | φ k (cid:105) A | G (cid:105) B | and similarly α → α + (cid:15) (cid:104) G | A (cid:104) G | B dω | G (cid:105) A | G (cid:105) B + (cid:88) | φ m (cid:105)(cid:54) = | G (cid:105) A | G (cid:105) B (cid:15) α − E m |(cid:104) φ m | dω | G (cid:105) A | G (cid:105) B | Using the above, we find that the eigenvectors transform up to first order as follows: | φ k (cid:105) A | G (cid:105) B → | φ k (cid:105) A | G (cid:105) B + (cid:15) | G (cid:105) A | G (cid:105) B a k − α + O ( (cid:15) ) , (A17) | G (cid:105) A | G (cid:105) B → | G (cid:105) A | G (cid:105) B + (cid:15) | φ k (cid:105) A | G (cid:105) B α − a k + O ( (cid:15) ) . 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