^{1}, Sabre Kais

^{1}, Alán Aspuru-Guzik

^{2}, Sam Rodriques

^{3}, Ben Brock

^{3}and Peter J. Love

^{3}

### Abstract

We investigate the evolution of entanglement in the Fenna-Matthew-Olson (FMO) complex based on simulations using the scaled hierarchical equations of motion approach. We examine the role of entanglement in the FMO complex by direct computation of the convex roof. We use monogamy to give a lower bound for entanglement and obtain an upper bound from the evaluation of the convex roof. Examination of bipartite measures for all possible bipartitions provides a complete picture of the multipartite entanglement. Our results support the hypothesis that entanglement is maximum primary along the two distinct electronic energy transfer pathways. In addition, we note that the structure of multipartite entanglement is quite simple, suggesting that there are constraints on the mixed state entanglement beyond those due to monogamy.

This project is supported by NSF CCI center, “Quantum Information for Quantum Chemistry (QIQC),” Award No. CHE-1037992, and by NSF Award No. PHY-0955518.

I. INTRODUCTION

II. METHOD: SCALED HIERARCHICAL EQUATIONS OF MOTION (HEOM)

III. ENTANGLEMENT ANALYSIS

A. Entanglement measures

B. Monogamy of entanglement

C. Convex roof extension of entanglement monotones

IV. RESULTS

A. Two site subsystems

B. Three site subsystems

C. Four site subsystems

D. Five site subsystems

E. Seven site calculations

F. Beyond the single exciton manifold

V. CONCLUSIONS

### Key Topics

- Entanglement measures
- 54.0
- Excitons
- 36.0
- Quantum entanglement
- 30.0
- Qubits
- 29.0
- Subspaces
- 18.0

## Figures

Evolution of pairwise concurrences in the FMO complex when site one is initially excited at T = 77 K. All 21 pairwise concurrences computed by the convex roof – these are equal to for each subsystem of two sites, computed across the single bipartition of the pair are also shown. For entanglements 1|2 and 1|3 we also plot the exact concurrence – the agreement is good enough that the difference between the convex roof and the exact calculation is not visible.

Evolution of pairwise concurrences in the FMO complex when site one is initially excited at T = 77 K. All 21 pairwise concurrences computed by the convex roof – these are equal to for each subsystem of two sites, computed across the single bipartition of the pair are also shown. For entanglements 1|2 and 1|3 we also plot the exact concurrence – the agreement is good enough that the difference between the convex roof and the exact calculation is not visible.

Entanglement evolution in the FMO complex when site six is initially excited at T = 77 K. All 21 pairwise entanglements computed by the convex roof are also shown. For entanglements 5|6 and 4|5 we also plot the exact concurrence – the agreement is good enough that the difference between the convex roof and the exact calculation is not visible.

Entanglement evolution in the FMO complex when site six is initially excited at T = 77 K. All 21 pairwise entanglements computed by the convex roof are also shown. For entanglements 5|6 and 4|5 we also plot the exact concurrence – the agreement is good enough that the difference between the convex roof and the exact calculation is not visible.

Monogamy bound and convex roof computation of entanglements 1|34, 2|34, 12|3, and 12|4. Particularly in the first 200 fs the convex roof closely matches the monogamy bound.

Monogamy bound and convex roof computation of entanglements 1|34, 2|34, 12|3, and 12|4. Particularly in the first 200 fs the convex roof closely matches the monogamy bound.

Entanglement evolution of FMO complex when site 1 is initially excited at cryogenic temperature T = 77 K. The triplet site entanglement among site 1, 2, and 3 and also the pairwise site entanglement between any two of site 1, 2, and 3 are plotted. The left panel shows the dynamics of the entanglement for the system alone while the right considers the effect of the environment.

Entanglement evolution of FMO complex when site 1 is initially excited at cryogenic temperature T = 77 K. The triplet site entanglement among site 1, 2, and 3 and also the pairwise site entanglement between any two of site 1, 2, and 3 are plotted. The left panel shows the dynamics of the entanglement for the system alone while the right considers the effect of the environment.

Time evolution of entanglement for multiple sites at T = 300 K. In the upper panel the entanglement measures across the indicated bipartitions among sites 1, 2, and 3 are shown when site 1 is initially excited. For the lower panel, site 6 is initially excited.

Time evolution of entanglement for multiple sites at T = 300 K. In the upper panel the entanglement measures across the indicated bipartitions among sites 1, 2, and 3 are shown when site 1 is initially excited. For the lower panel, site 6 is initially excited.

Measures of entanglement and monogamy bounds in a four qubit system when site 1 is initially excited at temperature T = 77 K. The measures computed for bipartitions 4|123, 7|123, and 7|456 by the convex roof and together with the monogamy bound are shown here. We see a larger variation in performance of the convex roof optimization here, with a smaller difference between the upper (convex roof) and lower (monogamy) bounds for 7|456 and 7|123 than for 4|123.

Measures of entanglement and monogamy bounds in a four qubit system when site 1 is initially excited at temperature T = 77 K. The measures computed for bipartitions 4|123, 7|123, and 7|456 by the convex roof and together with the monogamy bound are shown here. We see a larger variation in performance of the convex roof optimization here, with a smaller difference between the upper (convex roof) and lower (monogamy) bounds for 7|456 and 7|123 than for 4|123.

Time evolution of various entanglement measures for subsystem 1234 for site 1 initially excited at T = 77 K. The concurrence for subsystem 34 across bipartition 3|4 and the measures for subsystems 123 and 124 across bipartitions 3|12 and 4|12, respectively, are also shown. The measure across bipartition 4|123 is also shown.

Time evolution of various entanglement measures for subsystem 1234 for site 1 initially excited at T = 77 K. The concurrence for subsystem 34 across bipartition 3|4 and the measures for subsystems 123 and 124 across bipartitions 3|12 and 4|12, respectively, are also shown. The measure across bipartition 4|123 is also shown.

Entanglement measures for the four chromophore subsystem 1234 when site 1 is initially excited at temperature T = 77 K. The measure across bipartition 12|34 was computed via the convex roof procedure and is shown here, together with the concurrences for pairs of chromophores 13, 14, 23, and 24. We note that in this case, we see that the entanglement 12|34 evolves similarly to both the 1|3 and 2|3 concurrences.

Entanglement measures for the four chromophore subsystem 1234 when site 1 is initially excited at temperature T = 77 K. The measure across bipartition 12|34 was computed via the convex roof procedure and is shown here, together with the concurrences for pairs of chromophores 13, 14, 23, and 24. We note that in this case, we see that the entanglement 12|34 evolves similarly to both the 1|3 and 2|3 concurrences.

Time evolution of concurrences and measures for the FMO complex when site 6 is initially excited at temperature T = 77 K. The measures are shown are for subsystem 4567 across bipartitions 4|567, 5|467, 6|457, and 7|456. We also show the concurrences among the pairs of sites that determine the concurrence bounds for the measures across bipartitions 4|567, 5|467, 6|457, and 7|456, and the concurrence bounds themselves. The left panel shows the isolated system evolution and the right panel shows the open system dynamics with environment.

Time evolution of concurrences and measures for the FMO complex when site 6 is initially excited at temperature T = 77 K. The measures are shown are for subsystem 4567 across bipartitions 4|567, 5|467, 6|457, and 7|456. We also show the concurrences among the pairs of sites that determine the concurrence bounds for the measures across bipartitions 4|567, 5|467, 6|457, and 7|456, and the concurrence bounds themselves. The left panel shows the isolated system evolution and the right panel shows the open system dynamics with environment.

Evolution of entanglement measures for the subsystems of chromophores 12345, 12346, and 12347 in the FMO complex at cryogenic temperature T = 77 K for site 1 is initially excited. The measures are computed across bipartitions 5|1234, 6|1234, and 7|1234 and the corresponding monogamy bounds are also shown. Site 1, 2, 3, and 4 are sites evolved in the population pathway under this initial condition, and this data indicates that the entanglement of this subset (1234) of chromophores with the other three chromophores is small.

Evolution of entanglement measures for the subsystems of chromophores 12345, 12346, and 12347 in the FMO complex at cryogenic temperature T = 77 K for site 1 is initially excited. The measures are computed across bipartitions 5|1234, 6|1234, and 7|1234 and the corresponding monogamy bounds are also shown. Site 1, 2, 3, and 4 are sites evolved in the population pathway under this initial condition, and this data indicates that the entanglement of this subset (1234) of chromophores with the other three chromophores is small.

Time evolution of entanglement measures in the FMO complex for site 6 initially excited at cryogenic temperature 77 K. The measures are shown for subsystem 14567 across bipartition 1|4567 and subsystem 24567 across bipartition 2|4567.

Time evolution of entanglement measures in the FMO complex for site 6 initially excited at cryogenic temperature 77 K. The measures are shown for subsystem 14567 across bipartition 1|4567 and subsystem 24567 across bipartition 2|4567.

Time evolution of entanglement measures in the FMO complex for site 6 initially excited at 77 K. (cf. Figure 7 ). The measures are computed for subsystems 3567 across bipartition 3|567, subsystem 4567 across bipartition 4|567 and subsystem 34567 across bipartition 3|4567.

Time evolution of entanglement measures in the FMO complex for site 6 initially excited at 77 K. (cf. Figure 7 ). The measures are computed for subsystems 3567 across bipartition 3|567, subsystem 4567 across bipartition 4|567 and subsystem 34567 across bipartition 3|4567.

Entanglement measures for the full FMO system at 77 K with site one initially excited. The measures are shown for the partitions 1|23456, 4|123567, 5|123467, 7|123456 (solid lines), together with the corresponding monogamy bounds (dotted lines). These results illustrate the performance of the convex roof optimization and also show that the largest of these measures is that which gives the entanglement of chromophore 1 with the rest, 1|23456.

Entanglement measures for the full FMO system at 77 K with site one initially excited. The measures are shown for the partitions 1|23456, 4|123567, 5|123467, 7|123456 (solid lines), together with the corresponding monogamy bounds (dotted lines). These results illustrate the performance of the convex roof optimization and also show that the largest of these measures is that which gives the entanglement of chromophore 1 with the rest, 1|23456.

Entanglement measures for the full FMO system at 77 K with site one initially excited. The measures are shown for the partitions 2|13456, 3|124567, 6|123457 (solid lines), together with the corresponding monogamy bounds (dotted lines). These results illustrate the performance of the convex roof optimization and also show that the largest of these measures is that which gives the entanglement of chromophore 2 with the rest, 1|23456.

Entanglement measures for the full FMO system at 77 K with site one initially excited. The measures are shown for the partitions 2|13456, 3|124567, 6|123457 (solid lines), together with the corresponding monogamy bounds (dotted lines). These results illustrate the performance of the convex roof optimization and also show that the largest of these measures is that which gives the entanglement of chromophore 2 with the rest, 1|23456.

Entanglement measures for the full FMO system at 77 K with site one initially excited. These four plots show measures computed across all 21 bipartitions of the seven chromophore system into a pair of chromophores and the remaining quintuplet. Any measure that includes either chromophore 1 or chromophore 2 (but not both) on one side of the bipartition exhibits oscillations and the value of the measure is large. Any measure that has both chromophore 1 and 2 on the same side of the bipartition takes lower values and exhibits rapid growth in the first 100 fs, but never exceeds 0.5 in value.

Entanglement measures for the full FMO system at 77 K with site one initially excited. These four plots show measures computed across all 21 bipartitions of the seven chromophore system into a pair of chromophores and the remaining quintuplet. Any measure that includes either chromophore 1 or chromophore 2 (but not both) on one side of the bipartition exhibits oscillations and the value of the measure is large. Any measure that has both chromophore 1 and 2 on the same side of the bipartition takes lower values and exhibits rapid growth in the first 100 fs, but never exceeds 0.5 in value.

Entanglement measures for the full FMO system at 77 K with site one initially excited. These six plots show measures computed across all 35 bipartitions of the seven chromophore system into three of chromophores and the remaining four. Any measure that includes either chromophore 1 or chromophore 2 (but not both) on one side of the bipartition exhibits oscillations and the value of the measure is large. Any measure that has both chromophore 1 and 2 on the same side of the bipartition takes lower values and exhibits rapid growth in the first 100 fs, but never exceeds 0.5 in value.

Entanglement measures for the full FMO system at 77 K with site one initially excited. These six plots show measures computed across all 35 bipartitions of the seven chromophore system into three of chromophores and the remaining four. Any measure that includes either chromophore 1 or chromophore 2 (but not both) on one side of the bipartition exhibits oscillations and the value of the measure is large. Any measure that has both chromophore 1 and 2 on the same side of the bipartition takes lower values and exhibits rapid growth in the first 100 fs, but never exceeds 0.5 in value.

A comparison of the effects of adding in the two-exciton subspace for different values of |α^{2}|. The concurrence between sites one and two is plotted for the density matrix in Eq. (21) , with |α^{2}| = 0.5, 0.1, 0.01.

A comparison of the effects of adding in the two-exciton subspace for different values of |α^{2}|. The concurrence between sites one and two is plotted for the density matrix in Eq. (21) , with |α^{2}| = 0.5, 0.1, 0.01.

A comparison of the effects of adding in the two-exciton subspace γ|α|^{2}|0000011⟩⟨0000011| for different values of γ, with |α|^{2} = 0.5. The concurrence between sites one and two is plotted for the density matrix in (22) , with ρ2 = |0000011⟩⟨0000011|.

A comparison of the effects of adding in the two-exciton subspace γ|α|^{2}|0000011⟩⟨0000011| for different values of γ, with |α|^{2} = 0.5. The concurrence between sites one and two is plotted for the density matrix in (22) , with ρ2 = |0000011⟩⟨0000011|.

A curve showing how the amplitude of the concurrence between sites 1 and 2 at 21 fs varies as a function of γ, with the density matrix from (22) , with ρ2 = |0000011⟩⟨0000011| and |α|^{2} = 0.5.

A curve showing how the amplitude of the concurrence between sites 1 and 2 at 21 fs varies as a function of γ, with the density matrix from (22) , with ρ2 = |0000011⟩⟨0000011| and |α|^{2} = 0.5.

## Tables

Subsystems and bipartite cuts relevant to the FMO system. One may take a subsystem reduced density matrix of any m ⩽ 7 and consider all the bipartite cuts of each subsystem. This leads to a combinatoric explosion of different bipartite measures. Evidently it would be simpler to consider all cuts of the total system. We perform such convex roof calculations for the full seven chromophore system by restricting the convex roof to the single exciton manifold.

Subsystems and bipartite cuts relevant to the FMO system. One may take a subsystem reduced density matrix of any m ⩽ 7 and consider all the bipartite cuts of each subsystem. This leads to a combinatoric explosion of different bipartite measures. Evidently it would be simpler to consider all cuts of the total system. We perform such convex roof calculations for the full seven chromophore system by restricting the convex roof to the single exciton manifold.

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