Abstract
We introduce the concept of spatiotemporal steering (STS), which reduces, in special cases, to EinsteinPodolskyRosen steering and the recentlyintroduced temporal steering. We describe two measures of this effect referred to as the STS weight and robustness. We suggest that these STS measures enable a new way to assess nonclassical correlations in an open quantum network, such as quantum transport through nanostructures or excitation transfer in a complex biological system. As one of our examples, we apply STS to check nonclassical correlations among sites in a photosynthetic pigmentprotein complex in the FennaMatthewsOlson model.
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Introduction
Quantum steering is an intriguing quantum phenomenon, which enables one party (usually referred to as Alice) to use her different measurement settings to remotely prepare the set of quantum states of another spatiallyseparated party (say Bob). This ability, which is not achievable without quantum resources, was first described by Schrödinger^{1} in his response to the work of Einstein, Podolsky, and Rosen (EPR)^{2} on quantum entanglement and the related question about the completeness of quantum mechanics. As recently shown^{3}, quantum steering (also refereed to as EPR steering) is, in general, weaker than Bell’s nonlocality^{4, 5} but stronger than quantum entanglement^{6}. After eighty years, quantum steering has been gradually formulated mathematically^{3, 7,8,9,10} and observed experimentally^{7, 11,12,13,14,15,16,17,18,19,20}. Other developments include: using steering as a resource for quantuminformation processing, quantifying steering^{9, 10, 21,22,23}, clarifying its relationship to the problem of the incompatibility of measurements^{24,25,26,27,28}, connecting steering with quantum computation^{29, 30}, and multipartite quantum steering^{30,31,32,33,34}, among various other generalizations and applications (see ref. 35 and references therein).
Nonclassical temporal correlations (like photon antibunching) play a fundamental role in quantum optics research, since the HanburyBrown and Twiss experiments^{36} and the Glauber theory of quantum coherence^{37}. While there is as yet no clear temporal analog of quantum entanglement, attempts at defining such have led to new ideas about quantum causality (see, e.g., refs 38–40 and references therein). Recently, temporal steering^{41} was introduced as a temporal analog of EPR steering, which refers to a nonclassical correlation of a single object at different times. Contrary to temporal entanglement, temporal steering has a clear operational meaning^{29, 41,42,43,44,45,46,47}. In particular, temporal steering was used for testing the security of quantum key distribution protocols^{41, 46} and for quantifying the nonMarkovian dynamics of open systems^{44}. Recently, temporal steering was also experimentally demonstrated^{47} by measuring the violation of the temporal inequality presented in ref. 41. Moreover, a measure of temporal steering was proposed^{44, 46} and experimentally determined^{47}.
Here, we introduce the concept of spatiotemporal steering (STS) as a natural unification of the EPR and temporal forms of steering. In addition, we propose two measures of STS, specifically, its robustness and weight. We also show the usefulness of STS in testing and quantifying nonclassical correlations of quantum networks by analyzing two examples, including the decay of nonclassical correlations in quantum excitation transfers in the FennaMatthewsOlson (FMO) protein complex, which is one of the most widely studied photosynthetic complexes^{48}. Note that STS can also be applied to test quantumstate transfer in quantum networks like those described in refs 49, 50.
Results
Temporal steering: From temporal hiddenvariable model to temporal hiddenstate model
Let us briefly review the socalled temporal hiddenstate model for a single system at two moments of time^{29, 41, 44}. Consider that, during the evolution of the system from time 0 to time t, one can perform measurements using different settings {x} and {y} to obtain outcomes {a} and {b} at times 0 and t, respectively. If one makes two assumptions: (A1) noninvasive measurability at time 0, which means that one can obtain a measurement outcome without disturbing the system, and (A2) macrorealism (macroscopic realism)^{51}, which means that the outcome of the system preexists, no matter if a measurement has been performed or not. Under these conditions, there exist some hidden variables λ, which a priori determine the joint probability distributions^{52,53,54,55,56,57}
Now, if one replaces the assumption (A2) with (A2’), which means that during each moment of time the system can be described by a quantum state σ _{ λ }, which is determined by some hidden variables λ independent of the measurements performed before, then the hidden variables determine not only the observed data table \(p(ax)={\sum }_{\lambda }\,p(\lambda )p(ax,\lambda )\) at time t = 0, but also a priori the quantum state \(\rho ={\sum }_{\lambda }\,p(\lambda ){\sigma }_{\lambda }\) at time t. It is convenient to define the temporal assemblage
where \({\tilde{\sigma }}_{ax}(t)\) is the observed quantum state at time t conditioned on the earlier measurement event ax at time 0. Thus, the temporal assemblage is a set of subnormalized states, which characterizes the joint behaviour: (1) \(p(ax)={\rm{tr}}[{\sigma }_{ax}^{{\rm{T}}}(t)]\) and (2) \({\tilde{\sigma }}_{ax}(t)={\sigma }_{ax}^{{\rm{T}}}(t)/{\rm{tr}}[{\sigma }_{ax}^{{\rm{T}}}(t)]\). Furthermore, the formulation of the temporal hiddenstate model can be written as
Quantum mechanics predicts some assemblages, which do not admit the temporal hiddenstate model, and we refer to this situation as temporal steering ^{44}. Note that since the hiddenstate model is a strict subset of the hiddenvariable model, using the former model may admit an easier detection of the nonclassicality of the quantum dynamics than using the hiddenvariable model.
Spatiotemporal steering
Similarly, we can also generalize the hiddenstate model to the hybrid spatio and temporal scenario. That is, we would like to consider the hiddenstate model for a system B at time t, after the local measurement has been performed on a system A at time 0. Then, under the assumptions of noninvasive measurement for the system A at time 0 and the hidden state for the system B at time t, the spatiotemporal hiddenstate model is written as (for brevity, the term “spatiotemporal” will be sometimes omitted hereafter).
where \({\sigma }_{ax}^{\mathrm{ST},B}(t)\equiv {p}_{{\rm{A}}}(ax){\tilde{\sigma }}_{ax}^{{\rm{ST}},{\rm{B}}}(t)\), with \({\tilde{\sigma }}_{ax}^{{\rm{ST}},{\rm{B}}}(t)\) being the observed quantum state of the system B at time t, conditioned on the measurement event ax [with corresponding data table p _{A}(ax)] of the system A at time 0. When there is no risk of confusion, we will abbreviate \({\sigma }_{ax}^{{\rm{ST}},{\rm{B}}}(t)\) as \({\sigma }_{ax}^{{\rm{ST}}}(t)\), p _{A}(ax) as p(ax), and \({\sigma }_{\lambda }^{{\rm{B}}}\) as σ _{ λ }. The set of subnormalized states \({\{{\sigma }_{ax}^{{\rm{ST}}}(t)\}}_{a,x}\) is refereed to as a spatiotemporal assemblage having the property \(p(ax)={\rm{tr}}[{\sigma }_{ax}^{{\rm{ST}}}(t)]\) and \({\tilde{\sigma }}_{ax}^{{\rm{ST}}}(t)={\sigma }_{ax}^{{\rm{ST}}}(t)/{\rm{tr}}[{\sigma }_{ax}^{{\rm{ST}}}(t)]\), and can be certified if it admits the model, given by equation (3), via the following semidefinite programming (SDP) (see ref. 58 for SDP, and refs 8, 9, 27 for dealing with the certification of the hiddenstate model for a given assemblage):
where \({\rho }_{\lambda }\equiv p(\lambda ){\sigma }_{\lambda }\), and the notation ρ _{ λ } ≥ 0 denotes that ρ _{ λ } is a positivesemidefinite operator. Quantum mechanics predicts that
with ρ _{0} being the initial quantum state shared by the systems A and B at time 0, {F _{ ax }}_{ a } being the positiveoperatorvalued measure representing the measurement x. The quantum channel Λ describes the time evolution of the postmeasurement composite system from time 0 to time t see the schematic diagram in Fig. 1(a).
With an appropriately designed ρ _{0}, {F _{ ax }}_{ a,x }, and Λ, the assemblage cannot be written in the form of equation (3) i.e., there is no feasible solution of the SDP problem given in equation (4). In this situation, the assemblage is said to be spatiotemporal steerable. To quantify the degree of such steerability, we would like to introduce the quantifier called the STS weight (ST SW), which is defined as
(the same techniques have been demonstrated in refs 9, 44). \({\{{\sigma }_{ax}^{\mathrm{ST},\mathrm{US}}(t)\}}_{a,x}\) stands for the unsteerable (US) assemblage i.e., one admits equation (3), \({\{{\sigma }_{ax}^{\mathrm{ST},S}(t)\}}_{a,x}\) represents the steerable assemblage, and 0 ≤ μ ≤ 1. This can be formulated as the following SDP problem:
In addition, we would like to introduce another measure, referred to as the STS robustness (ST SR), which can be viewed as a generalization of the EPR steering robustness^{10} to the present spatiotemporal scenario. The STS robustness ST SR can be defined as the minimum noise \({\tau }_{ax}^{{\rm{ST}}}(t)\) to be added to \({\sigma }_{ax}^{{\rm{ST}}}(t)\), such that the mixed assemblage is unsteerable. That is, \(ST\,SR=\,{\rm{\min }}\,\alpha \,{\rm{subject}}\,{\rm{to}}\,{\{\frac{1}{1+\alpha }{\sigma }_{ax}^{{\rm{ST}}}(t)+\frac{\alpha }{1+\alpha }{\tau }_{ax}^{{\rm{ST}}}(t)={\sigma }_{ax}^{\mathrm{ST},\mathrm{US}}\}}_{a,x}\). This can also be formulated as an SDP problem. Specifically,
The STS robustness and weight, analogously to their EPR counterparts, have different operational meanings and properties. For example, one could expect that these measures can imply different orderings of states, analogously to this property exhibited by various measures of entanglement^{59,60,61}, Bell nonlocality^{62}, and nonclassicality^{63}. A detailed comparison of these two STS measures will be given elsewhere^{64}. Here, we have calculated the STS weight for Example 1, and the STS robustness for Example 2 in the following sections, just to show that these measures can easily be computed and interpreted.
Examining nonclassical correlations within a quantum network
A possible application of STS is that it can be used to witness whether two nodes of a quantum network are nonclassically correlated (or quantum connected). Consider two qubits on the opposite ends of a quantum network, as shown in Fig. 1(b). There may be a damage somewhere in the network, such that the quantum coherent interaction between distant nodes may be inhibited. To verify this, one can initially perform measurements at time t _{A} = 0 on siteA. On siteB, one performs measurements at a later time t. If the value of the STS weight (or, equivalently, the STS robustness) is always zero for the whole range of time t, one can say that the influence of the quantum measurement at siteA is not transmitted to siteB in a steerable way.
Example 1: The spatiotemporal steering weight in a threequbit network
As an example of STS in a quantum network, let us apply a simplified model of two qubits coherently coupled via a third qubit Fig. 2(b). The interaction Hamiltonian of the entire system is
where \({\sigma }_{+}^{i}\) (\({\sigma }_{}^{i}\)) is the raising (lowering) operator of the ith qubit respectively, while J _{12} (J _{23}) is the coupling strength between qubits 1 (2) and 2 (3). To simulate the damage in the network, and quantify it, we assume qubit 2 may suffer noiseinduced dephasing. For simplicity, the two coupling strengths are equal, i.e., \({J}_{12}={J}_{23}\equiv J\). The STS weight, calculated as described above, is plotted in Fig. 2(b). We can see that if the dephasing rate γ is very small, the STS weight oscillates with time t, revealing the coherent interaction between qubits 1 and 3 via the middle qubit. If γ is large (i.e. the middle node is damaged), one sees the growth of the STS weight at a later time. One can imagine that if the dephasing is very strong, it can inhibit the appearance of the STS weight. However, several caveats arise in that the apparent correlations may be transmitted via other means than the network itself (via some environment or eavesdropper). A possible opening for future research in this area is to consider a multipartite extension, and whether it can be used as a measure of quantum communities in networks^{65}.
Example 2: The spatiotemporal steering robustness in the FennaMatthewsOlson complex
Much attention has been devoted to the possible functional role of quantum coherence^{66, 67} in photosynthesis bacteria, since the observation of possible quantum coherent motion of an excitation within the FMO complex – a photosynthetic pigmentprotein complex^{68,69,70}. A simple treatment of the excitation transfer in the FMO complex normally considers seven coupled sites (chromophores), as shown in Fig. 3, and their interaction with the environment. The hierarchy method^{71,72,73,74,75} or other openquantum system models^{76, 77} can be used to explain the presence of quantum coherence and predict the physical quantities observed in experiments.
Empowered by STS, one can ask the following questions for a network like the FMO protein complex: When an excitation arrives at site6, and propagates through the network, how large is its quantum influence, if any, to other sites? When do such nonclassical correlations vanish? Previously, quantum entanglement in the FMO complex has been theoretically analyzed^{78}. Given the fact that the excitation transfer is dynamic in nature, with a specific starting site (site1 or site6), it is more natural to examine the nonclassical correlation between sites at different times by using the STS measures. However, we point out that evaluating these measures requires measurements in different “excitation” bases at both source and target sites. Thus, evaluating these measures represents an analysis of the network itself, and how quantum correlations propagate through it, much akin to the approach taken in ref. 79.
The model Hamiltonian of the single FMO monomer containing N sites can be written as (see, e.g. ref. 80 and references therein):
where the state Pauli operators represent an electronic excitation at site n, (n \(\in \) 1, …, 7), such that \({\sigma }_{z}^{(n)}={e}^{(n)}\rangle \langle {e}^{(n)}{g}^{(n)}\rangle \langle {g}^{(n)}\), ε _{ n } is the site energy of chromophore n, and J _{ n,n′} is the excitonic coupling between the nth and n′th sites. In the literature, because of the rapid recombination of multiple excitations in such a complex, it is common to simplify drastically this model by assuming that the whole complex only contains a single excitation. In that case the 2^{7} dimensional Hilbert space is reduced to a 7 dimensional Hilbert space. Here, while we also assume only a singleexcitation, we keep the full 2^{7} dimensional Hilbert space to enable us to consider measurements in a basis which represent superpositions of excitations at various sites. (Note that for simplicity, we omit the recently discovered eighth site^{81}).
In the regime that the excitonic coupling J _{ n,n′} is large compared with the reorganization energy, the electronnuclear coupling can be treated perturbatively^{82}, and the opensystem dynamics of the system can be described by the HakenStrobl mastertype equation^{83, 84},
where ρ is the system density matrix, and L[ρ] denotes the Lindblad operators
where the Lindblad superoperator L _{sink} describes the irreversible excitation transfer from site3 to the reaction center:
where \(s={\sigma }_{+}^{(R)}{\sigma }_{}^{\mathrm{(3)}}\), with \({\sigma }_{+}^{(R)}\) representing the creation of an excitation in the reaction center, and Γ denotes the transfer rate. The other Lindblad superoperator, L _{deph}, describes the temperaturedependent dephasing with the rate γ _{dp}:
where \({A}_{n}={\sigma }_{z}^{(n)}\). This dephasing Lindblad operator leads to the exponential decay of the coherences between different sites in the system density matrix. The puredephasing rate γ _{dp} can be estimated by applying the standard BornMarkov systemreservoir model^{85, 86}. We assume an Ohmic spectral density, which, combined with the BornMarkov approximations, leads to a dephasing rate directly proportional to the temperature^{86}. While more complex treatments are necessary to fully describe the true dynamics of the FMO complex, here we restrict ourselves to this weakcoupling Lindblad form for numerical efficiency and easier interpretation of results. Note that there exists a factor 1/8 between the dephasing rate γ _{dp} here and the orthodox one in the 7site model.
In the FMO monomer, the excitation transferring from site3 to the reaction center takes place on a time scale of ~1 ps, and the dephasing occurs on a time scale of ~100 fs^{86}. These two time scales are both much faster than that of the excitonic fluorescence relaxation (~1 ns), which is, thus, omitted here for simplicity. Here we present the values used for the system Hamiltonian in calculating the excitation transfer^{87}:
Here the diagonal elements correspond to ε _{ n }, and the offdiagonals to J _{ n,n′}. We omit the large groundstate offset, as it does not influence the results. This FMO dynamics description is based on our former work^{80}.
In Fig. 4, we numerically calculated the STS robustness of site6 to other sites by using the HakenStrobl equation of motion^{80, 84}. In plotting this figure, the temperature is chosen to be T = 15 K with the corresponding dephasing rate γ _{dp} = 7.7 cm^{−1} and the decay rate (into the reaction center from site3 only) Γ = 5.3 cm^{−1}. As seen from this figure, the largest STS robustness occurs from site6 to site5. This is because site6 and site5 have the second largest intersite coupling (≈89.7 cm^{−1}) in the whole network. Another interesting fact is that the robustness of site6 to site7 has the second largest magnitude (with a time delay) and the longest vanishing time (death time) of the STS robustness. In view of the coupling strength of the Hamiltonian, this may be due to the relative strong couplings of site5 to site4 (≈70.7 cm^{−1}) and site4 to site7 (≈61.5 cm^{−1}), such that the influence from site6 is transferred through these sites with a time delay. In other words, the STS robustness not only gives the magnitude of the nonclassical correlations between two sites, but also gives the information of how long the nonclassical correlation takes to arrive, and how long it can be sustained.
Conclusions
Although the concept of spatiotemporal quantum entanglement is fundamentally difficult to be described consistently, we showed that STS, describing a certain type of spatiotemporal nonclassical correlations, can indeed be defined and quantified in an operational way. We hope that this may provide a wider view than the purely spatial or temporal correlations separately. In addition, we showed that STS, with its measures, including the STS weight and STS robustness, can be useful to assess nonclassical correlations in quantum networks or other open quantum systems. As an application, we described two examples of testing nonclassical correlations in a toy model of a threequbit quantum network and in a more realistic model of the excitation transfer in the sevensite FMO complex. We believe that STS can be useful also for testing nonclassical correlations of more complex biological systems^{66, 79} and for describing quantum transport through artificial nanostructures^{88,89,90,91}. Finally, we mention that a possible experimental demonstration of STS can be based on a delayedtime modified version of the experiment on temporal steering reported in ref. 47.
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Acknowledgements
The authors acknowledge fruitful discussions with HuanYu Ku and Karol Bartkiewicz. We acknowledge the support of a grant from the John Templeton Foundation. This work is supported partially by the National Center for Theoretical Sciences and Ministry of Science and Technology (MOST), Taiwan, grant number MOST 1032112M006017MY4. S.L.C. acknowledges the support of the DAAD/MOST Sandwich Program 2016 No. 57261473. C.M.L. and G.Y.C. are supported by the Ministry of Science and Technology, Taiwan, under the Grants Numbers MOST 1042112M006016MY3 and 1052112M005008MY3, respectively. F.N. was also partially supported by the RIKEN iTHES Project, the MURI Center for Dynamic MagnetoOptics via the AFOSR award number FA95501410040, the IMPACT program of JST, CREST, and a GrantinAid for Scientific Research (A).
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Y.N.C., N.L., C.M.L., and A.M. conceived the idea. S.L.C. carried out the calculations under the guidance of Y.N.C. and G.Y.C. All authors contributed to the interpretation of the work and the writing of the manuscript.
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Chen, SL., Lambert, N., Li, CM. et al. SpatioTemporal Steering for Testing Nonclassical Correlations in Quantum Networks. Sci Rep 7, 3728 (2017). https://doi.org/10.1038/s41598017037894
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DOI: https://doi.org/10.1038/s41598017037894
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