Non-Markovianity over ensemble averages in quantum complex networks
NNon-Markovianity over ensemble averages inquantum complex networks
Johannes Nokkala ∗ , Sabrina Maniscalco , and Jyrki Piilo Turku Centre for Quantum Physics, Department of Physics andAstronomy, University of Turku, FI-20014, Turun Yliopisto, Finland Centre for Quantum Engineering, Department of Applied Physics,School of Science, Aalto University, P.O. Box 11000, FIN-00076Aalto, FinlandDecember 25, 2017
Abstract
We consider bosonic quantum complex networks as structured fi-nite environments for a quantum harmonic oscillator and investigatethe interplay between the network structure and its spectral density,excitation transport properties and non-Markovianity. After a reviewof the formalism used, we demonstrate how even small changes to thenetwork structure can have a large impact on the transport of excita-tions. We then consider the non-Markovianity over ensemble averagesof several different types of random networks of identical oscillatorsand uniform coupling strength. Our results show that increasing thenumber of interactions in the network tends to suppress the averagenon-Markovianity. This suggests that tree networks are the randomnetworks optimizing this quantity.
Understanding the dynamics of open quantum systems is important in sev-eral fields of physics and chemistry including problematics dealing, e.g., withquantum to classical transition and decoherence with its harmful effects for ∗ jsinok@utu.fi a r X i v : . [ qu a n t - ph ] D ec uantum information processing and communication. In general, formu-lating or deriving a suitable equation of motion for the density matrix pltfor the open system is often a daunting task. Perhaps the most celebratedand most used theoretical result in this context is the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation [1, 2] dρ s ( t ) dt = − i [ H s , ρ s ( t )] + (cid:88) k γ k (cid:18) C k ρ s ( t ) C † k − (cid:110) C † k C k , ρ s ( t ) (cid:111)(cid:19) , (1.1)with the associated completely positive and trace preserving dynamical mapwith semigroup property. Above, H s is the open system Hamiltonian, γ k are positive constant rates, and C k are the jump operators with k indexingthe different decoherence channels. Indeed, this master equation and thecorresponding publications had recently 40th anniversary celebrations inthe Symposium on Mathematical Physics in Toru´n in June 2016.GKSL master equation (1 .
1) describes Markovian memoryless open sys-tem dynamics and during the last 10-15 years there has been an increasingamount of research activities in understanding memory effects and quan-tifying non-Markovianity for open systems beyond the semigroup property[3, 4, 5]. A pair of complementary approaches here include a descriptionbased on quantifying the information flow between the open system andits environment [6] or the characterization of dynamical maps in terms oftheir divisibility properties [3, 8] while a large number of other ways to char-acterize non-Markovianity also exist, see e.g. [9, 10, 11, 12, 13]. Most ofthe research so far has focused on non-Markovianity using discrete variableopen systems as examples while in the current work we are interested in thememory effects in a continuous variable (CV) open system with controlledenvironmental structure.Indeed, here we consider structured finite environments modeled bybosonic quantum complex networks. While this and other kinds of quantumcomplex networks have recieved increasing attention in recent years in thecontext of perfect state transfer [14, 15], quantum random walks [16, 17],efficient entanglement distribution [18, 19, 20] and the unification of classi-cal and quantum network theory [21, 22], here the focus is on the interplaybetween the network structure and the reduced dynamics of an open quan-tum system attached to it. To this end, we investigate the impact of thestructure on the network spectral density, excitation transport propertiesand non-Markovianity of the reduced dynamics.The paper is organized as follows. Section 2 concerns the network itself.Here we present the microscopic model and briefly discuss the connection2etween the network Hamiltonian and certain matrix representations of ab-stract graphs in classical graph theory. The dynamics of the network is givenin terms of a symplectic matrix acting on the vector of operators at initialtime. In Section 3, we describe how complex quantum networks can betreated in the framework of the theory of open quantum systems as tunablestructured environments. We demonstrate how small changes in the networkstructure can have a large impact on its excitation transport properties. InSection 4, we consider the non-Markovianity of the reduced dynamics usinga recently introduced witness based on non-monotonicity of the evolution ofGaussian interferometric power. Finally, conclusions are drawn in Section5.
We set (cid:126) = 1 and work with position and momentum operators defined as q = ( a † + a ) / √ ω and p = ( a † − a ) i (cid:112) ω/
2, satisfying the commutation re-lation [ q, p ] = i . We consider networks of N unit mass quantum harmonicoscillators coupled by springlike couplings. The general form of a Hamilto-nian for such networks is H E = p T p q T Aq , (2.1)where we have introduced the vectors of position and momentum opera-tors q T = { q , ..., q N } and p T = { p , ..., p N } , and where A is the ma-trix containing the coupling terms and frequencies. It has elements A ij = δ ij ˜ ω i / − (1 − δ ij ) g ij /
2, where g ij is the strength of the springlike coupling g ij ( q i − q j ) / i and j , and˜ ω i = ω i + (cid:80) j g ij is the effective frequency of oscillator i resulting from ab-sorbing the quadratic parts of the coupling terms into the free Hamiltoniansof the oscillators.The matrix A , which completely determines the network Hamiltonian,can be related to some of the typical matrix representations of weightedgraphs, i.e. abstract networks of nodes connected by weighted edges. Byweighted, we mean that a magnitude is assigned to each connection. Thiscan be used to establish a link between the properties of the network andresults from graph theory. A paradigmatic example is the adjacency matrix V having elements V ij = w ij , where w ij is the weigth of the connection3etween nodes i and j ; a weigth of 0 corresponds to the nodes being dis-connected. Another matrix that arises very naturally is the Laplace matrix L , related to the adjacency matrix as L = D − V , where D is diagonalwith elements D ii = (cid:80) j w ij . In terms of them, matrix A can be writ-ten as A = ∆ ω / − V / A = ∆ ω / L /
2, where ∆ ˜ ω and ∆ ω arediagonal matrices of the effective and bare frequencies of the network oscil-lators, respectively, and weights are given by the coupling strengths. Thegraph aspect of this and other kinds of quantum networks have been veryrecently used to, e.g., develop a local probe for the connectivity and couplingstrength of a quantum complex network by using results of spectral graphtheory [24], and constructing Bell-type inequalities for quantum communi-cation networks by mapping the task to a matching problem of an equivalentunweighted bipartite graph [25].The Hamiltonian (2 .
1) is a special case of the quadratic Hamiltonian H = x T Mx , where the vector x contains both the position and momentumoperators and M is a 2 N × N matrix such that H is Hermitian. It can beshown [23] that quadratic Hamiltonians can be diagonalized to arrive at anequivalent eigenmode picture of uncoupled oscillators provided that M ispositive definite. Since H is Hermitian, this is equivalent with the positivityof the eigenvalues of M . In the case at hand, H E may be diagonalized withan orthogonal matrix K such that K T AK = ∆ , where the diagonal matrix ∆ holds the eigenvalues of A . By defining new operators (cid:40) Q = K T qP = K T p , (2.2)the diagonal form of H E reads H E = P T P Q T ∆Q , (2.3)which is the Hamiltonian of N decoupled oscillators with frequencies Ω i = √ ∆ ii . A bosonic quantum complex network is also an interesting system to study inits own right. Below, we review the mathematical tools useful for the task,adopting the definitions for a commutator and anti-commutator betweentwo operator-valued vectors used in [26]. While we will be later concernedwith networks initially in the thermal state, we will also briefly discuss the4ase of an initial Gaussian state without displacement. For a more detailedreview of Gaussian formalism in phase space, see [27]. What is presentedhere is straightforward to apply to the case where interactions with externaloscillators is considered, and we will do so in Section 3.Let x be a vector containing the position and momentum operators of thenetwork oscillators, and define the commutator between two operator valuedvectors as [ x , x T ] = x x T − ( x x T ) T . Now canonical commutation relationsgive rise to a symplectic form J , determined by [ x , x T ] = i J . Let x (cid:48) = Sx ,where S is a 2 N × N matrix of real numbers. In order for S to be a canonicaltransformation of x , the commutation relations must be preserved. This re-quirement gives i J = [ x (cid:48) , x (cid:48) T ] = [ Sx , ( Sx ) T ] = S [ x , x T ] S T = i SJS T , imply-ing that SJS T = J . Such a matrix is called symplectic with respect to sym-plectic form J . Symplectic matrices form the symplectic group Sp (2 N, R )with respect to matrix multiplication, which can be used to define a sym-plectic representation of the Gaussian unitary group, meaning that (up toan overall phase factor) the two groups are bijective.We fix x T = { q T , p T } = { q , ..., q N , p , ..., p N } throughout the rest of thepresent work. Then the symplectic form becomes J = (cid:16) I N − I N (cid:17) , where I N is the N × N identity matrix. By defining the vector of eigenmode operatorsto be X T = { Q T , P T } = { Q , ..., Q N , P , ..., P N } , we can express the trans-formation that diagonalizes the network Hamiltonian as X = (cid:16) K T K T (cid:17) x ;a direct calculation shows that the matrix diagonalizing the Hamiltonian isboth symplectic and orthogonal.In the eigenmode picture, the equations of motion are those of noninter-acting oscillators. By defining the auxiliary diagonal matrices with elements D Ωcos ii = cos(Ω i t ), D Ωsin ii = sin(Ω i t ) and ∆ Ω ii = Ω i , we can express them as (cid:18) Q ( t ) P ( t ) (cid:19) = (cid:18) D Ωcos ∆ − D Ωsin − ∆ Ω D Ωsin D Ωcos (cid:19) (cid:18) Q (0) P (0) (cid:19) , (2.4)where the block matrix acting on the vectors is again symplectic. To recoverthe dynamics of the network oscillators, we may use Eq. (2 .
2) to express x ( t ) in terms of either X (0) as (cid:18) q ( t ) p ( t ) (cid:19) = (cid:18) KD Ωcos K∆ − D Ωsin − K∆ Ω D Ωsin KD Ωcos (cid:19) (cid:18) Q (0) P (0) (cid:19) , (2.5)or in terms of x (0) as (cid:18) q ( t ) p ( t ) (cid:19) = (cid:18) KD Ωcos K T K∆ − D Ωsin K T − K∆ Ω D Ωsin K T KD Ωcos K T (cid:19) (cid:18) q (0) p (0) (cid:19) . (2.6)5otice that the group properties of symplectic matrices quarantees that inboth cases the block matrix remains symplectic.If we now restrict our attention to Gaussian states with zero mean, wemay define the covariance matrix of the initial state ascov( x (0)) = (cid:104) [ x (0) , x T (0)] + (cid:105) , (2.7)where the anti-commutator is defined as [ x , x ] + = x x T + ( x x T ) T . If x ( t ) = Sx (0), then the covariance matrix at time t becomescov( x ( t )) = cov( Sx (0)) = (cid:104) [( Sx (0) , ( Sx (0)) T ] + (cid:105) = S (cid:104) [ x (0) , x (0)) T ] + (cid:105) S T = S cov( x (0)) S T . (2.8)In the present case of symplectic matrices appearing in Eqs. (2 .
5) and(2 . x (0)), corresponds to the case where theinteractions are suddenly switched on at t = 0+. As here the state is notthe stationary state with respect to the Hamiltonian (2 . To implement an oscillator network, the basic requirements to meet are astatic topology, harmonic potential and quantum regime for the oscillators.To match the form of the Hamiltonian (2 . . .
5) or (2 . We consider as the open quantum system a single additional quantum har-monic oscillator interacting with one of the network oscillators. While thisis sufficient to our present purposes, what follows is straightforward to ex-tend to the case of multiple external oscillators or interactions with multiplenetwork nodes. Moreover, we will fix the states of the open system and thenetwork to be a Gaussian state and a thermal state of temperature T , re-spectively, assume factorizing initial conditions and work in such units thatthe Boltzmann constant k B = 1.The open system Hamiltonian is H S = ( p S + ω S q S ) /
2, and the formof the interaction Hamiltonian reads H I = − kq S q i , or equivalently, H I = − kq S (cid:80) Nj K ij Q j in the basis of eigenmodes, where k is the coupling strengthbetween the open system and the network. The total Hamiltonian is now H = H S + H E + H I . By including the operators of the open system as thefinal elements of the vectors of operators, we may express it analogously toHamiltonian (2 .
1) as H = { P , p S } T { P , p S } { Q , q S } T B { Q , q S } , (3.1)where the matrix B has diagonal elements B ii = Ω i / i < N + 1 and8 N +1 ,N +1 = ω S /
2, while B N +1 ,i = B i,N +1 = − k K li / i < N + 1; herethe index l is the index of the network oscillator directly interacting withthe open system. We may diagonalize the matrix B as O T BO = F where O is orthogonal and F diagonal with elements F ii = f i /
2, where f i will bethe frequencies of the modes in the fully diagonal picture. If we define thenew operators as (cid:40) Q = O T { Q , q S } P = O T { P , p S } , (3.2)the total Hamiltonian reads H E = P T P Q T F Q . (3.3)We are now in position to write down the symplectic matrix giving thedynamics of the total Hamiltonian. By following the steps leading fromHamiltonian (2 .
3) to Eq. (2 . Q ( t ) q ( t ) P ( t ) p ( t ) = (cid:18) OD cos O T O∆ − f D sin O T − O∆ f D sin O T OD cos O T (cid:19) Q (0) q (0) P (0) p (0) , (3.4)where we have introduced the diagonal matrices D cos ii = cos( f i t ), D sin ii =sin( f i t ) and ∆ fii = f i .As we will consider an initial thermal state for the network, throughoutthe rest of the present work we will consider as the initial basis the oneon the R.H.S. of the equation above, where the initial covariance matrixof the network is diagonal with elements (cid:104) Q i (0) (cid:105) = ( n i + 1 / / Ω i and (cid:104) P i (0) (cid:105) = ( n i + 1 / i , where n i = (exp(Ω i /T ) − − .If we are interested in the dynamics of the operators in the network basis,we may use Eq. (2 .
2) and define the symplectic and orthogonal N +1 × N +1matrix ˜ K with elements ˜ K N +1 ,i = K i,N +1 = 0 for i < N +1, ˜ K N +1 ,N +1 = 1,and ˜ K ij = K ij otherwise. Now q ( t ) q ( t ) p ( t ) p ( t ) = (cid:32) ˜ KOD cos O T ˜ KO∆ − f D sin O T − ˜ KO∆ f D sin O T ˜ KOD cos O T (cid:33) Q (0) q (0) P (0) p (0) . (3.5)9he dynamics can be readily determined from the initial covariance matrixof the total system, as outlined in Eq. (2 . t in the basis of either the eigenmodes or thenetwork oscillators, depending on which symplectic matrix is used. If theopen system has displacement, also the evolution of its first moments needsto be considered to determine the evolution of its state.While we are concerned with the dynamics of the open system as well asthe network oscillators, we mention here the possibility to treat the networkin the framework of Gaussian channels. For a general Gaussian state and for x (0) = (cid:16) q (0) p (0) (cid:17) , the elements of the covariance matrix of a single mode systemare cov( x (0)) ij = (cid:104) x (0) i x (0) j + x (0) j x (0) i (cid:105) / − (cid:104) x (0) i (cid:105)(cid:104) x (0) j (cid:105) . For anyGaussian channel taking the covariance matrix to time t , the transformationcan be written as cov( x ( t )) = C ( t )cov( x (0)) C ( t ) T + L ( t ) , (3.6)where C ( t ) and L ( t ) are real matrices and L ( t ) is symmetric. In terms of theelements of the symplectic matrix S of Eq. (3 . .
8) to be C ( t ) = (cid:18) S N +1 ,N +1 S N +1 , N +2 S N +2 ,N +1 S N +2 , N +2 (cid:19) , (3.7)and L ( t ) = (cid:88) i (cid:104) X i (0) (cid:105) (cid:18) S N +1 ,i S N +1 ,i S N +2 ,i S N +1 ,i S N +2 ,i S N +2 ,i (cid:19) , (3.8)where the sum is taken to 2 N + 1 excluding N + 1, such that L ( t ) is indepen-dent of the initial expectation values of the open system. The matrices C ( t )and L ( t ) now completely characterize the channel, allowing, e.g. to makecomparisons with channels defined by a master equation or to construct in-termediate channels taking the system from time t > s > t and checkingif the resulting channel is completely positive or not, as is done in a recentlyintroduced measure of non-Markovianity for Gaussian channels [39]. Thedifficulty in implementing this measure in the present case is neither in theconstruction of the intermediate map nor checking its complete positivity,but rather in the fact that it considers the limit s → t , and it is not clearhow to take such a limit in the case of numerical, rather than analytical,matrices C ( t ) and L ( t ). 10 .2 The spectral density One of the central concepts in the theory of open quantum systems is thespectral density of environmental couplings J ( ω ), which encodes the rele-vant information in the environment and interaction Hamiltonians into asingle function of frequency. The reduced dynamics of the open system canthen be determined once the initial state of the total system as well as thesystem Hamiltonian are fixed [40]. In particular, a heat bath is completelycharacterized by its spectral density and temperature. The definition ofthe spectral density, in terms of the environment eigenfrequencies Ω i andcoupling strengths to eigenmodes g i , reads J ( ω ) = π (cid:88) i g i Ω i δ ( ω − Ω i ) , (3.9)where δ is the Dirac’s delta function. The definition is rarely used in practice,since in the case of an infinite heat bath with a continuum of frequenciesthe spectral density becomes a continuous function, and phenomenologicalspectral densities are defined instead.In the case of finite environments it is convenient to use the relation be-tween J ( ω ) and the damping kernel γ ( t ), the latter appearing in the general-ized quantum Langevin equations giving the dynamics for the open systemoperators [40]. It is defined as γ ( t ) = (cid:88) i g i Ω i cos(Ω i t ) , (3.10)and the relation is given by J ( ω ) = ω (cid:90) ∞ γ ( t ) cos( ωt ) dt, (3.11)If the environment is finite, both Eq. (3 .
9) and Eq. (3 .
11) will result indelta spikes. However, by replacing the upper limit of integration by a finitetime t max , the intermediate form of the spectral density can be consideredinstead. If a quantum network defined by a Hamiltonian of the form (2 .
1) issufficiently symmetric, the reduced dynamics will have a regime where thesystem interacts with a continuum of frequencies as if the environment wasinfinite. This is evident from the damping kernel having a very small valueduring this transient, until finite size effects cause a revival of oscillations.The duration of this continuous regime of reduced dynamics depends on thestructure and size of the finite environment.11n the present case of quantum complex networks, the coupling strengthsto eigenmodes g i are determined by the interaction Hamiltonian H I and thematrix K diagonalizing the Hamiltonian (2 .
1) as g i = − k K li , where l isthe index of the network oscillator directly interacting with the system and k the interaction strength in the network basis. In Fig. 1, we show twoexamples of damping kernels and spectral densities for quantum networks.The symmetric network is a chain with nearest and next nearest couplings.Additionally, the chain is made homogeneous by setting the effective fre-quencies of the ends of the chain equal with the rest. The spectral density iscontinuous for the used value of t max . If the interaction time is sufficientlyshort, it would not be possible to tell from the reduced dynamics of an openquantum system coupled to the network alone that the environment is infact finite. In contrast, the disorder in the other network results in a highlystructured spectral density that does not have a continuous regime.In general, it may be asked whether J ( ω ) of a quantum complex networkcan be deduced from the reduced dynamics of the system. It can be shown[41] that, provided the coupling to the network k is weak and the networkis in a thermal state, the system excitation number is well approximatedby the expression (cid:104) n ( t ) (cid:105) = exp( − Γ t ) (cid:104) n (0) (cid:105) + n ( ω S )(1 − exp( − Γ t ), whereΓ = J ( ω S ) /ω S and n ( ω S ) = (exp( ω S /T ) − − , or the thermal averageboson number at system frequency ω S . The value of the spectral density atsystem frequency is then approximated by J ( ω S ) = ω S t ln (cid:18) ∆ n (0)∆ n ( t ) (cid:19) , (3.12)where ∆ n ( t ) = n ( ω S ) − (cid:104) n ( t ) (cid:105) . If T is known, the local value of the spectraldensity can be determined by performing measurements on the system only.This is demonstrated in Fig. 1, where the dots are probed values of thespectral density with each circle corresponding to one value of the systemfrequency. By keeping the interaction time fixed to the used value of t max ,it can be seen that even for networks with disorder, the probed values followthe shape of J ( ω ).It is also worth mentioning that the machinery introduced so far canbe used to approximate an infinite heat bath, determined by its spectraldensity, with a finite one. Together with its temperature, the finite bathis completely characterized by the coupling strengths g i and frequenciesΩ i . While there is considrebale freedom in choosing Ω i , they should coverthe non-vanishing parts of J ( ω ) and there should be enough of them topush the finite size effects to interaction times longer than what is beingconsidered. Next, the couplings are determined from the spectral density12 ymmetric network Disordered network ω k - J ( ω ) ω k - J ( ω )
200 400 600 800 t - k - γ ( t )
200 400 600 800 t - k - γ ( t ) Figure 1: (Color online) A comparison of the spectral densities and dampingkernels for a symmetric and a disordered network. The black dots are probedvalues of J ( ω ) extraced from the reduced dynamics of an open quantumsystem interacting with the networks. The symmetric network is a chain of N = 100 oscillators with nearest and next nearest neighbor couplings withmagnitudes of g = 0 . g = 0 .
02, respectively, while the disorderednetwork is a random network of N = 30 oscillators with a constant couplingstrength g = 0 .
05. For both, the bare frequency of the oscillators was ω = 0 .
25, the system-network interaction strength was k = 0 .
01 and thestates of the system and the network where a thermal state of T = 1 andvacuum, respectively. The system is coupled to the first oscillator in thechain and to a random oscillator of the disordered network.as follows. From Eq. (3 . (cid:82) ∞ π J ( ω ) ωdω = (cid:80) i g i .Approximating the integral on the left hand side with, e.g., a Riemann sum,and identifying the terms on both sides then gives g i = π J (Ω i )Ω i ∆Ω i , where∆Ω i = | Ω i − Ω i +1 | is the sampling interval. The range of interaction timeswhere the approximation is valid can be checked by comparing the dampingkernels calculated for the finite bath from Eq. (3 .
10) and for the infinitebath from the inversion of Eq. (3 . γ ( t ) = π (cid:82) ∞ J ( ω ) ω cos( ωt ) dω .The two will be similar up to the point where finite size effects manifest.13his can be of advantage when considering early or intermediate dynamicsin the case of a strong coupling, since the dynamics given by Eqs. (3 .
4) or(3 .
5) is exact.
Reservoir engineering aims to modify the properties of the environment of anopen quantum system, typically to protect non-classicality of the system orto increase the efficiency of some task. In the present case, the environment isa quantum network determined by the matrix A . To assess its properties asan environment, it is convenient to consider the effect of the structure on thespectral density J ( ω ), which can be returned to the effect of the structureon the eigenfrequencies Ω i and coupling strengths to eigenmodes g i . Bychanging the structure by, e.g., adding or removing links, one can try toeffectively decouple the system from the network by finding a configurationwhere J ( ω S ) has a small value, or alternatively to look for structures withincreases transport efficiency.In fact, assuming that the system can be freely coupled to any single nodein the network, a single network can produce as many spectral densities asit has nodes. This is because a coupling to a single node corresponds toa set of coupling strengths g i , which are in turn directly proportional to arow of the matrix K diagonalizing the network. On the other hand, the setof eigenfrequencies Ω i are completely determined by the eigenvalues of thematrix A and as such are independent of where in the network the systemis coupled.Even small changes to the network structure can have a large impacton both the network spectral density and excitation transport properties.Generally speaking, when the reduced dynamics has a continuous regime,the flow of energy is steady provided that the system is resonant with thenetwork. Furthermore, excitations can freely be exchanged between differentnodes in the network. On the other hand, when the degree of disorder inthe network is high, the excitations typically become locked to a subset ofthe network nodes and cannot spread effectively. We present examples ofthis in Fig. 2, where the same symmetric network is considered as in Fig.1. Rewiring randomly only a single coupling changes the path taken by themajority of excitations. Also shown is the excitation dynamics in a randomnetwork.While transport is inefficient in most random networks, a search can becarried out for exceptions, and indeed it can be shown that when samplingthe distribution of random networks, some rare cases have vastly superior14 Figure 2: (Color online) Examples of excitation transport in quantum net-works. In all examples, the interaction time is shown on the horizontal axiswhile the vertical axis corresponds to the index of the network oscillator.The color bar shows the difference between initial excitations and excita-tions at time t . On the left, the network is the symmetric network of Fig.1. Excitations propagate freely along the chain. In the middle, a single ran-domly chosen link in the symmetric network has been rewired, changing thetransport properties. On the right, evolution of excitations in the networkoscillators of a random network of N = 100 oscillators with bare frequency ω = 0 .
25 and coupling strengths g = 0 .
05 is shown. Excitations becomelocked to a subset of network oscillators.transport properties robust against ambient dephasing [42]. One may alsoask whether there is any connection between the excitation transport proper-ties and non-Markovianity. While in the spin-boson model non-Markovianityand the back-flow of excitations can behave similarly with respect to the en-vironment parameters [43], there does not seem to be such a connectionin the case of a continuous variable system [44]. Furthermore, even in thespin-boson model, information and excitation backflows can occur withoutthe other [45].
The dynamics of an open quantum system can significantly deviate from thememoryless Markovian case when the interaction between the open systemand the environment is strong, or if the environment is structured. Previousinvestigations [46] of harmonic chains with nearest neighbor couplings, hav-ing a Hamiltonian of the form (2 . J ( ω ) with a single band will then have two regimes of system fre-15uency where memory effects are strong while one with band-gaps will havemore. In this work, the Breuer-Laine-Piilo [47] and Rivas-Huelga-Plenio [48]measures were used.To the best of our knowledge, however, there have been no studies of non-Markovianity attempting to connect it to the structure of a complex network.While it is the case that any spectral density of an oscillator network withnon-regular structure can be replicated with an oscillator chain with nearestneighbor couplings [49, 50, 51], it is nevertheless of interest to ask whetherthe amount of non-Markovianity could be tied to the statistical propertiesof complex networks by comparing the average non-Markovianity over manyrealizations, and whether adding more structure typically increases the non-Markovianity or not.To this end, we considered three types of random networks presented inFigure 3. For all three cases, we fixed the size of the network to be N = 30and assumed that the network is connected, i.e. any node can be reachedfrom any other by following the links. The Erd˝os-R´enyi network G ( N, p )[52] is constructed from the completely connected network of N nodes byindependently selecting each link to be part of the final network with a prob-ability p . The Barab´asi-Albert network G ( N, l ) [53] is constructed from aconnected network of 3 nodes and repeatedly adding a new node with l links,connecting it randomly to existing nodes but favoring nodes which alreadyhave a high number of links, until the size N is reached. Setting l = 1 isan important special case, as the resulting network is a tree, i.e. it has thesmallest possible number of links that a connected network of size N canhave. Finally, a Watts-Strogatz network G ( N, p, n ) [54] is constructed start-ing from a circular network where all nodes are connected to n -th nearestneighbors, and then rewiring each link with the probability p . In this work,we fixed n = 2. The key concept used in several witnesses and measures of non-Markovianityis to track the dynamics of a quantity that can be shown to behave differentlyunder Markovian and non-Markovian evolutions. In this work, we considera recently introduced measure and a witness based on the non-monotonicityof Gaussian interferometric power under non-divisible dynamical maps [55].Gaussian inteferometric power Q quantifies the worst-case precision achiev-able in black-box phase estimation using a bipartite Gaussian probe com-posed of modes A and B . It is also a measure of discord-type correlations16 rd ő s - Rényi Barabási - Albert Watts - Strogatz
Figure 3: (Color online) Schematics for the used networks. Each col-umn corresponds to a network type while the rows correspond to differentparameter values. When the connection probability for the Erd˝os-R´enyi net-work is increased the number of links grows, but links are chosen randomly.This is at variance with the Barab´asi-Albert network where nodes with ahigher number of links are preferred when introducing new links, resultingin highly connected nodes when the connectivity parameter grows. Watts-Strogatz networks are constructed from a cycle graph by rewiring each linkwith a given probability. As this rewiring probability grows, the averagedistance between the nodes decreases, but the total number of links remainsconstant.between the two modes, as it vanishes for product states. For quantifyingnon-Markovianity, it is enough to consider the case where mode A is sub-jected to a local Gaussian channel while mode B remains unchanged. Thenthe expression for the Gaussian interferometric power Q has a closed form interms of the symplectic invariants of the two-mode covariance matrix σ AB [56].For Markovian channels, Q is a monotonically non-increasing functionof time, implying that ddt Q ( σ AB ) ≤
0. Any period of time where this doesnot hold is then a sign of non-Markovianity. Once the initial covariancematrix σ AB has been fixed, the degree of non-Markovianity of the reduceddynamics can then be quantified as 17 GIP = 12 (cid:90) ∞ ( |D ( t ) | + D ( t )) dt, (4.1)where D ( t ) = ddt Q ( σ AB ). While the related measure is defined with amaximization over all initial states for the bi-partite system, Eq. (4 . r = cosh − (5 /
2) and initial thermal excitations n A = n B = 1 / t = 50 andfix the frequency of the system to be the 15 th eigenfrequency of the networksto ensure that it is resonant.The results are shown in Fig. 4. For all considered cases, changingthe interaction strength affects the magnitude but not the behaviour ofnon-Markovianity against the network parameter. For Erd˝os-R´enyi andBarab´asi-Albert networks, the number of couplings between network oscil-lators grows with the parameter, reducing the amount of non-Markovianity.On the other hand, the number of couplings in the network is constant forthe Watts-Strogatz network. The results suggest that when the system isresonant with the network, non-Markovianity is highest for networks witha small amount of random couplings. For all considered coupling strengths,the highest non-Markovianity is achieved when the network is a tree. If thenetwork is highly symmetric, as is the case with Watts-Strogatz networkswith a low rewiring probability, the amount of non-Markovianity in theresonant case is very small. Non-Markovianity is increased by introducingdisorder into the network through rewiring of the couplings.Besides the results we present here, we also checked that increasingthe network temperature decreases the non-Markovianity. Furthermore, forcomparison we determined the non-Markovianity in the simple case of a ho-mogeneous chain with nearest-neighbor couplings only and found that evenat the edges of the spectral density, where memory effects are strongest, N GIP has a similar value than Erd˝os-R´enyi and Barab´asi-Albert networkshave in the resonant case. 18 = . p ⨯ GIP
Erd ő s - Rényi l ⨯ GIP
Barabási - Albert p ⨯ GIP
Watts - Strogatz k = . p ⨯ GIP l ⨯ GIP p ⨯ GIP k = . p ⨯ GIP l ⨯ GIP p ⨯ GIP
Figure 4: (Color online) A comparison of the non-Markovianity for threedifferent types of quantum complex networks. The columns correspond tothe type and the rows to interaction strength between the network and thesystem. The size of each network is fixed to N = 30 while a parameter con-trolling the structure of the network is varied. The parameters are connec-tion probability p , connectivity parameter l and rewiring probability p forErd˝os-R´enyi, Barab´asi-Albert and Watts-Strogatz networks, respectively.Refer to main text for details. Results are averaged over 1000 realizationsfor each parameter value. In this work, we have studied bosonic quantum complex networks in theframework of open quantum systems. After briefly investigating the effectof the network stucture on the spectral density and transport of excitations,we focused on the non-Markovianity in the reduced dynamics of an openquantum system interacting with the network.We considered non-Markovianity over ensemble averages of different typesof random networks of identical oscillators and constant coupling strengthbetween the network oscillators. Previous work shows that strong memoryeffects can occur in symmetric networks at the edges of the spectral den-sity and near band gaps. Here we have shown that increasing the disorder19f the network can lead to a high degree of non-Markovianity also whenthe system is resonant with the network, however increasing the number ofinteractions between network oscillators appears to suppress it, suggestingthat trees optimize the ensemble averaged non-Markovianity.While here we considered only the lower bound of a single non-Markovianitymeasure, it would be interesting to extend the investigations to other mea-sures such as the measure introduced by Torre, Roga and Illuminati [39].We expect that a systematic study could perhaps link some of the graphinvariants, such as the mean distance between nodes, to non-Markovianityand other non-classical properties of the quantum networks, such as the abil-ity to generate or transport entanglement. Such a link could pave way tostructural control of non-classical properties of quantum complex networks.Indeed, in the case of quantum walks on classical complex networks, it canbe shown that the quantumness of the walk is a function of both the initialstate and specific graph invariants. Furthermore, for a deeper understandingof quantum networks the introduction of purely quantum graph invariantswithout a classical counterpart would be needed.
Acknowledgments
The authors acknowledge financial support from the Horizon 2020 EU col-laborative projects QuProCS (Grant Agreenement No. 641277). J. N. ac-knowledges the Wihuri foundation for financing his graduate studies.
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