Multipartite entanglement transfer in spin chains
Tony J. G. Apollaro, Claudio Sanavio, Wayne Jordan Chetcuti, Salvatore Lorenzo
MMultipartite entanglement transfer in spin chains
Tony J. G. Apollaro a , Claudio Sanavio a , Wayne Jordan Chetcuti b , SalvatoreLorenzo c,a a Department of Physics, Faculty of Science, University of Malta, Msida MSD 2080, Malta b Dipartimento di Fisica e Astronomia “Ettore e Majorana” dell’ Universit`a di Catania, ViaS.Sofia 64, I-95123 Catania, Italy c Dipartimento di Fisica e Chimica - Emilio Segr`e, Universit`a degli Studi di Palermo, viaArchirafi 36, I-90123 Palermo, Italy
Abstract
We investigate the transfer of genuine multipartite entanglement across a spin- chain with nearest-neighbor XX -type interaction. We focus on the perturba-tive regime, where a block of spins is weakly coupled at each edge of a quan-tum wire, embodying the role of a multiqubit sender and receiver, respectively.We find that high-quality multipartite entanglement transfer is achieved at thesame time that three excitations are transferred to the opposite edge of thechain. Moreover, we find that both a finite concurrence and tripartite neg-ativity is attained at much shorter time, making GHZ -distillation protocolsfeasible. Finally, we investigate the robustness of our protocol with respect tonon-perturbative couplings and increasing lengths of the quantum wire.
Keywords: multipartite entanglement, quantum spin chains, perturbativedynamics
1. Introduction
Entanglement has become in the last few decades a central topic of manyapplications of quantum mechanics, ranging from quantum information [1] toquantum thermodynamics [2]. A great deal of work has been done to char-acterise its features and quantify the amount of entanglement shared amongquantum systems, see, e.g., Ref. [3] and references therein, with the focus ofdefining measures related to entanglement resource theories [4]. However, apartfrom low-dimensional bipartite systems [5], there are no necessary and sufficientcriteria to identify if a given quantum state is entangled, and the two qubitscase is the only quantum system for which currently a complete characteriza-tion of its entanglement has been achieved for both pure and mixed states [6].The multipartite entanglement in its simplest form, namely the tripartite en-tanglement shared among three qubits, is already so complex that no analytical
Email address: [email protected] (Tony J. G. Apollaro)
Preprint submitted to Journal of L A TEX Templates January 13, 2020 a r X i v : . [ qu a n t - ph ] J a n xpressions are know for its quantification. The main reason for this difficultycan be tracked back to the presence of two inequivalent SLOCC (stochasticlocal operations and classical communication) classes: the GHZ and the W class. Indeed, at variance with the two qubit scenario, where the Bell statesrepresent the maximally entangled states and every other two qubit state canbe generated from them, for three qubits, conversion between states belongingto the GHZ and the W class is impossible under SLOCC [7]. Notwithstandingthe conceptual and analytical difficulties related to three-qubit entanglement,the latter has found numerous applications both in fundamental physics, e.g., inexperimental tests of non-locality without relying on Bell’s inequality [8], andin proposed quantum information processing protocols [9], including cryptogra-phy [10], teleportation [11], and quantum error correction [12].Besides the characterization and quantification of tripartite qubit entangle-ment, an important task is also its generation, distribution, and protection. Aprototypical quantum channel is embodied by a quantum spin- chain, with thesender and receiver located at its edges [13, 14]. Whereas, the transfer of two-qubit entanglement has been extensively investigated via spin chain [15, 16, 17,18, 19], multipartite entanglement transfer between the edges of a spin chain hasnot yet been addressed. In this paper, we build on the perturbatively perfectexcitations transfer scheme [20] which has already been successfully adopted forone- and two-qubit quantum state transfer protocols [21, 22, 23]. In Ref. [20] ithas been shown that three excitations can be transferred between the edges ofa spin- XX -type chain provided that its length fulfills N = 4 n + 7, with n be-ing a non-negative integer. The excitation transfer occurs in the weak-couplingregime, of the sender and receiver block to the wire, and it approaches unity inthe limit of vanishing coupling, although at a price of a transfer time going toinfinity. Such a regime has been dubbed perturbatively perfect (PP) excitationtransfer. In this paper, we address the question as to whether the PP-excitationtransfer protocol is also efficient for the transfer of tripartite entangled states, inparticular for states belonging to the GHZ class. In the limit of weak, but finitecoupling the receiver state results to be in mixed state. Therefore, it is of theutmost importance to determine whether it is genuinely multipartite entangledand, eventually, be able to quantify its entanglement.The paper is organised as follows: in Sec. 2 we introduce the model andits dynamics, in Sec. 3 we revise some of the available tripartite entanglementwitnesses and monotones we will use for the analysis of the receiver state inSec. 3.1, where we report also the two-qubit concurrence. Finally, in Sec. 4 wedraw conclusions and outlooks.
2. The Model
In one of the most investigated quantum information transfer protocols, asender is coupled to a receiver by means of a quantum wire [13]. We rely on asimilar set-up, where both the sender and the receiver are embodied by a blockof three qubits, each weakly coupled to a quantum wire, see Fig. 1.2 ender receiverwire
Figure 1: Setup of the excitation transfer protocol. A sender (red) and receiver (blue) blockare weakly coupled by J at both edges of a quantum wire (green). Each part is made up bya 1D lattice described by the Hamiltonian in Eq. 1 with J i = J = 1, and they are coupled toeach other by J (cid:28) We consider an XX spin- Hamiltonian with nearest-neighbor interaction J i on a 1D lattice with open boundary conditions,ˆ H = N − (cid:88) i =1 J i (cid:0) ˆ σ xi ˆ σ xi +1 + ˆ σ yi ˆ σ yi +1 (cid:1) (1)where ˆ σ αi ( α = x, y ) represents the Pauli matrix of a spin- sitting on site i . Weassume couplings J i all uniform but for J i = J between the sender (receiver)block and the wire. We also set the coupling within the sender (receiver) blockand within the wire as our time and energy unit J i = J = 1.By the Jordan-Wigner transformation, the Hamiltonian in Eq. 1 can bemapped [24] into a spinless non-interacting fermion model,ˆ H = N − (cid:88) i =1 J i (cid:16) ˆ c † i +1 ˆ c i + ˆ c † i c i +1 (cid:17) , (2)where the ˆ c † i (ˆ c i ) now represents a creation (distruction) operator of a spinlessfermion on site i . Because of the quadratic nature of the Hamiltonian in Eq. 2,the diagonalisation is easily carried out and readsˆ H = N (cid:88) k =1 ω k ˆ c † k ˆ c k , (3)where { ω k , | φ k (cid:105) ≡ ˆ c † k | (cid:105)} are the eigenvalues and the eigenvectors of the N x N tridiagonal matrix with elements (cid:104) i | ˆ H | j (cid:105) = J i ( δ i,i +1 + δ i,i − ), describing thesingle-particle dynamics in the direct space basis, | i (cid:105) ≡ ˆ c i | (cid:105) . Here, and in thefollowing, | i (cid:105) ≡ | . . . i . . . (cid:105) represents a state with one excitation sitting onsite i .As in the following we are interested in the transfer, from the sender to thereceiver spins, of a | GHZ (cid:105) state | GHZ (cid:105) = 1 √ | (cid:105) + | (cid:105) ) , (4)we only need to explicate the dynamics in the 0- and 3-particle subspaces ofEq. 2, because the Hamiltonian in Eq. 1 conserves the total magnetisation inthe z -direction. 3he dynamics in the 3-particle sector is fully determined by the transitionamplitude matrix F nmrijk ( t ) between sites { i, j, k } and { n, m, r } , where i < j < k and n < m < r , having single-particle transition amplitudes f rs ( t ) as matrixelements [20] F nmrijk ( t ) = (cid:104) nmr | e − ı ˆ Ht | ijk (cid:105) = f ni ( t ) f mi ( t ) f ri ( t ) f nj ( t ) f mj ( t ) f rj ( t ) f nk ( t ) f mk ( t ) f rk ( t ) . (5)The single-particle transition amplitude is given by f rs ( t ) = (cid:104) r | e − it ˆ H | s (cid:105) = N (cid:88) k =1 e − iω k t (cid:104) r | φ k (cid:105)(cid:104) φ k | s (cid:105) = N (cid:88) k =1 e − iω k t φ rk φ ∗ ks , (6)evaluated via the eigenvalues and eigenstates of Eq. 3. Finally, the square mod-ulus of the determinant of the matrix in Eq. 5 gives the transition probabilityof the excitations between the selected sites { i, j, k } and { n, m, r } . As for the0-excitation sector, the fully polarised state | (cid:105) ≡ | ... (cid:105) is an eigenstate of theHamiltonian whose evolution can be neglected by rescaling its eigenenergy tozero | Ψ( t ) (cid:105) = e − it ˆ H | GHZ (cid:105) | (cid:105) w,r = 1 √ (cid:16) | (cid:105) + e − it ˆ H | (cid:105) (cid:17) , (7)where the initial state of our model consists of a | GHZ (cid:105) state of the first threespins and all the spins of the wire and the receiver spins in the | (cid:105) state.After a lengthy but straightforward calculation, the three qubits densitymatrix of the receiver block in the computational basis reads ρ r ( t ) = ρ ρ ρ ρ ρ ρ ∗ ρ ρ ρ ρ ρ ρ ∗ ρ ∗ ρ ρ ∗ ρ ρ
00 0 0 ρ ∗ ρ ∗ ρ ρ ∗ ρ . (8)Notice that the 1- and 2-excitations sector are block-diagonal as a consequenceof the excitation-conserving property of the Hamiltonian, whereas the 0- and3-particle sector are not because of the initial state of the sender. Each matrix4lement ρ ij can be expressed in terms of determinants of Eq. 5 as follows ρ = 12 + N − (cid:88) k 00 0 0 ρ + ρ , (10)with a similar expression holding for the other pairs in the block. Notice that allthe two-qubit density matrices are of X -type, and, consequently, the single-qubitdensity matrix will result diagonal.Let us now recap some results from Ref. [20] allowing us to express thesingle-particle transition amplitude of Eq. 6 in terms of just a few eigenvectorsof Eq. 3 exploiting the perturbative coupling regime.In Fig. 2 we express graphically the effect of the perturbative coupling be-tween the sender (receiver) block and the wire on the eigenenergies. It turns outthat, for lengths of the wire obeying n w = 4 n + 1, with n a non-negative integer,there are three resonances, one is at 1 st -order in perturbation theory, and twoare at 2 nd -order, symmetrically displaced around the former. As a consequence,only seven eigenstates give a perturbatively non-negligible contribution to Eq. 6,which can be reduced to just three taking into account the mirror-symmetry ofthe model, reflected by the symmetrical displacement of the 2 nd -order perturbedeigenenergies. Utilising elementary trigonometric identities, each single-particletransition amplitude entering Eqs. 9 via Eq. 5 is a function of only three fre-quencies ω ± = ω ± ω and ω . The former are the eigenenergies corrected at2 nd -order in perturbation theory, the latter at 1 st -order, corresponding to the5 igure 2: Perturbative analysis: the single-particle energy levels of Eq. 3 when sender, receiverand wire are uncoupled (left panel) and in the weak-coupling regime (right panel). For lengthsof the wire given by n w = 4 n + 1 there are one triple-degenerate level, resolved in energy at1 st -order perturbation theory (green circled) and two symmetric double-degenerate levels,resolved at 2 nd -order in J (violet circle). Clearly, the energy separation of the former is oforder J − , while the latter is of order J − . circles eigenergies in Fig. 2 ordered from below. For instance, the transitionamplitude of an excitation between site 1 and N − f N − ( t ) (cid:39) (cid:88) k =1 e − iω k t φ rk φ ∗ ks = 1 − ω +67 sin ω − − cos ω , (11)and between site 2 and N − f N − ( t ) (cid:39) (cid:88) k =1 e − iω k t φ rk φ ∗ ks = − sin ω +67 sin ω − . (12)Because of the different perturbation order corrections, the relation ω − (cid:28) ω (cid:28) ω +76 holds. This, as we will see, gives rise to two different time scales, T (cid:39) π ω − and (cid:101) T (cid:39) π ω dominating the oscillatory behaviour of the entanglementunder scrutiny in the following sections. Let us also stress that the 1 st -orderdoublet and the two 2 nd -order triplet perturbed eigenenergy each can supportonly one excitation. 3. Three qubit entanglement Having derived in the previous section 2 the tools to obtain the receiverthree qubit density matrix, in this section we will overview a few results aboutmultipartite entanglement we will use to tackle the multipartite entanglementtransfer problem. 6hereas two qubit entanglement criteria for an arbitrary density matrix havebeen derived [5] and entanglement monotones have a closed expression [25], forthe entanglement shared among three qubits the scenario is much more complex,and, for arbitrary mixed states no closed expression of an entanglement measureis known.One of the difficulties in characterising the entanglement shared among threequbits is the existence of six different SLOCC (stochastic local operations andclassical communication) classes for pure states: the GHZ - and W -class forgenuinely entangled states, three classes are composed by a two-qubit Bell stateand single qubit state embodying the bi-separable states with respect to eachpossible partition, and, finally, a product state of three qubits representing thefully separable state [7].This classification has been extended to mixed states [26] giving rise to ahierarchy of entanglement where local POVMs can transform states only from ahigher to a lower class, whereas each class is invariant under SLOCC, see Fig.3for the schematic structure. However, while pure states that are biseparablewith respect to each partition are also fully separable, the same does not holdfor mixed states because of the existence of PPT entangled states.For three qubits a pure state is called fully separable if it can be written inthe form | Ψ fs (cid:105) = | ψ (cid:105) | ψ (cid:105) | ψ (cid:105) (13)and a mixed state belongs to the fully separable class S if it can be written asa convex combination of fully separable pure states | ρ fs (cid:105) = (cid:88) i p i | Ψ fs (cid:105)(cid:104) Ψ fs | . (14)A pure bi-separable state, belonging to the class B , is defined as being separableunder one, or more, bi-partitions, { | , | , | } , as, e.g., in | Ψ bs (cid:105) = | ψ (cid:105) | ψ (cid:105) , (15)with qubits 1 and 2 possibly entangled. Consequently, a bi-separable mixedstate reads | ρ bs (cid:105) = (cid:88) i p i | Ψ bs (cid:105)(cid:104) Ψ bs | . (16)If a mixed state can not be written as in Eqs. 14 or 16, it contains genuinemultipartite entanglement, which can be of the W or of the GHZ -type. It holdsthat S ⊂ B ⊂ W ⊂ GHZ [26].In order to determine to which SLOCC class a three qubit pure state belongs,one can rely on the three-tangle τ [27] and the concurrence C . The GHZ classcontains states with τ > 0, whereas states in the W class have τ = 0 butfinite C (12) , C (13) , and C (23) ; bi-separable states have only one of the aboveconcurrences different from zero, and, finally, for fully separable states both τ igure 3: Schematic diagram of the classification of three qubit states: S fully separable, B bi-separable, W and GHZ non-separable. and all C ( ij ) vanish. This classification extends to mixed states by consideringthe classes in its pure state decomposition and using the convex roof extension ofa corresponding pure state entanglement measure: for GHZ -type entanglement τ is finite, for W -type entanglement τ = 0 but the concurrence of genuinemultipartite entanglement [28] is finite; whereas, for bi-separability, the squareroot of the global entanglement [29] is finite, and both the three-tangle andconcurrence of genuine multipartite entanglement are zero. Finally, for statesin the fully separable class, all entanglement measures vanish.Generally, convex roof extensions of pure state entanglement measures aredifficult to calculate as they involve an optimisation over an infinite numberof convex decompositions into pure states of a mixed state. Although efficientnumerical algorithms have been developed for several multipartite entanglementmeasures, see, e.g.,Ref. [30], for full rank density matrices, as is the one given8y Eq. 8, there is no efficient algorithm available to date.An alternative to entanglement measures is given by entanglement witnesses(EW) [31]. An EW is an hermitian operator W such that T r [ W ρ ] ≥ ρ not belonging to the entanglement class the EW aims at detecting. Assuch, W is a witness in the sense it constitutes a sufficient, but not necessarycriterion for detecting entanglement. For the GHZ-class several witnesses havebeen devised and their decomposition into local projective measurements haveallowed to detect experimentally genuine multipartite entanglement [32].In our analysis of the transfer of multipartite entanglement, we will use theentanglement witnesses of Ref. [26] W = 12 − | GHZ (cid:105)(cid:104) GHZ | . (17)One has T r [ W ρ ] > − < T r [ W ρ ] < ρ can belong either to the W or the GHZ -class, while only statesbelonging to the GHZ class have − < T r [ W ρ ] < − . For states belonging tothe W \ B class, the following witness can be used W W = − | W (cid:105)(cid:104) W | .In Ref. [33] a semidefinite programming (SDP) approach has been put for-ward in order to detect multipartite entanglement, although without distin-guishing between the GHZ - and the W -type entanglement. Using convex op-timisation technique, one is able to solve, for an arbitrary multipartite state ρ ,the minimization problem min T r [W ρ ] , (18)where W is a fully decomposable witness with respect to every bipartition of themultipartite system. Interestingly, (the negative of) Eq. 18 is also a multipartiteentanglement monotone and can hence be used to quantify genuine multipartiteentanglement [34].Apart from entanglement witnesses, the quantification of entanglement ina three qubit mixed state via the tangle τ is possible only in a few specificlow-rank cases [35]. However, bipartite entanglement measures can be used onmultipartite states by considering every possible partitions [36], and we will usein the following the tripartite negativity N ABC proposed in Ref. [37]: N (3) = (cid:113) N A | BC N AB | C N AC | B (19)where N X | Y Z is the negativity [38] N X | Y Z = (cid:80) i | λ | − , (20)with λ being the eigenvalues of the partial transpose of ρ XY Z with respect tothe subsystem X . However, due to the Peres-Horodecki criterion [39, 5], fordimensions higher than 2x2 and 2x3, N X | Y Z > X Y Z . Notice that N (3) ( ρ ) > GHZ state from ρ [40].Finally, let us also report for completeness, the concurrence between twoqubits i and j , C ( ij ) , [25]. Because all the two-qubit density matrices ρ ( ij ) inEq. 10 are of X -type, with a single non-zero off-diagonal element, the concur-rence reduces to [41] C ( ij ) = 2 max (cid:20) , (cid:12)(cid:12)(cid:12) ρ ( ij )12 (cid:12)(cid:12)(cid:12) − (cid:113) ρ ( ij )00 ρ ( ij )33 (cid:21) . (21) Let us now finally illustrate the main results of this work: the transfer ofmultipartite entanglement via perturbative couplings between a sender and areceiver block connected by a quantum wire. As we are interested only inthe receiver block, we renumber, for the sake of readability, the spins thereincontained n = 1 , , 3, starting from the edge. In Fig. 3.1 we report the resultsfor two entanglement witnesses, respectively given by Eq. 17 and Eq. 18, thetripartite negativity N (3) , Eq. 20, and the concurrence C (13) , Eq 21, betweenqubit 1 and 3 for a chain of length N = 19 and J = 0 . 01 both on a timescale of T and (cid:101) T . Being the concurrence between neighboring qubits C (12) = C (23) = 0, and the witness W W detecting W -class states positive at all timeswhen evaluated on the receiver density matrix, T r [ W ρ r ] > 0, we argue that no W -entanglement is present at any time in the receiver spins.Whereas the witness based on the fidelity with a GHZ state, Eq. 17, detectsgenuine multipartite entanglement at finite time-intervals, the witness in Eq. 18detects genuine multipartite entanglement at any time but for discrete timepoints. The latter coincide with the times when the tripartite negativity N (3) vanishes. There are regions where the witness − < T r [ W ρ ] < 0, hence thetripartite entanglement could be either GHZ or W -type. However, the fact that C (12) = C (23) = 0 is an indication that ρ belongs to the GHZ -class. The sameholds for the regions where the multipartite entanglement monotone derivedEq. 18 gives a non-zero value.We also observe that both N (3) and − T r [W ρ ] oscillate with period (cid:101) T , be-coming vanishingly small when the concurrence between qubit 1 and 3 reachesits maximum value, C (13) = . At these points t ∗ in time, the density matrixof the receiver block reads ρ ( t ∗ ) (cid:39) | (cid:105)(cid:104) | + 12 (cid:18) | (cid:105) − | (cid:105)√ (cid:19)(cid:18) (cid:104) | − (cid:104) |√ (cid:19) (22)= 12 | A C (cid:105)(cid:104) A C | ⊗ | B (cid:105)(cid:104) B | + 12 (cid:18) | A C (cid:105) − | A C (cid:105)√ (cid:19)(cid:18) (cid:104) A C | − (cid:104) A C |√ (cid:19) ⊗ | B (cid:105)(cid:104) B | , which is a biseparable state under the partition AC | B . Therefore, we can con-clude that these are the only (isolated) points in time where the state does not10 igure 4: Witness W , Eq 17, (blue dotted line) and W, Eq. 18, (black dotted line); tripartitenegativity N (3) , Eq. 19, (orange dotted line), and concurrence between qubit 1 and 3, C (13) ,Eq 21, (blue line) on a time scale of T = πω − (upper panel) and a few (cid:101) T = πω (lower panel)around the maximum of the fidelity given by τ = T (green vertical line). The horizontal redline is set at − to detect GHZ -class entanglement via the witness W . Notice that, around t = τ , C (13) and N (3) are oscillating in phase opposition. have any genuinely multipartite entanglement. Clearly, the reason for theseoscillations is that one of the excitations is travelling with frequency ω back11nd forth between the sender and the receiver block through the quantum wireexploiting the 1 st -order triplet.Analysing the short-time behaviour, we notice that qubits 1 and 3 get en-tangled with C (13) = already on a time-scale of (cid:101) T , whereas the tripartitenegativity N (3) , as well as the entanglement monotone − T r [W ρ ], is very small,Fig. 3.1. The reason still being the presence of the 1 st -order triplet, enteringthe transition amplitudes f ji with i = 1 , j = N − , N . Whereas, in orderto have finite genuinely tripartite entanglement one needs a finite probabilityto find three excitations on the receiver block, thus involving the two 2 nd -orderdoublets, which is the only term entering the transition amplitude in Eq. 12. Figure 5: Short-time behaviour of the concurrence C (13) , Eq 21, (left panel) and of thetripartite negativity N (3) , Eq. 19 (right panel). Notice that, at variance with the time scaleof T , on a times scale of the order of (cid:101) T , the two entanglement quantifiers are oscillating inphase. In Fig. 3.1 we test our protocol for increasing values of J and report a goodtransfer of genuine multipartite entanglement to the sender block in the weak-couling regime, say up to J (cid:39) . 1, after which a quick decay of the quality ofthe transfer is observed. Similarly, the transfer time τ at which the maximumis obtained follows τ ∝ J − in the perturbative regime, before breaking downafter J (cid:39) . C (13) = C (13) = , where C (13) is the concurrence ofassistance [42], evaluated by C ( ij ) = (cid:88) n =1 (cid:112) λ n , (23)where λ n are the eigenvalues of the matrix R = ρ (ˆ σ y ⊗ ˆ σ y ) ρ ∗ (ˆ σ y ⊗ ˆ σ y ). Thisquantity is the maximum entanglement achievable between two qubits by meansof LOCC operations on the complementary qubits, that is the sender and thewire qubits. 4. Conclusions In this paper we have shown how, by exploiting the weak-coupling dynam-ical regime, one is able to transfer maximally entangled three-qubit states be-tween the edges of a spin- chain with nearest-neighbor XX -interactions. We12 igure 6: (left panel) Maximum tripartite negativity N (3) , Eq. 19, blue dots, and witnessW, Eq. 18, red square, as a function of the coupling J for a chain of length N = 23. (rightpanel) Time τ at which the maximum are achieved vs J for the same parameters as in theleft panel. Notice how the two entanglement monotones change outside the weak-couplingregime. Similarly, around the same values, the power law τ ∼ J − , obtained from 2 nd -orderperturbation theory in Ref. [20], starts to fail. Lines are for guiding the eyes. have used a witness based on the fidelity with a GHZ state and one based onsemidefinite programming. The negative of the latter, as well as the tripartitenegativity, constitute also valid entanglement monotones and their dynamicsshows that genuine multipartite entanglement of the GHZ -class is efficientlytransferred with our protocol. Interestingly, although the multipartite entan-glement transfer peaks at a time scale determined by the inverse of the 2 nd -orderenergy gap in perturbation theory, a finite concurrence between a pair of spinsin the receiver block and a non-zero tripartite negativity, the latter a sufficientcondition for GHZ -distillability, is retrieved on the sender block already ontimes scales determined by the much faster inverse of the 1 st -order energy gapin perturbation theory. Moreover, while in the limit of vanishing couplings ofthe sender and the receiver block to the quantum wire, J → 0, the transferof the multipartite entanglement approaches one, but with the transfer timeapproaching infinity, we obtain that also for couplings J ∼ . 1, a significativeamount of multipartite entanglement is retrieved on the receiver spins in muchshorter time. Finally, although we were not able to evaluate the tangle, becauseof the full rank of the receivers density matrix, both the witnesses and the en-tanglement monotones considered indicate that for a large time interval aroundthe transfer time, the receivers state remains genuinely multipartite entangled,but for isolated points in time.Taking into consideration that genuine multipartite entanglement, despitethe analytical, and even numerical, difficulty of its characterisation and quan-tification, is a precious resource in many applications, ranging from cryptogra-phy to quantum error correction, we believe that a thorough investigation of itsdynamical properties may result useful and more studies in this direction areneeded. 13 ckowledgements The authors thank Prof. Andr´e Xuereb and Dr. Zsolt Bern´ad for usefuldiscussions. C. S. acknowledges funding by the European Union’s Horizon 2020research and innovation programme under Grant Agreement No. 732894 (FETProactive HOT). 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