Manipulating and protecting entanglement by means of spin environments

We study the dynamical behavior of two initially entangled qubits, each locally coupled to an environment embodied by an interacting spin chain. We consider energy-exchange qubit-environment couplings resulting in a rich and highly non trivial entanglement dynamics. We obtain exact results for the time-evolution of the concurrence between the two qubits and find that, by tuning the interaction parameters, one can freeze the dynamics of entanglement, therefore inhibiting its relaxation into the spin environments, as well as activate a sudden-death phenomenon. We also discuss the effects of an environmental quantum phase transition on the features of the two-qubit entanglement dynamics.


I. INTRODUCTION
The interplay between coherent and incoherent processes is key in the quantum mechanical processing of information. Systems designed in order to perform a given communication or computational task have to cope with the detrimental effects of the surrounding world, which can affect an otherwise coherent process in many distinct ways [1]. In the context of distributed quantum information processing (QIP), where networks of spatially remote quantum nodes are used in order to process information in a delocalized way, a good assumption is that each local processor is affected by its own environment. Such an architecture for a QIP device is currently at the focus of extensive and multifaceted theoretical and experimental endeavors [2].
Exstensive work has been performed on the incoherent dynamics resulting from the coupling of QIP systems with baths weakly perturbed by the system-induced back-action. The loss of quantum correlations due to the environment has recently received considerable attention [3], with particular emphasis on the phenomenon of environment-induced entanglement sudden death [4] (i.e., complete disentanglement in a finite time), which has also been experimentally tested for the case of electromagnetic environments [5]. And yet, especially for solid-state implementations of quantum processors, the case of structured environments is extremely relevant [6]. In this framework, the dynamics leading to complete disentanglement of two qubits coupled with a common spin environments (the so-called "central-qubit model") has been extensively studied [7][8][9]. Exponential decay of the concurrence [10] between two qubits initially prepared in pure states has been observed, the decay rate being enhanced for working points close to the quantum phase transition of the environment [8].
In this paper, we study the dynamical evolution of the entanglement between two remote qubits coupled with mutually independent spin environments. The exact time dependence of their reduced density matrix is obtained using an original approach [12,13] which allows us to track the dynamics of quantum averages of one-and two-spin observables and entanglement properties. In stark contrast with most of the available literature, here we consider a "transversal" qubitenvironment coupling that allows for energy transfer, resulting in a dissipative-like behavior of the qubits and in a much richer entanglement dynamics.
The transversal nature of the qubit-environment coupling allows us to reveal the occurrence of entanglement sudden death even when starting from initially pure states of the qubits, there included the maximally entangled ones which are extremely relevant for QIP. The above feature cannot emerge in the case of longitudinal couplings as reported in Refs. [7][8][9]11].
We show that, by setting the environment in different operating regime, one can induce either entanglement sudden death or a freezing effect. Moreover, we shed light onto the differences in behavior experienced by the parallel and antiparallel concurrence introduced in [17], whose interplay is crucial in determining the properties of the quantum correlations within the two-qubit state, in particular at the environmental quantum critical point where the antiparallel entanglement appears to be better preserved than the parallel one.
The paper is organized as follows. In Sec. II we define the physical system and the relevant interactions. In Sec. III we provide exact expressions for the dynamics of a single qubit coupled with a spin-environment. These results are used in Sec. IV in order to get the dynamics of the entanglement between two qubits, each coupled with a spin-environment via a local isotropic (Subsection IV A) or anisotropic (Subsection IV B) exchange interaction. Finally, in Sec. V we draw our conclusions and suggest a possible scenario where the main features of our physical model can be embodied.
FIG. 1: Sketch of the physical model. Each of a pair of entangled qubits, Q A and Q B , is locally coupled with a spin chain, Γ A and Γ B , via the HamiltoniansĤ 0 A andĤ 0 B , respectively. The dynamics within each chain is ruled by the intra-chain Hamiltonians,Ĥ Γ A andĤ Γ B . The wavy line indicates initial entanglement.

II. THE SYSTEM
We consider two non-interacting subsystems, A and B, each consisting of a qubit Q κ (κ = A, B) coupled with a chain Γ κ of N κ interacting S = 1/2 particles. Whenever useful we will hereafter use the index κ = A, B so as to generically refer to either of the two subsystems A or B. As usual, the qubits Q κ are described in terms of S = 1/2 spin operators, which we indicate asŝ 0κ . Operatorŝ n κ (n κ = 1, ..., N κ ) corresponds to the spin located at site n of the chain Γ κ . Notice that although the above notation suggests the spin describing Q κ to sit at site 0 of the respective chain Γ κ , this is just a useful convention, but has no implication about the physical nature of Q A and Q B .
The intra-chain interaction is of XY-Heisenberg type with local magnetic fields possibly applied in the z-direction (1) where h n κ is the field applied at site n κ and J x,y n κ are the coupling strengths of the intra-chain interactions. Each Γ κ is open-ended, while neither the J x,y n κ 's nor the magnetic fields h n κ need to be uniform along the chains. Qubit Q κ is coupled with the first spin of its environment, embodied by Γ κ , via an exchange interaction of strengths J x,y 0 κ and can be subjected to a local magnetic field h 0 κ directed along the z-direction. The corresponding Hamiltonian readŝ The Hamiltonian of the total system κ is thus given byĤ κ = H 0 κ +Ĥ Γ κ . In Fig. 1 we provide a sketch of the model considered. As stated previously, although A and B do not interact directly, they experience joint dynamics due to the possibility of sharing initial entanglement. Depending on the choice of the local interaction parameters and magnetic fields inĤ κ the efforts required to tackle the model greatly changes. In Section III we describe the approach used in order to achieve an exact solution for the dynamics of local observables.

III. EXACT SINGLE-QUBIT DYNAMICS
We first concentrate on the dynamics of one subsystem only (thus dropping the index κ throughout this Section). In particular, we are interested in the evolution of a given initial state of the qubit Q, as determined by its coupling with the chain Γ and under the influence of the local magnetic field. We resort to the Heisenberg picture, which has been recently shown to provide a convenient framework for the analysis of quantum many-body systems of interacting particles [12]. The key step is the use of the Campbell-Baker-Hausdorff (CBH) formula in the management of the time-evolution operator of the system. For an operatorÔ associated with an observable of a physical system with HamiltonianĤ, the CBH formula reads (we set = 1 throughout the paper) In virtue of the algebra satisfied by the Pauli matrices, we find that upon application of Eq. (3), the time evolution of the components ofŝ 0 readŝ whereσ α n (α = x, y, z) are the Pauli operators for the spin at site n andP n = n−1 i=1σ z i . The time-dependent coefficients Π x n (t) and ∆ x n (t) are the components of the (N + 1)-dimensional vectors Π x (t) and ∆ x (t) defined by where T stands for transposition, the vector v has components v i = δ i0 [14] and the tri-diagonal matrix τ has elements The coefficients Π y (t) and ∆ y (t) are obtained from Eqs. (5) and (6) by replacing τ with τ T . As τ τ T is real and symmetric, there is an orthogonal matrix U that diagonalizes it, so that (τ τ T ) p = U Λ 2p U T , with Λ the diagonal matrix whose elements λ i j = λ i δ i, j are the (positive) square roots of the eigenvalues of τ τ T . Similarly, there is an orthogonal matrix V that diagonalizes τ T τ and such that (τ T τ ) p = V Λ 2p V T with the same diagonal matrix Λ as above. As a consequence, after straightforward matrix algebra, one can sum up the timedependent series in Eqs. (5) and (6) to get where Ω(t) and Σ(t) are diagonal matrices with elements Ω i j (t) = cos(λ i t)δ i j and Σ i j (t) = sin(λ i t)δ i j . The above equations hold regardless of the local magnetic fields or the couplings J x,y n and J x,y 0 . Adopting the language of Refs. [12,13], the components of Π x,y (t) and ∆ x,y (t) embody the fluxes of information from the qubit Q to the spin chain Γ.
By means of Eqs. (4) one can determine the time evolution of the single-qubit density matrix where1 1 is the identity operator and · indicates the expectation value over the initial state of the system. Once the diagonalizations required for determining the vectors Π x(y) (t) and ∆ x(y) (t) are performed, we need to evaluate the expectation values entering Eq. (9). Such a task can be performed within two different scenarios. In the first, Γ has a small number of spins, so that one can design the precise structure of its initial state. In the second, Γ consists of a large number of spins, which puts the analysis in the thermodynamic limit, where we can benefit of specific symmetry properties ofĤ Γ . Here, we concentrate on the latter situation, which has been the subject of several recent papers, due to the fact that it can be used for describing a proper qubit-environment system. We assume that Q and Γ are initially uncorrelated and set the former in an arbitrary single-qubit stateρ 0 and the latter in an eigenstate |Ψ Γ ofĤ Γ . The initial state of the total system will thus beρ =ρ 0 ⊗ |Ψ Γ Ψ Γ |. Under such conditions, the procedure described in the above section is most conveniently implemented as Eqs. (4) greatly simplify due to the properties ofĤ Γ . In particular, the conservation rule [Ĥ, Using Eqs. (9)-(12) one can finally evaluate the single-qubit density matrixρ 0 (t).

IV. EXACT TWO-QUBIT DYNAMICS
We now consider the complete system A∪B. Under the assumption of non-interacting subsystems, the time propaga- Despite the absence of interaction, A and B might still display dynamical correlations depending on the initial state of the total system. In fact, if A∪B is prepared in an entangled state, its dynamical properties will depend on the structure of such initial state and the entanglement evolution will follow from the interactions ruling A and B separately. The values of the parameters enteringĤ κ might thus be considered as knobs for the entanglement dynamics.
We prepare the total system at time t = 0 in so that we can use the results of Sec. III and write the twoqubit density matrix at time t in terms of the time-evolved single-qubit one. In fact, using the results of Ref. [15], we have with ρ Q κ i j (t) the elements of the single-qubit density matrix at time t and K a tensor of time-dependent coefficients. The two-qubit state can then be written as where lower-case indices take values 1 and 2, while capital ones are defined according to by preparing both the chains in any of the eigenstate of their respective Hamiltonians, we can explicitely evaluate Eqs. (10)-(12) and hence Eq. (9). By comparing the latter with Eq. (14) we finally determine the coefficients K pr i j (t), thus fully specifying the dynamics of the two-qubit stateρ Q A Q B (t).
Our approach is fully general and can be used in a variety of different situations. Here we concentrate on the case where the initial state of the two qubits is one of the Bell states which we dub parallel (|φ ± ) and antiparallel (|ψ ± ) Bell states [17]. In virtue of the symmetries of such states, the concurrence of the two-qubit state can be written as C ↑↑ (t) = |ρ 14 |− ρ 22ρ33 (18) with C ↑↑ (0) = 1 forρ Q A Q B 0 (0) = |φ ± φ ± | and C ↑↓ (0) = 1 for ρ Q A Q B 0 (0) = |ψ ± ψ ± |. Here the notationρ IJ is used, for the sake of clarity, to indicate the matrix elements ρ Q A Q B IJ (t) defined in Eq. (15). In what follows C a (t) [C p (t)] is the concurrence corresponding to the case where the qubits are initially prepared in an antiparallel [parallel] Bell state.
As for the environments, we take two identical chains of an (equal) even number of spins N with homogeneous intrachain couplings and field, i.e. J x,y n κ =J and h n κ =h. Both chains are prepared in the ground state of H Γ κ , which is found via Jordan-Wigner and Fourier transformations [18,19]. Straightforward calculations yield the relevant mean values entering Eqs. (4) as where, for convenience of notation, we omit the index κ. We have introduced ϕ j,k = √ 2/(N + 1) sin( jϑ k ), ϑ k = kπ/(N + 1), k ∈ [1, N] and the Fermi wave number k F is determined by the magnetic field [19]. Moreover, due to the absence of symmetry breaking in Ψ Γ κ , it is σ x,y n κ = 0. We consider N finite but large enough to avoid finite-size effects to influence our results. For the range of parameters considered here, N = 50 is found to fulfill such condition and it is then chosen to set the length of both chains. Finally, we take the same coupling strength between each qubit and its chain, that is J α 0 κ = J α 0 .

A. Isotropic coupling between the qubit and the chain
We now consider an isotropic coupling between each qubit and its respective environment, defined by J x 0 κ = J y 0 κ = J 0 in Eq. (2) In the theory of open quantum systems, such coupling typically corresponds to a dissipative interaction treated in the rotating wave approximation [20]. For isotropic coupling, the total HamiltoniansĤ κ have rotational invariance along the zaxis. This implies τ = τ T , and thus Π x n (t) = Π y n (t) ≡ Π n (t), ∆ x n (t) = ∆ y n (t) ≡ ∆ n (t), which allows us to write From the above expressions we see that single-qubit states initially directed along the z-axis of the Bloch-sphere maintain such alignment regardless of the Hamiltonian parameters.
On the other hand, initial "equatorial" states with ŝ z 0 (0) = 0 evolve in time and remain on the equatorial plane only for zero overall magnetic field. The rotational invariance around the z-axis of the total HamiltoniansĤ A andĤ B has relevant consequences also on the evolution of the entanglement, as shown in Section IV B.
In the fully homogeneous case, i.e. for h 0 = h and J 0 = J, it is λ q = −2(h − cos qπ N+2 ) and the j th component of the corresponding eigenvector has the same form as ϕ j,q with N + 1 replaced by N + 2. Hereafter, time will be measured in units of J −1 . In the thermodynamic limit where N → ∞, the summations can be replaced by integrals yielding and, for the z component where J n (x) are the Bessel functions. Long-time expansions show that the mean value of the single-spin x(y) component decays as t − 3 2 . If h 0 = h = 0, Eq. (22) reduces to s z 0 (t) = γ(t) σ z 0 with γ(t) = J 2 1 (2t)/2t 2 , yielding a t −3 scaling at long times. Eqs. (21) and (22) give an excellent approximation also for finite N ( 50) within a time range where finite-size effects have not yet occurred. As the latter are caused by reflection of propagating excitations at the boundary of the chain, we can neglect finite-size effects for times up to ∼ N. In fact, as the maximum one-excitation group velocity is 2, it takes at least a time N for an excitation to leave Q and travel back to it.
Let us now analyze the evolution of the entanglement between the two qubits. We first notice that in the present case of isotropic coupling between Q κ and Γ κ and given that the exchange interaction along the chain is set to be of XX type, both HamiltoniansĤ κ have rotational invariance around the z axis. This enforces a disjoint dynamics of the off-diagonal matrix elements (ρ 23 andρ 14 ) entering Eq. (17) and (18). As a consequence, if C ↑↓ (0) = 0 it is C ↑↓ (t) = 0 at any time [the same holds for C ↑↑ (t)]. Moreover, we find that C a (t)≥C p (t) for any symmetric τ , irrespective of the Hamiltonian parameters. This means that antiparallel entanglement is more resilient to the effects of the spin environment. Using the analytical solutions given above we finally obtain the time evolution of the concurrence. For h=0 and in the fully omogeneous case of J 0 =J, we find from which, by using the definition of γ(t) in terms of Bessel functions, we infer that at t ESD ≃ 0.9037 ESD occurs. In fact, our results show that sudden death occurs also for J 0 J, exhibiting coupling-dependent characteristics. It is remarkable that disentanglement at finite time occurs, here, for pure initial states of the two qubits, a feature due to the specific form of qubit-environment coupling considered here.
In the weak coupling regime J 0 ≪ J, Fig. 2 shows that the concurrence relaxation-time grows as J 0 decreases. On the other hand, for strong coupling J 0 ≫ J, the non-Markovian character of the environments becomes evident [as shown in Fig. 3] and entanglement revivals occur due to the finite memory-time of the spin chain. These revivals can be intuitively understood as the result of the strong coupling between the two-qubit system and the spins at the first site of each chain. A large J 0 gives rise to almost perfectly coherent interactions within such qubit-spin pair, only weakly damped by leakage into the rest of the chains. A revival time (the time after which the revival exhibits a maximum) of about π/J 0 is inferred from Fig. 3, for J 0 /J ≫ 1. In Fig. 4 (a) such findings are summarized in terms of the dependence of log 10 [t ESD ] on log 10 [J 0 /J]. The quasi-linear trend shown there reveals that the growing rate of t ES D for J 0 /J < 1 is slightly larger than the decreasing rate at J 0 /J > 1. In Fig. 4 (b) we show the behavior of the revival time log 10 [t rev ] against log 10 [J 0 /J], which also exhibits a quasi-linear trend. The presence of finite magnetic fields significantly changes the dynamics of the entanglement. If h 0 > 0 and h = 0, i.e. the field is only applied to the qubits, we expect an effective decoupling of each qubit from the dynamics of its environment, such that bothŝ 0 κ precess with a Larmor frequency that depends mostly on h 0 , though it is subjected to small quantitative corrections due to the interaction with Γ κ at rate J 0 . In fact, in Fig. 5 we see that a larger amount of entanglement is mantained for considerably longer times as h 0 grows. Due to the above decoupling, as well as the condition h 0 A = h 0 B , correlations between Q A and Q B are preserved and so is their concurrence. In Fig. 5 (b) the same effect is illustrated by the timeaveraged concurrence C a,p = (1/δt) δt C a,p (t ′ )dt ′ (the average is calculated over a time window δt that excludes the oscillatory transients observed in Fig. 5 (a) for Jt 10 and Jt 45). The average entanglement grows with h 0 . We further notice that parallel and antiparallel concurrences are almost identical (with C a > C p as expected), though their difference vanishes only as h 0 → ∞. We have also found that for h 0 A h 0 B the phase relation between individual precessions is lost and no entanglement preservation is consequently observed.
We now switch off the magnetic field on both qubits (that is, we take h 0 = 0) and apply a finite field h > 0 on the environments. In this case a particularly interesting effect is observed as h becomes larger than the saturation value h = J and the dynamics of both chains slows down. As a consequence, after the transient, the dynamics of the correlations between the two qubits is considerably suppressed, due to the difficulty of the qubits to exchange excitations with saturated environments. A long-time entanglement memory effect results from this, which is evident in Fig. 6 (a). There, we also notice a reduction of the wiggling, which further witnesses the freezing of the entanglement dynamics. It should be remarked that such effect is profoundly different from the decoupling mechanism highlighted previously, where C a −C p was a monotonic function of the magnetic field. Here, in fact, a peak occurs in the difference between time-averaged con- currence components when h=J, as shown in Fig. (6) (b), revealing a drastic change in the entanglement behavior at the onset of an environmental QPT [21]. Clearly, at the environmental critical point, the antiparallel entanglement is favored against the parallel one, which is at the origin of the peculiar behavior observed in Fig. 6 (b) for the dashed line. The drastic change in the behavior of the average concurrence observed at h/J = 1 is unique of the mechanism discussed here and, as already stressed, clearly distinguished from the freezing effect due to mismatched frequencies at each qubit-environment subsystem. For h>J, the effect is fully established and the average concurrence increases, while C a and C p get closer to each other. Moreover, by defining Z(t) = (1/2t 2 ) N n=1 (n + 1) 2 J 2 n+1 (2t) and using the exact analytical expressions (valid for h > J) we see that when the environments are saturated (i.e. all the spins of the chains are aligned along the z axis) the concurrence dynamics does not depend on the magnetic field.

B. Anisotropic coupling
We finally consider the case of anisotropic coupling J x 0 J y 0 between each qubit and its respective environment (the chain). Differently from the case of isotropic coupling studied in Subsection IV A and as a direct consequence of the fact that the total magnetization of A and B is not conserved, the off-diagonal elementsρ 23 andρ 14 of the two-qubit reduced density matrix are not dynamically disjoint. This implies the possibility for the concurrence of the two-qubit state to switch from the parallel to the antiparallel type and viceversa.
For the sake of clarity, we consider extremely anisotropic conditions, setting J y 0 κ = 0. In the case of no magnetic field on both qubits, h 0 = 0, a very simple expression for the concurrence is found, due to the fact that ŝ x 0 κ (t) is a constant of motion. In particular, if the two qubits are initially prepared in The magnetic fields are set to zero everywhere but on the chain Γ A , where the field is let to change within the saturation region, h A = 1, 2, 10 (bottom to top). Solid (Dashed) lines are for for C ↑↑ (C ↑↓ ). All quantities are dimensionless. a combination of the two antiparallel Bell state, their concurrence will evolve as C On the other hand, if parallel Bell states are used to build up the initial entangled state, it is C , the two-qubit concurrence cannot switch between C ↑↑ and C ↑↓ . On the contrary, if Π y 0 A (t) Π y 0 B (t), one can drive the twoqubit system from parallel-type to antiparallel and viceversa. In fact, the switching between parallel and antiparallel entanglement is observed whenŝ 0 A andŝ 0 B are flipped, under the effect of the coupling with the first spin of their respective chain, at different frequencies (for instance when J x A J x B ), or when the dynamics of one subsystem is slowed down with respect to the other (for instance due to the fact that the field on one of the two chains is larger than the saturation value, as seen in Sec. IV A). However, as discussed in Ref. [3], a twochannel entanglement evolution has an upper bound given by the product of the one-channel dynamics. Therefore, for such an "entanglement switching" to occur, the Hamiltonian parameters of subsystems A and B should be set so as to retain high entanglement values. By fixing J x B , while setting h A ≫ J x A in order to slow down the entanglement relaxation in the corresponding channel, the efficiency of the switching increases. This is clearly seen in Fig. 7. On the other hand, we can decrease J x A so that channel B is far more responsible for the entanglement evolution. In this case too a very efficient switching mechanism is achieved, suggesting that the saturation region of the chain is associated to an effective decoupling of the qubit from its corresponding environment. Finally, we notice that, being the coefficients defined by Eqs. (8) regular oscillating functions of time, the concurrence can only vanish on a countable number of points on the temporal axis and cannot remain null for finite intervals of time. Therefore, entanglement sudden death is not observed.

V. CONCLUSIONS
We have analyzed the dynamics of an entangled qubit-pair connected to two structured environments composed of openended and finite interacting spin chains. The intra-chain inter-action has been modeled by an XX Heisenberg-like Hamiltonian, while the coupling between each qubit and its respective environment has been realized via an XY exchange term with the first spin of the chain. Application of uniform magnetic fields has been also considered. We have exactly determined the time-dependent two-qubit density matrix, starting from information gathered on the single-qubit dynamics.
We have then provided analytical solutions both for the case of finite even N and in the thermodynamic limit, thus getting access to a full-comprehensive and general analysis of entanglement evolution. Particular emphasis has been given to the relaxation-like dynamics implied by the specific type of coupling considered, which gives rise, under suitable conditions, to a sudden death of the entanglement that we have analyzed. Interestingly, we have unveiled the occurrence of ESD also when starting from initially pure two-qubit states, a peculiarity of our model that does not emerge under "longitudinal" qubit-environment couplings.
By manipulating the transverse magnetic field on the initially entangled qubits, we have shown the possibility of decoupling their dynamics from that of their respective environments, thereby allowing for a dynamical entanglement protection. On the other hand, when the magnetic fields applied to both chains are larger than the saturation value, the dynamics of the environments slows down and entanglement sudden death is not observed. Interesting features are observed when the environments undergo a quantum phase transition, which in this case happens when the field applied to the chain gets the saturation value, in particular as far as the the behaviour of parallel and antiparallel concurrence is concerned.
Our work provides the analytical characterization of the transverse (i.e. energy exchanging) coupling between a simple and yet interesting out-of-equilibrium system (the two qubits) and a non-trivial spin environments (the two chains). As spin models are now understood as effective descriptions of many different physical systems, our results hold the premises to find fertile application to a variety of cases. As a particularly interesting situation, one can think about the engineering of an effective spin environment by using unidimensional arrays of small-capacitance superconducting Josephson junctions [22], which show a sharp phase transition from Josephson-type behavior to Cooper-pair tunneling Coulomb blockade analogous to that of an XY model. This implementation thus constitutes a nearly ideal test for our predictions, since the effective environmental parameters can be modified through the use of gate voltages and external magnetic fluxes. It would be very interesting to study the applicability and relevance of a study such as the one performed here to the investigation of the properties of intrinsically open systems in quantum chemistry and biology exposed to finite-sized environments. In this context, it is particularly significant that the mathematical model used in order to describe the radical pair mechanism [23] bears some analogies with the central-qubit model.