Robust control of long distance entanglement in disordered spin chains

We derive temporally shaped control pulses for the creation of long-distance entanglement in disordered spin chains. Our approach is based on a time-dependent target functional and a time-local control strategy that permits to ensure that the description of the chain in terms of matrix product states is always valid. With this approach, we demonstrate that long-distance entanglement can be created even for substantially disordered interaction landscapes.

Many elementary tasks of quantum information processing can be performed on small scales with existing technology. For example, state tomography [1] on a single qubit is routinely done in many laboratories, and the number 15 has been factorized with a quantum device [2]. Realizing these tasks on larger scale is one of the most pressing challenges in todays research on engineered quantum systems: characterizing the state of many qubits is an actively pursued problem even on the theoretical side [3], and factorizing 77 or 187 is still impossible with our available technology. Similarly, entangled states of two qubits can be prepared with many systems [4,5], but most setups are not scalable; the number of entangled photons is limited by the increasingly low probabilities of spontaneous events [6], and satisfactory scaling has so far been demonstrated for trapped ions only [7].
A central difference between trapped ions and other systems is that ions interact via long-range interactions, what facilitates the creation of strongly entangled states. Most other systems, however, are limited by rapidly decreasing interactions, and this disadvantage easily compensates the added value of the long coherence times [8] that can be found e.g. in impurities of solid state lattices like N V -centers. These unfavorable interaction properties can be improved if auxiliary quantum systems are available to mediate interactions [9][10][11], and the establishment of long-distance entanglement via chains of auxiliary spins seems to be a very promising route [12,13].
Most approaches, however, rely on perfectly ordered chains, whereas any implantation of auxiliary spins is likely to result in a slightly disordered chains with non-uniform interactions. Our goal is to devise temporally shaped control fields that permit to establish long-distance entanglement independently of specific realization of such disorder. Compensation of disorder through suitably designed control fields is well established and has been demonstrated abundantly. In particular numerical pulse design [14,15] has proven very successful against various of disorders. In our current goal, however, numerical approaches suffer from the inherent growth of complexity of composite quantum systems with the number of constituents. We therefore resort to a description in terms of matrix-product-states (MPS) that permit to treat systems of several hundreds of spins [16,17]. Based on the MPS description and the underlying variational ansatz [18], the most advanced numerical algorithms being able to treat time evo-lution for large systems involving small amount of entanglement have been put forward [19,20]. Despite their limitation to describe weakly entangled states, MPS are a viable option for our purpose since we target the creation of strong entanglement among few distant spins. Whereas MPS fail to describe strongly entangled states of N spins without loosing their favourable scaling in N , they are perfectly capable to describe the state of an N spin system in which only a subset M ≪ N is strongly entangled.
Despite the favorable scaling of the computational effort with N , simulating a system with N ≫ 1 spins is a numerically expensive endeavor. Typical pulse shaping algorithms, however, rely on an iterative refinement that requires many repeated propagations [14,15], which pushes the problem from hard to practically impossible. In order to be able to treat sufficiently large spin chains, we therefore resort to a variation of Lyapunov control [21] that permits to identify a good pulse only with a single propagation. Normally Lyapunov control is based on the identification of the control field that maximizes the increment of the selected goal at each instance of time. The present goal is the creation of entanglement, and since entanglement is independent of local spin orientations, we can always choose our target functional to be independent of single-spin dynamics. As realistically available means of control like microwave or laser-fields induce such singlespin dynamics, regular Lyapunov control is not applicable. It is, however, possible to consider the curvatureτ rather than the increaseτ of a target functional τ to read off an instantaneously optimal control Hamiltonian and to establish welldefined entanglement properties in this fashion [21]. The desired robustness against disorder will finally be obtained with a control Hamiltonian that is constructed via an average over an ensemble of spin chains with different interaction parameters.
If we want the first and last spin of a chain to become entangled, we may choose with the reduced density matrices 1 N for the first and last spin, and the joint density matrix 1N for both spins as target functional. Since this quantity is independent of all but the two end spins, substantial entanglement among the remaining N − 2 spins might be building up in time so that the validity of the description in terms of MPS might break down. We will Suppose initially the system is in a completely separable state and there is no direct interaction between the end spins, the present time-local control scheme will not identify any control Hamiltonian that results in an increase of τ 1N ; only at a later stage when some entanglement has been built up, will a suitable control Hamiltonian be identified. We will therefore define a time-dependent target that is such that one can always find a suitable control Hamiltonian, and that will eventually coincide with τ 1N . If we consider a chain with nearest neigbour interaction only, then the only goal that is initially achievable is entanglement between an end-spin and its direct neigbor, say spin 1 and 2. Once, this goal is achieved, one may strive for the creation of entanglement between spin 1 and 3. Such an entanglement swapping scheme can be realized with the target functionals and our control strategy shall rely on a sequence of N −1 timeintervals with τ 1j as target functional in the j − 1 st interval whose end is reached once a satisfactory value for τ 1j has been reached.
To be specific, let us consider a spin chain with an Ising interaction that is characterized by a vector ⃗ J that contains the coupling constants for the N − 1 nearest neighbour interactions. The tunable control Hamiltonian is of the form and both H ⃗ J and H c (t) are defined in terms of the usual Pauli matrices. The reduced density matrices s = Trs Ψ⟩⟨Ψ are obtained through a partial trace over the projector onto the system state Ψ⟩, where s denotes the components that are characterized by s ands are all other components. The second time derivative of τ 1j is given in terms of the first and second time derivative of the reduced density matrices, which read˙ in terms of H = H ⃗ J + H c (t) [23]. Since τ 1j is invariant under singe-spin rotations,τ 1j is linear in the control parameters g x i and g y i despite the fact thatτ 1j is bilinear in˙ s and despite the fact that¨ s is bilinear in H c (t). Similarly,τ 1j does not depend on the time derivativesġ x i andġ y i . Given the linear dependencë one readily maximizesτ 1j under the constraint that the magnitude (g x i ) 2 + (g y i ) 2 of a local control Hamiltonian is limited by some maximally admitted value. The optimum is obtained by with the normalization constant Z i chosen such that the control does not exceed its admitted strength.
The state Ψ⟩ of the N spins is described with an MPS ansatz via the matrices A ij . The dimensions of these matrices (bond dimension) limit the overall entanglement of states that can be described with this ansatz, but a bond dimension between 20 and 30 is enough for the present purposes. Once the optimal control parameters (g x y i ) opt are determined as functions of the matrices A ij one may pursue the numerical propagation. In order to avoid treating time-dependent Hamiltonians it is appropriate to choose the control parameters constant during some short time interval ∆, and to update this choice after each multiple of this period [19].   (9)) that are reached sequentially. The solid lines depict the average entanglement for an ensemble of 10 spin chains with disordered coupling constants. The peaks are the result of control that is optimized for this ensemble. The dashed lines show the average entanglement dynamics for a test-ensemble of 20 disordered spin chains resulting from the same control sequence. There is only an essentially negligible drop of pairwise entanglement, i.e. the performance of the control sequence is largely independent of the specific properties of a spin chain. σ x with eigenvalue +1, and the target τ 1j is replaced by τ 1,j+1 if τ 1j saturates.
The value of τ 1j is reduced if spins other than 1 and j participate in any entanglement. This is merely due to the necessity to keep many-body entanglement sufficiently small for an efficient simulation, but the original goal of creating long-distant entanglement is not jeopardised by an unintentional creation of additional entanglement. One should therefore characterize the performance by the pairwise entanglement, i.e. the entanglement of the reduced density matrix 1j of spins 1 and j, rather than τ 1j . We characterize the pairwise entanglement via which is a lower bound of concurrence [22,24]. Indeed, one observes a sequential growth and decline of the different selections of pairwise entanglement E 1j similar to the behavior of τ 1j depicted in Fig. 1. There is an essentially negligible decrease of the peak height as j increases, and substantial entanglement of E 1,10 ≃ 0.9974 is established between the two spins at the end of the chain.
This behavior is by no means specific for uniform chains or chains of this length. Repeating this procedure for disordered chains with J i = r i J where r i are random numbers drawn spin-interactions, the performance of control can not be enhanced arbitrarily through larger control amplitudes; it is thus advisable to choose an amplitude that is sufficiently larger than J, but sufficiently small so that an integration based on finite time-steps is reliable. In the choice of a value for ∆ one can choose a compromise between efficiency and accuracy. from a uniform distribution [0.9, 1.1] resulted in similar values of τ 1N , and tests with longer chains gave τ 1N ≃ 0.993 for N = 20 and τ 1N = 0.974 for N = 80. There is thus only a negligible decay of the achievable entanglement with increasing systems size, and we can address the central question of whether this methods permits to identify a control pulse that works independently of the specific realization of the coupling constants J i .
For this endeavour we consider an ensemble of 10 chains with different realizations of coupling, and construct the optimal control Hamiltonians via the ensemble average of Eq.(7). Fig. 2 shows the entanglement dynamics around the peaks of E 1j for the ensemble average of different realizations. Only a minor tribute is payed to the disorder, as the maximally reached value of τ 1N is 0.98; that is, there is a loss of about 2%. To address the question of wether this pulse is applicable to this specific ensemble only, or, wether it will perform equally well for any other random realization of coupling constants, we can applying the pulse to a test ensemble of 20 randomly chosen spin chains. As the dash line in Fig. 2 shows, the behavior on this test ensemble is hardly different than that on the original ensemble, and an essentially perfect maximal value of 0.976 of pairwise entanglement between the end spin has been reached. This substantiates that the present method indeed permits to extend the distances over which entanglement can be created in a well-scalable fashion.
It is interesting to notice that our approach is quite different from the notion of 'perfect state transfer' (PST) or 'almost perfect state transfer' (APST), where Alice and Bob employ certain spin chain with perfect coupling strength as quantum wire and Alice replaces her spin in this wire with another spin encoding the state to transfer then waiting for certain amount of time such that the state could transfer to the site of Bob with non-zero fidelity [25][26][27]. This certainly also permits to create some entanglement between Alice and Bob, but our control scheme has the advantages that: i) It is robust against disorder in the coupling of the spins and against the lengths of the spin chain, whereas (A)PST greatly depends on the coupling and the length of the spin wire [26,27]. ii) The amount of entanglement we could achieve is much higher than that of (A)PST, and iii) the time cost to achieve such high entanglement is much less. For example, the maximally achieved entanglement between site 1 and site 10 within time cost 4000 J in perfect spin chain by (A)PST is 0.95 [25] as compared to 0.9974 for perfect spin chain and 0.988 for disordered spin chain that can be created within 12.76 J with our method. In particular with increasing system size, the advantage of the present method become apparent: entanglement established over 80 sites within time cost 4000 J in perfect spin chain with (A)PST methods does not exceed 0.5 [25], whereas using our control strategy permits to create entanglement between site 1 and site 80 amounting to 0.974 with the time cost roughly equal 130 J.
The creation of long-distance entanglement is also by no means limited to bipartite entanglement, but may also be employed for the creation of entanglement between three or more distant spins. The demonstration of the usability of MPS for the control of large spin systems, in particular, also suggests that other goals like the implementation of multi-qubit quantum gates involving distant spins can be realized in a similar fashion. In all such situations, the applicability of MPS for a given situation can be ensured through a suitably extended target functional that makes sure that many-body entanglement remains sufficiently limited. As simple control strategies like Lyapunov control might fail to identify goals whose realization requires many elementary interactions, it helps to define a sequence of intermediate goals. In the present case we did so by sudden changes of the target functional, but also smooth, continuous modulations of targets (which itself might become object of optimization) is conceivable. Such well-designed dynamical goals together with advanced numerical techniques like MPS promise to help us make the step from small scale proof-of-principle demonstrations towards large-scales.
Acknowledgments.-The numerical work was strongly supported by Mari Carmen Bañuls. J.C acknowledges the hospitality by the Institute for Interdisciplinary Information Sciences Tsinghua University and the Institute of Physics, Chinese Academy of Sciences. Computational resources in Rechenzentrum Garching and financial support by the European Research Council within the project 'Optimal dynamical control of quantum entanglement' is gratefully acknowledged.