## Splitting the entanglement

When particles in a quantum mechanical system are entangled, a measurement performed on one part of the system can affect the results of the same type of measurement performed on another part—even if these subsystems are physically separated. Kunkel *et al.*, Fadel *et al.*, and Lange *et al.* achieved this so-called distributed entanglement in a particularly challenging setting: an ensemble of many cold atoms (see the Perspective by Cavalcanti). In all three studies, the entanglement was first created within an atomic cloud, which was then allowed to expand. Local measurements on the different, spatially separated parts of the cloud confirmed that the entanglement survived the expansion.

## Abstract

A key resource for distributed quantum-enhanced protocols is entanglement between spatially separated modes. However, the robust generation and detection of entanglement between spatially separated regions of an ultracold atomic system remain a challenge. We used spin mixing in a tightly confined Bose-Einstein condensate to generate an entangled state of indistinguishable particles in a single spatial mode. We show experimentally that this entanglement can be spatially distributed by self-similar expansion of the atomic cloud. We used spatially resolved spin read-out to reveal a particularly strong form of quantum correlations known as Einstein-Podolsky-Rosen (EPR) steering between distinct parts of the expanded cloud. Based on the strength of EPR steering, we constructed a witness, which confirmed genuine 5-partite entanglement.

Quantum mechanics poses a fundamental limit on the simultaneous knowledge of two noncommuting observables, and . This limit is given by the Heisenberg uncertainty relation for the variances of the observables, . At the same time, quantum mechanics allows for nonlocal correlations between two systems, A and B, such that it is possible to infer from the measurement results obtained in system B the outcomes for the same observable in system A more accurately than predicted by the Heisenberg uncertainty constraint for system A (*1*). This phenomenon was first discussed in the seminal paper by Einstein, Podolsky, and Rosen to show that quantum mechanics is at odds with the assumptions of local realism (*2*). Schrödinger later termed this steering (*3*), because in this scenario the state in system A is influenced by the chosen measurement in B (A is steered by B). This is possible only if A and B are strongly entangled, which makes steering a witness for entanglement (*4*). Originally intended by Einstein, Podolsky, and Rosen to question the completeness of quantum mechanics, entanglement and nonlocality are now regarded as a resource for quantum technologies, such as quantum metrology (*5*), quantum cryptography (*6*, *7*), and quantum information processing (*8*).

Pioneering work on spatially distributed entanglement has been reported in pure photonic systems and in hot atomic vapors building on atom-light interaction (*9*–*13*). Ultracold atomic gases offer additional possibilities owing to the high coherence of the internal and motional degrees of freedom. In these systems, various schemes for generating nonlocal entanglement between spatially distinct subsystems have been realized using quantum gate operations in optical lattices (*14*, *15*) and long-range interactions in Rydberg systems (*16*, *17*). However, the reliable generation and detection of entanglement in discrete-variable systems so far has been limited to small atom numbers. In the continuous-variable limit considered in this work, the generation of Einstein-Podolsky-Rosen (EPR) entanglement through nonlinear dynamics of spatial multimode systems has been proposed (*18*–*20*). Here we present a robust method to spatially distribute entanglement generated by local interactions (*21*, *22*) in the spin degree of freedom of a Bose-Einstein condensate (BEC) by subsequent expansion of the atomic cloud. This constitutes an explicit experimental implementation of the recently formulated mapping of indistinguishable-particle entanglement in one mode to individually addressable subsystems (*23*, *24*).

Experimentally, we prepared a BEC of *N* ≈ 11,000 ^{87}Rb atoms in the *F* = 1 hyperfine manifold in the magnetic substate *m*_{F} = 0 (*F*, total spin quantum number; *m*_{F}, spin projection quantum number). We initiated spin dynamics that coherently populated the states *m*_{F} = ±1 with correlated particle pairs (*25*), leading to spin-nematic squeezing (*26*). This led to entanglement shared among all atoms in the condensate. Self-similar expansion for distributing the entanglement was initiated by switching off the longitudinal confinement. The atomic cloud spread in the remaining waveguide potential. After imaging with high optical resolution, we analyzed partitions of the resulting absorption signal to reveal entanglement and EPR steering between the corresponding atomic subsystems (Fig. 1).

As noncommuting observables and , we chose the spin operators and , where . Here, is the creation operator for a particle in the spin state *m*_{F} = *j*, h.c. denotes the Hermitian conjugate, and ϕ_{0} is an offset phase. In the case of negligible populations in *m*_{F} = ±1 compared with the total atom number *N*, which is fulfilled in the experiment, these operators obey the commutation relation (*27*)

The energy difference between the *m*_{F} = ±1 and *m*_{F} = 0 states caused by the second-order Zeeman shift leads to an evolution of the phase ϕ. By adjusting the hold time before expansion, ϕ was precisely controlled (*27*).

After the expansion, we applied a resonant radiofrequency pulse corresponding to a π/2 spin rotation and read out the population difference *N*^{–}(ϕ) = *N*_{+1} – *N*_{–1}, which realizes a measurement of (*27*). Because of the commutation relation (Eq. 1), the observed variances of the population differences after spin rotation fulfill the uncertainty relation(2)This inequality also applies locally to any subsystem where *N* represents the corresponding subsystem particle number.

We analyzed the absorption images by first partitioning each of them into two halves, A and B. In subsystem A, we detected reduced (enhanced) fluctuations of at phase ϕ = 0 (ϕ = π/2) compared with the case of a fully separable initial state (Fig. 2A). The minimum fluctuations are below the separable-state limit (nematic squeezing), and the variance product exceeds the uncertainty limit. To reveal EPR steering of A, we demonstrate that a measurement in subsystem B can be used to infer the outcome in A with an accuracy beating the local uncertainty limit (*28*).

The outcome in A can be estimated by an arbitrary function of the measurement result in B. We constructed an estimator based on five subdivisions of B. The corresponding values of were used to infer the result in A through the linear combination . The real numbers *g _{k}*(ϕ) were chosen to minimize the inference variance(3)which is depicted in Fig. 2B. The inference variance quantifies the accuracy with which can be inferred by the estimator . To compare the achieved accuracy with the local uncertainty relation, we evaluated the steering product

*S*_{A|B} < 1 signals EPR steering of A by B. In our experiment, we obtained a value of *S*_{A|B} = 0.62 ± 0.12 and *S*_{A|B} = 0.51 ± 0.19 after 60 and 150 ms of spin-mixing dynamics, respectively, verifying bipartite EPR steering in our system. The given errors correspond to the statistical estimation of 1 standard deviation (SD), applying jackknife resampling. For all given variances, the independently characterized photon shot noise contribution to the absorption signal has been subtracted (*27*). To underscore the spatial separation between the steered subsystems, we discarded a fraction η of the atoms in a region between A and B. Figure 2C shows that EPR steering can be verified up to a discarded fraction of ~30% of the atoms, which corresponds to a separation of ~13 μm between the two subsystems. This is consistent with monogamy of steering (*29*), which implies that by discarding more than a third of the whole system, no steering between equal partitions of the remaining system is possible.

For indistinguishable particles, one expects that the entanglement is uniformly distributed over the whole system. We illustrated this by partitioning the absorption signal into three parts of equal length. Analogous to the analysis above, we evaluated the inference variance for all permutations of ABC. Figure 3 shows that each part is steered by the remaining atomic cloud, confirming three-way steering (*30*). For partitions that are too small, spurious effects of the imaging technique become relevant. Because the position of each atom is mapped onto a spatially distributed absorption signal, classical correlations are dominant below a certain length scale. By analyzing a fully separable coherent spin state, we confirmed that for the partitions chosen in this work, classical correlations are negligible (*27*).

The observation of EPR steering allows for statements about multipartite entanglement. Specifically, the steering product can be used to construct a witness for genuine *m*-partite entanglement. For this, we partitioned the system into a subsystem A and the remainder B, which we divided into *m *– 1 parts with equal atom numbers. Generalizing the derivation in (*30*), we find that genuine *m*-partite entanglement (*31*) is present if the inequality(5)is fulfilled, given that *g*(0) · *g*(π/2) < 0 (*27*). Here, η_{A} = *N*_{A}/*N* denotes the fraction of atoms in system A, and the inferences are . If Eq. 5 holds, the quantum state of the system cannot be written as a mixture of states that are separable with respect to all possible bipartitions. This implies that each part, or conjunction of parts, is entangled with the rest of the system. In the limit *m* → ∞, Eq. 5 cannot hold owing to the Heisenberg uncertainty limit of the full system. Experimentally, we partitioned the absorption data of the atomic cloud into two parts and varied the fraction η_{A} (Fig. 4, inset). In this way, we verified genuine 5-partite entanglement (Fig. 4).

Our results, combined with the well-developed toolbox for the manipulation of ultracold gases, offer new perspectives for applications, as well as for fundamental questions. Retrapping and storage of the produced states in tailored potentials enable quantum enhanced sensing of spatially varying external fields. With the possibility of local control, the deterministic generation of more general classes of entangled states, including cluster states that are useful for continuous-variable quantum computation, is in reach (*32*). Our general strategy for the detection of entanglement between spatially separated regions can be applied to fundamental questions concerning the role of entanglement in long-time dynamics and thermalization of quantum many-particle systems (*33*).

Complementarily to our work, the detection of spatial entanglement patterns is reported in (*34*), and the observation of entanglement of spatially separated modes is described in (*35*).

## Supplementary Materials

www.sciencemag.org/content/360/6387/413/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S8

Data S1

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**We thank M. Reid and P. Hauke for discussions.

**Funding:**This work was supported by the Heidelberg Graduate School of Fundamental Physics; the Heidelberg Center for Quantum Dynamics; the European Commission, within the Horizon-2020 program, through the FET-Proactive grant AQuS (project no. 640800) and the European Research Commission Advanced Grant EntangleGen (project ID 694561); and the DFG (German Research Foundation) Collaborative Research Center SFB1225 (ISOQUANT).

**Author contributions:**The concept of the experiment was developed in discussion among all authors. P.K. and M.P. controlled the experimental apparatus. P.K., M.P., H.S., D.L., and M.K.O. discussed the measurements and analyzed the data. M.G. derived the witness for multipartite entanglement and, together with T.G., elaborated the theoretical framework. All authors contributed to the discussion of the results and the preparation of the manuscript.

**Competing interests:**The authors declare no competing interests.

**Data and materials availability:**The data presented in this paper are available in the supplementary materials.