# Deterministic distribution of multipartite entanglement and steering in a quantum network by separable states - arXiv.org

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Deterministic distribution of multipartite entanglement and steering in a quantum network by separable states Meihong Wang1,3 , Yu Xiang2,3,4 , Haijun Kang1,3 , Dongmei Han1,3 , Yang Liu1,3 , Qiongyi He2,3,4 ,∗ Qihuang Gong2,3,4 , Xiaolong Su1,3 ,† and Kunchi Peng1,3 1 State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan, 030006, China 2 State Key Laboratory for Mesoscopic Physics, School of Physics, Frontiers Science Center for Nano-optoelectronics & Collaborative Innovation Center of Quantum Matter, Peking University, Beijing 100871, China 3 Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China 4 Beijing Academy of Quantum Information Sciences, Beijing 100193, China As two valuable quantum resources, Einstein-Podolsky-Rosen entanglement and steering play important roles in quantum-enhanced communication protocols. Distributing such quantum resources among multiple remote arXiv:2101.01422v1 [quant-ph] 5 Jan 2021 users in a network is a crucial precondition underlying various quantum tasks. We experimentally demonstrate the deterministic distribution of two- and three-mode Gaussian entanglement and steering by transmitting sepa- rable states in a network consisting of a quantum server and multiple users. In our experiment, entangled states are not prepared solely by the quantum server, but are created among independent users during the distribu- tion process. More specifically, the quantum server prepares separable squeezed states and applies classical displacements on them before spreading out, and users simply perform local beam-splitter operations and ho- modyne measurements after they receive separable states. We show that the distributed Gaussian entanglement and steerability are robust against channel loss. Furthermore, one-way Gaussian steering is achieved among users that is useful for further directional or highly asymmetric quantum information processing. Quantum entanglement is an important resource for quan- tanglement among users, e.g., distributing entanglement by tum communication and computation [1]. Besides entangle- performing joint measurement (entanglement swapping) [34– ment, Einstein-Podolsky-Rosen (EPR) steering has also been 36], or by transmitting separable states [37–44]. In the scheme identified as a valuable resource for secure quantum informa- of distributing entanglement via separable ancilla, instead of tion tasks [2–5]. The states exhibiting steering are a strict preparing entanglement directly by the quantum server, en- subset of the entangled states, and a strict superset of the tanglement between two users is created by local operations, Bell-nonlocal states [6]. Distinct from both inseparability classical communication, and transmission of a separable an- and Bell nonlocality, the steerability of two directions be- cillary mode. It has been shown that this indirect method has tween the entangled parties could be asymmetric [7, 8] even advantages for distribution of mixed Werner states with de- it can only present in one direction [9], which has been suc- polarizing and dephasing noise [42, 45]. While significant cessfully demonstrated in the pioneer works using continu- progress has been reported in recent years [37–41], as well as ous variable (CV) Gaussian states [9–13], discrete variable first experiments implemented between two qubits [42] and (DV) systems [14–17], and a hybrid CV-DV system [18]. Re- between two Gaussian modes [43, 44], the study of this effi- markably, EPR steering has been created recently in mas- cient scheme is still in its infancy, and it is fair to say that our sive [19, 20] and high-dimensional systems [21–25]. The understanding of how powerful nonlocality can be provided concept of steering is important to quantum networks since by this method remains very limited so far. For instance, a it provides a way to verify entanglement, without the trust- generalized scheme was proposed to distribute Gaussian EPR worthy requirement of the equipment at all nodes of the net- steering by separable states [46], however, by reanalyzing data work. This has abundant applications to one-sided device- from those pioneer experiments [43, 44], we find that none independent (1SDI) quantum key distribution [26–28], quan- of them were able to demonstrate the shared EPR steering. tum secret sharing (QSS) [10, 29], secure quantum teleporta- As steerability is stronger than inseparability, in general it is tion [30, 31], and subchannel discrimination [32, 33]. harder to distribute steerability than inseparability. Moreover, towards a quantum network, it becomes an even more worth- At the current technology level, it is practical to establish a while objective to deeply explore the experimental feasibility network consisting of a quantum server, which has the abil- of distributing multipartite entanglement and steering between ity to prepare and manipulate quantum states, and two or more than two users with separable states. In addition, con- more users who are merely able to perform local measure- sidering the practical channel loss, how to distribute as large ments on their states [Fig. 1(a)]. Consequently, how to dis- as possible steerability at minimal cost is another important tribute entanglement by the quantum server to make it shared problem. among remote users becomes a crucial issue. The conven- tional method is to directly generate multipartite entangled In this Letter, we experimentally demonstrate the determin- states by a quantum server locally and then send to remote istic distribution of Gaussian entanglement and steering with nodes. Alternatively, there are indirect ways to build en- separable ancillary states both in two-user and multiuser sce-

2 FIG. 1. Schematic of the distribution experiment. (a) Schematic of the quantum network. Quantum server produces quantum states and sends separable states to users. The quantum resource is shared by users after local operations. (b) Schematic of the experimental setup. Two squeezed states with −3 dB squeezing (V s = 0.50) and +5.5 dB antisqueezing (Va = 3.55) are produced by two nondegenerate optical parametric amplifiers (NOPA1 and NOPA2). Displacements for all modes are implemented by coupling modulated coherent beams with quantum states on 99:1 beam splitters. The correlated noise is added by amplitude (AM) and phase (PM) modulators, respectively. The distributed states are measured by balanced homodyne detectors for partial reconstruction of the covariance matrix. The lossy channel is simulated by a half-wave plate and a polarization beam splitter. narios. In the experiment, a quantum server prepares inde- storage oscilloscope at the sampling rate of 500 KS/s. pendent squeezed states and applies classical displacements In the experimental process, a series of correlated displace- on them, which makes initial states fully separable, and then ments (Gaussian noises) need to be optimized and added to distributes them to users; each user performs a local beam- realize this indirect distribution. Consequently, much effort is splitter operation on the received states and transmits one out- made to make sure the classical noises in same quadratures put state of the beam splitter to the next user, where the clas- are canceled at the users’ stations, that is, the added Gaus- sical displacements ensure the separability between the trans- sian noises must be synchronized. To do so, all of the dis- mitted mode and the rest of the states in the network. In- placements added on the amplitude and phase modulators are stead of providing a particular example to show the entangle- taken from two independent noise sources, respectively. With ment distribution via separable ancilla for two users [43, 44], the increase of the number of users involved in the network, we rather experimentally implement the distribution of max- comes the requirement of even more effort to synchronize the imal steerability in general by optimizing the displacements added noises on all amplitude and phase modulators. In ad- according to the initial squeezing level, transmittance of the dition, more relative phases on the beam splitters need to be beam splitter, and transmission efficiencies in the channels. controlled precisely in the distribution of the three-mode state. The distributed Gaussian entanglement and steerability are ro- bust against channel loss. Furthermore, moving beyond two The process for distributing Gaussian entanglement and parties brings up richer steerability structures including one- steering to three users, which is the smallest instance of a way and one-to-multimode steering by mere transmission of true quantum network, contains three steps. In the first step, separable ancillas, which could be used for providing unprece- the quantum server prepares a position (or amplitude quadra- dented security for a future quantum internet [47, 48]. ture) squeezed state Ĉin and a momentum (or phase quadra- ture) squeezed state Âin generated from two NOPAs, and two We demonstrate the distribution of multipartite Gaussian coherent (or vacuum) states B̂in and D̂in (see Appendix A). entanglement and steering where entangled states are gener- Then appropriate local classical displacements are applied to ated deterministically and information is encoded in the po- all modes according to the following relations: sition or momentum quadratures of photonic harmonic oscil- lators [1]. In our experiment, two bright squeezed states are x̂C0 → x̂Cin + FC xdis , p̂A0 → p̂Ain + FA pdis , generated by two nondegenerate optical parametric amplifiers x̂B0 → x̂Bin + FB xdis , p̂B0 → p̂Bin − FB pdis , (NOPAs). Each of the NOPAs consists of a potassium titanyl x̂D0 → x̂Din + FD xdis , p̂D0 → p̂Din − FD pdis , (1) phosphate (KTP) crystal and an output coupling mirror. The schematic of the experimental setup is illustrated in Fig. 1(b), where x̂ j and p̂ j represent the position and momentum observ- and the details of the experiment can be found in Appendix A. ables of the state corresponding to the subscript j, satisfying The output states are measured in the time domain when the the canonical commutation relation [ x̂ j , p̂ j ] = 2i. The classi- signals of the homodyne detectors are demodulated at a side- cal displacements are determined by xdis and pdis which obey band frequency of 3 MHz with a bandwidth of 30 kHz. The Gaussian distribution with the same variance, and coefficients demodulated signals are recorded simultaneously by a digital Fk (k = A, B, C, D) corresponding to each mode. The coef-

3 ficient Fk (T i , η, V s,a ) is a function of transmittance of beam- Here, ν̄Nj M\N ( j = 1, ..., m) denote the symplectic eigenvalues splitter T i , transmission efficiency η in the channel, variances of the Schur complement σ̄N M\N = M − γT N −1 γ of subsys- of squeezing V s and antisqueezing Va of the input squeezed tem N, with diagonal blocks N and M corresponding to the states. Since Âin , B̂in , Ĉin , D̂in are prepared independently reduced states of subsystems and the off-diagonal matrices γ and the added displacements are local operations and classical and γT encoding the intermodal correlations between subsys- communication, the resulting states Â0 , B̂0 , Ĉ0 , D̂0 sent from tems. A nonzero GN→M > 0 denotes the presence of steering the quantum server to users are fully separable. from N to M, and a higher value means stronger steerability. In the second step, optical modes Â0 and Ĉ0 are transmit- The steerability in the opposite direction G M→N can be ob- ted to Alice. We assume that Alice is close to the quantum tained by swapping the roles of N and M. The covariance server, i.e., ηS A = 1, while optical modes B̂0 and D̂0 are trans- matrices of generated states after each step are detailed in Ap- mitted to Bob and David through lossy channels (the case for pendix C. ηS A , 1 is discussed in Appendix E). In the two-user scenario, only optical mode B̂0 is transmitted to Bob, while David is not involved. In the third step, all users perform beam-splitter operations on their received optical modes and measure the obtained states with homodyne detectors. Alice couples modes Â0 and Ĉ0 on a balanced beam splitter with T 1 = 1/2, then keeps one output mode Â and sends the other one Ĉ1 to Bob. The dis- placement operations on initial input modes ensure the sepa- rability across Ĉ1 |Â B̂0 and B̂0 |ÂĈ1 splittings but entanglement between Â and B̂0Ĉ1 , which is essential for the present pro- FIG. 2. Experimental results for two users. (a) The minimum sym- tocol. Bob couples the ancillary mode Ĉ1 and his mode B̂0 plectic eigenvalues PPTA with respect to Â| B̂ splitting is always on the beam splitter T 2 . Up to this stage, two-mode entangle- smaller than 1. (b) The steerability GA→B is obtained and robust against loss in channels. Error bars represent one standard deviation ment and steering between modes Â and B̂ (one of the output and are obtained based on the statistics of measured noise variances. modes of Bob’s beam splitter) are established. Meanwhile, the distributed steerability GA→B can be maximized by opti- In this scheme, the crucial idea is that the ancillary modes mizing displacement coefficient FB , which was not uncovered (Ĉ1 , Ĉ2 ) in the channels are separable from the other modes. by previous studies. The conditions for separability depend on the parameters In the distribution for three users, the other output mode of Fk (T i , η, V s,a ), xdis , and pdis . Without losing generality, we fix Bob’s beam splitter Ĉ2 is sequentially transmitted to David. A the variances of xdis and pdis to 1.50, T 1 = 1/2, FA = FC = 1, further challenge, apart from the requirement for separability then the condition for separability across Ĉ1 |Â B̂0 splitting in across Ĉ1 |Â B̂0 splitting, is that we need to carefully design the two-user scenario only depends on the parameter FB , and the displacement on mode D̂in to keep the second ancillary that for the separability across Ĉ2 |Â B̂D̂0 splitting in the three- mode Ĉ2 separable from all the users’ modes Â B̂D̂0 . David user scenario depends on the parameters FB and FD . Addi- couples the received mode Ĉ2 with his displaced mode D̂0 on tionally, on the basis of satisfying the above separable condi- the beam splitter T 3 , and hence quantum entanglement and tions, we optimize Fk to achieve the highest distributed steer- steering among three users, including modes Â, B̂, and D̂, can abilities for each desired distribution direction. be built. Similarly, the Gaussian steerability GA→BD can be To evaluate the performance of the present entanglement maximized by adjusting the displacement coefficient FD . and steering distribution network, we investigate the effect of The distributed entangled states and the measurements both channel loss in our experiment since the transmission distance have Gaussian nature, thus, to detect Gaussian entanglement of the quantum state is limited by inevitable loss in a prac- between subsystems N and M (each subsystem contains n and tical quantum network. In the case of two users, in order m modes, respectively) we adopt the positive partial transpo- to achieve the highest Gaussian steerability GA→B , the opti- sition (PPT) criterion [49] which is necessary and sufficient mized displacementh coefficient on mode B̂in is set to FB = √ i 2ηAB (1 − T 2 )Va / ηS B T 2 (Va + V s ) , where T 2 is the trans- p when n = 1 and m ≥ 1. The separable condition is that all symplectic eigenvalues of the covariance matrix after the par- mittance of Bob’s beam splitter, and ηAB and ηS B are the trans- tially transposition σ>NNM are not smaller than 1 (see Appendix mission efficiencies for the channels from Alice to Bob and B). from quantum server to Bob, respectively. Thus, the maximal distributed steerability GA→B is given by The steerability between two partitions (N → M) is quanti- fied by the criterion from Ref. [50], where it was given by " Va + V s # GA→B = ln . X (1 − ηAB + ηAB T 2 )(Va + V s ) + 2ηAB (1 − T 2 )V s Va GN→M (σN M ) = max 0, − ln(ν̄Nj M\N ) . (2) (3) j:ν̄Nj M\N

4 then the largest distributed steerability GA→B is 2(Va + V s ) " # G A→B = ln (4) (2 − η)(Va + V s ) + 2ηV s Va with FB ≈ 1.24. When all channels are ideal, i.e., η = 1, we measure the covariance matrix σAB0 C1 and verify the condi- tions for separability across Ĉ1 |Â B̂0 splitting and B̂0 |ÂĈ1 split- ting according to their minimum PPT values 1.264 > 1 and 1.182 > 1, respectively, while Â is entangled with group of B̂0Ĉ1 due to its minimum PPT value 0.701 < 1, under the above optimized displacement (see Appendix D). Note that when modes Ĉ1 , B̂0 are transmitted in lossy channels, i.e., η < 1, the requirement for the separable conditions will be relaxed. As shown in Fig. 2, the distributed entanglement be- tween Alice and Bob and one-way Gaussian steerability from Alice to Bob GA→B always exist when η > 0 , which means this indirect distribution protocol is robust against channel loss. After the successful distribution between two users, we extend this protocol to a three-user case. Figure 3 shows that the distributed three-mode entanglement and steerabil- ity are also robust against loss in quantum channels. As an example, the transmission efficiencies from quantum server to Bob, quantum server to David, Alice to Bob, and Bob to David are assumed to be the same. To achieve the maxi- mum steerability GA→BD , we optimize the displacements for √ FIG. 3. Experimental results for three users. (a) All of the minimum modes B̂in and D̂in by FB ≈ 1.24 and FD = 2 ηVa /(Va + symplectic eigenvalues PPTA (black), PPTB (red), and PPTD (blue) V s ) with T 2 = T 3 = 1/2. Meanwhile, an additional con- with respect to Â| B̂D̂, B̂|ÂD̂, and D̂|Â B̂ splittings are always smaller dition for separability across Ĉ2 |Â B̂D̂0 splitting needs to be than 1. (b) The steerabilities GA→BD , GA→B , and GA→D are obtained satisfied. Hence, we experimentally reconstruct the covari- and robust against channel losses. Error bars represent one standard ance matrix σABC2 D0 (see Appendix D), then verify that the deviation and are obtained based on the statistics of measured noise minimum PPT value for splitting across Ĉ2 |Â B̂D̂0 is 1.177 > variances. 1 when η = 1. Similarly, when modes Ĉ2 , D̂0 are transmitted in lossy channels, the separable condition required by splitting across Ĉ2 |Â B̂D̂0 is more easily satisfied. ability comes from the mixture of two initial squeezed states It is clearly shown in Fig. 3(a) that three-mode entangle- at Alice’s station by a balanced beam splitter, Alice holds half ment is shared among Alice, Bob, and David after the distri- of the information of the whole state, while Bob, David, and bution. Different from entanglement, only the one-way steer- the ancillary mode together hold the other half, which makes abilities GA→BD > 0, GA→B > 0 and GA→D > 0 are achieved, it much harder for Bob himself, and even together with David, and the collective steerability (GA→BD ) is always higher than to steer Alice. This means that with the current parameters our the individual steerabilities (GA→B and GA→D ), as shown in experiment presents a highly asymmetric network with direc- Fig. 3(b). We also note that the steering from Bob to David tional steerability from Alice to other users. does not exist in any case (i.e., GB→D = 0). This result can be After successful distribution, the quantum resources shared understood as a consequence of the monogamy relation pro- among distant users are widely available for real-world ap- posed in Ref. [51] where two independent parties cannot steer plications to networked quantum information tasks. For in- a third party simultaneously under Gaussian measurements. stance, the hierarchical structure presented in this network, Thus, GA→D > 0 prohibits the possibility of GB→D > 0. where Alice acts as a superior who can always steer (pilot) Note that the present experimental results show the abil- any of the subordinate users (GA→B, D, BD > 0), can be ap- ity to distribute the Gaussian steerability from Alice to other plied to implement secure directional quantum key distribu- users (including the individual user and the group of them) by tion and quantum teleportation from Alice to Bob (David, or transmitting separable modes. This is because squeezed states their group). are transmitted to Alice firstly, and then separable modes are Furthermore, by adjusting the displacements and the trans- transmitted from Alice to other users sequentially, i.e., it has a mittances of beam splitters, our protocol can also distribute sequential property in such a distribution scheme. It can also on-demand quantum resources for specific quantum infor- be understood in the following way: the final distributed steer- mation tasks. For example, the steerabilities GBD→A >

5 0 and GB→A = GD→A = 0 are required for 1SDI QSS, where The NOPA with a semimonolithic structure, which is simi- the dealer Alice sends a secret and players (Bob and David) lar to that in our previous experiment [52–54], is used to gen- are able to decode the information only with their collabo- erate squeezed states. Each of NOPAs consists of an α-cut ration [29]. To distribute such a resource via separable an- type-II KTiOPO4 (KTP) crystal (3 × 3 × 10 mm3 ) and a con- cillas, we need to adjust the displacement coefficients FB = cave mirror with curvature 50 mm, which is mounted on a 0.92 and FD = 1.70 with initial −10 dB squeezing and +11 dB piezo-electric transducer (PZT) for locking actively length of antisqueezing, such that the steerability GBD→A can be dis- NOPAs. The front face of KTP crystal is coated to be used for tributed for 0.80 < η ≤ 1 where GB→A = GD→A = 0. This the input coupler and the concave mirror serves as the output means that 1SDI QSS can be implemented in the range of coupler of squeezed states. The transmittances of the front 4.90 km with a fiber loss of 0.2 dB/km (see Appendix C). face of KTP crystal at 540 nm and 1080 nm are 40% and In summary, we present deterministic distribution of mul- 0.04%, respectively. The end-face of KTP is antireflection tipartite quantum resources by combining quantum channels coated for both 1080 nm and 540 nm. The transmittances of and classical communications in a network consisting of a output coupler at 540 nm and 1080 nm are 0.5% and 12.5%, quantum server and multiple users. We demonstrate that it respectively. The cavity length of each NOPA is 53.8 mm. is feasible to distribute not only Gaussian entanglement but The cavity is locked by using Pound-Drever-Hall method with also EPR steering among two and three users via separable a phase modulation (New Focus, 4004) of 53 MHz on injected ancillas. Moreover, the maximum steerability allowed by signal [55]. the present network structure is distributed by optimizing the When a NOPA is operating at amplification status (the rel- experimental parameters. The distributed entanglement and ative phase between injected signal and pump beam is locked steerability are robust against channel losses, which further to zero), the coupled modes at +45◦ and −45◦ polarization di- confirms the significance and practical feasibility of the pre- rections are the phase quadrature (momentum) squeezed state sented method. This work provides a distinct approach for and the amplitude quadrature (position) squeezed state, re- distributing precious multipartite quantum resources and takes spectively. Conversely, when a NOPA is operating at deampli- a step forward in studying potential applications of this kind fication status (the relative phase between injected signal and of protocols in a quantum network. pump beam is locked to (2n + 1)π), the coupled modes at +45◦ This work was financially supported by National Natu- and −45◦ polarization directions are the amplitude quadrature ral Science Foundation of China (Grants No. 11834010, (position) and the phase quadrature (momentum) squeezed No. 61675007, No. 11975026, No. 62005149, and No. states, respectively [52–54]. In the experiment, we choose 12004011), National Key R&D Program of China (Grants the bright squeezed state of each NOPA (+45◦ polarization No. 2016YFA0301402, No. 2018YFB1107205 and No. directions) as the signal mode, because the relative phase be- 2019YFA0308702). X.S. thanks the program of Youth Sanjin tween squeezed state and assistant coherent beam modulating Scholar, and the Fund for Shanxi “1331 Project” Key Subjects Gaussian noise needs to be controlled, so NOPA1 and NOPA2 Construction. Q.H. acknowledges the Beijing Natural Science are operating at deamplification and amplification conditions, Foundation (Z190005) and the Key R&D Program of Guang- respectively. dong Province (Grant No. 2018B030329001). In the protocol, the displacements for all modes are imple- M. W. and Y. X. contributed equally to this work. mented by the electro-optical modulators. The displacements on the amplitude quadrature are realized by amplitude modu- lator (AM), and the orthogonality displacements on the phase Appendix A: Details of experiment quadrature are added by phase modulator (PM). The coher- ent beams carrying the Gaussian noise are mixed with cor- The experimental setup is shown in Fig. A1. In responding input mode on 99:1 beam-splitters, and the rela- our experiment, the x̂-squeezed and p̂-squeezed states are tive phase differences between the input modes of 99:1 beam- produced by non-degenerate optical parametric amplifiers splitters are locked to zero. The displaced modes are coupled (NOPA1 and NOPA2) pumped by a common laser source, on the beam-splitters T 1 , T 2 , T 3 located at the user’s stations, which is a continuous wave intracavity frequency-doubled and the relative phases of them are also locked to zero. It and frequency-stabilized Nd:YAP-LBO (Nd-doped YAlO3 is important to lock the relative phases of optical modes pre- perovskite-lithium triborate) laser. Two mode cleaners are cisely in the distribution protocol, especially in the case of inserted between the laser source and the NOPAs to filter real networks applications with optical fibers. In order to lock noise and higher order spatial modes of the laser beams at the relative phase between two input modes of each beam- 540 nm and 1080 nm. The fundamental wave at 1080 nm splitter, the transmitted mode of high reflection mirror after wavelength is used for the injected signals of the NOPAs and the beam-splitter is detected and fed back to a piezo mirror by the local oscillators for the homodyne detectors. The second- a microcontroller unit [56]. In our experiment, the phase fluc- harmonic wave at 540 nm wavelength serves as pump field of tuation on each beam-splitter is controlled to be around 1◦ by the NOPAs, in which a pair of signal and idler modes with adjusting parameters of the phase locking system. The inter- orthogonal polarizations at 1080 nm are generated through an ference efficiencies between two input beams coupled on each intracavity frequency-down-conversion process. beam-splitter are about 99%.

6 FIG. A1. Detailed schematic of experiment setup. MC: mode cleaner. NOPA: non-degenerated optical parametric amplifier. EOM: electro- optical modulator for locking NOPAs (New Focus, 4004). PM: phase modulator. AM: amplitude modulator. HD: homodyne detector. HD-A, HD-B and HD-D corresponding to Alice’s, Bob’s and David’s homodyne detector. In the distribution of two-mode state, only two displaced signals of homodyne detectors are demodulated at sideband squeezed states and a displaced coherent state are transmitted frequency of 3 MHz with bandwidth of 30 kHz. The demodu- to Alice and Bob, respectively. Alice couples two displaced lated signals are recorded simultaneously by a digital storage squeezed states on a beam-splitter T 1 , then keeps one output oscilloscope at the sampling rate of 500 KS/s. The squeezing mode Â at her station and transmits the other output mode and anti-squeezing levels of the two input states (modes Âin Ĉ1 to Bob. Bob couples modes Ĉ1 and B̂0 on a beam-splitter and Ĉin ) are measured by homodyne detectors in the time do- T 2 . One of output mode B̂ is obtained and the other output main at the quantum server’s station, when the displacements mode Ĉ2 is abandoned. In this way, Gaussian entanglement are not added. and steerability between modes Â and B̂ are obtained. In the distribution of three-mode state, another displaced coherent state D̂0 is also transmitted to David. David couples modes Appendix B: The criteria of Gaussian entanglement Ĉ2 and D̂0 on a beam-splitter T 3 . One of output mode D̂ is achieved, and the other output mode Ĉ3 is abandoned. In this way, Gaussian entanglement and steerability among modes Â, The properties of a (n + m)-mode Gaussian state of a bipar- B̂ and D̂ are established. tition system can be determined by its covariance matrix For satisfying the separable condition, the variances of xdis N γ ! σN M = > (A1) and pdis are fixed to 1.50, which can be determined according γ M to Eq. A7a in the experiment. When the modes Âin and Ĉin are replaced by coherent states (Va = V s = 1 in Eq. A7a in this with matrix element σi j = hξ̂i ξ̂ j + ξ̂ j ξ̂i i/2 − hξ̂i ihξ̂ j i, where case), we measure variances of amplitude and phase quadra- ξ̂ ≡ ( x̂1 , p̂1 , ..., x̂n , p̂n , x̂n+1 , p̂n+1 ..., x̂n+m , p̂n+m )| is the vector tures of mode Â. When measured variances are 2.43 dB higher of the amplitude and phase quadratures of optical modes. The than normalized shot noise limit (corresponding to variance of submatrices N and M are corresponding to the reduced states vacuum state), the variances of xdis and pdis will be 1.50. of subsystems N and M, respectively. The properties of quantum states are measured by partially In the main text, the PPT value is used to quantify the en- reconstructed covariance matrix with the balanced homodyne tanglement. Based on above covariance matrix, the symplec- detectors. The interference efficiencies between signal and lo- tic eigenvalue can be calculated. If the covariance matrix after cal oscillator fields in detection system are 99% and the quan- partial transposition fulfills the inequality [49] tum efficiencies of photodiodes are 99.6%. In our experiment, the output states are measured in the time domain when the σ>NNM + iΩ ≥ 0 (A2)

7 Cov( x̂A , x̂B0 ) Cov( x̂A , p̂B0 ) " # the state is separable with respect to N − M splitting, σAB0 = (A6d) where σ>NNM = T N σN M T N> is the partially transposed matrix for Cov( p̂A , x̂B0 ) Cov( p̂A , p̂B0 ) subsystem N, T N is a unit diagonal matrix except for the ele- Cov( x̂A , x̂C1 ) Cov( x̂A , p̂C1 ) " # ment T 2n, 2n = −1, and Ω is the symplectic matrix described σAC1 = (A6e) Cov( p̂A , x̂C1 ) Cov( p̂A , p̂C1 ) as Cov( x̂B0 , x̂C1 ) Cov( x̂B0 , p̂C1 ) " # σB0 C1 = (A6f) Cov( p̂B0 , x̂C1 ) Cov( p̂B0 , p̂C1 ) " # 0 1 Ω = ⊕m+n k=1 (A3) −1 0 The elements in the matrices are given by This criterion is equivalent to finding the symplectic eigenval- Va + V s + Vdis ues of the covariance matrix after partial transposition in Ref. 42 x̂A = 42 p̂A = (A7a) 2 [43]. 42 x̂B0 = 42 p̂B0 = ηS B (1 + Vdis FB2 ) + 1 − ηS B (A7b) When one subsystem holds only one mode n = 1 and the ηAB (Va + V s + Vdis ) other subsystem holds the rest of modes, the state is insepara- 42 x̂C1 = + 1 − ηAB (A7c) 2 ble if the minimum symplectic eigenvalue of σ>NNM is smaller p 2ηS B Vdis FB than 1. For the case of two modes, i.e., m = n = 1, the min- Cov( x̂A , x̂B0 ) = −Cov( p̂A , p̂B0 ) = (A7d) imum PPT value (minimum symplectic eigenvalue after par- 2 √ ηAB (Va − V s − Vdis ) tially transpose) is expressed as Cov( x̂A , x̂C1 ) = −Cov( p̂A , p̂C1 ) = 2 (A7e) q 1 p µ= √ C − C2 − 4 detσN M (A4) 2ηAB ηS B Vdis FB p 2 Cov( x̂B0 , x̂C1 ) = Cov( p̂B0 , p̂C1 ) = − (A7f) 2 where C = detN + detM − 2 detγ. Cov( x̂A , p̂B0 ) = Cov( p̂A , x̂B0 ) = 0 (A7g) Cov( x̂A , p̂C1 ) = Cov( p̂A , x̂C1 ) = 0 (A7h) Appendix C: The distribution in lossy channels Cov( x̂B0 , p̂C1 ) = Cov( p̂B0 , x̂C1 ) = 0 (A7i) where Va , V s are the variances of squeezing and anti-squeezing of initial squeezed states, Vdis = The distribution of two-mode state. In the distribution pro- h(∆xdis )2 i = h(∆pdis )2 i is the variance of displacements tocol with separable states, it is crucial to make sure the sep- on modes Âin and Ĉin , FB is the corresponding coefficient for arable condition is maintained in the whole process. In the mode B̂in , and ηAB and ηS B are the transmission efficiencies distribution of two-mode state, the displacement operations from Alice to Bob and that from quantum server to Bob, applied on initial input modes in the quantum server ensure respectively. the separability across Ĉ1 |Â B̂0 splitting and the inseparabil- ity between Â and B̂0Ĉ1 . After mixing modes Â0 and Ĉ0 by In the distribution of two-mode state, the separability beam-splitter T 1 at Alice’s station, the modes Â and Ĉ1 are ob- across Ĉ1 |Â B̂0 splitting and inseparability between Â and the tained. Mode B̂0 is transmitted from quantum server to Bob. group ( B̂0Ĉ1 ) are required. By classical displacement opera- After the transmission of optical mode ô over a lossy p channel, tions, the separability across splitting Ĉ1 |Â B̂0 is satisfied when √ the output optical mode is given by ôL = ηô + 1 − ηôvac , where η and ôvac represent the transmission efficiency of lossy 2(1 − V s ) Vdis > Vsep = (A8) channel and optical vacuum mode induced by loss into the 2 − ηS B FB2 (1 − V s ) quantum channel, respectively. The covariance matrix includ- ing modes Â, B̂0 , and Ĉ1 is given by where the minimum requirement for the variance of displace- ment Vsep is named as the separable boundary. The insep- σA σAB0 σAC1 arable condition for Â| B̂0Ĉ1 splitting is Vdis > 0, which is | σ = σAB0 σB0 σB0 C1 (A5) more relax than separable boundary Vsep . Therefore, the clas- | | σAC1 σB0 C1 σC1 sical displacements on modes Âin and Ĉin should be larger than Vsep in order to satisfy the separable condition. where " # 42 x̂A 0 After the ancillary mode Ĉ1 is transmitted to Bob, the σA = (A6a) 0 42 p̂A modes B̂0 and Ĉ1 are coupled on Bob’s beam-splitter T 2 . " # Thus, the covariance matrix of the output modes Â and B̂ in 42 x̂B0 0 σB0 = (A6b) the distribution between two users is expressed by 0 42 p̂B0 σA σAB " # " # 42 x̂C1 0 σC1 = (A6c) σ= | (A9) 0 42 p̂C1 σAB σB

8 FIG. A2. Details of Fig. 3 in the main text. (a) PPT value PPTD for the splitting D̂|Â B̂ and (b) steering parameter GA→D for the very small value of transmission efficiencies in the distribution of three-mode state, when the displacement coefficients FB and FD are optimized. where from Alice to Bob, when the rest of parameters are fixed. " 2 # When the transmittance of Bob’s beam-splitter is chosen 4 x̂A 0 σA = (A10a) to T 2 = 1/2 and transmission efficiency ηAB = ηS B = η = 1, 0 42 p̂A " # the separable boundary Vsep = 0.808. In our experiment, we 42 x̂B 0 choose Vdis = 1.50 which satisfies the separable condition. σB = (A10b) 0 42 p̂B The dependence of maximum steerability GA→B on the Cov( x̂A , x̂B ) Cov( x̂A , p̂B ) " # σAB = (A10c) transmission efficiency η in the distribution of two-mode state Cov( p̂A , x̂B ) Cov( p̂A , p̂B ) through lossy channels (blue curve in Fig. 2 in the main text) The elements in the matrices are given by is expressed by Va + V s + Vdis 2(Va + V s ) 42 x̂A = 42 p̂A = (A11a) GA→B = ln[ ] (A13) 2 (2 − η)(Va + V s ) + 2ηV s Va 42 x̂B = 42 p̂B where the transmittances T 1 = T 2 = 1/2 and optimal displace- ηAB (1 − T 2 )(Va + V s + Vdis ) √ = + ηS B T 2 Vdis FB2 ment FB = 2Va /(Va + V s ) are chosen. 2 − 2ηS B ηAB T 2 (1 − T 2 )Vdis FB + 1 − ηAB + ηAB T 2 p (A11b) The distribution of three-mode state. The output state co- variance matrix in case of the distribution between three users, Cov( x̂A , x̂B ) = −Cov( p̂A , p̂B ) in the presence of lossy channels can be expressed as ηAB (1 − T 2 )(Va − V s − Vdis ) + 2ηS B T 2 Vdis FB p p = 2 (A11c) σA σAB σAD | Cov( x̂A , p̂B ) = Cov( p̂A , x̂B ) = 0 (A11d) σ = σAB σB σBD (A14) | | σAD σBD σD In order to obtain the maximum steerability GA→B , the where displacement coefficient p FB needs to be √ optimized. Here, the optimal FB = 2ηAB (1 − T 2 )Va /[ ηS B T 2 (Va + V s )], " 2 4 x̂A 0 # which is the function of transmittance T 2 , transmission ef- σ A = (A15a) 0 42 p̂A ficiencies ηS B , ηAB , and the variances of squeezing and " 2 # 4 x̂B 0 anti-squeezing of squeezed states. By substituting opti- σB = (A15b) mized coefficient FB into Eq. A8, the separable boundary 0 42 p̂B " 2 # across Ĉ1 |Â B̂0 splitting between two users is expressed by 4 x̂D 0 σD = (A15c) 0 42 p̂D T 2 (1 − V s )(Va + V s )2 Vsep = Cov( x̂A , x̂B ) Cov( x̂A , p̂B ) " # (A12) T 2 (Va + V s ) − ηAB (1 − T 2 )(1 − V s )Va 2 2 σ = (A15d) AB Cov( p̂A , x̂B ) Cov( p̂A , p̂B ) Cov( x̂A , x̂D ) Cov( x̂A , p̂D ) " # From this expression, we can see that the separable bound- σAD = (A15e) ary decreases with the decrease of transmission efficiency ηAB Cov( p̂A , x̂D ) Cov( p̂A , p̂D )

9 Cov( x̂B , x̂D ) Cov( x̂B , p̂D ) " # σBD = (A15f) the maximun steerability GA→BD in lossy channels. Hence the Cov( p̂B , x̂D ) Cov( p̂B , p̂D ) steerabilities GA→BD , GA→B , and GA→D are expressed by The elements in the matrices are given by 4(Va + V s ) GA→BD = ln[ ] (4 − − 2η)(Va + V s ) + (4η + 2η2 )V s Va η2 Va + V s + Vdis 42 x̂A = 42 p̂A = (A16a) (A18a) 2 η(Va + V s + Vdis ) 2(Va + V s ) 4 x̂B = 4 p̂B = 2 2 +f GA→B = ln[ ] (A18b) 4 (A16b) (2 − η)(Va + V s ) + 2ηV s Va 4(Va + V s ) GA→D = ln[ ] (A18c) η (Va + V s + Vdis ) 2 (4 − η2 )(Va + V s ) + 2η2 V s Va 42 x̂D = 42 p̂D = + g (A16c) 8 respectively. Cov( x̂A , x̂B ) = −Cov( p̂A , p̂B ) The optimal displacements in the presence of lossy channel √ 2η(Va − V s − Vdis + 2Vdis FB ) with different transmission efficiencies η are listed in the Table p = A1. We measure all the covariance matrices of the distributed 4 (A16d) modes experimentally, and then calculate the steerability and PPT values to quantify the EPR steering and entanglement. In Cov( x̂A , x̂D ) = −Cov( p̂A , p̂D ) the distribution of three-mode state, the blue curves of Fig. 3 η(Va − V s − Vdis ) = +j (A16e) in the main text are shown in Fig. A2 for very small value of 4 transmission efficiencies, which clearly shows that PPT value Cov( x̂B , x̂D ) = Cov( p̂B , p̂D ) for splitting D̂|Â B̂ and steerability GA→D are robust against 2η3 (Va + V s + Vdis ) p loss. = +k (A16f) 8 Furthermore, our protocol can be also applied to distribute Cov( x̂A , p̂B ) = Cov( p̂A , x̂B ) = 0 (A16g) steerability in the opposite direction that GBD→A > 0 while Cov( x̂A , p̂D ) = Cov( p̂A , x̂D ) = 0 (A16h) GB→A = GD→A = 0 by appropriately adjusting experimental Cov( x̂B , p̂D ) = Cov( p̂B , x̂D ) = 0 (A16i) parameters. The three-mode state with this steering proper- ties is a necessary resource for one-sided device-independent where quantum secret sharing (1sDI QSS), a protocol used to send a η √ highly important message to two players (need not assume f = (Vdis FB2 − 1 − 2Vdis FB ) + 1 (A17a) reliable devices) who must collaborate to obtain the infor- 2 √ mation sent by the dealer. To distribute such resource via 4 + η2 (Vdis FB2 + 2Vdis FB − 1) + 2ηVdis FD2 g= separable ancillas, we keep the transmittances of all beam- 4 splitters T 1 = T 2 = T 3 = 1/2 and displacements xdis = pdis = √ 2 η3 Vdis FD ( 2FB + 1) p − (A17b) 1.50 unchanged, and then choose optimal displacement coeffi- 4 √ cients FB = 0.92 and FD = 1.70 for the initial −10 dB squeez- √ 2 ηVdis FD − 2ηVdis FB ing and +11 dB anti-squeezing to distribute as large as possi- j= (A17c) ble steerability from the secrete receivers B̂D̂ to the dealer Â. 4 √ √ − 2η3 (Vdis FB2 + 1) + 2ηVdis FD ( 2FB − 1) p k= 4 (A17d) and η = ηS B = ηAB = ηS D = ηBD is the transmission efficiency of all lossy channels, FD is the displacement coefficient for mode D̂in . When three users are involved in the quantum network and T 1 , T 2 , T 3 are fixed to 1/2, the optimal displace- ments √ for modes B̂in and D̂in are optimized as FB = √ 2Va /(Va + V s ) and FD = 2 ηVa /(Va + V s ) for achieving Table A1 Optimal displacements in lossy channels. η 1 0.8 0.6 0.4 0.2 FIG. A3. The dependences of steerabili- FB 1.239 1.239 1.239 1.239 1.239 ties GBD→A , GB→A and GD→A on transmission efficiency. The steerability GBD→A > 0 when the transmission efficiency is large FD 1.752 1.567 1.357 1.108 0.784 than 0.80, and neither Bob nor David has ability to steer Alice.

10 Simultaneously, the separable conditions during distribution secure QSS can be obtained within the transmission efficiency process are guaranteed, which are evidenced by the minimum range of 0.94 < η ≤ 1, which means this state is a useful re- PPT values for splittings across Ĉ1 |Â B̂0 and Ĉ1 |Â B̂D̂0 being source in the range of 1.34 km. This confirms that our scheme 1.02 > 1 and 1.01 > 1, respectively. is feasible to successfully distribute various incarnations of quantum nonlocality by effectively classical means. From Fig. A3 we can see that steerability GBD→A > 0 ex- ists when 0.80 < η ≤ 1 (ηAB = ηS B = ηBD = ηS D = η), and steerabilities GB→A and GD→A are always equal to zero. If we Appendix D: Verification of separable conditions consider transmission in a fiber with a loss of α=0.2 dB/km (η = 10−αL/10 ), the achievable transmission distance for im- In the distribution of two-mode state, we experimentally plementing 1sDI QSS task L will be about 4.90 km. If the key verify the separable condition for splitting Ĉ1 |Â B̂0 by calcu- is encoded on Alice’s state, then it can only be unlocked by lating the PPT value of the covariance matrix σAB0 C1 . When Bob and David with high accuracy if they combine measure- the variances of the displacements added on modes Âin , Ĉin ment outcomes. A guaranteed secret key rate for providing se- are fixed to 1.50, T 1 = 1/2 and η = 1, the optimal displace- curity against eavesdropping is given by K ≥ GBD→A − ln(e/2) ment FB ≈ 1.24 as given in the Table A1. The experimentally [46]. As shown in Fig. A3, the nonzero key rate (K > 0) for reconstructed covariance matrix is 2.754 0 1.296 0 0.764 0 0 2.759 0 −1.294 0 −0.767 1.296 0 3.282 0 −1.296 0 σAB0 C1 = (A19) 0 −1.294 0 3.276 0 −1.291 0.764 0 −1.296 0 2.768 0 0 −0.767 0 −1.291 0 2.786 Based on the covariance matrix σAB0 C1 , the minimum of transmission efficiency, the chosen variances of displace- PPT values for the splittings Â| B̂0Ĉ1 , B̂0 |ÂĈ1 , and Ĉ1 |Â B̂0 ments (Vdis = 1.50) in case of η = 1 make sure the separable are 0.701, 1.182, and 1.264, respectively. This verifies that conditions for smaller transmission efficiencies are also satis- the mode Â is inseparable with the group ( B̂0Ĉ1 ), but mode B̂0 fied. is separable from the group (ÂĈ1 ), mode Ĉ1 is separable from In order to verify the separability for splitting the group (Â B̂0 ), respectively. When modes Ĉ1 , B̂0 are trans- across Ĉ2 |Â B̂D̂0 in the case of distributing three-mode mitted in lossy channels, i.e., η < 1, the requirement for the entangled state, we experimentally reconstructed the separable condition will be relaxed, see Eq. A12. Because the covariance matrix σABC2 D0 in the case of Vdis = 1.50, requirement for the variance of displacements corresponding T 1 = T 2 = 1/2, η = 1 (which determine the optimal to separable boundary (Eq. A12) decrease with the decrease displacements FB ≈ 1.24 and FD ≈ 1.75), which is given by 2.757 0 1.483 0 0.318 0 1.809 0 0 2.753 0 −1.462 0 −0.312 0 −1.817 1.483 0 1.774 0 0.277 0 0.985 0 0 −1.462 0 1.777 0 0.297 0 1.061 σABC2 D0 = (A20) 0.318 0 0.277 0 4.277 0 3.643 0 0 −0.312 0 0.297 0 4.251 0 3.573 1.809 0 0.985 0 3.644 0 5.592 0 0 −1.817 0 1.061 0 3.573 0 5.606 Based on the covariance matrix σABC2 D0 , the mini- Similarly, when modes Ĉ2 , D̂0 are transmitted in lossy chan- mum PPT values for the splittings Â| B̂Ĉ2 D̂0 , B̂|ÂĈ2 D̂0 , nels, the requirement for separable condition will also be re- Ĉ2 |Â B̂D̂0 and D̂0 |Â B̂Ĉ2 are 0.589, 0.686, 1.177, and 1.183, laxed, such that it is always satisfied at different transmission respectively. From the calculating results, we can see that the efficiencies. second ancillary mode Ĉ2 is separable from the group (Â B̂D̂0 ).

11 FIG. A4. Distributed entanglement and steerability when there is channel loss between the server and Alice for two users’ case. FIG. A5. Distributed entanglement and steerabilities when there is channel loss between the server and Alice for three users’ case. Appendix E. The effect of loss between Alice and the server In the case of distribution for three users, the distributed entanglement is still robust against channel losses, as shown in Fig. A5(a), while the steerabilities from mode Â to B̂, D̂, In this section, we analyze the effect of the loss between the collaboration of B̂ and D̂ only exist when η > 0.81, as Alice and the server on the distribution of steering and entan- shown in Fig. A5(b). The optimal displacements on modes B̂ glement, which corresponds to a network structure different and D̂ considering general losses are given by from our experiment. √ In brief, the loss between Alice and the server will decrease 2 η[1 + (Va − 1)η] FB = (A22a) the distance of steering distribution. In the current squeez- 2 + (Va + V s − 2)η √ ing levels (-3/+5.5 dB), when the transmission efficiency from 2 2η[1 + (Va − 1)η] p quantum server to Alice is assumed to be same with that from FD = = 2ηFB (A22b) 2 + (Va + V s − 2)η quantum server to Bob, i.e. ηS A = ηS B = ηAB = η, Gaussian entanglement is always robust against channel losses in the Here, η = ηS A = ηS B = ηAB = ηS D = ηBD . distribution for two users, as shown in Fig. A4(a). However, different from entanglement, EPR steering from Alice to Bob According to the above analysis, when the distances be- exists only when the transmission efficiency η > 0.81. This tween the server and each user are the same, the distribution means that the distributed steerability is more sensitive to the of entanglement is always robust against the channel losses, channel loss between Alice and server than entanglement, as while the distribution of quantum steering is achievable in a shown in Fig. A4(b). The optimal displacement on mode B̂ limited transmission distance. For instance, the transmission considering general losses is given by distance is less than 4.9 km in a fiber channel with loss 0.2 √ dB/km as the transmission efficiency needs to be higher than 2 ηS A ηAB [1 + (Va − 1)ηS A ] 0.81. Note that, the robustness of steering shown in our ex- FB = √ (A21) ηS B [2 + (Va + V s − 2)ηS A ] periment is because we considered the structure that Alice is

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