Multipartite Einstein Podolsky Rosen steering and genuine tripartite entanglement with optical networks

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1 Multipartite Einstein Podolsky Rosen steering and genuine tripartite entanglement with optical networks Seiji Armstrong 1, Meng Wang 2, Run Yan Teh 3, Qihuang Gong 2, Qiongyi He 2,3,, Jiri Janousek 1, Hans-Albert Bachor 1, Margaret D. Reid 3,, Ping Koy Lam 1,4 1 Centre for Quantum Computation and Communication Technology, Department of Quantum Science, The Australian National University, Canberra, ACT 0200, Australia 2 State Key Laboratory for Mesoscopic Physics, School of Physics, Peking University, and Collaborative Innovation Center of Quantum Matter, Beijing, China 3 Centre for Quantum and Optical Science, Swinburne University of Technology, Melbourne, Victoria 3122, Australia 4 College of Precision Instrument and Opto-Electronics Engineering, Tianjin University, Key Laboratory of Opto-Electronics Information Technology, Ministry of Education, Tianjin, , China. seiji.armstrong@gmail.com; qiongyihe@pku.edu.cn; mdreid@swin.edu.au. NATURE PHYSICS 1

2 I. CRITERIA FOR GENUINE TRIPARTITE ENTANGLEMENT Three systems are said to be genuinely tripartite entangled iff the density operator for the tripartite system cannot be represented in the biseparable form ρ BS = P 1 η (1) R ρr 23ρ R 1 + P 2 η (2) R ρr 13 ρ R 2 + P 3 η (3) R ρr 12 ρ R 3. (1) R R R Here ρ R ij and ρr k are arbitrary quantum density operators for the composite system i and j, and for k, respectively. The P 1, P 2 and P 3 are probabilities for the system being in a state with a given bipartition. Thus, k P k =1, and R η(k) R =1[1, 2]. The negation that the system can be described by any one of the bipartitions ρ km,n = i η(n) i ρ i km ρi n is a demonstration of full tripartite inseparability [3 5]. For pure quantum states, where only one bipartition is possible, full tripartite inseparability is equivalent to genuine tripartite entanglement. For more general mixed states however, it is readily evident by counterexample that full tripartite inseparability does not imply genuine tripartite entanglement. These points have been explained by Shalm et al [6]. Our aim is to derive criteria involving position/ momentum observables that can be used to demonstrate genuine tripartite entanglement. Such criteria have been derived by Shalm et al [6] but these are not directly useful for our experiment, which generates continuous variable (CV) multipartite EPR-type states that are not simple Greenberger- Horne-Zeilinger (GHZ) states [3, 5]. First, we consider the tripartite system described by ρ km,n = i η (n) i ρ i kmρ i n, (2) where two but not three of the systems can be entangled (though may not be). We assume each system is a single mode with boson operator a j (j =1, 2, 3) and define the quadrature amplitudes as x j =(a j +a j ) and p j =(a j a j )/i. Criterion (1): The violation of the inequality u v 1 (3) is sufficient to confirm genuine tripartite entanglement, where u = x 1 (x 2 + x 3 )/ 2 and v = p 1 +(p 2 + p 3 )/ 2. Proof: The uncertainty relation x j p j 1 implies that the inequality ( u) 2 ( v) 2 1 (4) holds for all three types of states given by the form ρ 12,3, ρ 13,2, and ρ 23,1. To prove this, let us assume that the system is described by one of the mixtures ρ km,n = i η(n) i ρ i km ρi n. Then on using the Cauchy-Schwarz inequality, we find ( u) 2 ( v) 2 [ i η (n) i ( u) 2 i ][ i η (n) i ( v) 2 i ] [ i η (n) i ( u) i ( v) i ] 2, (5) where ( u) i ( v) i is the product of the variances for a pure product state of type ψ km ψ n denoted by i. Generally, let us consider a system in a product state of type ψ a ψ b and define the linear combinations x a + gx b and p a + gp b of the operators x a, p a and x b, p b for the systems described by wavefunctions ψ a and ψ b respectively. It is always true that the variances for such a product state satisfy [ (x a +gx b )] 2 = ( x a ) 2 +g 2 ( x b ) 2 and [ (p a +gp b )] 2 = ( p a ) 2 +g 2 ( p b ) 2. This implies that [ (x a + gx b )] 2 [ (p a + gp b )] 2 = [( x a ) 2 + g 2 ( x b ) 2 ] [( p a ) 2 + g 2 ( p b ) 2 ] [( x a )( p a )+g 2 ( x b )( p b )] 2, (6) where we use that for any real numbers x and y, x 2 + y 2 2xy. We can apply this result to deduce that for a product state of type ψ 12 ψ 3, it is true that ( u) i ( v) i [ (x 1 x 2 / 2)][ (p 1 + p 2 / 2)] ( x 3)( p 3 ) =1. A similar result holds for product states of type ψ 13 ψ 2. For states of type ψ 23 ψ 1, we find ( u) i ( v) i ( x 1 )( p 1 )+ 1 2 [ (x 2 + x 3 )][ (p 2 + p 3 )] 2. This proves the result (4). We now see that for any mixture (1), ( u) 2 ( v) 2 (P 1 R η (1) R ( u)2 R + P 2 η (2) R ( u)2 R + P 3 η (3) R ( u)2 R ) R R 2 NATURE PHYSICS

3 SUPPLEMENTARY INFORMATION (P 1 η (1) R ( v)2 R + P 2 η (2) R ( v)2 R + P 3 η (3) R ( v)2 R ) R R R [P 1 η (1) R ( u) R( v) R + P 2 η (2) R ( u) R ( v) R + P 3 η (3) R ( u) R ( v) R ]2 R R R 1, (7) where we use the Cauchy-Schwarz inequality and that R P R =1. Criterion (1) can be generalized. First, one can prove that the separability assumption (2) implies the inequality: u v h n g n + h k g k + h m g m (8) where u = h n x n + h k x k + h m x m and v = g n p n + g k p k + g m p m. Proof: Let us assume that the system is described by the mixture ρ km,n = i η(n) i ρkm i ρi n. Then on using the Cauchy-Schwarz inequality, from above we find u v i η(n) i ( u) i ( v) i, where ( u) i ( v) i is the product of the variances for a pure product state of type ψ km ψ n denoted by i. Generally, let us consider a system in a product state of type ψ a ψ b and define the linear combinations x a + gx b and p a + gp b of the operators x a, p a and x b, p b for the systems described by wavefunctions ψ a and ψ b respectively. It is always true that the variances for such a product state satisfy [ (x a + gx b )] 2 = ( x a ) 2 + g 2 ( x b ) 2 and [ (p a + gp b )] 2 = ( p a ) 2 + g 2 ( p b ) 2. This implies the result (6), which we can then use to deduce that for a product state of type ψ km ψ n, it is true that ( u) i ( v) i [ (h k x k + h m x m )][ (g k p k + g m p m )] + h n g n ( x n )( p n ) h k g k + h m g m + h n g n. Criterion 1.b: Genuine tripartite entanglement is observed if the inequality is violated. Proof: This follows directly from the result (8). u v min{ h 1 g 1 + h 2 g 2 + h 3 g 3, h 2 g 2 + h 1 g 1 + h 3 g 3, h 3 g 3 + h 1 g 1 + h 2 g 2 } (9) II. CRITERIA FOR GENUINE TRIPARTITE EINSTEIN-PODOLSKY-ROSEN STEERING We can readily modify the proof of Criterion (1) to derive a criteria sufficient to demonstrate genuine tripartite steering. Similar criteria have been derived in the supplementary material of Ref. [7] and we follow that approach. To prove genuine tripartite steering [7], it is sufficient to falsify a description of the statistics based on a hybrid Local-Nonlocal Hidden State (LHS) model, as defined by Refs. [7 10], where the averages are given as: e.g. X 1 X 2 X 3 = P 1 η (1) R X 2X 3 R X 1 R,ρ + P 2 η (2) R X 1X 3 R X 2 R,ρ + P 3 η (3) R X 1X 2 R X 3 R,ρ. R R R (10) Here the ρ subscript denotes that the averages are consistent with those of a quantum density matrix. In the first instance, no such constraint is made for the moments X k X m R, written without the subscript ρ. These latter moments are not assumed to arise from a local quantum state, e.g. they do not necessarily obey two-party quantum uncertainty relations. We now derive a criterion based on the model, with minimal assumptions about the nature of the two-party moments X k X m R. The Local Hidden State model (10) is one in which the system is in a probabilistic mixture of the three bipartitions of the system, where the relative probabilities are given as P 1, P 2, P 3. We denote the three bipartitions as {23, 1} st, {13, 2} st, {12, 3} st where the subscript st reminds us that to test for steering, the state of the set written after the comma is assumed to be a local quantum state (The first combined state written before the comma is not necessarily representable as a local quantum state). We look for a criterion for genuine tripartite steering based on the inequality (4). The key point to the derivation is to remember we follow the assumptions of separability but we use the quantum uncertainty relation only for the site that has been assumed to be a Local Quantum State. For the other two sites, we do not assume anything about the variances of the local states (except that they are positive). We restrict to where each system is a single mode, with boson operators a 1, a 2 and a 3, and define the quadrature amplitudes as above, which implies an uncertainty relation for each mode given by x n p n 1. First consider the system to be in a special sort of bipartition based on the model (10) that we denote {km, n} st, where we assume that only system n is constrained to be a quantum state. NATURE PHYSICS 3

4 4 Now, we derive the EPR steering version of the product inequality given by Criterion (1). Criterion (2): The violation of the inequality u v 0.5, (11) is sufficient to confirm genuine tripartite EPR steering, where u = x 1 (x 2 + x 3 )/ 2 and v = p 1 +(p 2 + p 3 )/ 2. Proof: The proof proceeds as for Criterion (1), except that the states with a definite bipartition are now less constrained. For the bipartition {12, 3} st, we can only assume quantum uncertainty constraints for state 3, which implies ( u) i ( v) i (x 1 x 2 / 2) (p 1 + p 2 / 2) x 3 p 3 1 2, and similarly for bipartition {13, 2} st. For bipartition {23, 1} st we find ( u) i ( v) i x 1 p (x 2 + x 3 ) (p 2 + p 3 ) 1. Following the method of the proof of Criterion (1), this allows us to deduce Criterion (2). The inequality of Criterion (2) is sufficient to confirm genuine tripartite steering, but requires noise reduction at a much greater level than Criterion (1). Our current experiment meets the condition of Criterion (1). It does not meet the criterion (2) but we are able in Section V to give predictions for it with suitably modified input-state squeezing levels. III. MONOGAMY RELATION AND QUANTUM SECRET SHARING A. Monogamy inequality Here, we explain and derive the monogamy result S A C S A B 1, (12) where S A B = inf X A B inf P A B. The average conditional inference variances are defined as: [ inf X A B ] 2 = P (x B )[ (X A x B )] 2, (13) x B and [ inf P A B ] 2 = p B P (p B )[ (P A p B )] 2. (14) where x B (p B ) are the possible results of a measurement performed on system B. The usual choice of measurement for x B is that which optimizes the inference of X A i.e. the one that will minimize the average conditional variance, though this is not essential to the validity of the monogamy result. Similarly, p B are the possible results for a second measurement performed at B, that are usually chosen to optimize the inference of P A. The monogamy result is proven in Refs. [7, 11], but the proof we give here is more detailed. We note that the choice of scaling used to define the quadrature phase amplitudes X A/B, P A/B leads to the Heisenberg uncertainty relation X A P A 1. To derive the relation, we note that the observer (Bob) at B can make a local measurement O B to infer a result for an outcome of X A at A. The set of values denoted by x B are the spectrum of results for the measurement O B, and P (x B ) is the probability for the outcome x B. The conditional distribution P (X A x B ) has a variance which we denote by [ (X A x B )] 2. The ( inf X A B ) 2 is thus the average conditional variance. Similarly, Bob can make another measurement, denoted Q B, to infer a result for the outcome of P A at A. Denoting the results of this measurement by the set p B, we define the conditional variances as for X A. Now, a third observer C ( Charlie ) can also make such inference measurements, with uncertainty inf X B C and inf P B C. Let us denote the outcomes of Charlie s measurements, for inferring Alice s X A or P A, by x C and p C respectively. Since Bob and Charlie can make the measurements simultaneously, a conditional quantum density operator ρ A {xb,p C } for system A, given the outcomes x B and p C for Bob and Charlie s measurements, can be defined. The P (x B,p C ) is the joint probability for these outcomes. The moments predicted by this conditional quantum state must satisfy the Heisenberg uncertainty relation. That is, we can define the variance of X A conditional on the joint measurements as (X A x B,p C ) and (P A x B,p C ) and these must satisfy (X A x B,p C ) (P A x B,p C ) 1. We also note that [ inf X A B ] 2 x B,p C P (x B,p C )[ (X A x B,p C )] 2 and [ inf P A C ] 2 x B,p C P (x B,p C )[ (P A x B,p C )] 2 (proved in the Lemma below). Hence, using the Cauchy-Schwarz inequality, we can write [ inf X A B ] 2 [ inf P A C ] 2 P (x B,p C )[ (X A x B,p C )] 2 x B,p c 4 NATURE PHYSICS

5 SUPPLEMENTARY INFORMATION x B,p c P (x B,p C )[ (P A x B,p C )] 2 [ x B,p C P (x B,p C ) (X A x B,p C ) (P A x B,p C )] 2 1. (15) Similarly, Bob can measure to infer P A and Charlie can measure to infer X A, and it must also be true that Hence, it must be true that S A B S A C 1. Lemma: We follow the steps: [ (X A x B )] 2 = X A P (X A x B )(X A µ xb ) 2 inf P A B inf X A C 1. (16) 1 = P (X A,x B )(X A µ xb ) 2 P (x B ) X A 1 = P (X A,x B,p C )(X A µ xb ) 2 P (x B ) X A p C 1 p(x B,p C ) P (X A x B,p C )(X A µ xb,p P (x B ) C ) 2 p C X A 1 = p(x B,p C )[ (X A x B,p C )] 2, P (x B ) p C where we have denoted µ xb to be the mean of P (X A x B ) and µ xb,p C to be the mean of P (X A x B,p C ). We have also used that the value of the constant µ that minimizes (x µ) 2 will be the mean of the associated probability distribution. In the current paper, the values for the inference variances as defined in (13-14) are determined by linear optimization. It is explained in Ref. [12] how the value determined this way cannot be less than the (smallest) value given by the definition (13-14). Furthermore, it is explained in Ref. [12] that for Gaussian states, the values according to the two definitions become equal. We also make the point that the derivation of the monogamy relation requires three distinct systems (as given by three independent modes), but does not require that the systems be spatially separated. Hence, the monogamy relation is applicable to the current experiment. We note that the base-value for the product of the steering parameters as measured in the experiment is higher than the value corresponding to the Heisenberg bound, as expected due to the impurity of the input squeezed states. B. Gaussian assumption for no-way and one-way steering, and secret sharing The monogamy relation follows as a result of the Heisenberg uncertainty relation, and is true for any experimental realization i.e. there is no assumption, for example, that the states generated in the experiment are Gaussian states. This is also true of the steering inequalities. If we satisfy S A B < 1 for any possible g, then we have confirmed steering of system A by B. Our depictions of the no-way and one-way steering need clarification however. If we show that S A B 1 for all g, then we have confirmed the impossibility of what we will call two-mode Gaussian bipartite steering of A by B. This follows because for such bipartite Gaussian systems, it has been shown by Jones et al [9] that the bipartite EPR criterion S A B < 1 (when optimized by the appropriate choice of the linear constant g) is a necessary and sufficient condition for steering of A by B. Two-mode Gaussian systems imply two-mode Gaussian states and Gaussian measurements (as may be made using homodyne detection). We have not strictly negated however that steering can take place via e.g. other types of measurements, or systems. Hence, while the two-way steering result is without assumption, our meaning of one-way and no-way steering is strictly for Gaussian steering only i.e. is within the assumption of Gaussian systems. For the quantum secret sharing illustrated in Figure 3, the measurements made by the collaborating parties Bob and Charlie are device-independent (which means without the assumption of reliable calibrated quantum measurements). However, we emphasize that the negation that any one of Bob or Charlie can infer the amplitudes to low uncertainty is not fully device-independent, since we negate the possibility of steering based on the assumption (as explained above) of two-mode Gaussian systems. NATURE PHYSICS 5

6 C. Three-way quantum secret sharing The Figure 3 of the main text illustrates the principle of Gaussian secret sharing [13 15], where Bob and Charlie infer Alice s amplitude, as in the arrangement shown in the schematic. It is possible to manipulate the two beam splitter reflectivities R 1 and R 2, so that the Gaussian secret sharing is a property that can be shared by any combination of three of the observers i.e. any two parties can secret share to uncover the amplitude of the third party. We call this three-way secret sharing. The three-way secret sharing is predicted if one adjusts the first beamsplitter to R 1 : (1 R 1 )=1:4, and the second to a value 0.44 <R 2 < Denoting the parties by i, j, k where each can be 1, 2 or 3 and i j k, we predict: for 0.07 R , we find S and S while S 1 23 < 1, so that 3 and 2 can secret share the amplitudes of 1; for 0 <R , 1 and 3 can secret share the amplitudes of 2; and for 0.44 <R 2 1, 2 and 1 can secret share the amplitudes of 3. The values specified here are based on the input states of the experiment (squeezing for the first input is 8.9dB for X and 3.6dB for P ; squeezing for the second input is 4.1dB for X and 9.5dB for P ). D. Directional steering: N +1 regimes exist for an N-party state created with asymmetry The precise experimental thresholds required for manipulating the steering in the 7-qumode case depend on both the asymmetric losses on the various channels, and the stochastic experimental noises that arise for example from the impurity of the vacuum inputs to the beam splitters. To illustrate the principle, we examine how the steering of the tripartite state generated as depicted in Figure 1 can be manipulated. In the Figures 2 and 3, we present the results of a theoretical analysis for the tripartite case based on the input states of the experiment. Figure 1. The asymmetric system in the tripartite case: The asymmetries of the system are created by adjusting the reflectivities R 1 and R 2 of BS1 and BS2, and the amount of loss on each channel. Here, we manipulate the loss for mode k by applying a beam splitter with efficiency of transmission denoted η k as depicted. 6 NATURE PHYSICS

7 SUPPLEMENTARY INFORMATION way 3 way 0.6 Η no way 2 way Η 1 Figure 2. Four different regimes of directional steering in the tripartite case: Here, the regimes are created by adjusting the efficiencies η 1 and η 2. We choose η 3 =1(no loss) and R 1 = R 2 =1/2 but we vary η 1, η 2. A requirement for steering is that the steering mode has losses no greater than 50%. We note that mode 3 can be steered by 1 alone, and is hence insensitive to the loss on mode 2. The black contour lines give the steering (S 1 K ) 2 of mode 1; the red contour lines give the steering (S 2 K ) 2 of mode 2; and the blue contour lines give the steering (S 3 K ) 2 of mode 3. The boundaries for steering are indicated by the colored rectangles. The three-way regime means here that each of the single modes 1, 2 and 3 can be steered by the other two; two-way means that only two of the modes can be steered; one-way that only one mode can be steered, and no-way that no single mode can be steered. We do not examine the reverse directions of steering in this paper. 1.1 EPR steering parameter no way way way way Η Figure 3. Four different regimes of directional steering in the tripartite case: Here, the regimes are created by adjusting the reflectivities of BS1 and BS2, for the symmetric case where η 1 = η 2 = η. We choose R 1 =3/4, R 2 =1/4 and η 3 =1(no loss). The black, red and blue lines give the values of the EPR steering parameter (S 1 K ) 2, (S 2 K ) 2 and (S 3 K ) 2, respectively. IV. THE OPTIMAL g VALUES FOR 3-MODE AND 7-MODE Optimal values for the quadrature gains are found analytically for each linear optics circuit, and set of unique inputs. The gains are then fed into the experimental setup so that the experiment and theoretical predictions both NATURE PHYSICS 7

8 8 employ the same set of parameters. Numerical optimisation algorithms were employed in order to compensate for experimental imperfections but differences were negligible, indicating a remarkable fit between the experimental and theory. Table IV displays the relevant gain values and inferred variances for tripartite steering. Inferred variance EP R value g 1,x g 2,x g 3,x g 1,p g 2,p g 3,p ( inf ˆx 1 ) / / / / ( inf ˆx 2 ) / / / / ( inf ˆx 3 ) / / / / ( inf ˆp 1 ) / / / / ( inf ˆp 2 ) / / / / ( inf ˆp 3 ) / / / / ( ) 2 S ( ) 2 S ( ) 2 S Table I. (Theoretical result) Tripartite steering g k,x and g k,p values as defined in equations (2-4) of the main paper. Here the gains correspond the schematic Fig1 here: mode 1 is on the left side of BS1 and 2, 3 are located on the right side of BS1. The beam-splitters are set to R 1 =0.511, and R 2 =0.5. Inferred variance EP R value g k (k i) g k (k i) (i is on the left side of BS1) all k on the same side as i all k on the different side to i ( inf ˆx i) g k,x =0.086 g k,x =0.325 ( inf ˆp i) g k,p =0.047 g k,p =0.302 ( ) 2 Si K Inferred variance EP R value g k (k i) g k (k i) (i is on the right side of BS1) all k on the same side of i all k on the different side to i ( inf ˆx i ) g k,x =0.104 g k,x =0.319 ( inf ˆp i ) g k,p =0.070 g k,p =0.295 ( ) 2 Si K Table II. (Theoretical result) 7-partite steering g values. remaining N 1 inferred systems. Here we label the mode i as the steered system and k are the V. PREDICTIONS FOR GENUINE TRIPARTITE ENTANGLEMENT AND EPR STEERING We analyze predictions for the inequality (1) and (4) of the main paper using the theoretical model (Fig. 4(a)). Assuming pure squeezed inputs with a squeeze parameter r (to give variances e 2r and e 2r in the squeezed and antisqueezed quadratures respectively), we find that as r, the left side of inequality (1) becomes 0 (Fig. 4(b)). This result was pointed out by van Loock and Furusawa [3]. Current experimental data corresponds to inputs with 4.1dB and 3.6dB squeezing, which is sufficient to produce the genuine tripartite entanglement shown as the solid line in Fig. 4(c). The theoretical predictions are consistent with experimental data. Genuine tripartite steering as given by Criterion (2) is predicted when the squeezing of the inputs is greater than 6dB (dashed line in Fig. 4 (c)). We note however that the Criterion (2) has been derived with fewer assumptions about the nature of the observations made by the different observers at the different nodes. This makes it a useful result, if observed. In particular, the confirmation of the single genuine steering inequality Criterion(2) is sufficient to certify that in a network of three nodes, any two observers can determine the amplitudes of the remaining single mode to high precision (below the standard quantum limit for both position and momentum) without the assumption of trustworthy equipment at 8 NATURE PHYSICS

9 SUPPLEMENTARY INFORMATION the nodes of the two observers. Thus, each individual on a network, can set up entanglement verification with the remaining two sites, with no assumption of trust of devices for all others on the network. The procedure works three-way, no matter which site becomes that of the trusted observer. entanglement parameter entanglement parameter b r c Figure 4. (a) Schematic for the generation of the genuine tripartite entangled state. (b) Prediction for the genuine tripartite entanglement EPR parameter (ˆx 1 (ˆx 2+ˆx 3 ) 2 ) (ˆp 1 + (ˆp 2+ˆp 3 ) 2 ) where 1, 2, 3 are produced as in the schematic (a). We assume R 1 =1/2 and that the inputs are two ideal squeezed input states with squeeze parameter r. According to Criteria 1 and 2 (inequalities (1) and (4) of the main paper) genuine tripartite entanglement and steering are confirmed when the parameter is less than 1 and 0.5 respectively. (c) Predictions for the experimental inputs with 4.1dB and 3.6dB squeezing (solid line), and for inputs with 6dB (dashed line), versus R 1. R 1 We can also analyze the predictions for the more general inequality given by the generalized Criterion (1b) with NATURE PHYSICS 9

10 1.00 entanglement parameter Figure 5. Genuine tripartite entanglement EPR parameter (ˆx 1 +h 2ˆx 2 +h 3ˆx 3 ) (ˆp 1 +g 2 ˆp 2 +g 3 ˆp 3 ) versus reflectivity R 1 of first beam splitter when R 2 =1/2. Dashed curve corresponds to the case with optimal gains h 2 = h 3 = , g 2 = g 3 =0.6411, which give the minimum value of the parameter Solid curve corresponds to the case where the gains are fixed with h 2 = h 3 = 1 2, g 2 = g 3 = 1 2, for which the minimum value is R 1 optimal gains, i.e. (h 1ˆx 1 + h 2ˆx 2 + h 3ˆx 3 ) (g 1 ˆp 1 + g 2 ˆp 2 + g 3 ˆp 3 )<min{ g 3 h 3 + h 1 g 1 + h 2 g 2, g 2 h 2 + h 1 g 1 + h 3 g 3, g 1 h 1 + g 2 h 2 + h 3 g 3 }. (17) We examine the case where we fix the the reflectivity of the second beam splitter to be R 2 =1/2, but vary the reflectivity R 1 of the first beam splitter. R 2 = 1/2 means that the gains for mode 2 and 3 are the same, i.e. h 2 = h 3 = h, g 2 = g 3 = g, where g, h are usually selected to be smaller than 1. Then selecting h 1 = g 1 =1, we can analyze the right side of inequality (17) and find that the minimum becomes 1. The minimum value of the left side is then found to be with optimal gains h = , g = The minimum value is located at the reflectivity R (specifically R 1 = ). By comparison, if we fix the gains h = 1/ 2 and g =1/ 2, then the minimum value of the left side is when R 1 =0.5. The results are summarized in the Figure 5. The choice of beam splitter reflectivity R is to an excellent approximation optimal to satisfy the inequalities. [1] Bancal, J. et al. Device-Independent Witnesses of Genuine Multipartite Entanglement. Phys. Rev. Lett. 106, (2011). [2] Bourennane, M. et al., Experimental Detection of Multipartite Entanglement using Witness Operators. Phys. Rev. Lett. 92, (2004). [3] Van Loock, P. & Furusawa, A. Detecting genuine multipartite continuous-variable entanglement. Phys. Rev. A 67, (2003). [4] Aoki, T. et al. Experimental Creation of a Fully Inseparable Tripartite Continuous-Variable State. Phys. Rev. Lett. 91, (2003). [5] Armstrong, S. et al. Programmable multimode quantum networks. Nature Commun. 3, 1026 (2012). [6] Shalm, L. K. et al. Three-photon energy time entanglement. Nature Phys. 9, (2012). [7] He, Q. Y. & Reid, M. D. Genuine Multipartite Einstein-Podolsky-Rosen Steering. Phys. Rev. Lett. 111, (2013). [8] Wiseman, H. M., Jones, S. J. & Doherty, A. C. Steering, Entanglement, Nonlocality, and the Einstein-Podolsky-Rosen Paradox. Phys. Rev. Lett. 98, (2007). [9] Jones, S. J. et al. Entanglement, Einstein-Podolsky-Rosen correlations, Bell nonlocality, and steering. Phys. Rev. A 76, (2007). [10] Svetlicnhy, G. et al. Distinguishing three-body from two-body nonseparability by a Bell-type inequality. Phys. Rev. D 35, 3066 (1987). [11] Reid, M. D. Monogamy inequalities for the Einstein-Podolsky-Rosen paradox and quantum steering. Phys. Rev. A 88, (2013). 10 NATURE PHYSICS

11 SUPPLEMENTARY INFORMATION [12] Reid, M. D. et al. Colloquium: The Einstein-Podolsky-Rosen paradox: From concepts to applications. Rev. Mod. Phys. 81, (2009). [13] Hillery, M., Buzek, V. & Berthiaume, A. Quantum secret sharing. Phys. Rev. A 59, (1999). [14] Bogdanski, J. et al. Experimental quantum secret sharing using telecommunication fiber. Phys. Rev. A 78, (2008). [15] Lance, A. M. et al. Continuous variable (2, 3) threshold quantum secret sharing schemes. New J. Phys. 5, 4 (2003). NATURE PHYSICS 11