December 1, 2014 / Vol. 39, No. 23 / OPTICS LETTERS
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Collective multipartite Einstein–Podolsky–Rosen steering: more secure optical networks Meng Wang,1 Qihuang Gong,1,2 and Qiongyi He1,2,* 1
State Key Laboratory of Mesoscopic Physics, School of Physics, Peking University, Beijing 100871, China 2 Collaborative Innovation Center of Quantum Matter, Beijing 100871, China *Corresponding author: [email protected] Received May 22, 2014; revised October 30, 2014; accepted October 30, 2014; posted October 31, 2014 (Doc. ID 212349); published November 26, 2014
Collective multipartite Einstein–Podolsky–Rosen (EPR) steering is a type of quantum correlation shared among N parties, where the EPR paradox of one party can only be realized by performing local measurements on all the remaining N − 1 parties. We formalize the collective tripartite steering in terms of local hidden state model and give the steering inequalities that act as signatures and suggest how to optimize collective tripartite steering in specific optical schemes. The special entangled states with property of collective multipartite steering may have potential applications in ultra-secure multiuser communication networks where the issue of trust is critical. © 2014 Optical Society of America OCIS codes: (270.5585) Quantum information and processing; (270.6570) Squeezed states; (000.2658) Fundamental tests. http://dx.doi.org/10.1364/OL.39.006703
It is a major challenge in modern optical communication networks to guarantee ultra-secure information transmission between users at different terminals so that any eavesdropper with access to the communication channel is incapable of acquiring useful information. Many quantum techniques have been developed for secure quantum cryptographic communications, such as quantum key distribution [1] and quantum secret sharing [2,3], where high security is guaranteed by fundamental laws of quantum mechanics. One of the key phenomena of quantum mechanics is the known Einstein–Podolsky–Rosen (EPR) paradox [4]. It established a link between entanglement and nonlocality [5], by showing that there are correlated quantum states that demonstrate an inconsistency between the completeness of quantum mechanics and the concept of local realism. Schrödinger introduced the term “steering” to describe the apparent nonlocality manifested by EPR paradox, where measurements made by one observer at location A can immediately affect the state of another observer at a remote location B [6]. Reid proposed a simple form of an EPR paradox (considering nonideal states) based on Heisenberg uncertainty relation [7,8]. In this scheme, by performing local measurements on system A, Alice can predicate the measurement results xB and pB of remote system B to a given accuracy based on her outcomes xA and pA . The uncertainties associated with the collapsed wave functions are quantified by the conditional variances Δinf xB ΔxB − gx xA and Δinf pB ΔpB − gp pA , where gx and gp are real scaling factors chosen to minimize the inferred errors. For some choice of measurements xA and pA , if the conditional variances are sufficiently small such that Δinf xB Δinf pB < 12 jhxB ; pB ij, an EPR paradox arises. This is because the predetermination for xB and pB in this case is more accurate than allowed by the uncertainty principle. Recent work of Wiseman and co-workers [9,10] formalized the idea of steering in terms of violation of local hidden state (LHS) model, and revealed that the EPR paradox is a realization of quantum steering. For two-mode Gaussian systems, the Reid criterion has 0146-9592/14/236703-04$15.00/0
been shown to be necessary and sufficient to detect bipartite steering [9]. Bipartite steering has been confirmed for continuous variable (CV) [11–13] and discrete regimes [14–16]. EPR steering has now been extended to multipartite scenarios [17–19]. In this Letter, we generalize the concept of collective multipartite steering: a quantum system consisting of N parties shows collective steering of a given party i, if party i can be steered by all the remaining N − 1 parties, but not by any N − 2 or fewer parties. This means the group of N − 1 parties must collaborate after performing local measurements on each individual system in order to extract the information of party i. This nonlocality is the natural multipartite extension of the original EPR paradox, and has the potential application to achieve ultrasecure N-party quantum secret sharing [17]. We first explicitly formalize the meaning of collective tripartite EPR steering in terms of LHS model and clarify the signatures to confirm the existence of such correlation. Then we test a tripartite CV Greenberger–Horne– Zeilinger (GHZ)-like state [20–22] and show that it always possesses the property of collective tripartite steering. In addition, we suggest how to achieve collective tripartite steering in a specific optical scheme by simply employing the reported experimental data [23] and adjusting reflectivities of beamsplitters (BS). The interesting feature is noticed that the impure squeezing, which is often thought to be detrimental, is indeed favorable to realize collective steering in a fairly large parameter regime in experiments. We consider two spatially separated measurements X i and X j , with outcomes xi and xj . The system can be formulated as a LHS model [9] if the joint probability for outcomes of these measurements satisfies Z Pxi ; xj
λ
dλPλPxj jλP Q xi jλ:
(1)
It is an intermediate case between the local hidden variable and quantum separable models. Here, λ labels Bell’s © 2014 Optical Society of America
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“local hidden variables” [5], and Pxk jλ is the probability of xk (k i; j) given the values of λ. The R function Pλ is the probability distribution for λ with λ Pλdλ 1. The subscript Q in function P Q xi jλ indicates that the system i is a quantum state, such that the probability distribution is constrained by the Heisenberg uncertainty principle Δxi Δpi ≥ 12 jhxi ; pi ij. The other system j, however, is not assumed to be a quantum state, e.g., it does not necessarily obey quantum uncertainty relation, and therefore, one can have a complete classical knowledge about the outcomes of xj and pj . In Einstein–Podolsky– Rosen’s language, they are predetermined “elements of reality” for the state of system j. Failure of model (1) implies steering of subsystem i by measurements performed on j [9]. Now we turn to the case of collective tripartite steering. For a quantum system consisting of three spatially separated parties labeled by fi; j; kg, we impose the asymmetric constraint that there exists a quantum density operator ρλi such that hxi iλ Trρλi xi for site i, but not for the group of sites fj; kg. To demonstrate steering of system i by local measurements performed on j and k, we need to falsify a description of the statistics based on a hybrid LHS model, where the joint probability distribution is given as Z Pxi ; xj ; xk
λ
dλPλPxj ; xk jλP Q xi jλ:
(2)
It can be easily seen that collective tripartite steering of system i can be translated to the failure of model (2), while model (1) remains valid for both systems j and k. This means that the information of system i can only be extracted by a joint measurement of the group fj; kg, but not by any measurement of j or k alone. In order to negate the bipartite steering, the criterion used to test steering of one party by another must be necessary and sufficient. Reid inequality formalized in Ref. [7] has been proved in Ref. [9] as a necessary and sufficient criterion to detect bipartite steering for Gaussian systems. In the following discussion, we will use the Reid inequality to test the collective tripartite steering in three-mode Gaussian systems. Consider three harmonic oscillators (fields) at sites i, j, and k, with boson operators ai;j;k . The quadratures x and p are given by x a a† , p −ia − a† with the uncertainty relation ΔxΔp ≥ 1. From Eqs. (1) and (2), collective tripartite EPR steering can be obtained when S ijj Δxi − gxj xj Δpi − gpj pj ≥ 1; S ijk Δxi − gxk xk Δpi − gpk pk ≥ 1;
(3)
and
minimized by optimizing gain factors g via ∂S∕∂g 0. If the minimized S ijj and S ijk satisfy inequality (3), we guarantee that all bipartite steering is removed, as other steering parameters with arbitrary gain factors must be larger than the minimized value. The special quantum states with property of collective multipartite steering may provide a solution to the challenges of secure quantum communication network. In model (2), the state of systems j and k are not assumed to be a quantum state. This means that both sets of values, xj and pj , xk and pk , could be false outcomes of measurements due to hacking attacks or untrustworthy receivers and their apparatus. The collective steering criterion does not require the trust of devices used by steering parties, whereas the measurements on system i must be trusted [24]. As an example, Alice first prepares a three-mode quantum state. This state satisfies the collective steering criterion and is steerable from the group j, k to i. Alice holds the mode i and encrypts the secret information into the quadratures of mode i. Then she distributes the modes j and k to Bob and Charlie. To decipher the secret information, Bob and Charlie must collaborate by making local measurement on their own mode and communicate their results to infer the quadratures of Alice’s mode with high accuracy. Collective tripartite steering removes the possibility of extracting the secret by a single receiver. Moreover, Alice can make sure the secret message is transmitted securely to Bob and Charlie by validating the criterion without the trust assumption on the measurements outcomes reported by Bob and Charlie. Those properties can be useful to the one-sided device-independent quantum secret sharing protocol [25–28]. Next, we show that a tripartite CV GHZ-like state always possesses the property of collective tripartite steering. The CV GHZ-like states are introduced by van Loock and his coworkers [20–22] as the continuous variable analogues to the multipartite GHZ states of qubits. The tripartite CV GHZ-like state can be generated by first superimposing a pure squeezed mode with a vacuum mode at an asymmetric BS (1∶2), then mixing one of the output modes with another vacuum mode at second balanced BS [20–22]. The three output modes are locally measured by balanced homodyne detection, as shown in Fig. 1. For such a state, we find that bipartite steering parameter given by Eq. (3) is always equal to quantum shot noise with optimal gains, i.e., S ijj S ijk 1;
which means any output mode i cannot be steered by other output mode j or k alone. However, steering of i can be demonstrated by measurements involving a group of fj; kg, evidenced by the tripartite steering parameter given by Eq. (4),
S ijfj;kg Δxi − gxj xj gxk xk × Δpi − gpj pj gpk pk < 1:
(4)
Note that inequalities (3) represent party i cannot be steered by any other party j or k alone, and inequality (4) means party i can be steered by the collaboration of the remaining two parties. EPR steering parameters S can be
(5)
S ijfj;kg
s 9 < 1; 5 4 cosh2r
(6)
for any squeezing parameter r > 0 with optimal gains gxj gxk e2r − 1∕2e2r 1, gpj gpk 1 − e2r ∕ 2 e2r , where i; j; k 1; 2; 3 and i ≠ j ≠ k. Thus we
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⃗xT x1 ; x2 ; x3 T , ⃗pT p1 ; p2 ; p3 T , ⃗xTin where T T T ⃗p x1;in ; x2;in ; x3;in , in p1;in ; p2;in ; p3;in , and the BS operation matrix T 123 is given as 2
T 123
3 p p r1 0 1 − r1 6 p p 7 p 6 − r1 r2 1 − r 1 r 2 1 − r2 7 4 p 5: p p 1 − r 1 1 − r 2 − r 1 1 − r 2 − r 2 (9)
Fig. 1. Schematic diagram to test steering of mode 1 by modes 2 and 3 using homodyne detection techniques. Each output mode is combined at a balanced beamsplitter (BS) with an intense LO field, whose phase shift ϕLO determines the measurement of X or P quadratures. The output fields of each balanced BS are detected by photodiodes, and their difference current is proportional to the measured quadratures. Three modes are electronically combined in an appropriate way. The fluctuations of the current give a measure proportional to the inferred errors of the quadratures.
conclude that a tripartite CV GHZ-like state possesses the collective tripartite steering correlation regardless the value of the squeezing parameter of the input mode. Despite theoretical importance, the ideal CV GHZ-like state requires stringent experimental conditions, including pure squeezed input states, ideal BS, and detectors. However, in a realistic experiment setup, noise will inevitably inflate variance of the squeezed vacuum in either quadrature. In the following discussion, we will show that noise is more effective in destroying the bipartite steering rather than the tripartite steering, as has been shown in a hybrid three-mode optomechanical system [29]. Next, we discuss how to utilize the reported experimental data of impure squeezed state to create the special states with property of collective steering. We check the squeezed state produced by type-I optical parametric amplification based on periodically poled potassium titanyl phosphate at the telecommunication wavelength of 1550 nm with a noise reduction of 5.3 dB in the squeezed quadrature and an increase of 9.8 dB in the antisqueezed quadrature [23]. We find that if we use this impure squeezed state as the input mode x1;in ; p1;in in Fig. 1 and properly adjust the reflectivities r 1 and r 2 of BS1 and BS2 , respectively, it is possible to make the three output modes exhibit the collective EPR steering correlation. Applying the BS operations p p T 123 BS2 cos−1 r 2 BS1 cos−1 r 1
(7)
to the input modes yields for the following output modes ⃗xT T 123 ⃗xTin ;
⃗pT T 123 ⃗pTin ;
(8)
From the output modes in Eq. (8), we can get the inferred variances and therefore steering parameters S ijj , S ijk and S ijfj;kg of mode i. To demonstrate the collective tripartite steering, we need to adjust the reflectivities of BS1 and BS2 appropriately to take into account asymmetric squeezing values of the impure squeezed input x1;in ; p1;in . We first fix BS1 , and then adjust BS2 to find the region to achieve collective tripartite steering. Figure 2 shows that by setting BS1 at r 1 ∶1 − r 1 1∶3, we can make steering parameters of mode 1 satisfying S 1j2 > 1; S 1j3 > 1; S 1jf2;3g < 1 when 0.30 ≲ r 2 ≲ 0.70 (region between two red points). This condition guarantees the steering of mode 1 by modes 2 and 3, but negates the steering of mode 1 by single mode 2 or 3. Similarly, S 2j1 > 1; S 2j3 > 1; S 2jf1;3g < 1 can be achieved when 0.27 ≲ r 2 < 1 (right region of solid square), which demonstrates the collective steering of mode 2, and S 3j1 > 1; S 3j2 > 1; S 3jf1;2g < 1 when 0 < r 2 ≲ 0.73 (left region of empty square) that gives the collective steering of mode 3. Comparing with tripartite CV GHZ-like state that strictly requires fixing the first BS at 1∶2 and the second BS at r 2 0.5, interestingly, we find that the impure squeezing input results in a wider range for the collective steering to be valid. The impact of impurity is often thought to be unfavorable to accomplish quantum tasks. However, we show the performance of the asymmetric squeezing has advantages in this application. With highly improved techniques for generating squeezed state and
Fig. 2. EPR steering parameters S ijj 2 (labeled ijj) and S ijfj;kg 2 (labeled ijfj; kg) versus the reflectivity r 2 of BS2 when BS1 is fixed at 1∶3. The red dashed, dotted, and solid lines represent S 1j2 2 , S 1j3 2 , and S 1jf2;3g 2 , respectively. The blue dashed, dotted, and solid lines represent S 2j1 2 , S 2j3 2 , and S 2jf1;3g 2 , respectively. The black dashed, dotted, and solid lines represent S 3j1 2 , S 3j2 2 , and S 3jf1;2g 2 , respectively.
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homodyne measurement [30–32], this special states can be generalized to the multipartite case. In conclusion, we have explicitly formalized the meaning of collective tripartite steering in terms of LHS model, and clarified the signatures as an experimental witness to demonstrate this property for Gaussian CV states. Our results show that a tripartite CV GHZ-like state always exhibits collective tripartite steering. In addition, we present a practical protocol for achieving that special states based on impure squeezing input. We show that it is possible to verify collective tripartite steering within realistic experimental conditions by optimizing the reflectivities of optical beamsplitters. The method introduced here can be straightforwardly generalized to other multipartite systems to produce collective EPR steering, which are the central resource for ultra-secure messages transmission in quantum networks with many users where the issue of trust is of importance. This work was supported by the National Natural Science Foundation of China under Grant Nos. 11274025 and 11121091. References 1. C. H. Bennett and G. Brassard, in Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing (IEEE, 1984), pp. 175–179. 2. M. Hillery, V. Bužek, and A. Berthiaume, Phys. Rev. A 59, 1829 (1999). 3. R. Cleve, D. Gottesman, and H.-K. Lo, Phys. Rev. Lett. 83, 648 (1999). 4. A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935). 5. J. S. Bell, Physics 1, 195 (1965). 6. E. Schrödinger, Proc. Cambridge Philos. Soc. 31, 555 (1935). 7. M. D. Reid, Phys. Rev. A 40, 913 (1989). 8. M. Reid, P. Drummond, E. Cavalcanti, W. Bowen, P. Lam, H. Bachor, U. Andersen, and G. Leuchs, Rev. Mod. Phys. 81, 1727 (2009). 9. H. M. Wiseman, S. J. Jones, and A. C. Doherty, Phys. Rev. Lett. 98, 140402 (2007). 10. E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, Phys. Rev. A 80, 032112 (2009). 11. Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, Phys. Rev. Lett. 68, 3663 (1992).
12. Y. Wang, H. Shen, X. L. Jin, X. l. Su, C. D. Xie, and K. C. Peng, Opt. Express 18, 6149 (2010). 13. V. Händchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R. F. Werner, and R. Schnabel, Nat. Photonics 6, 598 (2012). 14. D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, Nat. Phys. 6, 845 (2010). 15. D. H. Smith, G. Gillett, M. P. de Almeida, C. Branciard, A. Fedrizzi, T. J. Weinhold, A. Lita, B. Calkins, T. Gerrits, H. M. Wiseman, S. Nam, and A. G. White, Nat. Commun. 3, 625 (2012). 16. A. J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, Phys. Rev. X 2, 031003 (2012). 17. Q. Y. He and M. D. Reid, Phys. Rev. Lett. 111, 250403 (2013). 18. E. G. Cavalcanti, Q. Y. He, M. D. Reid, and H. M. Wiseman, Phys. Rev. A 84, 032115 (2011). 19. Q. Y. He, P. D. Drummond, and M. D. Reid, Phys. Rev. A 83, 032120 (2011). 20. P. van Loock and S. L. Braunstein, Phys. Rev. Lett. 84, 3482 (2000). 21. P. van Loock and S. L. Braunstein, Phys. Rev. A 63, 022106 (2001). 22. P. van Loock and A. Furusawa, Phys. Rev. A 67, 052315 (2003). 23. M. Mehmet, S. Steinlechner, T. Eberle, H. Vahlbruch, A. Thuring, K. Danzmann, and R. Schnabel, Opt. Lett. 34, 1060 (2009). 24. B. Opanchuk, L. Arnaud, and M. D. Reid, Phys. Rev. A 89, 062101 (2014). 25. C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, Phys. Rev. A 85, 010301(R) (2012). 26. A. Acin, N. Gisin, and L. Masanes, Phys. Rev. Lett. 97, 120405 (2006). 27. A. Acin, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, Phys. Rev. Lett. 98, 230501 (2007). 28. T. Moroder, J. D. Bancal, Y. C. Liang, M. Hofmann, and O. Gühnel, Phys. Rev. Lett. 111, 030501 (2013). 29. M. Wang, Q. H. Gong, Z. Ficek, and Q. Y. He, Phys. Rev. A 90, 023801 (2014). 30. Y. Takeno, M. Yukawa, H. Yonezawa, and A. Furusawa, Opt. Express 15, 4321 (2007). 31. T. Eberle, V. Händchen, and R. Schnabel, Opt. Express 21, 11546 (2013). 32. H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, and R. Schnabel, Phys. Rev. Lett. 100, 033602 (2008).
6703
Collective multipartite Einstein–Podolsky–Rosen steering: more secure optical networks Meng Wang,1 Qihuang Gong,1,2 and Qiongyi He1,2,* 1
State Key Laboratory of Mesoscopic Physics, School of Physics, Peking University, Beijing 100871, China 2 Collaborative Innovation Center of Quantum Matter, Beijing 100871, China *Corresponding author: [email protected] Received May 22, 2014; revised October 30, 2014; accepted October 30, 2014; posted October 31, 2014 (Doc. ID 212349); published November 26, 2014
Collective multipartite Einstein–Podolsky–Rosen (EPR) steering is a type of quantum correlation shared among N parties, where the EPR paradox of one party can only be realized by performing local measurements on all the remaining N − 1 parties. We formalize the collective tripartite steering in terms of local hidden state model and give the steering inequalities that act as signatures and suggest how to optimize collective tripartite steering in specific optical schemes. The special entangled states with property of collective multipartite steering may have potential applications in ultra-secure multiuser communication networks where the issue of trust is critical. © 2014 Optical Society of America OCIS codes: (270.5585) Quantum information and processing; (270.6570) Squeezed states; (000.2658) Fundamental tests. http://dx.doi.org/10.1364/OL.39.006703
It is a major challenge in modern optical communication networks to guarantee ultra-secure information transmission between users at different terminals so that any eavesdropper with access to the communication channel is incapable of acquiring useful information. Many quantum techniques have been developed for secure quantum cryptographic communications, such as quantum key distribution [1] and quantum secret sharing [2,3], where high security is guaranteed by fundamental laws of quantum mechanics. One of the key phenomena of quantum mechanics is the known Einstein–Podolsky–Rosen (EPR) paradox [4]. It established a link between entanglement and nonlocality [5], by showing that there are correlated quantum states that demonstrate an inconsistency between the completeness of quantum mechanics and the concept of local realism. Schrödinger introduced the term “steering” to describe the apparent nonlocality manifested by EPR paradox, where measurements made by one observer at location A can immediately affect the state of another observer at a remote location B [6]. Reid proposed a simple form of an EPR paradox (considering nonideal states) based on Heisenberg uncertainty relation [7,8]. In this scheme, by performing local measurements on system A, Alice can predicate the measurement results xB and pB of remote system B to a given accuracy based on her outcomes xA and pA . The uncertainties associated with the collapsed wave functions are quantified by the conditional variances Δinf xB ΔxB − gx xA and Δinf pB ΔpB − gp pA , where gx and gp are real scaling factors chosen to minimize the inferred errors. For some choice of measurements xA and pA , if the conditional variances are sufficiently small such that Δinf xB Δinf pB < 12 jhxB ; pB ij, an EPR paradox arises. This is because the predetermination for xB and pB in this case is more accurate than allowed by the uncertainty principle. Recent work of Wiseman and co-workers [9,10] formalized the idea of steering in terms of violation of local hidden state (LHS) model, and revealed that the EPR paradox is a realization of quantum steering. For two-mode Gaussian systems, the Reid criterion has 0146-9592/14/236703-04$15.00/0
been shown to be necessary and sufficient to detect bipartite steering [9]. Bipartite steering has been confirmed for continuous variable (CV) [11–13] and discrete regimes [14–16]. EPR steering has now been extended to multipartite scenarios [17–19]. In this Letter, we generalize the concept of collective multipartite steering: a quantum system consisting of N parties shows collective steering of a given party i, if party i can be steered by all the remaining N − 1 parties, but not by any N − 2 or fewer parties. This means the group of N − 1 parties must collaborate after performing local measurements on each individual system in order to extract the information of party i. This nonlocality is the natural multipartite extension of the original EPR paradox, and has the potential application to achieve ultrasecure N-party quantum secret sharing [17]. We first explicitly formalize the meaning of collective tripartite EPR steering in terms of LHS model and clarify the signatures to confirm the existence of such correlation. Then we test a tripartite CV Greenberger–Horne– Zeilinger (GHZ)-like state [20–22] and show that it always possesses the property of collective tripartite steering. In addition, we suggest how to achieve collective tripartite steering in a specific optical scheme by simply employing the reported experimental data [23] and adjusting reflectivities of beamsplitters (BS). The interesting feature is noticed that the impure squeezing, which is often thought to be detrimental, is indeed favorable to realize collective steering in a fairly large parameter regime in experiments. We consider two spatially separated measurements X i and X j , with outcomes xi and xj . The system can be formulated as a LHS model [9] if the joint probability for outcomes of these measurements satisfies Z Pxi ; xj
λ
dλPλPxj jλP Q xi jλ:
(1)
It is an intermediate case between the local hidden variable and quantum separable models. Here, λ labels Bell’s © 2014 Optical Society of America
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“local hidden variables” [5], and Pxk jλ is the probability of xk (k i; j) given the values of λ. The R function Pλ is the probability distribution for λ with λ Pλdλ 1. The subscript Q in function P Q xi jλ indicates that the system i is a quantum state, such that the probability distribution is constrained by the Heisenberg uncertainty principle Δxi Δpi ≥ 12 jhxi ; pi ij. The other system j, however, is not assumed to be a quantum state, e.g., it does not necessarily obey quantum uncertainty relation, and therefore, one can have a complete classical knowledge about the outcomes of xj and pj . In Einstein–Podolsky– Rosen’s language, they are predetermined “elements of reality” for the state of system j. Failure of model (1) implies steering of subsystem i by measurements performed on j [9]. Now we turn to the case of collective tripartite steering. For a quantum system consisting of three spatially separated parties labeled by fi; j; kg, we impose the asymmetric constraint that there exists a quantum density operator ρλi such that hxi iλ Trρλi xi for site i, but not for the group of sites fj; kg. To demonstrate steering of system i by local measurements performed on j and k, we need to falsify a description of the statistics based on a hybrid LHS model, where the joint probability distribution is given as Z Pxi ; xj ; xk
λ
dλPλPxj ; xk jλP Q xi jλ:
(2)
It can be easily seen that collective tripartite steering of system i can be translated to the failure of model (2), while model (1) remains valid for both systems j and k. This means that the information of system i can only be extracted by a joint measurement of the group fj; kg, but not by any measurement of j or k alone. In order to negate the bipartite steering, the criterion used to test steering of one party by another must be necessary and sufficient. Reid inequality formalized in Ref. [7] has been proved in Ref. [9] as a necessary and sufficient criterion to detect bipartite steering for Gaussian systems. In the following discussion, we will use the Reid inequality to test the collective tripartite steering in three-mode Gaussian systems. Consider three harmonic oscillators (fields) at sites i, j, and k, with boson operators ai;j;k . The quadratures x and p are given by x a a† , p −ia − a† with the uncertainty relation ΔxΔp ≥ 1. From Eqs. (1) and (2), collective tripartite EPR steering can be obtained when S ijj Δxi − gxj xj Δpi − gpj pj ≥ 1; S ijk Δxi − gxk xk Δpi − gpk pk ≥ 1;
(3)
and
minimized by optimizing gain factors g via ∂S∕∂g 0. If the minimized S ijj and S ijk satisfy inequality (3), we guarantee that all bipartite steering is removed, as other steering parameters with arbitrary gain factors must be larger than the minimized value. The special quantum states with property of collective multipartite steering may provide a solution to the challenges of secure quantum communication network. In model (2), the state of systems j and k are not assumed to be a quantum state. This means that both sets of values, xj and pj , xk and pk , could be false outcomes of measurements due to hacking attacks or untrustworthy receivers and their apparatus. The collective steering criterion does not require the trust of devices used by steering parties, whereas the measurements on system i must be trusted [24]. As an example, Alice first prepares a three-mode quantum state. This state satisfies the collective steering criterion and is steerable from the group j, k to i. Alice holds the mode i and encrypts the secret information into the quadratures of mode i. Then she distributes the modes j and k to Bob and Charlie. To decipher the secret information, Bob and Charlie must collaborate by making local measurement on their own mode and communicate their results to infer the quadratures of Alice’s mode with high accuracy. Collective tripartite steering removes the possibility of extracting the secret by a single receiver. Moreover, Alice can make sure the secret message is transmitted securely to Bob and Charlie by validating the criterion without the trust assumption on the measurements outcomes reported by Bob and Charlie. Those properties can be useful to the one-sided device-independent quantum secret sharing protocol [25–28]. Next, we show that a tripartite CV GHZ-like state always possesses the property of collective tripartite steering. The CV GHZ-like states are introduced by van Loock and his coworkers [20–22] as the continuous variable analogues to the multipartite GHZ states of qubits. The tripartite CV GHZ-like state can be generated by first superimposing a pure squeezed mode with a vacuum mode at an asymmetric BS (1∶2), then mixing one of the output modes with another vacuum mode at second balanced BS [20–22]. The three output modes are locally measured by balanced homodyne detection, as shown in Fig. 1. For such a state, we find that bipartite steering parameter given by Eq. (3) is always equal to quantum shot noise with optimal gains, i.e., S ijj S ijk 1;
which means any output mode i cannot be steered by other output mode j or k alone. However, steering of i can be demonstrated by measurements involving a group of fj; kg, evidenced by the tripartite steering parameter given by Eq. (4),
S ijfj;kg Δxi − gxj xj gxk xk × Δpi − gpj pj gpk pk < 1:
(4)
Note that inequalities (3) represent party i cannot be steered by any other party j or k alone, and inequality (4) means party i can be steered by the collaboration of the remaining two parties. EPR steering parameters S can be
(5)
S ijfj;kg
s 9 < 1; 5 4 cosh2r
(6)
for any squeezing parameter r > 0 with optimal gains gxj gxk e2r − 1∕2e2r 1, gpj gpk 1 − e2r ∕ 2 e2r , where i; j; k 1; 2; 3 and i ≠ j ≠ k. Thus we
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⃗xT x1 ; x2 ; x3 T , ⃗pT p1 ; p2 ; p3 T , ⃗xTin where T T T ⃗p x1;in ; x2;in ; x3;in , in p1;in ; p2;in ; p3;in , and the BS operation matrix T 123 is given as 2
T 123
3 p p r1 0 1 − r1 6 p p 7 p 6 − r1 r2 1 − r 1 r 2 1 − r2 7 4 p 5: p p 1 − r 1 1 − r 2 − r 1 1 − r 2 − r 2 (9)
Fig. 1. Schematic diagram to test steering of mode 1 by modes 2 and 3 using homodyne detection techniques. Each output mode is combined at a balanced beamsplitter (BS) with an intense LO field, whose phase shift ϕLO determines the measurement of X or P quadratures. The output fields of each balanced BS are detected by photodiodes, and their difference current is proportional to the measured quadratures. Three modes are electronically combined in an appropriate way. The fluctuations of the current give a measure proportional to the inferred errors of the quadratures.
conclude that a tripartite CV GHZ-like state possesses the collective tripartite steering correlation regardless the value of the squeezing parameter of the input mode. Despite theoretical importance, the ideal CV GHZ-like state requires stringent experimental conditions, including pure squeezed input states, ideal BS, and detectors. However, in a realistic experiment setup, noise will inevitably inflate variance of the squeezed vacuum in either quadrature. In the following discussion, we will show that noise is more effective in destroying the bipartite steering rather than the tripartite steering, as has been shown in a hybrid three-mode optomechanical system [29]. Next, we discuss how to utilize the reported experimental data of impure squeezed state to create the special states with property of collective steering. We check the squeezed state produced by type-I optical parametric amplification based on periodically poled potassium titanyl phosphate at the telecommunication wavelength of 1550 nm with a noise reduction of 5.3 dB in the squeezed quadrature and an increase of 9.8 dB in the antisqueezed quadrature [23]. We find that if we use this impure squeezed state as the input mode x1;in ; p1;in in Fig. 1 and properly adjust the reflectivities r 1 and r 2 of BS1 and BS2 , respectively, it is possible to make the three output modes exhibit the collective EPR steering correlation. Applying the BS operations p p T 123 BS2 cos−1 r 2 BS1 cos−1 r 1
(7)
to the input modes yields for the following output modes ⃗xT T 123 ⃗xTin ;
⃗pT T 123 ⃗pTin ;
(8)
From the output modes in Eq. (8), we can get the inferred variances and therefore steering parameters S ijj , S ijk and S ijfj;kg of mode i. To demonstrate the collective tripartite steering, we need to adjust the reflectivities of BS1 and BS2 appropriately to take into account asymmetric squeezing values of the impure squeezed input x1;in ; p1;in . We first fix BS1 , and then adjust BS2 to find the region to achieve collective tripartite steering. Figure 2 shows that by setting BS1 at r 1 ∶1 − r 1 1∶3, we can make steering parameters of mode 1 satisfying S 1j2 > 1; S 1j3 > 1; S 1jf2;3g < 1 when 0.30 ≲ r 2 ≲ 0.70 (region between two red points). This condition guarantees the steering of mode 1 by modes 2 and 3, but negates the steering of mode 1 by single mode 2 or 3. Similarly, S 2j1 > 1; S 2j3 > 1; S 2jf1;3g < 1 can be achieved when 0.27 ≲ r 2 < 1 (right region of solid square), which demonstrates the collective steering of mode 2, and S 3j1 > 1; S 3j2 > 1; S 3jf1;2g < 1 when 0 < r 2 ≲ 0.73 (left region of empty square) that gives the collective steering of mode 3. Comparing with tripartite CV GHZ-like state that strictly requires fixing the first BS at 1∶2 and the second BS at r 2 0.5, interestingly, we find that the impure squeezing input results in a wider range for the collective steering to be valid. The impact of impurity is often thought to be unfavorable to accomplish quantum tasks. However, we show the performance of the asymmetric squeezing has advantages in this application. With highly improved techniques for generating squeezed state and
Fig. 2. EPR steering parameters S ijj 2 (labeled ijj) and S ijfj;kg 2 (labeled ijfj; kg) versus the reflectivity r 2 of BS2 when BS1 is fixed at 1∶3. The red dashed, dotted, and solid lines represent S 1j2 2 , S 1j3 2 , and S 1jf2;3g 2 , respectively. The blue dashed, dotted, and solid lines represent S 2j1 2 , S 2j3 2 , and S 2jf1;3g 2 , respectively. The black dashed, dotted, and solid lines represent S 3j1 2 , S 3j2 2 , and S 3jf1;2g 2 , respectively.
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homodyne measurement [30–32], this special states can be generalized to the multipartite case. In conclusion, we have explicitly formalized the meaning of collective tripartite steering in terms of LHS model, and clarified the signatures as an experimental witness to demonstrate this property for Gaussian CV states. Our results show that a tripartite CV GHZ-like state always exhibits collective tripartite steering. In addition, we present a practical protocol for achieving that special states based on impure squeezing input. We show that it is possible to verify collective tripartite steering within realistic experimental conditions by optimizing the reflectivities of optical beamsplitters. The method introduced here can be straightforwardly generalized to other multipartite systems to produce collective EPR steering, which are the central resource for ultra-secure messages transmission in quantum networks with many users where the issue of trust is of importance. This work was supported by the National Natural Science Foundation of China under Grant Nos. 11274025 and 11121091. References 1. C. H. Bennett and G. Brassard, in Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing (IEEE, 1984), pp. 175–179. 2. M. Hillery, V. Bužek, and A. Berthiaume, Phys. Rev. A 59, 1829 (1999). 3. R. Cleve, D. Gottesman, and H.-K. Lo, Phys. Rev. Lett. 83, 648 (1999). 4. A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935). 5. J. S. Bell, Physics 1, 195 (1965). 6. E. Schrödinger, Proc. Cambridge Philos. Soc. 31, 555 (1935). 7. M. D. Reid, Phys. Rev. A 40, 913 (1989). 8. M. Reid, P. Drummond, E. Cavalcanti, W. Bowen, P. Lam, H. Bachor, U. Andersen, and G. Leuchs, Rev. Mod. Phys. 81, 1727 (2009). 9. H. M. Wiseman, S. J. Jones, and A. C. Doherty, Phys. Rev. Lett. 98, 140402 (2007). 10. E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, Phys. Rev. A 80, 032112 (2009). 11. Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, Phys. Rev. Lett. 68, 3663 (1992).
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