OPEN Test of Einstein-Podolsky-Rosen Steering

SUBJECT AREAS: Based on the All-Versus-Nothing Proof QUANTUM Chunfeng Wu1,2, Jing-Ling Chen1,3, Xiang-Jun Ye1,3, Hong-Yi Su1,3, Dong-Ling Deng4, Zhenghan Wang5 INFORMATION &C.H.Oh1,6 QUANTUM MECHANICS

1Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, 2Pillar of Engineering Received Product Development, Singapore University of Technology and Design, 20 Dover Drive, Singapore 138682, 3Theoretical Physics 5 November 2013 Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071, People’s Republic of China, 4Department of Physics and Michigan Center for Theoretical Physics, University of Michigan, Ann Arbor, Michigan 48109, USA, 5Microsoft Research, Accepted Station Q, University of California, Santa Barbara, CA 93106, USA, 6Department of Physics, National University of Singapore, 2 13 February 2014 Science Drive 3, Singapore 117551. Published 6 March 2014 In comparison with entanglement and Bell nonlocality, Einstein-Podolsky-Rosen steering is a newly emerged research topic and in its incipient stage. Although Einstein-Podolsky-Rosen steering has been explored via violations of steering inequalities both theoretically and experimentally, the known inequalities Correspondence and in the literatures are far from well-developed. As a result, it is not yet possible to observe requests for materials Einstein-Podolsky-Rosen steering for some steerable mixed states. Recently, a simple approach was presented to identify Einstein-Podolsky-Rosen steering based on all-versus-nothing argument, offering a should be addressed to strong condition to witness the steerability of a family of two-qubit (pure or mixed) entangled states. In this C.F.W. work, we show that the all-versus-nothing proof of Einstein-Podolsky-Rosen steering can be tested by (chunfeng_wu@sutd. measuring the projective probabilities. Through the bound of probabilities imposed by local-hidden-state edu.sg); J.-L.C. model, the proposed test shows that steering can be detected by the all-versus-nothing argument ([email protected]) experimentally even in the presence of imprecision and errors. Our test can be implemented in many physical systems and we discuss the possible realizations of our scheme with non-Abelian anyons and or C.H.O. (phyohch@ trapped ions. nus.edu.sg)

n 1935, Einstein, Podolsky, and Rosen (EPR) questioned the completeness of quantum mechanics (QM) based on local realism1. Many efforts have been devoted to a deeper understanding of QM in the form of three types I of quantum nonlocalities: quantum entanglement, EPR steering, and Bell nonlocality2. Within the hierarchy of nonlocalities, the set of EPR steerable states is a subset of entangled states and a superset of Bell nonlocal states. Quantum entanglement and Bell nonlocality have attained flourishing developments since 1964. However, EPR steering is a newly emerged research topic and, to date, is far from being completely understood. Steering inequalities for EPR steering are the analog of Bell inequalities for Bell nonlocality. Their violations, predicted by quantum mechanics, reveal EPR steering. Such a violation rules out the existence of a local-hidden-state (LHS) model, the same way the violation of a Bell inequality rules out the existence of a local-hidden-variable (LHV) model. In comparison to the development of Bell nonlocality, the research on EPR steering is in its developing stages, even though Schro¨dinger discussed the concept in 19353. A reason for this is the absence of a rigorous formulation of the concept of EPR steering, which did not appear until the work of Wiseman, Jones, and Doherty2 in 2007. Indeed, EPR steering answers a question of fundamental quantum physics as well as opens new possibilities for quantum communication, thus it has inspired some recent research in quantum information theory4. For a pure entangled state shared by two separated observers Alice and Bob, Bob’s qubit can be ‘‘steered’’ into different states although Alice has no access to the qubit. Schro¨dinger adopted the word steering to describe this type of nonlocality. This means that Alice has the ability to remotely prepare Bob’s particle in different states by A measuring her particle using different settings, and here we use r~a to denote the conditional state Bob gets if Alice measures her particle with measurement A^ and obtains result a. While Bob suspects that Alice may send him some non-entangled particles and fabricate the results based her knowledge of LHS. If Bob’s system admits a LHS model {2jrj}, where rj’s are states that Bob does not know (but Alice knows), and 2j . 0 is the probability of rj, thenX Alice could attempt toX fabricate the results using her knowledge of j, in other words, r~A~ 2 ajA^,j 2 r , with 2 ajA^,j ¼ 1. If Bob finds there is a LHS model which can describe his a j j j a

SCIENTIFIC REPORTS | 4 : 4291 | DOI: 10.1038/srep04291 1 www.nature.com/scientificreports conditional states after he asks Alice to perform the measurement on predicted projective probabilities as stated in the following. For her particle, then he is not convinced the existence of EPR steering. QM, Bob obtains zero probabilities after he performs some appro- Very recently many results have been achieved to show violations priate projective measurements on his qubit of steering inequalities both theoretically and experimentally, thus hi QM ^z,0 5–11 ~ ~ rendering LHS model untenable . However, the existed steering P1 Trji 1 Bhj1 r~B 0, inequalities in the literatures are far from well-developed, and there- hi QM~ ^z,1 ~ fore it is not yet possible to observe EPR steering for some steerable P2 Trji 0 Bhj0 r~B 0, 12 mixed states . Another elegant approach to explore the contradic- hi ð2Þ tion between QM and LHS model is the all-versus-nothing (AVN) QM~ y\ y\r~^x,0 ~ , P3 Tr B B 0 proof of the existence of EPR steering. This can be considered as the hi QM \ \ ^x,1 steering analog of Greenberger-Horne-Zeilinger (GHZ) argument P ~Tr Q Q r~ ~0, without inequalities for Bell nonlocality13. Currently such an AVN 4 B B proof for EPR steering has been shown to be a strong condition to where jyHæ 5 sin hj0æ 2 cos hj1æ and jQHæ 5 sin hj0æ 1 cos witness the steerability of a family of two-qubit (pure or mixed) B B B B B hj1æ are orthogonal to jyæ and jQæ , respectively. However, for a entangled states and have the ability of detecting asymmetric steer- B B B LHS model, it predicts the corresponding probabilities as follows, ing12. This also offers an effective way to detect EPR steering for two "# qubits experimentally. X In this work, we investigate the test of EPR steering according to its LHS~ ~ P1 Trji 1 Bhj1 2ðÞa 0j^z,j 2jrj , AVN argument and demonstrate directly the contradiction between j "# LHS model and QM. We show that by observing projective prob- X abilities, the existence of steering can be verified by defining a prob- PLHS~Trji 0 hj0 2ðÞa~1j^z,j 2 r , ability bound imposed by LHS model. Our test is the first one 2 B j j j proposed to detect EPR steering based on the AVN proof and it is "#ð3Þ suitable for all the two-qubit entangled states specified in Ref. 12, X LHS~ y\ y\ 2 ~ ^,j 2 r , both pure and mixed. The possible implementation of our test is P3 Tr B ðÞa 0jx j j j discussed by using non-Abelian Fibonacci anyons and trapped ions, "# but it is not limited to these systems. Our test is also applicable to X many other physical systems, such as photons, atoms as well as LHS~ Q\ Q\ 2 ~ ^,j 2 r : P4 Tr B ðÞa 1jx j j superconductors, etc. In a system of non-Abelian Fibonacci anyons, j each logical qubit is encoded into triplet of Fibonacci anyons and the 12 corresponding operations are carried out by braiding the anyons. As From the AVN proof , we know that the state jYæAB possesses EPR braids are performed by taking an anyon either around another or ? steering if h 0orp/2, andX this tells us there exists no LHS model of not, which will not cause small errors from slight imprecisions in the the state such that r~A~ 2 ajA^,j 2 r . When h 5 0orp/2, the way that anyons are moved. Therefore, the test is fault-tolerant to a j j j errors and offers high experimental precision. In an ion-trap experi- state is separable, and hence it is possible to find a LHS model to ment, present experimental achievements on high-fidelity state initi- describe it. Therefore, we know that the probabilities (3) cannot be alization, quantum gates and state readout make our scheme of zero simultaneously except h 5 0orp/2. detecting steering possibly testable. In an ideal test for EPR steering, after Alice performs projective measurement on her qubit of the state jYæAB, Bob then measures the H H probabilities by projecting the states j0æB, j1æB, jy æB and jQ æB on his Results B ~ qubits. If he finds the four probabilities Pi ðÞi 1,2,3,4 are all zero, First let Alice and Bob share a pure entangled state jYæAB 5 cos hj00æ 1 sin hj11æ . In the steering scenario, Alice adopts the then EPR steering is demonstrated. Nevertheless, in real experiments AB AB (Exp), measurement results are inevitably affected by experimental following settings: A^[ P^z ,P^x , where P^z and P^x denote Alice’s a a a z precision and errors. It is possible that the probabilities obtained ^ ^ 5 projective measurements in z-and x-directions, and a (with a 0, experimentally may deviate from the theoretical values slightly, i.e., 1) is measurement result. After Alice’s measurements, Bob’s con- Exp~ QMze e ditional states become Pi Pi i (here i are small numbers caused by errors). We then investigate how close a LHS model could be to simulate Eq. ^z,0~ 2 r~B cos hji0 Bhj0 , (2). We have shown that for the state jYæAB the probabilities LHS~ ~ , , , ^z,1~ 2 Pi 0 ðÞi 1 2 3 4 only if the state shared by Alice and Bob is r~B sin hji1 Bhj1 , LHS not steerable, otherwise, some of Pi cannot be zero. For the sake of ^, 1 ð1Þ simplicity, consider the inevitable errors e 5 e for all i, we can detect r~x 0~ jiy hjy , i B B LHSw 2 steerability if Pi e for some i. Therefore, the optimal LHS model 1 for this experiment is the one making PLHS approach to PQM as r~x^,1~ jiQ hjQ , i i B 2 B closely as possible for all i. We define ^ where r~A,a describes Bob’s state after Alice performs measurement A^ ~ LHS{ QM B D min max Pi Pi , ð4Þ LHS i[fg1,2,3,4 and obtains result a, and jyæB 5 cos hj0æB 1 sin hj1æB, jQæB 5 cos hj0æ 2 sin hj1æ . If there exists a LHS model can fake the results (1), B B where D describes the bound of probabilities imposed by the optimal i.e., there exists a suitable ensembleX {2jrj} and a stochastic map ^ LHS model. In our test, EPR steering can be detected when D . e. 2 ajA^,j satisfying r~A,a~ 2 ajA^,j 2 r , then Bob is not B j j j Fig. 1(a) shows the relation between parameter h and D obtained convinced that Alice can steer his conditional states. Otherwise the numerically (see the Methods section). We find that D is of order LHS model contradicts with QM. 1022 when h is not closed to 0 or p/2. This implies that the EPR 12 According to the AVN proof , the entangled state jYæAB cannot steering of state jYæAB can be revealed by the experiments with be described by any LHS model except h 5 0orp/2. The incisive precision e , 1023. In Fig. 1(a), it is observed that D changes sym- contradiction between QM and LHS model is due to different metrically with respect to h and approaches to its maximal value

SCIENTIFIC REPORTS | 4 : 4291 | DOI: 10.1038/srep04291 2 www.nature.com/scientificreports when h 5 p/4. This shows the more entangled the state is, the easier universal topological quantum computation14. Follow Freedman to detect EPR steering in the experiment. et al.’s work15, we encode logical qubits into triplets of anyons with We next consider a two-qubit mixed state total topological charge 1: j0æL 5 j((?, ?) , ?) æ and j1æL 5 j((?, ?) , ?) æ I t t t ~ 2 z zz 2 z z (here L denotes ‘‘logical’’). The so-called noncomputational state rAB cos hy y sin hQjiABhjQ : ð5Þ AB jNCæ 5 j((?, ?)t, ?)Iæ is the only state of three anyons that has total topological charge 0. Quantum operations can be constructed by z 1 Here y ~ pffiffiffi ji00 zji11 and using two elementary braiding operations R , R acting on the AB 2 AB AB 1 2 Hilbert space of three Fibonacci anyons and their inverses16,17. The z ~ 1ffiffiffi z jiQ AB p ji01 AB ji10 AB are two Bell states. The measure- obtained quantum gates, together with the controlled-NOT gate 2 obtained in Refs. 16–18 are useful in the construction of EPR steering ^ ^z ^x ment settings of Alice are still A[ Pa,Pa , and after Alice performs test by preparing logical-qubit states and achieving required opera- ^ measurement A andX obtains result a, Bob’s conditional states can be tions (see the Methods section). Several candidates for realizing non- A^,a~ ^ Abelian anyons have been suggested in physical systems, such as expressed as r~B 2 ajA,j 2jr provided with a LHS model j j 19 ^ fractional quantum Hall liquid , rotating Bose-Einstein conden- {2jrj} and a stochastic map 2 ajA,j . Similarly, for the state rAB, sates20, as well as quantum spin systems21,22. Bob obtains quantum probabilities of measuring his qubit in the Another possible system to explore the realization of our test states j0æB, j1æB, j1æB and j2æB as experimentally is trapped ion. Refs. 23–25 have reported experi- hi2 QM ^z,0 sin h mental results of high-fidelity state preparation, quantum gate opera- P ~Trji 1 hj1 r~ ~ , 40 1 1 B B 2 tions, and state measurement for optical qubits stored in Ca held hi in a trap. State preparation is usually done by precisely manipulating sin2 h PQM~Trji 0 hj0 r~^z,1 ~ , the internal levels of ion utilizing laser pulses and the Blatt group 2 B B 24 hi2 ð6Þ realized state initialization with fidelity more than 99.8% .Bya 26,27 ^, Mølmer-Sørensen-type gate operation , a Bell-type entangled PQM~Tr ji{ hj{ r~x 0 ~0, 3 B B state of ions with a fidelity of 99.3(1)% was realized in the same hi 24 QM ^x,1 work . The Blatt group also presented single-qubit gates with fidelity ~ z z r~ ~ 23 P4 Tr jiBhjB 0, exceeding 99.9% in trapped ions . As for state measurement cap- 25 ability in an ion-trap experiment, A. H. Myerson et al. achieved ~ 1ffiffiffi ^z,a 99.991(1)% readout fidelity, sufficient for fault-tolerant quantum where ji+ B p ji0 B+ji1 B , and r~B is Bob’s conditional state 2 computation by measuring population of states using time-resolved after Alice performs projective measurement in ^z-direction, etc. It photon counting. For the entangled state realized in Ref. 24, the has been proved that there does not exist any LHS model for rAB with probability bound is found to be D 5 0.0732, and this means that p h=+ 12 such that probability equations in (6) can be satisfied the EPR steering of the entangled state can be verified experimentally 4 with precision e , 0.0732. The experimental achievements in the simultaneously. If Bob observes experimentally these four probabil- literatures23–25 tell us that our test of EPR steering based on the AVN QM’ ities Pi s, then EPR steering of the state is exhibited, or there exists proof is possibly realizable with current techniques in ion-trap no LHS model. Consider experimental imprecision and errors, we experiments. r also investigate the condition to detect EPR steering of AB by plot- To summarize, we have presented a test to identify EPR steering D h ting the variation of LHS bound versus , see Fig. 1 (b). It can be based on the AVN argument by measuring projective probabilities. e , 23 found that for the experiments with precision 10 , the EPR Our test is applicable to the family of two-qubit entangled states r h p steering of AB can be observed when is not close to /4. It is worthy specified in Ref. 12 regardless of pure or mixed. We have provided of pointing out that our test of EPR steering is not limited to the states the condition on experimental implementation of our scheme Y r j æAB and AB, but also applicable to the family of two-qubit entang- through expression (4) that EPR steering can be observed in the led states specified in Ref. 12, regardless of pure or mixed. presence of experimental imprecision and errors. Our result is the first experimental test presented to detect EPR steering by resorting Discussions to the AVN proof, and it can be implemented in systems such as non- Let us make some discussions on the possible realization of our test in Abelian anyons and trapped ions. The primary advantage of our test physical systems. We first consider non-Abelian Fibonacci anyons based on non-Abelian anyons is that it is fault-tolerant, or the logical which are shown to be the simplest non-Abelian quasiparticles for quantum state used is robust against local perturbations. Specifically,

Figure 1 | Numerical results of the bound D imposed by the optimal LHS model versus h. For (a) | YæAB and (b) rAB.

SCIENTIFIC REPORTS | 4 : 4291 | DOI: 10.1038/srep04291 3 www.nature.com/scientificreports

X N m ~ 2N{ia z1, where a denote the measurement results of A (a 5 0, 1). It a i~1 i i i i N is not difficult to find that ma ranges from 1 to 2 depending on ai. We then do the decomposion by choosing

ðÞma ~ rk rk, ð7Þ

N ðÞma 2 ~ P 2 ai A^i,k 2 , ð8Þ k i~1 k

 1ifai~0 ^ ðÞma 2 0jAi,k ~ ð9Þ 0 else

ðÞma 2ðÞma where rk is the hidden state and k is its weight. By direct calculations it can be X N 2 ðÞm ðÞm ^ ~ ^ ðÞma a a verified that 2 ajA,k 2krk 2 ajA,k 2 r . Eq. (9) shows the ma ~1 k k reconstructed stochastic mapsX are all deterministic. Thus by this way, we get a dLHS A~ ^ model that satisfies r~ 2 ajA,j 2jrj. Figure 2 | Approximating quantum gates Uh by braiding Fibonacci a j anyons. In the plotting, time flows from left to right, U represents U and X 1 p/6 Lemma 2. For a dLHS model, r~A~ 2 ajA^,j 2 r can be rewritten as X a j j j U represents U . ^ ^ ^ 2 2p/3 PArA~ 2 r , where HA stands for the set of hidden states that contribute to a a j[ A^ j j a Ha ^ rA indicating the corresponding 2 ajA^,j ~1. The equality holds if and only if the it has been proven that these logical qubits might be robust to ran- a 28 following equalities are fulfilled, dom perturbations . Our test can also be realizable in ion-trap 8 P > PA^ ~ 2 experiments based on current experimental techniques as recent <> a j j[ A^ progress in trapped ion offers high-fidelity state preparation, Ha ^ P ð10Þ > A^ !A~ ! quantum gate operations, and state measurement for optical qubits :> Pa ra 2j rj j[ A^ stored in it. Let us point out that the realization of our test is not Ha limited to the two systems but also applicable to many other physical ^ !A ! rA^ r systems like photons, atoms and superconductors, etc. We expect where ra and rj are the Bloch vectors of a and j respectively.. ^ A^ Let us look at the proof of the lemma. We have rA~ 1z!r :!s 2 and further investigations in this direction, both theoretically and  a a !:! experimentally. rj~ 1zrj s 2, where 1 describes identity matrix. So the equality of X ^ X X A^ A^ ~ A^ z A^ !A:!~ z !:! Pa ra A^ 2jrj gives Pa 1 Pa ra s A^ 2j1 A^ 2j rj s . j[Ha j[Ha j[Ha Methods Thus we obtain Eq. (10). To find the optimal LHS model. We here present a theorem which is used to find the We would like to point out that Eq. (10) is similar to the problem describing center optimal LHS model for a given two-qubit state. Theorem – For any given two-qubit A^ ! of mass if we treat the probabilities 2j and Pa as masses, as well as Bloch vectors (rj state rAB in a N -setting protocol, if there is a LHS model for the state, then there is a !A^ LHS model with the number of hidden states no larger than 2N. The proof of the and ra ) as the position vectors of various masses. Lemma 2 shows that the task to A^ A^ Theorem needs two lemmas associated with the concept of deterministic LHS (dLHS) find a dLHS model for a state ra with probability Pa is equivalent to find a distri- 2 ^,j [ , V^ j A^ model which is a LHS satisfying ajA fg0 1 , A, , a. We also briefly restate the bution of masses in the Bloch sphere with total mass Pa and center of mass being r~A^ ^ !A^ notations to be used, a is the conditional states of Bob after Alice measures A and located at ra . We show in the following that with the aid of Eq. (10), we can impose gets result a g {0, 1}, the tilde here denotes this state is unnormalized and its norm is constraints on measurement settings to find a dLHS model. If we cannot find a dLHS A^ Pa , the probability associated with the output a. model for rAB, Lemma 1 shows that we can neither find a LHS model, and this thus affirms the steerability of rAB. For any given rj, we can always assign a N-length bit ^ ^ ^ Lemma 1. For any given two-qubit state rAB, if there is a LHS model for rAB then there string constructed from 2 ajA1,j 2 ajA2,j 2 ajAN ,j considering is a dLHS model for r . ^ AB 2 ajAi,j [fg0,1 . Next let us describe the LHS model by dividing hidden states {rj} ^[ ^ ,^ ,^ , ...,^ r In a general N-setting protocol, we have A A1 A2 A3 AN . Suppose AB has a into many subsets with each subset containing all of the rj that has the same N-length LHS description thus thereX is an ensemble {2jrj} and an associated probability bit string. Thus in this way, each subset is unique, or not overlapping with others. We 2 ajA^,j fulfilling r~A~ 2 ajA^,j 2 r . We note that can take each of the subsets as one new hidden state by resorting to Lemma 2. We use a j j j the fact that hidden state can be treated as mass point so we can consider the centre of 2 1jA^,j ~1{2 0jA^,j . Now if V j,A^ , 2 0jA^,j [fg0,1 , then it is a dLHS model. mass of each subset as the new state and the weight of the new state is the corre- ^ We next check each j to see whether 2 0jA,j [fg0,1 . For any j with spoinding total mass. It is not difficult to find that there are totally 2N such new states, 2 0jA^,j [fg0,1 , we keep these terms unchanged. For j~k 2 0jA^,k [=fg0,1 ,we and thus the LHS model has only 2N hidden states. This ends our proof of the decompose this term into 2N separate terms as follows. First we define a new term Theorem.

Figure 3 | Variations of D for | YæAB versus n/h. (a) h 5 p/8 and n ranges from 20 to 120, (b) h 5 p/6 and n ranges from 20 to 120, as well as (c) n 5 46, 50, 100 and h ranges from 0 to p/2. From (a) and (b), we find that the variation of D is negligibly small when n . 45 since the variation is in the ten- thousandths place. From (c) it is clear that the three curves are almost overlapped and the results show that n 5 46 is large enough to obtain a reasonable value of D.

SCIENTIFIC REPORTS | 4 : 4291 | DOI: 10.1038/srep04291 4 www.nature.com/scientificreports

Therefore we conclude that the optimal LHS model contains an ensemble with four 10. He, Q. Y., Drummond, P. D. & Reid, M. D. Entanglement, EPR steering, and Bell- pure hidden states in the two-setting protocol, and more hidden states make no nonlocality criteria for multipartite higher-spin systems. Phys. Rev. A 83, improvement.X The optimal LHS model is numerically obtained by minimizing 032120(2011). 4 QM function F ~ vn for a large n, where v ~ PLHS{P . In this approach, we 11. Bennet, A. J., Evans, D. A., Saunders, D. J., Branciard, C., Cavalcanti, E. G., n i~1 i i i i utilize the knowledge of vector norm. First we have (F )1/n which is the l -norm of Wiseman, H. M. & Pryde, G. J. Arbitrarily Loss-Tolerant Einstein-Podolsky- n n Rosen Steering Allowing a Demonstration over 1 km of Optical Fiber with No ~~ , , , T vector v ðÞv1 v2 v3 v4 . We know that l‘ -norm of a vector is just its maximum Detection Loophole. Phys. Rev. X 2, 031003 (2012). element and hence by definition D equals to the minimum of l -norm of the vector ‘ 12. Chen, J. L. et al. All-Versus-Nothing Proof of Einstein-Podolsky-Rosen Steering. ~ T ~v ðÞv1,v2,v3,v4 . So for a large enough n we can get a good approximation of D from Sci. Rep. 3, 2143 (2013). 1/n minimizing (Fn) with varied rj. In our calculations, we use n 5 46 since we find 13. Greenberger, D. M., Horne, M. A. & Zeilinger, A. ‘‘Going beyond Bell’s theorem’’ numerically the improvement of D by choosing a number larger than 46 is negligibly in Bell’s Theorem, Quantum Theory, and Conceptions of the Universe (eds Kafatos, small. As shown in Fig. 3 (a) and (b), we find the values of D by choosing h 5 p/8, p/6 M.) 69 (Kluwer, Dordrecht, Holland, 1989). Yæ D for j AB with different n (ranging from 20 to 120). It is clear that the values of do 14. Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Sarma, S. D. Non-Abelian not change substantially and the change is in the ten-thousandths place when n is anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083 (2008). greater than 45. The results show us that n 5 46 is large enough to obtain a reasonable 15. Freedman, M. H., Larsen, M. & Wang, Z. A Modular Functor Which is Universal value of D. Apparemently we can choose other values of n as long as n . 45 and the for Quantum Computation. Commun. Math. Phys. 227, 605–622 (2002). choice will not affect the value of D much. We also plot the variation of D versus h by 16. Bonesteel, N. E., Hormozi, L., Zikos, G. & Simon, S. H. Braid Topologies for choosing different n for jYæ in Fig. 3 (c). Seen from Fig. 3 (c), the three curves AB Quantum Computation. Phys. Rev. Lett. 95, 140503 (2005). corresponding to n 5 46, n 5 50, n 5 100 respectively are almost overlapped. Hence we know that n 5 46 is large enough to obtain a reasonable value of D. 17. Hormozi, L., Zikos, G., Bonesteel, N. E. & Simon, S. H. Topological quantum compiling. Phys. Rev. B 75, 165310 (2007). Approximation of quantum gates in non-Abelian Fibonacci anyons. Quantum 18. Deng, D. L., Wu, C., Chen, J. L. & Oh, C. H. Fault-Tolerant Greenberger-Horne- Zeilinger Paradox Based on Non-Abelian Anyons. Phys. Rev. Lett. 105, 060402 operations can be constructed by using two elementary braiding operations R1, R2 acting on the Hilbert space of three Fibonacci anyons and their inverses16,17. In Fig. 2, (2010). we plot the braids that approximate the quantum gate 19. Xia, J. S. et al. Electron Correlation in the Second Landau Level: A Competition  Between Many Nearly Degenerate Quantum Phases. Phys. Rev. Lett. 93, 176809 cos h sin h (2004). Uh~ ð11Þ sin h {cos h 20. Cooper, N. R., Wilkin, N. K. & Gunn, J. M. F. Quantum Phases of Vortices in Rotating Bose-Einstein Condensates. Phys. Rev. Lett. 87, 120405 (2001). with U1 5 Up/6 and U2 5 U2p/3. Any other quantum gate Uh can be obtained in a 21. 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Nature 443, 316–319 (2006). controlled-NOT gate obtained in Refs. 16–18 are useful in the construction of EPR 24. Benhelm, J., Kirchmair, G., Roos, C. F. & Blatt, R. Towards fault-tolerant quantum g steering test by preparing logical-qubit states. We apply the operation Uh (with h computing with trapped ions. Nature Phys. 4, 463–466 (2008). L~ L 6 L (0, p/2)) on the logical qubit A of initial state jiY 0 ji0 A ji0 B and a controlled- 25. Myerson, A. H. et al. High-Fidelity Readout of Trapped-Ion Qubits. Phys. Rev. NOT gate is followed on the two logical qubits, then we have the two-logical-qubit Lett. 100, 200502 (2008). L pure states jiY AB. To prepare mixed state, we need an ancilla logical qubit C, and 26. Sørensen, A. & Mølmer, K. Quantum Computation with Ions in Thermal Motion. L~ L 6 L 6 L Phys. Rev. Lett. 82, 1971 (1999). initially assume that the logical qubits are in the state jiY 0 ji0 A ji0 B ji0 C.We apply Hadamard gate on logical qubit A, Uh on logical qubit C, then a controlled- 27. Sørensen, A. & Mølmer, K. Entanglement and quantum computation with ions in NOT gate on logical qubits A and B, and finally a controlled-NOT gate on logical thermal motion. Phys. Rev. A 62, 022311 (2000). L z L L z L L qubits C and B, we then have jiY ~cos hy ji0 zsin hQjiji1 . Look at 28. Tsomokos, D. I., Osborne, T. J. & Castelnovo, C. Interplay of topological order and ABC AB C AB C spin glassiness in the toric code under random magnetic fields. Phys. Rev. B 83, rL the first two qubits only, we successfully have the state ABas in (5). All the operations L 075124 (2011). ^z ^x y\ y\ Q\ L Q\ involved in our scheme, such as Pa and Pa for Alice, B and B for 29. Kitaev, A. Yu., Shen, A. H. & Vyalyi, M. N. Classical and Quantum Computation Bob, can be carried out by braiding the Fibonacci anyons. For instance, the single- (American Mathematical Society, Providence, 1999). L y\ Q\ L logical-qubit states B and B of Bob can be realized by using the action of U6h L onji 1 B (up to a global phase). Acknowledgments 1. Einstein, A., Podolsky, B. & Rosen, N. Can Quantum-Mechanical Description of We thank V. Vedral for valuable discussions. J.L.C. is supported by National Basic Research Physical Reality Be Considered Complete? Phys. Rev. 47, 777 (1935). Program (973 Program) of China under Grant No. 2012CB921900 and NSF of China 2. Wiseman, H. M., Jones, S. J. & Doherty, A. C. Steering, entanglement, nonlocality, (Grant No. 11175089). This work is also partly supported by National Research Foundation and the Einstein-Podolsky-Rosen paradox. Phys. Rev. Lett. 98, 140402 (2007). and Ministry of Education, Singapore (Grant No. WBS: R-710-000-008-271). 3. Schro¨dinger, E. Discussion of probability relations between separated systems. Proc. Cambridge Philos. Soc. 31, 555–562 (1935). 4. Branciard, C., Cavalcanti, E. G., Walborn, S. P., Scarani, V. & Wiseman, H. M. Author contributions One-sided device-independent quantum key distribution: Security, feasibility, All authors contributed to this work including developing the scheme and preparing the and the connection with steering. Phys. Rev. A 85, 010301(R) (2012). manuscript. C.W. and J.L.C. initiated the idea. C.W., J.L.C., X.J.Y., H.Y.S. and D.L.D. 5. Reid, M. D. Demonstration of the Einstein-Podolsky-Rosen paradox using proposed the scheme. C.W., J.L.C., Z.W. and C.H.O. wrote the main manuscript text. X.J.Y. nondegenerate parametric amplification. Phys. Rev. A 40, 913 (1989). and H.Y.S. prepared the figures. All authors reviewed the manuscript. 6. Reid, M. D., Drummond, P. D., Bowen, W. P., Cavalcanti, E. G., Lam, P. K., Bachor, H. A., Anderson, U. L. & Leuchs, G. Colloquium: The Einstein-Podolsky- Additional information Rosen paradox: From concepts to applications. Rev. Mod. Phys. 81, 1727 (2009). Competing financial interests: The authors declare no competing financial interests. 7. Cavalcanti, E. G., Jones, S. J., Wiseman, H. M. & Reid, M. D. Experimental criteria for steering and the Einstein-Podolsky-Rosen paradox. Phys. Rev. A 80, 032112 How to cite this article: Wu, C.F. et al. Test of Einstein-Podolsky-Rosen Steering Based on (2009). the All-Versus-Nothing Proof. Sci. Rep. 4, 4291; DOI:10.1038/srep04291 (2014). 8. Saunders, D. J., Jones, S. J., Wiseman, H. M. & Pryde, G. J. Experimental EPR- steering using Bell-local states. Nature Phys. 6, 845–849 (2010). This work is licensed under a Creative Commons Attribution- 9. Smith, D. H. et al. Conclusive quantum steering with superconducting transition NonCommercial-ShareAlike 3.0 Unported license. To view a copy of this license, edge sensors. Nature Comm. 3, 625 (2012). visit http://creativecommons.org/licenses/by-nc-sa/3.0

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