All-Versus-Nothing Proof of Einstein-Podolsky-Rosen Steering

All-Versus-Nothing Proof of Einstein-Podolsky-Rosen Steering Jing-Ling Chen 1,2 , Xiang-Jun Ye 1,2 , Chunfeng Wu 2,6 , Hong-Yi Su 1,2 , Ada ´n Cabello 3 , L. C. Kwek 2,4 &C.H.Oh 2,5 1 Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071, People’s Republic of China, 2 Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, 3 Departamento de Fı ´sica Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain, 4 National Institute of Education and Institute of Advanced Studies, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616, 5 Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542, 6 Pillar of Engineering Product Development, Singapore University of Technology and Design, 20 Dover Drive, Singapore 138682. Einstein-Podolsky-Rosen steering is a form of quantum nonlocality intermediate between entanglement and Bell nonlocality. Although Schro ¨dinger already mooted the idea in 1935, steering still defies a complete understanding. In analogy to ‘‘all-versus-nothing’’ proofs of Bell nonlocality, here we present a proof of steering without inequalities rendering the detection of correlations leading to a violation of steering inequalities unnecessary. We show that, given any two-qubit entangled state, the existence of certain projective measurement by Alice so that Bob’s normalized conditional states can be regarded as two different pure states provides a criterion for Alice-to-Bob steerability. A steering inequality equivalent to the all-versus-nothing proof is also obtained. Our result clearly demonstrates that there exist many quantum states which do not violate any previously known steering inequality but are indeed steerable. Our method offers advantages over the existing methods for experimentally testing steerability, and sheds new light on the asymmetric steering problem. Quantum nonlocality is an invaluable resource in numerous quantum information protocols. It is part of a hierarchical structure 1 : quantum states that have Bell nonlocality 2 form a subset of Einstein-PodolskyRosen steerable states which, in turn, form a subset of entangled states. The concept of steering can historically be traced back to Schro¨dinger’s reply 3 to the Einstein-Podolsky-Rosen argument 4 , and it has since been rigorously formulated by Wiseman, Jones, and Doherty 1 . Within the steering scenario, Alice prepares a bipartite system, keeps one particle and sends the other one to Bob. She announces that the Bob’s particle is entangled with hers, and thus that she has the ability to ‘‘steer’’ the state of Bob’s particle at a distance. This means that she could prepare Bob’s particle in different states by measuring her particle using different settings. However, Bob does not trust Alice; Bob worries that she may send him some unentangled particles and fabricate the results using her knowledge about the local hidden state (LHS) of his particles. Bob’s task is to prove that no such hidden states exist. The study of Bell nonlocality have witnessed phenomenal developments to date with important widespread applications 5–7 . Its existence can be demonstrated through two different approaches: the first concerns the violations of Bell inequalities, and the second relies on an all-versus-nothing (AVN) proof without inequalities 8–11 . The AVN proof shows a logical contradiction between the local-hidden-variable models and quantum mechanics, and thus offers an elegant argument of the nonexistence of local-hidden-variable models. What is possible with Bell nonlocality and local hidden variables should also be possible with steering and local hidden states. In stark contrast to Bell nonlocality, the study of steering is still at its infancy. Recent works like Refs. 1,12 put steering on firmer grounds. Like Bell nonlocality, this topic is generally of broad interest, as it hinges on questions pertaining to the foundations of quantum physics 13 , and at the same time reveals new possibilities for quantum information 14 . Einstein-Podolsky-Rosen steering can be detected through the violation of a steering inequality, which rules out the LHS model in the same spirit in which the violation of a Bell inequality rules out the local-hidden-variable model. Recently, several steering inequalities have been proposed and experimentally tested 15–18 . Nevertheless, steering is far from being completely understood and the subject deserves further investigation. The AVN proof for Bell nonlocality 8–11 has been developed to rule out any local-hidden-variable models. Likewise, it is interesting to find out if there an analogous AVN proof which can rule out any LHS models for steering. The OPEN SUBJECT AREAS: QUANTUM INFORMATION THEORETICAL PHYSICS QUBITS QUANTUM MECHANICS Received 17 January 2013 Accepted 11 June 2013 Published 5 July 2013 Correspondence and requests for materials should be addressed to J.-L.C. (cqtchenj@nus. edu.sg) SCIENTIFIC REPORTS | 3 : 2143 | DOI: 10.1038/srep02143 1 purpose of this work is to present an affirmative answer to this question by showing that Einstein-Podolsky-Rosen steering without inequalities exists in a two-qubit system. This proof is an analogy of AVN argument for Bell nonlocality without inequalities, and offers advantages over the existing methods for experimentally testing steerability as well as shedding new light on the asymmetric steering problem. In addition, a steering inequality based on the AVN proof is also obtained. Results Steering without inequalities for two qubits.The two-setting steering scenario can be described as follows: at the beginning, Alice prepares a two-qubit state r AB . She keeps one qubit and sends the other to Bob. She then announces that it is entangled with the one she holds (see Fig. 1), and that she could remotely ‘‘steer’’ his state by projective measurements P^ n a~z{1ðÞ a^ n:~ s½ =2, with ^ nthe measurement direction, a(with a50, 1) the Alice’s measurement result, the 2 32 identity matrix, and ~ s~sx,sy,sz  the vector of the Pauli matrices. Bob then asks Alice to perform two projective measurements P^ n1 aand P^ n2 a(with ^ n1=^ n2) on her qubit and to tell him the measurement results of a. After Alice’s measurement has been done, Bob obtains the four conditional states ~ r^ nj a. Alice could cheat Bob if there exists an ensemble 2jrj no (see the gray box with colored particles in Fig. 1) and a stochastic map 2aj^ A,j  from jto a, such that the following equations hold, ~ r^ nj a~X j 2a^ nj,j   2jrj,a~0,1;j~1,2ðÞ:ð1Þ In order for Bob to be convinced that Alice can steer his state, Bob needs to be sure that no such hidden states are indeed possible. If we demand that Bob’s states possess an LHS description, then his density matrices should satisfy Eq. (1). A contradiction among the four equations, meaning that they cannot have a common solution of 2jrj no and 2a^ n,jjðÞÞ, convinces Bob that an LHS model does not exist and that Alice can steer the state of his qubit. It is worth mentioning that the set of equations (1) plays an analogous role to the one in the standard Greenberger-Horne-Zeilinger (GHZ) argument 8 . The principal difference between the arguments is that the set of equations in (1) deal with density matrices whereas in the GHZ argument, each equation pertains to the outcomes of measurements and therefore corresponds to real numbers. The constraints imposed by LHS model on density matrices are much stricter than constraints imposed by real numbers. This provides an intuitive explanation as to why AVN proof would work for the Einstein-Podolsky-Rosen steering of two-qubit states. Suppose that Alice initially prepares a product state r AB 5 jy A æÆy A jfljy B æÆy B j. It can be verified that, for any projective measurement P^ n a(with P^ n a=yA ji yA hj and y\ A y\ A ) performed by Alice, Bob always obtains two identical pure normalized conditional states as r^ n a~~ r^ n atr~ r^ n a~yB ji yB hj ,(a50, 1), which means that Alice cannot steer Bob’s state. Moreover, Bob can obtain two identical pure normalized conditional states if and only if r AB is a direct-product state. Hence, hereafter we assume that r^ n 0and r^ n 1are two different pure states, i.e., r^ n 0=r^ n 1. For a general r AB ,r^ n aare not pure. If they are pure, then r AB possesses the following uniform form: rAB~P^ n 06~ r^ n 0zP^ n 16~ r^ n 1zz^ n ji {^ n hj 6M z{^ n ji z^ n hj 6M{ , where +^ njiare eigenstates of ^ n:~ s,Mis a 2 32 complex matrix under the positivity condition of r AB , and M{is the Hermitian conjugation of M. For r AB , it is not difficult to find that M~0 if and only if r AB is separable, and the state r AB admits a LHS (which means that it is not steerable) if and only if M~0 (see the Methods section). In a twosetting steering protocol of ^ n1,^ n2 fg , if Bob can obtain two different pure normalized conditional states along Alice’s projective direction ^ n1(or ^ n2), the following three propositions are equivalent: (i) M=0. (ii) r AB is entangled. (iii) No LHS model exists for Bob’s states, so r AB is steerable (in the sense of Alice steering Bob’s state). We thus have our steering argument concluded, and that is given any two-qubit entangled state, the existence of certain projective measurement by Alice so that Bob’s normalized conditional states are two different pure states provides a criterion for Alice-to-Bob steerability. Although the standard GHZ argument is elegant for providing a full contradiction between local-hidden-variable model and quantum mechanics (with 100% success probability) , its validity is only limited to some pure states with high symmetry, such as N-qubit GHZ states and cluster states with N$3 19 . Hardy attempted to extend the GHZ argument to an arbitrary two-qubit system 9 . However, Hardy’s argument works for only 9% of the runs of a specially constructed experiment. Moreover, Hardy’s proof is not Figure 1 | The steering scenario illustration. Alice first prepares a two-qubit state and keeps one qubit. She then sends the other qubit to Bob and announces that it is entangled with the one she possesses (see the pair of red balls and green arrows). Thus she could remotely ‘‘steer’’ Bob’s state by projective measurements. However, Bob does not trust Alice and he worries that she may fabricate the results using her knowledge about LHS. In the twosetting steering scenario, Bob asks Alice to perform two specific projective measurements on her qubit (see the red dashed arrow) and to let him know the measurement results (see the blue dashed arrow). After Alice’s measurement (see the measurement device), Bob obtains four conditional states (see the dashed circle). Alice could cheat Bob if there exists an ensemble (see the gray box with colored particles) and a stochastic map, such that the set of equations (1) holds. To be convinced that Alice can steer his state, Bob needs to confirm that no such hidden states are possible. www.nature.com/scientificreports SCIENTIFIC REPORTS | 3 : 2143 | DOI: 10.1038/srep02143 2 valid for two-qubit maximally entangled state. To overcome this, Cabello proposed an AVN proof for two observers, each possessing a two-qubit maximally entangled state 10,11 . Nowadays, there is no AVN proof of Bell nonlocality for a genuine two-qubit state presented. However, we show that for any two-qubit entangled state r AB , if there exists a projective direction ^ nsuch that Bob’s normalized conditional states r^ n abecome two different pure states, then Alice can steer Bob’s state. Our steering argument is not only valid for twoqubit pure states, but it is also applicable to a wider class of states including mixed states. The AVN proof versus the known steering inequalities.Let us compare our result with the known steering inequalities. First, they play different roles in demonstrating steering: steering inequality follows a similar approach to the Bell inequality for Bell nonlocality, while steering without inequality serves as an analogous counterpart to the GHZ test of Bell nonlocality without Bell inequalities. Secondly, our argument shows that there are many quantum steerable states that do not violate any known steering inequalities. For an example, consider the state rh V~VYhðÞ ji YhðÞ hj z1{VðÞWhðÞ ji WhðÞ hj ,ð2Þ where jY(h)æ5cos hj00æ1sin hj11æ,jW(h)æ5cos hj10æ1sin hj01æ. It is entangled when Vg[0, 1/2) <(1/2, 1] and hg(0, p/2). It can be easily verified that, for state (2), after Alice performs an ^ x-direction measurement on her qubit, Bob’s normalized conditional states are just two different pure states, cos hj0æ1sin hj1æand cos hj0æ2sin hj1æ. Thus, based on our AVN proof of steering, Alice can always steer Bob’s state using just a two-setting protocol ^ x,^ z fg .On the other hand, a class of N-setting steering inequality SN~ 1 NXN k~1Ak~ sB k  {CNƒ0 has been introduced in Ref. 15 to show the ability of Alice steering Bob’s state. By running a numerical check of a 10-setting steering inequality of the above form, we observe that, for some regions of Vand h, the steering inequality cannot detect the steering of state (2)(as shown in Fig. 2 a). The colors denote different violation values, as shown in the legend. The blank region indicates that the steerability of state (2) cannot be detected by resorting to this inequality. Finally, unlike quantum entanglement and Bell nonlocality, the definition of steering is asymmetric 1,20 . Our AVN proof can shed light on this problem. The state (2) is not symmetric under a permutation of Alice and Bob (even with local unitary transformations acting on the state). The known steering inequalities in Ref. 15 do not reveal asymmetric steering (see Fig. 2 a). However, our argument presents a promising way to reveal asymmetric steering. According to our AVN proof, the state (2) exhibits two-setting asymmetric steering. On one hand, Alice can always steer Bob’s state using just the two-setting protocol ^ x,^ z fg . On the other hand, after Bob has performed a projective measurement along an arbitrary ^ n-direction on his qubit, Alice’s normalized conditional states can never be cast into two different pure states, allowing for the existence of LHS models. Take the state with parameters V53/5 and h5p/8 as an example (whose corresponding point is outside of the colored region in Fig. 2 a): Numerical results show that, for any two-setting protocol ^ n1,^ n2 fg , there is always a solution of LHS for Alice’s conditional states. In short, this example illustrates a state in which the steering scenario is not interchangeable. This result can be of practical importance, since asymmetric steering has applications in one-way quantum cryptography 21 and may have potential applications in other fields of quantum information processing. A steering inequality.It is known that a Bell inequality can be derived from the GHZ argument 22 . This is also the case for the steering without inequalities argument. The steering inequality equivalent to the AVN proof reads W3 hi {CLHSƒ0,ð3Þ subject to the constraint W1 hi ~W2 hi ~0. Here Wjare projectors as W1~P^ n 06r^ n\ 0,W2~P^ n 16r^ n\ 1,W3~z ji z hj 6^ nB ji ^ nB hj , with r^ n\ a orthogonal to r^ n a,zji~z ^ njiz{ ^ njiðÞ ffiffiffi 2 p,^ nB ji~cos hB 20jiz sin hB 2eiQB1 ji ,Wj  ~tr Wj%AB  , and CLHS~max^ nBtr ^ nB ji ^ nB hj ~ r^ n 0z  ~ r^ n 1Þ2Þ is the upper bound for the LHS model. Its physical implication can be described as follows: Suppose Alice performs a projective measurement in the ^ n-direction and finds that Bob can obtain two different pure normalized conditional states, then W1 hi ~W2 hi ~0. They then perform a joint-measurement W3(in which Alice’s measurement direction is perpendicular to ^ ndirection). According to Lemma 2 (see the Methods section), the LHS model requires M~0, thus the probability W3 hi is bounded by C LHS . However, with quantum mechanics, this bound is always exceeded due to a non-vanishing M. Figure 2 | (a) Detecting steerability of the state (2) using the ten-setting steering inequalities. We explore the steering of state (2) via violation of the ten-setting inequality presented in Ref. 15. The colors denote different values of quantum violation, as scaled in the legend. The blank region indicates that steerability of (2) cannot be detected by this inequality. With the replacement Ak? ~ sA kand ~ sB k?Bkin the above inequality, one obtains a similar steering inequality S’Nƒ0to show Bob’s ability of steering Alice’s state. The inequality S’N~10ƒ0yields the same violation region. This indicates that steering inequalities in Ref. 15 cannot reveal asymmetric steering. (b) Detecting steerability using the steering inequality (3). We show the steering of the state r col through violation of inequality (3). Quantum prediction of the left-hand-side of the inequality always succeeds 0 unless V50orh50, p/2. www.nature.com/scientificreports SCIENTIFIC REPORTS | 3 : 2143 | DOI: 10.1038/srep02143 3 As an instance, we investigate the steering of state rcol~ VYhðÞ ji YhðÞ hj z1{VðÞ col, with color noise col~00 ji 00 hj zð 11ji11hjÞ=2 by using our inequality (3). We find that Bob’s conditional states on Alice’s projective measurement in the z-direction are two different pure states j0æÆ0jand j1æÆ1j, and the upper bound is C LHS 5(1 1Vjcos 2hj)/4. The quantum prediction of the left-hand-side of inequality (3) reads 1 2Vsin2hfor hg[0, p/4], and 1 2Vcos2hfor hg[p/4, p/2], which do not vanish unless V50orh 50, p/2 (see Fig. 2 b). The violation of the inequality clearly demonstrates that the state r col possesses steerability except V50orh50, p/2. Discussion We have presented an AVN proof of Einstein-Podolsky-Rosen steering for two qubits without inequalities based on a two-setting steering protocol. The argument is valid for any two-qubit entangled state, both pure and mixed. We show that many quantum states that do not violate any known steering inequalities are indeed steerable states. This provides a new perspective for understanding steerability and offers an elegant argument for the nonexistence of LHS models without resorting to steering inequalities. The result also sheds new light on the asymmetric steerability – a phenomenon with no counterpart in quantum entanglement and Bell nonlocality. The result is testable through measurements of Bob’s conditional states and provides a simple alternative to the existing experimental method for detecting steerability 15–18 . Theoretically, a two-setting steering protocol can be used to show that no LHS models exist for r AB if the state satisfies the condition given in our AVN argument. Experimentally, the determination of the steerability of a quantum state can be done by performing quantum state tomography 23 on Bob’s qubit. Moreover, a steering inequality is obtained from our AVN argument, and this inequality offers another way to test steerability of states. Like Bell nonlocality whose importance has only been realized with the rapid development of quantum information science, we anticipate further developments in this exciting area. Methods We prove two Lemmas in the section. The steerability of r AB is equivalent to that of the state %AB~UA6ðÞrAB U{ A6  . It is always possible for Alice to choose an appropriate unitary matrix Uthat rotates the direction ^ nto the direction ^ z. Therefore, we can initially set ^ n~^ zby studying the state %AB instead of r AB . After Alice performs a projective measurement in the ^ z-direction, Bob’s unnormalized conditional states are ~ r^ z 0~trA0 ji0 hj 6ðÞ%AB ½~m1Q1 ji Q1 hj ,ð4aÞ ~ r^ z 1~trA1ji1hj6 ðÞ %AB ½ ~m2Q2 jiQ2 hj,ð4bÞ with m1~tr ~ r^ z 0  ,m2~tr ~ r^ z 1  ,r^ z 0~Q1 jiQ1 hj, and r^ z 1~Q2 jiQ2 hj. Then one has %AB~m10ji0hj6Q1 jiQ1 hjzm21ji1hj6Q2 jiQ2 hj z0ji1hj6Mz1ji0hj6M{: Lemma 1.M~0 if and only if %AB is separable. Proof.Look at the form of %AB, obviously M~0 implies %AB is separable. To prove the converse, one needs the definition of separability: %AB~PipitAi6tBi, where t Ai and t Bi are, respectively, Alice and Bob’s local density matrices, and p i .0 satisfy Pipi~1. For convenience, let tmn Ai m,n~1,2ðÞdenote the element of Alice’s density matrix t Ai . By calculating trA0 ji0 hj 6ðÞ%AB ½and trA1 ji1 hj 6ðÞ%AB ½, one has Pipit11 AitBi~m1Q1 ji Q1 hj ,Pipit22 AitBi~m2Q2 ji Q2 hj . Let Q\ 1 and Q\ 2 be two pure states that are orthogonal to jQ 1 æand jQ 2 æ, respectively. Notice that tr Pipitmm Ai tBi|Q\ m Q\ m   ~0,m~1,2ðÞ, thus, for any index i, we have tmm Ai tr tBi Q\ m Q\ m   ~0, which results in t11 Ait22 Ai tr tBi Q\ 1 Q\ 1   ztr tBi Q\ 2 Q\ 2   ~0:ð5Þ Since Q\ 1 =Q\ 2 , they cannot be simultaneously perpendicular to the state t Bi ,thus t11 Ait22 Ai~0, which yields t12 Ai~t21 Ai~0 due to positivity condition of t Ai .So M~Pipit12 AitBi~0. Lemma 1 is henceforth proved. Lemma 2.The state %AB admits a local-hidden-state (LHS) model (which means that it is not steerable) if and only if M~0. Proof.M~0 implies %AB is separable, thus %AB admits a LHS model. Now we focus on the proof of necessity. If Alice’s measurement setting is ^ z,^ x fg , then one has ~ r^ x 0~1 2m1Q1 jiQ1 hjzm2Q2 jiQ2 hjzMzM{  ,ð6aÞ ~ r^ x 1~1 2m1Q1 jiQ1 hjzm2Q2 jiQ2 hj{M{M{  :ð6bÞ Substitute Eqs. (4a)(4b)(6a)(6b) into Eq. (1) and due to Q\ 1 ~ r^ z 0Q\ 1 ~0 and Q\ 2 ~ r^ z 1Q\ 2 ~0, one immediately has r j g{jQ 1 æÆQ 1 j,jQ 2 æÆQ 2 j} for any j. Based on which, Eqs. (6a) (6b) are valid only if MzM{~axQ1 ji Q1 hj zbxQ2 ji Q2 hj ðÞ =2, with a x ,bx[R. Similarly, if Alice’s measurement setting is ^ z,^ yfg, then one has M{M{~iayQ1 jiQ1 hjzbyQ2 jiQ2 hj . 2, with a y ,by[R. If there exists a LHS model for Bob’s states, then M~aQ 1 ji Q1 hj zbQ 2 ji Q2 hj , with a5a x 1ia y ,b5b x 1ib y . Substitute Minto Eq. (5), we have %AB~m1Ta6Q1 ji Q1 hj zm2Tb6Q2 ji Q2 hj , with Ta~1a a0  and Tb~0b b1  . Now we construct the following two projectors: Q1~x1 ji x1 hj 6Q\ 2 Q\ 2 ,Q2~x2 ji x2 hj 6Q\ 1 Q\ 1 , where jx 1 æis the eigenvector of T a with eigenvalue v1~1{ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1z4a jj 2 q  2ƒ0, and jx 2 æis the eigenvector of T b with eigenvalue v2~1{ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1z4b jj 2 q  2ƒ0. Because %AB is a density matrix, one has tr %ABQ1 ðÞ~v1m1Q\ 2Q1   2§0, tr %ABQ2 ðÞ~v2m2Q\ 1Q2   2§0: This leads to M~0. Lemma 2 is henceforth proved. Three measurement settings were mentioned in the proof of Lemma 2. This does not mean that we need a three-setting protocol to show steering. For a given entangled state %AB, a two-setting protocol is enough to demonstrate steering. Lemma 2 shows that MzM{and M{M{cannot be linearly expanded of jQ 1 æÆQ 1 jand jQ 2 æÆQ 2 j simultaneously (because that means M~0 and r AB is separable). 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H.Y.S. and X.J.Y. prepared figures 1 and 2. All authors reviewed the manuscript. Additional information Reprints and permission information is available at www.nature.com/reprints. Competing financial interests: The authors declare no competing financial interests. How to cite this article: Chen, J.-L. et al. All-Versus-Nothing Proof of Einstein-Podolsky-Rosen Steering. Sci. Rep. 3, 2143; DOI:10.1038/srep02143 (2013). This work is licensed under a Creative Commons AttributionNonCommercial-ShareAlike 3.0 Unported license. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0 www.nature.com/scientificreports SCIENTIFIC REPORTS | 3 : 2143 | DOI: 10.1038/srep02143 5