Abstract

Quantum correlation is a fundamental resource for various quantum information tasks. It is thus of importance to share the correlation to utilize it for many parties, but sharing quantum correlation among multiple parties is strictly restricted by the well-known monogamy relations. Nonetheless, this restriction can be relaxed when weak measurements are employed. Here, we experimentally demonstrate multiple-observer quantum steering by exploiting sequential weak measurements. Specifically, we observe simultaneous triple violations of the quantum steering inequality among four observers for a bipartite entangled system. Our results not only provide fundamental insights into the relation between quantum steering and measurement disturbance, but also suggest that quantum steering might be repeatably exploited to find applications to, for example, unbounded randomness certification and sharing secret keys among multiple parties simultaneously.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

The correlation of measurement outcomes is said to be nonlocal if it cannot be explained by locally prepared systems. Entanglement is the basic example of such quantum correlation [1], and it has been regarded as the most prominent property of quantum systems. The significance of quantum correlation over classical comes from the fact that many applications in various quantum information tasks rely on quantum correlations [24]. It is thus of importance to share such quantum correlation with multiple parties so that an entangled system is fully exploited for many parties. However, sharing quantum correlations in multi-party systems is strictly restricted due to fundamental properties known as the monogamy relations for quantum correlations [57]. Nonetheless, it has been shown that multiple observers can share quantum correlations by exploiting the weak quantum measurement [811].

While experimental demonstrations of sharing nonlocal correlations have been reported very recently, these experiments focused on sharing Bell nonlocality among three observers [1214]. In particular, double Clauser–Horne–Shimony–Holt (CHSH)–Bell inequality violations were observed for bipartite entangled systems. As the Bell nonlocality is not the only form of quantum correlations, it is thus interesting to investigate sharing other forms of quantum correlations. In addition to the Bell nonlocality, another distinct form of nonlocal quantum correlation is Einstein–Podolsky–Rosen (EPR) quantum steering [1518]. Unlike the Bell nonlocality, the quantum steering has a clear operational meaning that one can remotely control a quantum state even when a measurement apparatus is untrusted [19]. This notable feature thus makes quantum steering a useful resource for various applications such as quantum key distribution [20], randomness certification [21,22], and secret sharing [23].

In this work, we experimentally demonstrate sharing EPR quantum steering among multiple parties by exploiting variable weak strength measurements [2427]. We consider a bipartite entangled photon pair, where one of the photons is subject to sequential weak quantum measurements. In particular, we observe simultaneous triple violations of EPR quantum steering inequality, thereby directly verifying that the quantum steering is sharable with more than four observers. This result could suggest that the number of observers to share the quantum correlation might be unbounded [28].

2. THEORY

Let us begin with considering the following game to describe sharing quantum steering with multiple parties. Here, two spin-1/2 particles are shared with separated parties. As shown in Fig. 1, one particle is sent to Alice (A), and the other one is subsequently sent to multiple Bobs, denoted as Bob1 (B1), Bob2 (B2), and Bob3 (B3). In this game, each Bob’s task is to remotely control (steer) the quantum state of Alice’s particle, simultaneously and independently. Alice will be convinced by Bob1 (or B2 or B3) if the correlation between Alice’s and Bob1’s (or B2’s or B3’s) measurement outcomes cannot be explained by the local-hidden-state model (LHS) [29]. More clearly, the state assemblage given by Alice is said to be unsteerable if the assemblage can be locally prepared. Similar to the Bell nonlocality, captured by violating the CHSH inequality for dichotomic observables, the EPR steering also can be identified by showing the violation of an inequality derived using the LHS model.

 figure: Fig. 1.

Fig. 1. Sharing EPR quantum steering scenario by multiple observers. A bipartite entangled pair is shared with Alice and multiple Bobs. In our scenario, three independent Bobs wish to claim that the quantum state given by Alice is steerable simultaneously. To verify the EPR steering correlation, each observer extracts dichotomic measurement outcomes {${a}$, ${b}$, ${c}$, ${d}$} for randomly chosen measurement directions {${\vec \alpha _j}$, ${\vec \beta _j}$, ${\vec \gamma _j}$, ${\vec \delta _j}$}. In particular, we consider three measurement settings, i.e., $j = \{1,2,3\}$.

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We investigate the quantum steering between Alice and $i$-th Bob with the linear steering inequality [29] with three randomly chosen measurement settings $j \in \{1,2,3\}$. The inequality is given by

$${S_i} = \frac{1}{{\sqrt 3}}\left| {\sum\limits_{j = 1}^3 \langle {{\cal A}_j} \otimes {\cal B}_j^i\rangle} \right| \le 1,$$
where ${{\cal A}_j}$ and ${\cal B}_j^i$ refer to the measurement for Alice and $i$-th Bob, respectively. To verify the violation, each observer performs the measurement for given setting $j$, where the chosen measurement directions are labeled as ${\vec \alpha _j}$, ${\vec \beta _j}$, ${\vec \gamma _j}$, and ${\vec \delta _j}$, for A, B1, B2, and B3, respectively. Here, the measurement outcomes for A, B1, B2, and B3 are denoted as $\{a,b,c,d\}$, respectively, and they will be used to calculate the steering parameter ${S_i}$.

In order to violate the steering inequality for A-B1, A-B2, A-B3 simultaneously, both Bob1 and Bob2 must perform their measurements weakly. Otherwise, the entanglement will be completely destroyed, i.e., if Bob1 performs a strong projective measurement, the steering inequality will not be violated for A-B2 and A-B3. The weak measurements for B1 and B2 are described by positive-operator valued measures (POVMs), of which elements are set as

$${\hat {\cal M}_{\pm |\vec n}} = \lambda {\hat \Pi _{\pm |\vec n}} + (1 - \lambda)\frac{{\hat {\mathbb I}}}{2},$$
where $\lambda \in [0,1]$ is the measurement strength, $\vec n$ is the measurement direction, $\hat {\mathbb I}$ is the identity operator, and ${\hat \Pi _{\pm |\vec n}}$ are the projectors associated with measurement outcomes for given measurement settings. For example, if the measurement direction is set as $\vec n = \vec z$, we have ${\hat \Pi _{+ |\vec z}} = |0\rangle \langle 0|$ and ${\hat \Pi _{- |\vec z}} = |1\rangle \langle 1|$. Here, $\lambda = 1$ corresponds to strong measurement, and $\lambda = 0$ represents no measurement. Weak measurement, which partially collapses the system, is achieved for $0 \lt \lambda \lt 1$.

The POVM elements are implemented with corresponding Kraus operators such that ${\hat {\cal M}_{\pm |\vec n}} = \hat {\cal K}_{\pm |\vec n}^\dagger {\hat {\cal K}_{\pm |\vec n}}$, where we choose the Kraus operators given as

$${\hat {\cal K}_{\pm |\vec n}} = \frac{1}{{\sqrt 2}}(\sqrt {1 \pm \lambda} {\hat \Pi _{+ |\vec n}} + \sqrt {1 \mp \lambda} {\hat \Pi _{- |\vec n}}).$$

It should be noted that these choices of Kraus operators are optimal in the sense that the tradeoff relation between the information gain and the state disturbance relation ${F^2} + {G^2} \le 1$ is saturated [8,30]. In other words, ${F^2} + {G^2} = 1$ is satisfied for the optimal measurement. Note that the information gain is quantified with the parameter $G$, and the amount of disturbance is quantified by the fidelity factor $F$ between the post-measurement state and the original input state.

3. EXPERIMENT

Now, we describe our experimental setup. Correlated photon pairs at 780 nm are produced via type-II spontaneous parametric down-conversion (SPDC) [31]. A beta-barium borate (BBO) crystal with a sandwich configuration is utilized to generate high-quality polarization entangled photons. The entangled photons are collected into single-mode fibers, and delivered to the experimental setup shown in Fig. 2(a). By using a set of waveplates [not shown in Fig. 2(a)], the input polarization state is prepared in a maximally entangled state ${\rho _{{\rm AB}}} = |{\Psi ^ -}\rangle \langle {\Psi ^ -}|$. Here, $|{\Psi ^ -}\rangle$ is the singlet state in $|{\Psi ^ -}\rangle \equiv (|{\rm HV}\rangle - |{\rm VH}\rangle)/\sqrt 2$, and $|H\rangle$ and $|V\rangle$ refer to horizontal and vertical polarization states, respectively.

 figure: Fig. 2.

Fig. 2. (a) Experimental setup. Polarization entangled photon pairs are prepared and distributed to Alice and Bobs. Alice performs the strong projective measurement with a set of waveplates and a polarizing beam splitter (PBS). Bob1 and Bob2 perform weak quantum measurement with the measurement strengths of ${\lambda _1}$ and ${\lambda _2}$, respectively. Finally, Bob3 also performs the projective measurement. BD, beam displacer; H, half-wave plate; ${Q}$, quarter-wave plate. (b) Schematic of the weak measurement with variable strength $\lambda$. See the main text for details.

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 figure: Fig. 3.

Fig. 3. Demonstrating the triple violations of EPR quantum steering. (a) Steering parameters ${{\cal S}_i}$ are experimentally measured for various measurement strengths. Here, ${\lambda _1}$ and ${\lambda _2}$ denote the measurement strengths for Bob1 and for Bob2, respectively. Experimentally measured steering parameters are shown as markers, where the blue, purple, and orange circles correspond to ${{\cal S}_1}$, ${{\cal S}_2}$, and ${{\cal S}_3}$. Theoretical predictions are also represented as surface plots, and compared with the experimental data. (b) Region of triple violations represented with shaded color for clarity. (c) Steering parameters as a function of ${\lambda _1}$ for fixed ${\lambda _2} = 0.76$. (d) Steering parameters as a function of ${\lambda _2}$ for fixed ${\lambda _1} = 0.64$. For (c) and (d), shaded areas with sky-blue color indicate the violation of steering inequality. In pink areas, the triple violations of EPR steering are clearly demonstrated. The errors in ${\cal S}$ are obtained by performing 100 Monte Carlo simulation runs by taking into account of the Poisson photon counting statistics. The error bars are too small to be visible.

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Bob1 performs the optimal weak measurement on his qubit by using a measurement apparatus depicted in Fig. 2(b). To show this apparatus correctly, implementing the aforementioned optimal weak measurement given in Eqs. (2) and (3), let us suppose an arbitrary pure state $|\psi {\rangle _{{\rm in}}} = \alpha |\phi \rangle + \beta |{\phi _ \bot}\rangle$, with $\langle \phi |{\phi _ \bot}\rangle = 0$ given as an input. A quarter-wave plate (Q1) and a half-wave plate (H1) rotate the polarization state in such a way that ${\hat R_ +}|\psi {\rangle _{{\rm in}}} \to \alpha |{\rm H}\rangle + \beta |{\rm V}\rangle$. The photon is then entered into the interferometer with polarizing beam displacers (BD1 and BD2). Inside the interferometer, both the horizontal and vertical polarization components at each path evolve to $\sin 2{\theta _\lambda}|{\rm H}\rangle + \cos 2{\theta _\lambda}|{\rm V}\rangle$. Then, the unnormalized output state after BD2 is given as $\alpha \cos 2{\theta _\lambda}|{\rm H}\rangle + \beta \sin 2{\theta _\lambda}|{\rm V}\rangle$. Finally, $\hat R_ + ^{- 1}$ operation is applied by using H2 and Q2, giving the final (unnormalized) output state as $|\psi {\rangle _{{\rm out}}} = \alpha \cos 2{\theta _\lambda}|\phi \rangle + \beta \sin 2{\theta _\lambda}|{\phi _ \bot}\rangle$. The transformation described here is identical to the evolution by the Kraus operator ${\hat {\cal K}_ +}$ in Eq. (3), provided that the measurement strength $\lambda$ is defined as $\lambda \equiv 2\mathop {\cos}\nolimits^2 (2{\theta _\lambda}) - 1$ for ${\theta _\lambda} \in [0,\pi /8]$. We note that only the case of extracting the measurement outcome of ${+}{1}$ is considered by post-selecting a particular output mode of the interferometer. Although the other output mode corresponds to the case of registering the outcome of ${-}{1}$, we consider only the particular output mode for simplicity of the experimental implementation. Instead, the case of ${-}{1}$ outcome is considered by simply changing the rotating operation associated with the waveplates (Q1, H1, H2, Q2) as ${\hat R_ -}|\psi {\rangle _{{\rm in}}} \to \beta |{\rm H}\rangle + \alpha |{\rm V}\rangle$. The half-wave plate at ${\theta _\lambda}$ determines the measurement strength. By using the other wave plates, the choice of measurement direction is set, and the measurement outcome is post-selected. Likewise, Bob2 performs the weak measurement. Here, the measurement strengths for B1 and B2 are denoted as $\lambda 1$ and $\lambda 2$, respectively. Finally, Alice and Bob3 choose their measurement directions with a combination of wave plates and a polarizing beam splitter. Four cases of coincidence detection events between two detectors (D1-D3, D1-D4, D2-D3, D2-D4) are registered with a home-built field programmable gate array (FPGA)-based coincidence counting unit [32], and each case corresponds to the case yielding four possible outcomes for Alice and Bob3.

In our experiment, four trials are necessary to consider four possible outcomes of $b$ and $c$ for given measurement directions $\{{\vec \alpha _k},{\vec \beta _l},{\vec \gamma _m},{\vec \delta _n}\}$ so that we obtain the joint measurement probability

$$p(a,b,c,d|{\vec \alpha _k},{\vec \beta _l},{\vec \gamma _m},{\vec \delta _n}),$$
where the probability is experimentally measured by normalizing the coincidence counting events to satisfy $\sum_{a,b,c,d} p$$(a,b,c,d|{\vec \alpha _k},{\vec \beta _l},{\vec \gamma _m},{\vec \delta _n}) = 1$ for any set of $\{{\vec \alpha _k},{\vec \beta _l},{\vec \gamma _m},{\vec \delta _n}\}$. Since there are ${3^4}$ choices of measurement settings, we performed a total of 324 trials. And for each trial, we recorded the coincidence counting events for 4 s.

Now, we are ready to investigate the EPR steering inequality given in Eq. (1) with the experimentally obtained joint measurement probability. Note that all the steering parameters among Alice-Bob1, Alice-Bob2, and Alice-Bob3 are simultaneously accessible. For example, the steering parameter ${{\cal S}_2}$ for Alice and Bob2 is calculated as

$${{\cal S}_2} = \frac{1}{{\sqrt 3}}\left| {\sum\limits_{j = 1}^3 \sum\limits_{a,c} ac \cdot p(a,c|{{\vec \alpha}_j},{{\vec \gamma}_j})} \right|,$$
where the conditional probability $p(a,c|{\vec \alpha _j},{\vec \gamma _j})$ is obtained from the experimentally measured joint probability given in Eq. (4) as
$$\!\!\!p(a,c|{\vec \alpha _j},{\vec \gamma _j}) = \frac{1}{9}\sum\limits_{l = 1}^3 \sum\limits_{n = 1}^3 \sum\limits_{b,d} p(a,b,c,d|{\vec \alpha _j},{\vec \beta _l},{\vec \gamma _j},{\vec \delta _n}).\!$$

Similarly, the other steering parameters such as ${{\cal S}_1}$ and ${{\cal S}_3}$ can be obtained simultaneously from the same measurement data. The measurement directions for Alice and Bob1 were chosen to maximize ${{\cal S}_1}$, where the condition is given as ${\vec \alpha _1} = {\vec \beta _1} = \vec x$, ${\vec \alpha _2} = {\vec \beta _2} = \vec y$, and ${\vec \alpha _3} = {\vec \beta _3} = \vec z$. We also chose the measurement directions for Bob2 and Bob3 as ${\vec \gamma _1} = {\vec \delta _1} = \vec x$, ${\vec \gamma _2} = {\vec \delta _2} = \vec y$, and ${\vec \gamma _3} = {\vec \delta _3} = \vec z$. Rather than these choices, ${\vec \gamma _m}$ and ${\vec \delta _n}$ can be optimized, but it is possible only when ${\lambda _1}$ is known for Bob2 and ${\lambda _1}$ and ${\lambda _2}$ are known for Bob3.

We measured ${{\cal S}_1}$, ${{\cal S}_2}$, and ${{\cal S}_3}$ for several values of ${\lambda _1}$ and ${\lambda _2}$. As shown in Fig. 3(a), the experimental data are compared with theoretical expectations, showing a good agreement. The maximum steering parameter is expected to be $\sqrt 3$ in our three measurement cases, which can be obtained at ${\lambda _1} \to 1$ for ${{\cal S}_1}$, ${\lambda _1} \to 0$ and ${\lambda _2} \to 1$ for ${{\cal S}_2}$, and ${\lambda _1} = {\lambda _2} \to 0$ and ${\lambda _3} \to 1$ for ${{\cal S}_3}$. As the measurement strength ${\lambda _1}$ becomes weaker, the steering parameter ${{\cal S}_1}$ decreases, whereas both ${{\cal S}_2}$ and ${{\cal S}_3}$ increase. For a fixed value of ${\lambda _1}$, there exists a trade-off relation between ${{\cal S}_2}$ and ${{\cal S}_3}$ depending on ${\lambda _2}$. An interesting region is highlighted in Fig. 3(b), where all the steering parameters ${{\cal S}_1}$, ${{\cal S}_2}$, and ${{\cal S}_3}$ show a violation of the steering inequality in Eq. (1). The triple violations of steering inequality are more clearly shown in Figs. 3(c) and 3(d), and summarized in Table 1. The measured and theoretical steering parameters are displayed for fixed values of ${\lambda _2} = 0.76$ in Fig. 3(c) and of ${\lambda _1} = 0.64$ in Fig. 3(d). In particular, we observe the triple violations of steering inequality at $\{{\lambda _1} = 0.64,{\lambda _2} = 0.76\}$ with more than 40 standard deviations. Although each Bob performed his measurement independently in our scenario, that does not mean the violation of the monogamy relation. In our case, each Bob’s measurement was realized sequentially. Therefore, while the no-signaling condition between Alice and multiple Bobs holds, signaling between each Bob is in principle possible.

Tables Icon

Table 1. Summary of Observed Steering Parameters at ${\lambda _1} = 0.64$ and ${\lambda _2} = 0.76, S_i\nleq 1$ (represents the quantum steering correlation)

4. SUMMARY

To summarize, we have experimentally demonstrated sharing quantum correlations with multiple parties. In particular, we have observed triple violations of EPR quantum steering inequality among multiple observers using sequential weak measurements with variable measurement strengths. Unlike previous experiments on sharing Bell nonlocality, where more than double violations of CHSH–Bell inequality with unbiased inputs were not observed, we directly show that quantum steering correlations are sharable with at least more than four parties with three measurement settings. It would be possible to further extend the number of parties for simultaneously sharing quantum steering correlations by using more measurement settings [9,17]. We further note that the number of independent observers for sharing Bell nonlocality might not be bounded as well [28]. This is in stark contrast to the previously known conjecture that more than double violations of CHSH–Bell inequality are impossible with independent observers [8,10]. Our results not only shed new light on our insights into fundamental properties of quantum correlations, but also allow us to fully harness an entangled system for various applications [33,34].

Funding

Korea Institute of Science and Technology (2E30620); National Research Foundation of Korea (2019M3E4A107866011, 2019M3E4A1079777, 2019R1A2C2006381); Institute for Information and Communications Technology Promotion (2020-0-00947, 2020-0-00972).

Acknowledgment

We would like to thank Byung-Kwon Park for technical support for the FPGA-based multi-channel coincidence counting unit.

Disclosures

The authors declare no conflicts of interest.

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References

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  1. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
    [Crossref]
  2. A. Acín and L. Masanes, “Certified randomness in quantum physics,” Nature 540, 213–219 (2016).
    [Crossref]
  3. V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
    [Crossref]
  4. H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, “Nonlocality and communication complexity,” Rev. Mod. Phys. 82, 665–698 (2010).
    [Crossref]
  5. V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A 61, 052306 (2000).
    [Crossref]
  6. T. Pramanik, M. Kaplan, and A. S. Majumdar, “Fine-grained Einstein-Podolsky-Rosen–steering inequalities,” Phys. Rev. A 90, 050305 (2014).
    [Crossref]
  7. B. Toner, “Monogamy of non-local quantum correlations,” Proc. R. Soc. A 465, 59–69 (2009).
    [Crossref]
  8. R. Silva, N. Gisin, Y. Guryanova, and S. Popescu, “Multiple observers can share the nonlocality of half of an entangled pair by using optimal weak measurements,” Phys. Rev. Lett. 114, 250401 (2015).
    [Crossref]
  9. D. Das, A. Ghosal, S. Sasmal, S. Mal, and A. S. Majumdar, “Facets of bipartite nonlocality sharing by multiple observers via sequential measurements,” Phys. Rev. A 99, 022305 (2019).
    [Crossref]
  10. S. Sasmal, D. Das, S. Mal, and A. S. Majumdar, “Steering a single system sequentially by multiple observers,” Phys. Rev. A 98, 012305 (2018).
    [Crossref]
  11. S. Mal, A. S. Majumdar, and D. Home, “Sharing of nonlocality of a single member of an entangled pair of qubits is not possible by more than two unbiased observers on the other wing,” Mathematics 4, 48 (2016).
    [Crossref]
  12. M.-J. Hu, Z.-Y. Zhou, X.-M. Hu, C.-F. Li, G.-C. Guo, and Y.-S. Zhang, “Observation of non-locality sharing among three observers with one entangled pair via optimal weak measurement,” npj Quantum Inf. 4, 63 (2018).
    [Crossref]
  13. M. Schiavon, L. Calderaro, M. Pittaluga, G. Vallone, and P. Villoresi, “Three-observer bell inequality violation on a two-qubit entangled state,” Quantum Sci. Technol. 2, 015010 (2017).
    [Crossref]
  14. T. Feng, C. Ren, Y. Tian, M. Luo, H. Shi, J. Chen, and X. Zhou, “Observation of nonlocality sharing via “strong” weak measurements,” arXiv: 1912.02979 [quant-ph].
  15. R. Uola, A. C. S. Costa, H. C. Nguyen, and O. Gühne, “Quantum steering,” Rev. Mod. Phys. 92, 015001 (2020).
    [Crossref]
  16. D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845–849 (2010).
    [Crossref]
  17. H. A. Shenoy, S. Designolle, F. Hirsch, R. Silva, N. Gisin, and N. Brunner, “Unbounded sequence of observers exhibiting Einstein-Podolsky-Rosen steering,” Phys. Rev. A 99, 022317 (2019).
    [Crossref]
  18. K. Sun, X.-J. Ye, Y. Xiao, X.-Y. Xu, Y.-C. Wu, J.-S. Xu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Demonstration of Einstein-Podolsky-Rosen steering with enhanced subchannel discrimination,” npj Quantum Inf. 4, 12 (2018).
    [Crossref]
  19. R. Gallego and L. Aolita, “Resource theory of steering,” Phys. Rev. X 5, 041008 (2015).
    [Crossref]
  20. C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, “One-sided device-independent quantum key distribution: security, feasibility, and the connection with steering,” Phys. Rev. A 85, 010301 (2012).
    [Crossref]
  21. E. Passaro, D. Cavalcanti, P. Skrzypczyk, and A. Acín, “Optimal randomness certification in the quantum steering and prepare-and-measure scenarios,” New J. Phys. 17, 113010 (2015).
    [Crossref]
  22. F. J. Curchod, M. Johansson, R. Augusiak, M. J. Hoban, P. Wittek, and A. Acín, “Unbounded randomness certification using sequences of measurements,” Phys. Rev. A 95, 020102 (2017).
    [Crossref]
  23. Q. Y. He and M. D. Reid, “Genuine multipartite Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 111, 250403 (2013).
    [Crossref]
  24. Y.-S. Kim, J.-C. Lee, O. Kwon, and Y.-H. Kim, “Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nat. Phys. 8, 117–120 (2012).
    [Crossref]
  25. Y.-W. Cho, Y. Kim, Y.-H. Choi, Y.-S. Kim, S.-W. Han, S.-Y. Lee, S. Moon, and Y.-H. Kim, “Emergence of the geometric phase from quantum measurement back-action,” Nat. Phys. 15, 665–670 (2019).
    [Crossref]
  26. Y. Kim, Y.-S. Kim, S.-Y. Lee, S.-W. Han, S. Moon, Y.-H. Kim, and Y.-W. Cho, “Direct quantum process tomography via measuring sequential weak values of incompatible observables,” Nat. Commun. 9, 192 (2018).
    [Crossref]
  27. G. Foletto, L. Calderaro, A. Tavakoli, M. Schiavon, F. Picciariello, A. Cabello, P. Villoresi, and G. Vallone, “Experimental certification of sustained entanglement and nonlocality after sequential measurements,” Phys. Rev. Appl. 13, 044008 (2020).
    [Crossref]
  28. P. J. Brown and R. Colbeck, “An unbounded number of independent observers can share the nonlocality of a single maximally entangled qubit pair,” arXiv:2003.12105v1 [quant-ph].
  29. E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein-Podolsky-Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
    [Crossref]
  30. H.-T. Lim, Y.-S. Ra, K.-H. Hong, S.-W. Lee, and Y.-H. Kim, “Fundamental bounds in measurements for estimating quantum states,” Phys. Rev. Lett. 113, 020504 (2014).
    [Crossref]
  31. O. Kwon, Y.-W. Cho, and Y.-H. Kim, “Single-mode coupling efficiencies of type-ii spontaneous parametric down-conversion: collinear, noncollinear, and beamlike phase matching,” Phys. Rev. A 78, 053825 (2008).
    [Crossref]
  32. B. K. Park, Y.-S. Kim, O. Kwon, S.-W. Han, and S. Moon, “High-performance reconfigurable coincidence counting unit based on a field programmable gate array,” Appl. Opt. 54, 4727–4731 (2015).
    [Crossref]
  33. H.-W. Li, Y.-S. Zhang, X.-B. An, Z.-F. Han, and G.-C. Guo, “Three-observer classical dimension witness violation with weak measurement,” Commun. Phys. 1, 10 (2018).
    [Crossref]
  34. X.-B. An, H.-W. Li, Z.-Q. Yin, M.-J. Hu, W. Huang, B.-J. Xu, S. Wang, W. Chen, G.-C. Guo, and Z.-F. Han, “Experimental three-party quantum random number generator based on dimension witness violation and weak measurement,” Opt. Lett. 43, 3437–3440 (2018).
    [Crossref]

2020 (2)

R. Uola, A. C. S. Costa, H. C. Nguyen, and O. Gühne, “Quantum steering,” Rev. Mod. Phys. 92, 015001 (2020).
[Crossref]

G. Foletto, L. Calderaro, A. Tavakoli, M. Schiavon, F. Picciariello, A. Cabello, P. Villoresi, and G. Vallone, “Experimental certification of sustained entanglement and nonlocality after sequential measurements,” Phys. Rev. Appl. 13, 044008 (2020).
[Crossref]

2019 (3)

Y.-W. Cho, Y. Kim, Y.-H. Choi, Y.-S. Kim, S.-W. Han, S.-Y. Lee, S. Moon, and Y.-H. Kim, “Emergence of the geometric phase from quantum measurement back-action,” Nat. Phys. 15, 665–670 (2019).
[Crossref]

D. Das, A. Ghosal, S. Sasmal, S. Mal, and A. S. Majumdar, “Facets of bipartite nonlocality sharing by multiple observers via sequential measurements,” Phys. Rev. A 99, 022305 (2019).
[Crossref]

H. A. Shenoy, S. Designolle, F. Hirsch, R. Silva, N. Gisin, and N. Brunner, “Unbounded sequence of observers exhibiting Einstein-Podolsky-Rosen steering,” Phys. Rev. A 99, 022317 (2019).
[Crossref]

2018 (6)

K. Sun, X.-J. Ye, Y. Xiao, X.-Y. Xu, Y.-C. Wu, J.-S. Xu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Demonstration of Einstein-Podolsky-Rosen steering with enhanced subchannel discrimination,” npj Quantum Inf. 4, 12 (2018).
[Crossref]

S. Sasmal, D. Das, S. Mal, and A. S. Majumdar, “Steering a single system sequentially by multiple observers,” Phys. Rev. A 98, 012305 (2018).
[Crossref]

M.-J. Hu, Z.-Y. Zhou, X.-M. Hu, C.-F. Li, G.-C. Guo, and Y.-S. Zhang, “Observation of non-locality sharing among three observers with one entangled pair via optimal weak measurement,” npj Quantum Inf. 4, 63 (2018).
[Crossref]

Y. Kim, Y.-S. Kim, S.-Y. Lee, S.-W. Han, S. Moon, Y.-H. Kim, and Y.-W. Cho, “Direct quantum process tomography via measuring sequential weak values of incompatible observables,” Nat. Commun. 9, 192 (2018).
[Crossref]

H.-W. Li, Y.-S. Zhang, X.-B. An, Z.-F. Han, and G.-C. Guo, “Three-observer classical dimension witness violation with weak measurement,” Commun. Phys. 1, 10 (2018).
[Crossref]

X.-B. An, H.-W. Li, Z.-Q. Yin, M.-J. Hu, W. Huang, B.-J. Xu, S. Wang, W. Chen, G.-C. Guo, and Z.-F. Han, “Experimental three-party quantum random number generator based on dimension witness violation and weak measurement,” Opt. Lett. 43, 3437–3440 (2018).
[Crossref]

2017 (2)

F. J. Curchod, M. Johansson, R. Augusiak, M. J. Hoban, P. Wittek, and A. Acín, “Unbounded randomness certification using sequences of measurements,” Phys. Rev. A 95, 020102 (2017).
[Crossref]

M. Schiavon, L. Calderaro, M. Pittaluga, G. Vallone, and P. Villoresi, “Three-observer bell inequality violation on a two-qubit entangled state,” Quantum Sci. Technol. 2, 015010 (2017).
[Crossref]

2016 (2)

S. Mal, A. S. Majumdar, and D. Home, “Sharing of nonlocality of a single member of an entangled pair of qubits is not possible by more than two unbiased observers on the other wing,” Mathematics 4, 48 (2016).
[Crossref]

A. Acín and L. Masanes, “Certified randomness in quantum physics,” Nature 540, 213–219 (2016).
[Crossref]

2015 (4)

R. Silva, N. Gisin, Y. Guryanova, and S. Popescu, “Multiple observers can share the nonlocality of half of an entangled pair by using optimal weak measurements,” Phys. Rev. Lett. 114, 250401 (2015).
[Crossref]

R. Gallego and L. Aolita, “Resource theory of steering,” Phys. Rev. X 5, 041008 (2015).
[Crossref]

E. Passaro, D. Cavalcanti, P. Skrzypczyk, and A. Acín, “Optimal randomness certification in the quantum steering and prepare-and-measure scenarios,” New J. Phys. 17, 113010 (2015).
[Crossref]

B. K. Park, Y.-S. Kim, O. Kwon, S.-W. Han, and S. Moon, “High-performance reconfigurable coincidence counting unit based on a field programmable gate array,” Appl. Opt. 54, 4727–4731 (2015).
[Crossref]

2014 (2)

H.-T. Lim, Y.-S. Ra, K.-H. Hong, S.-W. Lee, and Y.-H. Kim, “Fundamental bounds in measurements for estimating quantum states,” Phys. Rev. Lett. 113, 020504 (2014).
[Crossref]

T. Pramanik, M. Kaplan, and A. S. Majumdar, “Fine-grained Einstein-Podolsky-Rosen–steering inequalities,” Phys. Rev. A 90, 050305 (2014).
[Crossref]

2013 (1)

Q. Y. He and M. D. Reid, “Genuine multipartite Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 111, 250403 (2013).
[Crossref]

2012 (2)

Y.-S. Kim, J.-C. Lee, O. Kwon, and Y.-H. Kim, “Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nat. Phys. 8, 117–120 (2012).
[Crossref]

C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, “One-sided device-independent quantum key distribution: security, feasibility, and the connection with steering,” Phys. Rev. A 85, 010301 (2012).
[Crossref]

2010 (2)

H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, “Nonlocality and communication complexity,” Rev. Mod. Phys. 82, 665–698 (2010).
[Crossref]

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845–849 (2010).
[Crossref]

2009 (3)

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein-Podolsky-Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[Crossref]

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

B. Toner, “Monogamy of non-local quantum correlations,” Proc. R. Soc. A 465, 59–69 (2009).
[Crossref]

2008 (1)

O. Kwon, Y.-W. Cho, and Y.-H. Kim, “Single-mode coupling efficiencies of type-ii spontaneous parametric down-conversion: collinear, noncollinear, and beamlike phase matching,” Phys. Rev. A 78, 053825 (2008).
[Crossref]

2000 (1)

V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A 61, 052306 (2000).
[Crossref]

1935 (1)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

A. Shenoy, H.

H. A. Shenoy, S. Designolle, F. Hirsch, R. Silva, N. Gisin, and N. Brunner, “Unbounded sequence of observers exhibiting Einstein-Podolsky-Rosen steering,” Phys. Rev. A 99, 022317 (2019).
[Crossref]

Acín, A.

F. J. Curchod, M. Johansson, R. Augusiak, M. J. Hoban, P. Wittek, and A. Acín, “Unbounded randomness certification using sequences of measurements,” Phys. Rev. A 95, 020102 (2017).
[Crossref]

A. Acín and L. Masanes, “Certified randomness in quantum physics,” Nature 540, 213–219 (2016).
[Crossref]

E. Passaro, D. Cavalcanti, P. Skrzypczyk, and A. Acín, “Optimal randomness certification in the quantum steering and prepare-and-measure scenarios,” New J. Phys. 17, 113010 (2015).
[Crossref]

An, X.-B.

Aolita, L.

R. Gallego and L. Aolita, “Resource theory of steering,” Phys. Rev. X 5, 041008 (2015).
[Crossref]

Augusiak, R.

F. J. Curchod, M. Johansson, R. Augusiak, M. J. Hoban, P. Wittek, and A. Acín, “Unbounded randomness certification using sequences of measurements,” Phys. Rev. A 95, 020102 (2017).
[Crossref]

Bechmann-Pasquinucci, H.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Branciard, C.

C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, “One-sided device-independent quantum key distribution: security, feasibility, and the connection with steering,” Phys. Rev. A 85, 010301 (2012).
[Crossref]

Brown, P. J.

P. J. Brown and R. Colbeck, “An unbounded number of independent observers can share the nonlocality of a single maximally entangled qubit pair,” arXiv:2003.12105v1 [quant-ph].

Brunner, N.

H. A. Shenoy, S. Designolle, F. Hirsch, R. Silva, N. Gisin, and N. Brunner, “Unbounded sequence of observers exhibiting Einstein-Podolsky-Rosen steering,” Phys. Rev. A 99, 022317 (2019).
[Crossref]

Buhrman, H.

H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, “Nonlocality and communication complexity,” Rev. Mod. Phys. 82, 665–698 (2010).
[Crossref]

Cabello, A.

G. Foletto, L. Calderaro, A. Tavakoli, M. Schiavon, F. Picciariello, A. Cabello, P. Villoresi, and G. Vallone, “Experimental certification of sustained entanglement and nonlocality after sequential measurements,” Phys. Rev. Appl. 13, 044008 (2020).
[Crossref]

Calderaro, L.

G. Foletto, L. Calderaro, A. Tavakoli, M. Schiavon, F. Picciariello, A. Cabello, P. Villoresi, and G. Vallone, “Experimental certification of sustained entanglement and nonlocality after sequential measurements,” Phys. Rev. Appl. 13, 044008 (2020).
[Crossref]

M. Schiavon, L. Calderaro, M. Pittaluga, G. Vallone, and P. Villoresi, “Three-observer bell inequality violation on a two-qubit entangled state,” Quantum Sci. Technol. 2, 015010 (2017).
[Crossref]

Cavalcanti, D.

E. Passaro, D. Cavalcanti, P. Skrzypczyk, and A. Acín, “Optimal randomness certification in the quantum steering and prepare-and-measure scenarios,” New J. Phys. 17, 113010 (2015).
[Crossref]

Cavalcanti, E. G.

C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, “One-sided device-independent quantum key distribution: security, feasibility, and the connection with steering,” Phys. Rev. A 85, 010301 (2012).
[Crossref]

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein-Podolsky-Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[Crossref]

Cerf, N. J.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Chen, J.

T. Feng, C. Ren, Y. Tian, M. Luo, H. Shi, J. Chen, and X. Zhou, “Observation of nonlocality sharing via “strong” weak measurements,” arXiv: 1912.02979 [quant-ph].

Chen, J.-L.

K. Sun, X.-J. Ye, Y. Xiao, X.-Y. Xu, Y.-C. Wu, J.-S. Xu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Demonstration of Einstein-Podolsky-Rosen steering with enhanced subchannel discrimination,” npj Quantum Inf. 4, 12 (2018).
[Crossref]

Chen, W.

Cho, Y.-W.

Y.-W. Cho, Y. Kim, Y.-H. Choi, Y.-S. Kim, S.-W. Han, S.-Y. Lee, S. Moon, and Y.-H. Kim, “Emergence of the geometric phase from quantum measurement back-action,” Nat. Phys. 15, 665–670 (2019).
[Crossref]

Y. Kim, Y.-S. Kim, S.-Y. Lee, S.-W. Han, S. Moon, Y.-H. Kim, and Y.-W. Cho, “Direct quantum process tomography via measuring sequential weak values of incompatible observables,” Nat. Commun. 9, 192 (2018).
[Crossref]

O. Kwon, Y.-W. Cho, and Y.-H. Kim, “Single-mode coupling efficiencies of type-ii spontaneous parametric down-conversion: collinear, noncollinear, and beamlike phase matching,” Phys. Rev. A 78, 053825 (2008).
[Crossref]

Choi, Y.-H.

Y.-W. Cho, Y. Kim, Y.-H. Choi, Y.-S. Kim, S.-W. Han, S.-Y. Lee, S. Moon, and Y.-H. Kim, “Emergence of the geometric phase from quantum measurement back-action,” Nat. Phys. 15, 665–670 (2019).
[Crossref]

Cleve, R.

H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, “Nonlocality and communication complexity,” Rev. Mod. Phys. 82, 665–698 (2010).
[Crossref]

Coffman, V.

V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A 61, 052306 (2000).
[Crossref]

Colbeck, R.

P. J. Brown and R. Colbeck, “An unbounded number of independent observers can share the nonlocality of a single maximally entangled qubit pair,” arXiv:2003.12105v1 [quant-ph].

Costa, A. C. S.

R. Uola, A. C. S. Costa, H. C. Nguyen, and O. Gühne, “Quantum steering,” Rev. Mod. Phys. 92, 015001 (2020).
[Crossref]

Curchod, F. J.

F. J. Curchod, M. Johansson, R. Augusiak, M. J. Hoban, P. Wittek, and A. Acín, “Unbounded randomness certification using sequences of measurements,” Phys. Rev. A 95, 020102 (2017).
[Crossref]

Das, D.

D. Das, A. Ghosal, S. Sasmal, S. Mal, and A. S. Majumdar, “Facets of bipartite nonlocality sharing by multiple observers via sequential measurements,” Phys. Rev. A 99, 022305 (2019).
[Crossref]

S. Sasmal, D. Das, S. Mal, and A. S. Majumdar, “Steering a single system sequentially by multiple observers,” Phys. Rev. A 98, 012305 (2018).
[Crossref]

de Wolf, R.

H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, “Nonlocality and communication complexity,” Rev. Mod. Phys. 82, 665–698 (2010).
[Crossref]

Designolle, S.

H. A. Shenoy, S. Designolle, F. Hirsch, R. Silva, N. Gisin, and N. Brunner, “Unbounded sequence of observers exhibiting Einstein-Podolsky-Rosen steering,” Phys. Rev. A 99, 022317 (2019).
[Crossref]

Dušek, M.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Einstein, A.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

Feng, T.

T. Feng, C. Ren, Y. Tian, M. Luo, H. Shi, J. Chen, and X. Zhou, “Observation of nonlocality sharing via “strong” weak measurements,” arXiv: 1912.02979 [quant-ph].

Foletto, G.

G. Foletto, L. Calderaro, A. Tavakoli, M. Schiavon, F. Picciariello, A. Cabello, P. Villoresi, and G. Vallone, “Experimental certification of sustained entanglement and nonlocality after sequential measurements,” Phys. Rev. Appl. 13, 044008 (2020).
[Crossref]

Gallego, R.

R. Gallego and L. Aolita, “Resource theory of steering,” Phys. Rev. X 5, 041008 (2015).
[Crossref]

Ghosal, A.

D. Das, A. Ghosal, S. Sasmal, S. Mal, and A. S. Majumdar, “Facets of bipartite nonlocality sharing by multiple observers via sequential measurements,” Phys. Rev. A 99, 022305 (2019).
[Crossref]

Gisin, N.

H. A. Shenoy, S. Designolle, F. Hirsch, R. Silva, N. Gisin, and N. Brunner, “Unbounded sequence of observers exhibiting Einstein-Podolsky-Rosen steering,” Phys. Rev. A 99, 022317 (2019).
[Crossref]

R. Silva, N. Gisin, Y. Guryanova, and S. Popescu, “Multiple observers can share the nonlocality of half of an entangled pair by using optimal weak measurements,” Phys. Rev. Lett. 114, 250401 (2015).
[Crossref]

Gühne, O.

R. Uola, A. C. S. Costa, H. C. Nguyen, and O. Gühne, “Quantum steering,” Rev. Mod. Phys. 92, 015001 (2020).
[Crossref]

Guo, G.-C.

M.-J. Hu, Z.-Y. Zhou, X.-M. Hu, C.-F. Li, G.-C. Guo, and Y.-S. Zhang, “Observation of non-locality sharing among three observers with one entangled pair via optimal weak measurement,” npj Quantum Inf. 4, 63 (2018).
[Crossref]

K. Sun, X.-J. Ye, Y. Xiao, X.-Y. Xu, Y.-C. Wu, J.-S. Xu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Demonstration of Einstein-Podolsky-Rosen steering with enhanced subchannel discrimination,” npj Quantum Inf. 4, 12 (2018).
[Crossref]

X.-B. An, H.-W. Li, Z.-Q. Yin, M.-J. Hu, W. Huang, B.-J. Xu, S. Wang, W. Chen, G.-C. Guo, and Z.-F. Han, “Experimental three-party quantum random number generator based on dimension witness violation and weak measurement,” Opt. Lett. 43, 3437–3440 (2018).
[Crossref]

H.-W. Li, Y.-S. Zhang, X.-B. An, Z.-F. Han, and G.-C. Guo, “Three-observer classical dimension witness violation with weak measurement,” Commun. Phys. 1, 10 (2018).
[Crossref]

Guryanova, Y.

R. Silva, N. Gisin, Y. Guryanova, and S. Popescu, “Multiple observers can share the nonlocality of half of an entangled pair by using optimal weak measurements,” Phys. Rev. Lett. 114, 250401 (2015).
[Crossref]

Han, S.-W.

Y.-W. Cho, Y. Kim, Y.-H. Choi, Y.-S. Kim, S.-W. Han, S.-Y. Lee, S. Moon, and Y.-H. Kim, “Emergence of the geometric phase from quantum measurement back-action,” Nat. Phys. 15, 665–670 (2019).
[Crossref]

Y. Kim, Y.-S. Kim, S.-Y. Lee, S.-W. Han, S. Moon, Y.-H. Kim, and Y.-W. Cho, “Direct quantum process tomography via measuring sequential weak values of incompatible observables,” Nat. Commun. 9, 192 (2018).
[Crossref]

B. K. Park, Y.-S. Kim, O. Kwon, S.-W. Han, and S. Moon, “High-performance reconfigurable coincidence counting unit based on a field programmable gate array,” Appl. Opt. 54, 4727–4731 (2015).
[Crossref]

Han, Z.-F.

He, Q. Y.

Q. Y. He and M. D. Reid, “Genuine multipartite Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 111, 250403 (2013).
[Crossref]

Hirsch, F.

H. A. Shenoy, S. Designolle, F. Hirsch, R. Silva, N. Gisin, and N. Brunner, “Unbounded sequence of observers exhibiting Einstein-Podolsky-Rosen steering,” Phys. Rev. A 99, 022317 (2019).
[Crossref]

Hoban, M. J.

F. J. Curchod, M. Johansson, R. Augusiak, M. J. Hoban, P. Wittek, and A. Acín, “Unbounded randomness certification using sequences of measurements,” Phys. Rev. A 95, 020102 (2017).
[Crossref]

Home, D.

S. Mal, A. S. Majumdar, and D. Home, “Sharing of nonlocality of a single member of an entangled pair of qubits is not possible by more than two unbiased observers on the other wing,” Mathematics 4, 48 (2016).
[Crossref]

Hong, K.-H.

H.-T. Lim, Y.-S. Ra, K.-H. Hong, S.-W. Lee, and Y.-H. Kim, “Fundamental bounds in measurements for estimating quantum states,” Phys. Rev. Lett. 113, 020504 (2014).
[Crossref]

Hu, M.-J.

M.-J. Hu, Z.-Y. Zhou, X.-M. Hu, C.-F. Li, G.-C. Guo, and Y.-S. Zhang, “Observation of non-locality sharing among three observers with one entangled pair via optimal weak measurement,” npj Quantum Inf. 4, 63 (2018).
[Crossref]

X.-B. An, H.-W. Li, Z.-Q. Yin, M.-J. Hu, W. Huang, B.-J. Xu, S. Wang, W. Chen, G.-C. Guo, and Z.-F. Han, “Experimental three-party quantum random number generator based on dimension witness violation and weak measurement,” Opt. Lett. 43, 3437–3440 (2018).
[Crossref]

Hu, X.-M.

M.-J. Hu, Z.-Y. Zhou, X.-M. Hu, C.-F. Li, G.-C. Guo, and Y.-S. Zhang, “Observation of non-locality sharing among three observers with one entangled pair via optimal weak measurement,” npj Quantum Inf. 4, 63 (2018).
[Crossref]

Huang, W.

Johansson, M.

F. J. Curchod, M. Johansson, R. Augusiak, M. J. Hoban, P. Wittek, and A. Acín, “Unbounded randomness certification using sequences of measurements,” Phys. Rev. A 95, 020102 (2017).
[Crossref]

Jones, S. J.

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845–849 (2010).
[Crossref]

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein-Podolsky-Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[Crossref]

Kaplan, M.

T. Pramanik, M. Kaplan, and A. S. Majumdar, “Fine-grained Einstein-Podolsky-Rosen–steering inequalities,” Phys. Rev. A 90, 050305 (2014).
[Crossref]

Kim, Y.

Y.-W. Cho, Y. Kim, Y.-H. Choi, Y.-S. Kim, S.-W. Han, S.-Y. Lee, S. Moon, and Y.-H. Kim, “Emergence of the geometric phase from quantum measurement back-action,” Nat. Phys. 15, 665–670 (2019).
[Crossref]

Y. Kim, Y.-S. Kim, S.-Y. Lee, S.-W. Han, S. Moon, Y.-H. Kim, and Y.-W. Cho, “Direct quantum process tomography via measuring sequential weak values of incompatible observables,” Nat. Commun. 9, 192 (2018).
[Crossref]

Kim, Y.-H.

Y.-W. Cho, Y. Kim, Y.-H. Choi, Y.-S. Kim, S.-W. Han, S.-Y. Lee, S. Moon, and Y.-H. Kim, “Emergence of the geometric phase from quantum measurement back-action,” Nat. Phys. 15, 665–670 (2019).
[Crossref]

Y. Kim, Y.-S. Kim, S.-Y. Lee, S.-W. Han, S. Moon, Y.-H. Kim, and Y.-W. Cho, “Direct quantum process tomography via measuring sequential weak values of incompatible observables,” Nat. Commun. 9, 192 (2018).
[Crossref]

H.-T. Lim, Y.-S. Ra, K.-H. Hong, S.-W. Lee, and Y.-H. Kim, “Fundamental bounds in measurements for estimating quantum states,” Phys. Rev. Lett. 113, 020504 (2014).
[Crossref]

Y.-S. Kim, J.-C. Lee, O. Kwon, and Y.-H. Kim, “Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nat. Phys. 8, 117–120 (2012).
[Crossref]

O. Kwon, Y.-W. Cho, and Y.-H. Kim, “Single-mode coupling efficiencies of type-ii spontaneous parametric down-conversion: collinear, noncollinear, and beamlike phase matching,” Phys. Rev. A 78, 053825 (2008).
[Crossref]

Kim, Y.-S.

Y.-W. Cho, Y. Kim, Y.-H. Choi, Y.-S. Kim, S.-W. Han, S.-Y. Lee, S. Moon, and Y.-H. Kim, “Emergence of the geometric phase from quantum measurement back-action,” Nat. Phys. 15, 665–670 (2019).
[Crossref]

Y. Kim, Y.-S. Kim, S.-Y. Lee, S.-W. Han, S. Moon, Y.-H. Kim, and Y.-W. Cho, “Direct quantum process tomography via measuring sequential weak values of incompatible observables,” Nat. Commun. 9, 192 (2018).
[Crossref]

B. K. Park, Y.-S. Kim, O. Kwon, S.-W. Han, and S. Moon, “High-performance reconfigurable coincidence counting unit based on a field programmable gate array,” Appl. Opt. 54, 4727–4731 (2015).
[Crossref]

Y.-S. Kim, J.-C. Lee, O. Kwon, and Y.-H. Kim, “Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nat. Phys. 8, 117–120 (2012).
[Crossref]

Kundu, J.

V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A 61, 052306 (2000).
[Crossref]

Kwon, O.

B. K. Park, Y.-S. Kim, O. Kwon, S.-W. Han, and S. Moon, “High-performance reconfigurable coincidence counting unit based on a field programmable gate array,” Appl. Opt. 54, 4727–4731 (2015).
[Crossref]

Y.-S. Kim, J.-C. Lee, O. Kwon, and Y.-H. Kim, “Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nat. Phys. 8, 117–120 (2012).
[Crossref]

O. Kwon, Y.-W. Cho, and Y.-H. Kim, “Single-mode coupling efficiencies of type-ii spontaneous parametric down-conversion: collinear, noncollinear, and beamlike phase matching,” Phys. Rev. A 78, 053825 (2008).
[Crossref]

Lee, J.-C.

Y.-S. Kim, J.-C. Lee, O. Kwon, and Y.-H. Kim, “Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nat. Phys. 8, 117–120 (2012).
[Crossref]

Lee, S.-W.

H.-T. Lim, Y.-S. Ra, K.-H. Hong, S.-W. Lee, and Y.-H. Kim, “Fundamental bounds in measurements for estimating quantum states,” Phys. Rev. Lett. 113, 020504 (2014).
[Crossref]

Lee, S.-Y.

Y.-W. Cho, Y. Kim, Y.-H. Choi, Y.-S. Kim, S.-W. Han, S.-Y. Lee, S. Moon, and Y.-H. Kim, “Emergence of the geometric phase from quantum measurement back-action,” Nat. Phys. 15, 665–670 (2019).
[Crossref]

Y. Kim, Y.-S. Kim, S.-Y. Lee, S.-W. Han, S. Moon, Y.-H. Kim, and Y.-W. Cho, “Direct quantum process tomography via measuring sequential weak values of incompatible observables,” Nat. Commun. 9, 192 (2018).
[Crossref]

Li, C.-F.

K. Sun, X.-J. Ye, Y. Xiao, X.-Y. Xu, Y.-C. Wu, J.-S. Xu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Demonstration of Einstein-Podolsky-Rosen steering with enhanced subchannel discrimination,” npj Quantum Inf. 4, 12 (2018).
[Crossref]

M.-J. Hu, Z.-Y. Zhou, X.-M. Hu, C.-F. Li, G.-C. Guo, and Y.-S. Zhang, “Observation of non-locality sharing among three observers with one entangled pair via optimal weak measurement,” npj Quantum Inf. 4, 63 (2018).
[Crossref]

Li, H.-W.

Lim, H.-T.

H.-T. Lim, Y.-S. Ra, K.-H. Hong, S.-W. Lee, and Y.-H. Kim, “Fundamental bounds in measurements for estimating quantum states,” Phys. Rev. Lett. 113, 020504 (2014).
[Crossref]

Luo, M.

T. Feng, C. Ren, Y. Tian, M. Luo, H. Shi, J. Chen, and X. Zhou, “Observation of nonlocality sharing via “strong” weak measurements,” arXiv: 1912.02979 [quant-ph].

Lütkenhaus, N.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Majumdar, A. S.

D. Das, A. Ghosal, S. Sasmal, S. Mal, and A. S. Majumdar, “Facets of bipartite nonlocality sharing by multiple observers via sequential measurements,” Phys. Rev. A 99, 022305 (2019).
[Crossref]

S. Sasmal, D. Das, S. Mal, and A. S. Majumdar, “Steering a single system sequentially by multiple observers,” Phys. Rev. A 98, 012305 (2018).
[Crossref]

S. Mal, A. S. Majumdar, and D. Home, “Sharing of nonlocality of a single member of an entangled pair of qubits is not possible by more than two unbiased observers on the other wing,” Mathematics 4, 48 (2016).
[Crossref]

T. Pramanik, M. Kaplan, and A. S. Majumdar, “Fine-grained Einstein-Podolsky-Rosen–steering inequalities,” Phys. Rev. A 90, 050305 (2014).
[Crossref]

Mal, S.

D. Das, A. Ghosal, S. Sasmal, S. Mal, and A. S. Majumdar, “Facets of bipartite nonlocality sharing by multiple observers via sequential measurements,” Phys. Rev. A 99, 022305 (2019).
[Crossref]

S. Sasmal, D. Das, S. Mal, and A. S. Majumdar, “Steering a single system sequentially by multiple observers,” Phys. Rev. A 98, 012305 (2018).
[Crossref]

S. Mal, A. S. Majumdar, and D. Home, “Sharing of nonlocality of a single member of an entangled pair of qubits is not possible by more than two unbiased observers on the other wing,” Mathematics 4, 48 (2016).
[Crossref]

Masanes, L.

A. Acín and L. Masanes, “Certified randomness in quantum physics,” Nature 540, 213–219 (2016).
[Crossref]

Massar, S.

H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, “Nonlocality and communication complexity,” Rev. Mod. Phys. 82, 665–698 (2010).
[Crossref]

Moon, S.

Y.-W. Cho, Y. Kim, Y.-H. Choi, Y.-S. Kim, S.-W. Han, S.-Y. Lee, S. Moon, and Y.-H. Kim, “Emergence of the geometric phase from quantum measurement back-action,” Nat. Phys. 15, 665–670 (2019).
[Crossref]

Y. Kim, Y.-S. Kim, S.-Y. Lee, S.-W. Han, S. Moon, Y.-H. Kim, and Y.-W. Cho, “Direct quantum process tomography via measuring sequential weak values of incompatible observables,” Nat. Commun. 9, 192 (2018).
[Crossref]

B. K. Park, Y.-S. Kim, O. Kwon, S.-W. Han, and S. Moon, “High-performance reconfigurable coincidence counting unit based on a field programmable gate array,” Appl. Opt. 54, 4727–4731 (2015).
[Crossref]

Nguyen, H. C.

R. Uola, A. C. S. Costa, H. C. Nguyen, and O. Gühne, “Quantum steering,” Rev. Mod. Phys. 92, 015001 (2020).
[Crossref]

Park, B. K.

Passaro, E.

E. Passaro, D. Cavalcanti, P. Skrzypczyk, and A. Acín, “Optimal randomness certification in the quantum steering and prepare-and-measure scenarios,” New J. Phys. 17, 113010 (2015).
[Crossref]

Peev, M.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Picciariello, F.

G. Foletto, L. Calderaro, A. Tavakoli, M. Schiavon, F. Picciariello, A. Cabello, P. Villoresi, and G. Vallone, “Experimental certification of sustained entanglement and nonlocality after sequential measurements,” Phys. Rev. Appl. 13, 044008 (2020).
[Crossref]

Pittaluga, M.

M. Schiavon, L. Calderaro, M. Pittaluga, G. Vallone, and P. Villoresi, “Three-observer bell inequality violation on a two-qubit entangled state,” Quantum Sci. Technol. 2, 015010 (2017).
[Crossref]

Podolsky, B.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

Popescu, S.

R. Silva, N. Gisin, Y. Guryanova, and S. Popescu, “Multiple observers can share the nonlocality of half of an entangled pair by using optimal weak measurements,” Phys. Rev. Lett. 114, 250401 (2015).
[Crossref]

Pramanik, T.

T. Pramanik, M. Kaplan, and A. S. Majumdar, “Fine-grained Einstein-Podolsky-Rosen–steering inequalities,” Phys. Rev. A 90, 050305 (2014).
[Crossref]

Pryde, G. J.

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845–849 (2010).
[Crossref]

Ra, Y.-S.

H.-T. Lim, Y.-S. Ra, K.-H. Hong, S.-W. Lee, and Y.-H. Kim, “Fundamental bounds in measurements for estimating quantum states,” Phys. Rev. Lett. 113, 020504 (2014).
[Crossref]

Reid, M. D.

Q. Y. He and M. D. Reid, “Genuine multipartite Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 111, 250403 (2013).
[Crossref]

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein-Podolsky-Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[Crossref]

Ren, C.

T. Feng, C. Ren, Y. Tian, M. Luo, H. Shi, J. Chen, and X. Zhou, “Observation of nonlocality sharing via “strong” weak measurements,” arXiv: 1912.02979 [quant-ph].

Rosen, N.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

Sasmal, S.

D. Das, A. Ghosal, S. Sasmal, S. Mal, and A. S. Majumdar, “Facets of bipartite nonlocality sharing by multiple observers via sequential measurements,” Phys. Rev. A 99, 022305 (2019).
[Crossref]

S. Sasmal, D. Das, S. Mal, and A. S. Majumdar, “Steering a single system sequentially by multiple observers,” Phys. Rev. A 98, 012305 (2018).
[Crossref]

Saunders, D. J.

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845–849 (2010).
[Crossref]

Scarani, V.

C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, “One-sided device-independent quantum key distribution: security, feasibility, and the connection with steering,” Phys. Rev. A 85, 010301 (2012).
[Crossref]

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Schiavon, M.

G. Foletto, L. Calderaro, A. Tavakoli, M. Schiavon, F. Picciariello, A. Cabello, P. Villoresi, and G. Vallone, “Experimental certification of sustained entanglement and nonlocality after sequential measurements,” Phys. Rev. Appl. 13, 044008 (2020).
[Crossref]

M. Schiavon, L. Calderaro, M. Pittaluga, G. Vallone, and P. Villoresi, “Three-observer bell inequality violation on a two-qubit entangled state,” Quantum Sci. Technol. 2, 015010 (2017).
[Crossref]

Shi, H.

T. Feng, C. Ren, Y. Tian, M. Luo, H. Shi, J. Chen, and X. Zhou, “Observation of nonlocality sharing via “strong” weak measurements,” arXiv: 1912.02979 [quant-ph].

Silva, R.

H. A. Shenoy, S. Designolle, F. Hirsch, R. Silva, N. Gisin, and N. Brunner, “Unbounded sequence of observers exhibiting Einstein-Podolsky-Rosen steering,” Phys. Rev. A 99, 022317 (2019).
[Crossref]

R. Silva, N. Gisin, Y. Guryanova, and S. Popescu, “Multiple observers can share the nonlocality of half of an entangled pair by using optimal weak measurements,” Phys. Rev. Lett. 114, 250401 (2015).
[Crossref]

Skrzypczyk, P.

E. Passaro, D. Cavalcanti, P. Skrzypczyk, and A. Acín, “Optimal randomness certification in the quantum steering and prepare-and-measure scenarios,” New J. Phys. 17, 113010 (2015).
[Crossref]

Sun, K.

K. Sun, X.-J. Ye, Y. Xiao, X.-Y. Xu, Y.-C. Wu, J.-S. Xu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Demonstration of Einstein-Podolsky-Rosen steering with enhanced subchannel discrimination,” npj Quantum Inf. 4, 12 (2018).
[Crossref]

Tavakoli, A.

G. Foletto, L. Calderaro, A. Tavakoli, M. Schiavon, F. Picciariello, A. Cabello, P. Villoresi, and G. Vallone, “Experimental certification of sustained entanglement and nonlocality after sequential measurements,” Phys. Rev. Appl. 13, 044008 (2020).
[Crossref]

Tian, Y.

T. Feng, C. Ren, Y. Tian, M. Luo, H. Shi, J. Chen, and X. Zhou, “Observation of nonlocality sharing via “strong” weak measurements,” arXiv: 1912.02979 [quant-ph].

Toner, B.

B. Toner, “Monogamy of non-local quantum correlations,” Proc. R. Soc. A 465, 59–69 (2009).
[Crossref]

Uola, R.

R. Uola, A. C. S. Costa, H. C. Nguyen, and O. Gühne, “Quantum steering,” Rev. Mod. Phys. 92, 015001 (2020).
[Crossref]

Vallone, G.

G. Foletto, L. Calderaro, A. Tavakoli, M. Schiavon, F. Picciariello, A. Cabello, P. Villoresi, and G. Vallone, “Experimental certification of sustained entanglement and nonlocality after sequential measurements,” Phys. Rev. Appl. 13, 044008 (2020).
[Crossref]

M. Schiavon, L. Calderaro, M. Pittaluga, G. Vallone, and P. Villoresi, “Three-observer bell inequality violation on a two-qubit entangled state,” Quantum Sci. Technol. 2, 015010 (2017).
[Crossref]

Villoresi, P.

G. Foletto, L. Calderaro, A. Tavakoli, M. Schiavon, F. Picciariello, A. Cabello, P. Villoresi, and G. Vallone, “Experimental certification of sustained entanglement and nonlocality after sequential measurements,” Phys. Rev. Appl. 13, 044008 (2020).
[Crossref]

M. Schiavon, L. Calderaro, M. Pittaluga, G. Vallone, and P. Villoresi, “Three-observer bell inequality violation on a two-qubit entangled state,” Quantum Sci. Technol. 2, 015010 (2017).
[Crossref]

Walborn, S. P.

C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, “One-sided device-independent quantum key distribution: security, feasibility, and the connection with steering,” Phys. Rev. A 85, 010301 (2012).
[Crossref]

Wang, S.

Wiseman, H. M.

C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, “One-sided device-independent quantum key distribution: security, feasibility, and the connection with steering,” Phys. Rev. A 85, 010301 (2012).
[Crossref]

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845–849 (2010).
[Crossref]

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein-Podolsky-Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[Crossref]

Wittek, P.

F. J. Curchod, M. Johansson, R. Augusiak, M. J. Hoban, P. Wittek, and A. Acín, “Unbounded randomness certification using sequences of measurements,” Phys. Rev. A 95, 020102 (2017).
[Crossref]

Wootters, W. K.

V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A 61, 052306 (2000).
[Crossref]

Wu, Y.-C.

K. Sun, X.-J. Ye, Y. Xiao, X.-Y. Xu, Y.-C. Wu, J.-S. Xu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Demonstration of Einstein-Podolsky-Rosen steering with enhanced subchannel discrimination,” npj Quantum Inf. 4, 12 (2018).
[Crossref]

Xiao, Y.

K. Sun, X.-J. Ye, Y. Xiao, X.-Y. Xu, Y.-C. Wu, J.-S. Xu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Demonstration of Einstein-Podolsky-Rosen steering with enhanced subchannel discrimination,” npj Quantum Inf. 4, 12 (2018).
[Crossref]

Xu, B.-J.

Xu, J.-S.

K. Sun, X.-J. Ye, Y. Xiao, X.-Y. Xu, Y.-C. Wu, J.-S. Xu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Demonstration of Einstein-Podolsky-Rosen steering with enhanced subchannel discrimination,” npj Quantum Inf. 4, 12 (2018).
[Crossref]

Xu, X.-Y.

K. Sun, X.-J. Ye, Y. Xiao, X.-Y. Xu, Y.-C. Wu, J.-S. Xu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Demonstration of Einstein-Podolsky-Rosen steering with enhanced subchannel discrimination,” npj Quantum Inf. 4, 12 (2018).
[Crossref]

Ye, X.-J.

K. Sun, X.-J. Ye, Y. Xiao, X.-Y. Xu, Y.-C. Wu, J.-S. Xu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Demonstration of Einstein-Podolsky-Rosen steering with enhanced subchannel discrimination,” npj Quantum Inf. 4, 12 (2018).
[Crossref]

Yin, Z.-Q.

Zhang, Y.-S.

M.-J. Hu, Z.-Y. Zhou, X.-M. Hu, C.-F. Li, G.-C. Guo, and Y.-S. Zhang, “Observation of non-locality sharing among three observers with one entangled pair via optimal weak measurement,” npj Quantum Inf. 4, 63 (2018).
[Crossref]

H.-W. Li, Y.-S. Zhang, X.-B. An, Z.-F. Han, and G.-C. Guo, “Three-observer classical dimension witness violation with weak measurement,” Commun. Phys. 1, 10 (2018).
[Crossref]

Zhou, X.

T. Feng, C. Ren, Y. Tian, M. Luo, H. Shi, J. Chen, and X. Zhou, “Observation of nonlocality sharing via “strong” weak measurements,” arXiv: 1912.02979 [quant-ph].

Zhou, Z.-Y.

M.-J. Hu, Z.-Y. Zhou, X.-M. Hu, C.-F. Li, G.-C. Guo, and Y.-S. Zhang, “Observation of non-locality sharing among three observers with one entangled pair via optimal weak measurement,” npj Quantum Inf. 4, 63 (2018).
[Crossref]

Appl. Opt. (1)

Commun. Phys. (1)

H.-W. Li, Y.-S. Zhang, X.-B. An, Z.-F. Han, and G.-C. Guo, “Three-observer classical dimension witness violation with weak measurement,” Commun. Phys. 1, 10 (2018).
[Crossref]

Mathematics (1)

S. Mal, A. S. Majumdar, and D. Home, “Sharing of nonlocality of a single member of an entangled pair of qubits is not possible by more than two unbiased observers on the other wing,” Mathematics 4, 48 (2016).
[Crossref]

Nat. Commun. (1)

Y. Kim, Y.-S. Kim, S.-Y. Lee, S.-W. Han, S. Moon, Y.-H. Kim, and Y.-W. Cho, “Direct quantum process tomography via measuring sequential weak values of incompatible observables,” Nat. Commun. 9, 192 (2018).
[Crossref]

Nat. Phys. (3)

Y.-S. Kim, J.-C. Lee, O. Kwon, and Y.-H. Kim, “Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nat. Phys. 8, 117–120 (2012).
[Crossref]

Y.-W. Cho, Y. Kim, Y.-H. Choi, Y.-S. Kim, S.-W. Han, S.-Y. Lee, S. Moon, and Y.-H. Kim, “Emergence of the geometric phase from quantum measurement back-action,” Nat. Phys. 15, 665–670 (2019).
[Crossref]

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845–849 (2010).
[Crossref]

Nature (1)

A. Acín and L. Masanes, “Certified randomness in quantum physics,” Nature 540, 213–219 (2016).
[Crossref]

New J. Phys. (1)

E. Passaro, D. Cavalcanti, P. Skrzypczyk, and A. Acín, “Optimal randomness certification in the quantum steering and prepare-and-measure scenarios,” New J. Phys. 17, 113010 (2015).
[Crossref]

npj Quantum Inf. (2)

K. Sun, X.-J. Ye, Y. Xiao, X.-Y. Xu, Y.-C. Wu, J.-S. Xu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Demonstration of Einstein-Podolsky-Rosen steering with enhanced subchannel discrimination,” npj Quantum Inf. 4, 12 (2018).
[Crossref]

M.-J. Hu, Z.-Y. Zhou, X.-M. Hu, C.-F. Li, G.-C. Guo, and Y.-S. Zhang, “Observation of non-locality sharing among three observers with one entangled pair via optimal weak measurement,” npj Quantum Inf. 4, 63 (2018).
[Crossref]

Opt. Lett. (1)

Phys. Rev. (1)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

Phys. Rev. A (9)

D. Das, A. Ghosal, S. Sasmal, S. Mal, and A. S. Majumdar, “Facets of bipartite nonlocality sharing by multiple observers via sequential measurements,” Phys. Rev. A 99, 022305 (2019).
[Crossref]

S. Sasmal, D. Das, S. Mal, and A. S. Majumdar, “Steering a single system sequentially by multiple observers,” Phys. Rev. A 98, 012305 (2018).
[Crossref]

V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A 61, 052306 (2000).
[Crossref]

T. Pramanik, M. Kaplan, and A. S. Majumdar, “Fine-grained Einstein-Podolsky-Rosen–steering inequalities,” Phys. Rev. A 90, 050305 (2014).
[Crossref]

H. A. Shenoy, S. Designolle, F. Hirsch, R. Silva, N. Gisin, and N. Brunner, “Unbounded sequence of observers exhibiting Einstein-Podolsky-Rosen steering,” Phys. Rev. A 99, 022317 (2019).
[Crossref]

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein-Podolsky-Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[Crossref]

O. Kwon, Y.-W. Cho, and Y.-H. Kim, “Single-mode coupling efficiencies of type-ii spontaneous parametric down-conversion: collinear, noncollinear, and beamlike phase matching,” Phys. Rev. A 78, 053825 (2008).
[Crossref]

F. J. Curchod, M. Johansson, R. Augusiak, M. J. Hoban, P. Wittek, and A. Acín, “Unbounded randomness certification using sequences of measurements,” Phys. Rev. A 95, 020102 (2017).
[Crossref]

C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, “One-sided device-independent quantum key distribution: security, feasibility, and the connection with steering,” Phys. Rev. A 85, 010301 (2012).
[Crossref]

Phys. Rev. Appl. (1)

G. Foletto, L. Calderaro, A. Tavakoli, M. Schiavon, F. Picciariello, A. Cabello, P. Villoresi, and G. Vallone, “Experimental certification of sustained entanglement and nonlocality after sequential measurements,” Phys. Rev. Appl. 13, 044008 (2020).
[Crossref]

Phys. Rev. Lett. (3)

Q. Y. He and M. D. Reid, “Genuine multipartite Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 111, 250403 (2013).
[Crossref]

H.-T. Lim, Y.-S. Ra, K.-H. Hong, S.-W. Lee, and Y.-H. Kim, “Fundamental bounds in measurements for estimating quantum states,” Phys. Rev. Lett. 113, 020504 (2014).
[Crossref]

R. Silva, N. Gisin, Y. Guryanova, and S. Popescu, “Multiple observers can share the nonlocality of half of an entangled pair by using optimal weak measurements,” Phys. Rev. Lett. 114, 250401 (2015).
[Crossref]

Phys. Rev. X (1)

R. Gallego and L. Aolita, “Resource theory of steering,” Phys. Rev. X 5, 041008 (2015).
[Crossref]

Proc. R. Soc. A (1)

B. Toner, “Monogamy of non-local quantum correlations,” Proc. R. Soc. A 465, 59–69 (2009).
[Crossref]

Quantum Sci. Technol. (1)

M. Schiavon, L. Calderaro, M. Pittaluga, G. Vallone, and P. Villoresi, “Three-observer bell inequality violation on a two-qubit entangled state,” Quantum Sci. Technol. 2, 015010 (2017).
[Crossref]

Rev. Mod. Phys. (3)

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, “Nonlocality and communication complexity,” Rev. Mod. Phys. 82, 665–698 (2010).
[Crossref]

R. Uola, A. C. S. Costa, H. C. Nguyen, and O. Gühne, “Quantum steering,” Rev. Mod. Phys. 92, 015001 (2020).
[Crossref]

Other (2)

P. J. Brown and R. Colbeck, “An unbounded number of independent observers can share the nonlocality of a single maximally entangled qubit pair,” arXiv:2003.12105v1 [quant-ph].

T. Feng, C. Ren, Y. Tian, M. Luo, H. Shi, J. Chen, and X. Zhou, “Observation of nonlocality sharing via “strong” weak measurements,” arXiv: 1912.02979 [quant-ph].

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Figures (3)

Fig. 1.
Fig. 1. Sharing EPR quantum steering scenario by multiple observers. A bipartite entangled pair is shared with Alice and multiple Bobs. In our scenario, three independent Bobs wish to claim that the quantum state given by Alice is steerable simultaneously. To verify the EPR steering correlation, each observer extracts dichotomic measurement outcomes {${a}$, ${b}$, ${c}$, ${d}$} for randomly chosen measurement directions {${\vec \alpha _j}$, ${\vec \beta _j}$, ${\vec \gamma _j}$, ${\vec \delta _j}$}. In particular, we consider three measurement settings, i.e., $j = \{1,2,3\}$.
Fig. 2.
Fig. 2. (a) Experimental setup. Polarization entangled photon pairs are prepared and distributed to Alice and Bobs. Alice performs the strong projective measurement with a set of waveplates and a polarizing beam splitter (PBS). Bob1 and Bob2 perform weak quantum measurement with the measurement strengths of ${\lambda _1}$ and ${\lambda _2}$, respectively. Finally, Bob3 also performs the projective measurement. BD, beam displacer; H, half-wave plate; ${Q}$, quarter-wave plate. (b) Schematic of the weak measurement with variable strength $\lambda$. See the main text for details.
Fig. 3.
Fig. 3. Demonstrating the triple violations of EPR quantum steering. (a) Steering parameters ${{\cal S}_i}$ are experimentally measured for various measurement strengths. Here, ${\lambda _1}$ and ${\lambda _2}$ denote the measurement strengths for Bob1 and for Bob2, respectively. Experimentally measured steering parameters are shown as markers, where the blue, purple, and orange circles correspond to ${{\cal S}_1}$, ${{\cal S}_2}$, and ${{\cal S}_3}$. Theoretical predictions are also represented as surface plots, and compared with the experimental data. (b) Region of triple violations represented with shaded color for clarity. (c) Steering parameters as a function of ${\lambda _1}$ for fixed ${\lambda _2} = 0.76$. (d) Steering parameters as a function of ${\lambda _2}$ for fixed ${\lambda _1} = 0.64$. For (c) and (d), shaded areas with sky-blue color indicate the violation of steering inequality. In pink areas, the triple violations of EPR steering are clearly demonstrated. The errors in ${\cal S}$ are obtained by performing 100 Monte Carlo simulation runs by taking into account of the Poisson photon counting statistics. The error bars are too small to be visible.

Tables (1)

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Table 1. Summary of Observed Steering Parameters at λ 1 = 0.64 and λ 2 = 0.76 , S i 1 (represents the quantum steering correlation)

Equations (6)

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S i = 1 3 | j = 1 3 A j B j i | 1 ,
M ^ ± | n = λ Π ^ ± | n + ( 1 λ ) I ^ 2 ,
K ^ ± | n = 1 2 ( 1 ± λ Π ^ + | n + 1 λ Π ^ | n ) .
p ( a , b , c , d | α k , β l , γ m , δ n ) ,
S 2 = 1 3 | j = 1 3 a , c a c p ( a , c | α j , γ j ) | ,
p ( a , c | α j , γ j ) = 1 9 l = 1 3 n = 1 3 b , d p ( a , b , c , d | α j , β l , γ j , δ n ) .

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