## 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 [2–4]. 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 [5–7]. Nonetheless, it has been shown that multiple observers can share quantum correlations by exploiting the weak quantum measurement [8–11].

While experimental demonstrations of sharing nonlocal correlations have been reported very recently, these experiments focused on sharing Bell nonlocality among three observers [12–14]. 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 [15–18]. 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 [24–27]. 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.

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

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

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

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.

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

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

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.

## 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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