Observation of non-locality sharing among three observers with one entangled pair via optimal weak measurement

Bell non-locality plays a fundamental role in quantum theory. Numerous tests of the Bell inequality have been reported as the ground-breaking discovery of the Bell theorem. Up to now, however, most discussions of the Bell scenario have focused on a single pair of entangled particles distributed to only two separated observers. Recently, it has been shown surprisingly that multiple observers can share the non-locality from an entangled pair using the method of weak measurement without post-selection [Phys. Rev. Lett. 114, 250401 (2015)]. Here we report an observation of double CHSH-Bell inequality violations for a single pair of entangled photons with strength continuous-tunable optimal weak measurement in a photonic system. Our results shed new light on the interplay between non-locality and quantum measurements and our design of weak measurement protocol may also be significant for important applications such as unbounded randomness certification and quantum steering. A study shows that the nonlocality of quantum entangled particles can be shared among three observers. Since John Bell’s 1964 theorem, which provides experimentally testable inequalities, Bell tests have been used to probe quantum theory. In a typical Bell test scenario, one pair of quantum entangled particles is distributed to two separate observers, usually referred to as Alice and Bob. A team of researchers led by Yong-Sheng Zhang from the University of Science and Technology of China now show that the nonlocality of an entangled pair can be shared with more observers. The team do this by introducing a second Bob to the Bell test, who accesses the same particle from the entangled pair as the first Bob via weak measurements, enabling them to explore the interplay between nonlocality and quantum measurements.


INTRODUCTION
Non-locality, which was pointed out by Einstein, Podolsky and Rosen (EPR), 1 plays a fundamental role in quantum theory. It has been intensively investigated as the ground-breaking discovery of Bell theorem by John Bell in 1964. 2 Bell theorem states that any local-realistic theory cannot reproduce all the predictions of quantum theory and gives an experimental testable inequality 3 that later improved by Clauser, Horne, Shimony and Holt (CHSH). 4 Numerous tests of CHSH-Bell inequality have been realized in various quantum systems [5][6][7][8][9][10][11][12][13] and strong loophole-free Bell tests have been reported recently. [14][15][16] To date, however, most discussions of Bell scenario focus on one pair of entangled particles distributed to only two separated observers Alice and Bob. 17 It is thus an important and fundamental question whether or not multiple observers can share the non-locality from an entangled pair. Using the concept of weak measurement without post-selection, Silva et al. give a surprising-positive answer to above question and show a marvellous physical fact that measurement disturbance and information gain of a single system are closely related to nonlocality distribution among multiple observers in one entangled pair. 18 In this article, we report an experimental realization of sharing non-locality among three observers with strength continuoustunable optimal weak measurement in a photonic system. We produce pairs of polarization-entangled photons in our experiment and send it to Alice and Bob1, Bob2 separately, in this case two Bobs access the same single particle from the entangled pairs with Bob1 performs weak measurement. The realization of sharing non-locality is certified by the observed double violations of CHSH-Bell inequality among Alice-Bob1 and Alice-Bob2. The reason behind this is that weak measurement performed by Bob1 can be strong enough to obtain quantum correlations between Alice and Bob1, and weak enough to retain quantum correlations between Alice and Bob2. Our results not only shed new light on the interplay between non-locality and quantum measurements but also could find significant applications such as in unbounded randomness certification 19,20 and quantum steering. 21,22

RESULTS
As one of the foundations of quantum theory, the measurement postulate states that upon measurement, a quantum system will collapse into one of its eigenstates, with the probability determined by the Born rule. Whereas this type of strong measurement, which is projective and irreversible, obtains the maximum information about a system, it also completely destroys the system after the measurement. Weak measurement, i.e., the coupling between the system and the probe is weak, however, can be used to extract less information about the system with less disturbance. It should be noted that this kind of weak disturbance measurement combined with post selection usually refers to weak measurement, 23 which has been shown to be a powerful method in signal amplification, 24-26 state tomography 27,28 and in solving quantum paradoxes 29 over the past decades. Hereafter, we follow the definition in ref. 18 where weak measurement just refers to the measurement with intermediate coupling strength between the system and the probe. In contrast to strong projective measurement, weak measurement is non-destructive and retains some original properties of the measured system, e.g., coherence and entanglement. Because the entanglement is not completely destroyed by weak measurement, a particle that has been measured with intermediate strength can still be entangled with other particles, and therefore, sharing non-locality among multiple observers is possible.

Modified Bell test with weak measurement
In a typical Bell test scenario, one pair of entangled spin-1/2 particles is distributed between two separated observers, Alice and Bob (Fig. 1a), who each receive a binary input x, y∈ {0, 1} and subsequently give a binary output a, b ∈ {1, −1}. For each input x (y), Alice (Bob) performs a strong projective measurement of her (his) spin along a specific direction and obtains the outcome a (b). The scenario is characterized by a joint probability distribution P (ab|xy) of obtaining outcomes a and b, conditioned on measurement inputs x for Alice and y for Bob. The fixed measurement inputs x and y defines the correlations C ðx;yÞ ¼ P a;b abPðabjxyÞ. The CHSH-Bell test is focused on the so-called S value defined by the combination of correlations Whereas S ≤ 2 in any local hidden variable theory, 4 quantum theory gives a more relaxed bound of 2 ffiffi ffi 2 p . 31 Here, we consider a new Bell scenario in which there are two observers Bob1 and Bob2 access to the same one-half of the entangled state of spin-1/2 particles (Fig. 1b). Alice, Bob1 and Bob2 each receive a binary input x, y 1 , y 2 ∈ {0,1} and subsequently provide a binary output a, b 1 , b 2 ∈ {1, −1}. For each input y 1 , Bob1 performs weak measurement of his spin along a specific direction, whereas Alice and Bob2 perform strong projective measurements for their input x and y 2 . With the outcome b 1 , Bob1 sends the measured spin particle to Bob2. The scenario is now characterized by joint conditional probabilities P(ab 1 b 2 |xy 1 y 2 ), and an incisive question is raised whether Bob1 and Bob2 can both share nonlocality with Alice. The answer is surprisingly positive that the statistics of both Alice-Bob1 and Alice-Bob2 can indeed violate the CHSH-Bell inequality simultaneously. 18 The quantities G and F of weak measurement, respectively, determine the S values of Alice-Bob1 and Alice-Bob2 in the new Bell scenario. In the case that the Tsirelson's bound 2 ffiffi ffi 2 p of the CHSH-Bell inequality can be attained, the calculation gives (see more details in Methods) Realization of optimal weak measurement in a photonic system To observe significant double violations of the CHSH-Bell inequality, the realization of optimal weak measurement is a key and necessary requirement. In the original scheme proposed in ref. 18 the spatial degree of freedom of particle is used as the pointer. However, the particle with common used spatial distributions, e.g. Gaussian distribution, only realizes sub-optimal weak measurement, i.e., F 2 + G 2 < 1. Here, we propose and realize optimal weak measurement in a photonic system by using discrete pointer, i.e., path degree of freedom of photons instead of continuous pointer. 32 It should be noted here that whether or not the pointer is continuous or discrete do not change any results discussed above. Before illustration of the experimental realization, it should be emphasized first that weak measurement is mathematically equivalent to positive operator valued measures (POVMs) formalism 33 and this becomes our basis of experimental design. For the spin system discussed above, if Bob1 performs weak measurement and obtains outcome ±1, the states of measured system will accordingly collapse into with probability P(±1) = Tr(|Ψ ±1 〉 s 〈Ψ ±1 |). The weak measurement of Bob1 is actually to realize a two-outcome POVMs with Kraus operators 34 corresponding to outcome ±1. In our realization of weak measurement of Bob1 with photonic elements as shown in Fig Fig. 1 Bell test. a Typical Bell scenario in which one pair of entangled particles is distributed to only two observers: Alice and Bob. b Modified Bell scenario in which Bob1 and Bob2 access the same single particle from the entangled pair with Bob1 performs a weak measurement used as pointer. In order to perform weak measurement in specific polarization basis {|φ〉, |φ ⊥ 〉} with defined observable σ φ = |φ〉〈φ| − |φ ⊥ 〉〈φ ⊥ |, we first transform the measured basis {|φ〉, |φ ⊥ 〉} to basis {|H〉, |V〉} via half wave plate (HWP1), then realize weak measurement of observable σ H = |H〉〈H| − |V〉〈V| via optical elements between HWP1 and HWP4, HWP5 and finally transform back to {|φ〉, |φ ⊥ 〉} basis via HWP4 and HWP5. HWP1, HWP4 and HWP5 are rotated by the same angle φ/2.
In practical experiment, we use the setup shown in Fig. 2b instead of that shown in Fig. 2a. The setup shown in Fig. 2b can realize the same optimal weak measurement as that in Fig. 2a and the only difference is that specific outcome can be selected by rotating HWP1 and HWP4. When Bob1 performs weak measurement of observable σ φ with HWP1 and HWP4 rotated at φ/2 (or π/ 4 − φ/2) degree, photons comes out of setup have state |Ψ +1 〉 = M +1 |Φ〉 (or |Ψ −1 〉 = M −1 |Φ〉) corresponding to outcome +1 (or −1). Here, M +1 = cosθ|φ〉〈φ| − sinθ|φ ⊥ 〉〈φ ⊥ |, M −1 = sinθ|φ〉〈φ| − cosθ|φ ⊥ 〉〈φ ⊥ | are Kraus operators and Bob1 extracts his measurement outcomes by final coincidence detection given that the rotation angles of HWP1 and HWP4 are known to him. It should be emphasized here that the outcomes of Bob1 are actually obtained by Bob2 in our photonic experiment. This is because that the measurement of Bob1 is realized by coupling polarization of photons to its path and the outcomes are encoded in the path after measurement.

Experimental observation of double Bell inequality violations
In our Bell test experiment (Fig. 3), polarization-entangled pairs of photons in state ðjHijVi À jVijHiÞ= ffiffi ffi 2 p are generated by pumping a type-II apodized periodically poled potassium titanyl phosphate (PPKTP) crystal to produce photon pairs at a wavelength of 798 nm. A 4.5 mW pump laser centered at a wavelength of 399 nm is produced by a Moglabs ECD004 laser, and a PPKTP crystal is embedded in the middle of a Sagnac interferometer to ensure the production of high-quality, high-brightness entangled pair. 35,36 The maximum coincidence counting rates in the horizontal/ vertical basis are~3200 s −1 . The visibility of coincidence detection for the maximally entangled state is measured to be 0.997 ± 0.006 in the horizontal/vertical polarization basis {|H〉, |V〉} and 0.993 ± 0.008 in the diagonal/antidiagonal polarization basis fðjHi ± jViÞ= ffiffi ffi 2 p g, achieved by rotating the polarization analyzers for two photons.
Alice, Bob1 and Bob2 each have two measurement choices, and for each choice, two trials are needed, corresponding to two different outcomes. For each fixed θ, which determines the strength of the weak measurement F = sin2θ, we have implemented 64 trials for calculating S A−B1 and S A−B2 . To ensure that the Tsirelson's bound 2 ffiffi ffi 2 p can be approached, Alice chooses measurement along direction Z or X, while Bobs choose measurement along ðÀZ þ XÞ= ffiffi ffi 2 p or ÀðZ þ XÞ= ffiffi ffi 2 p direction. In this experiment, HWP6 is set at (0°, 45°) or (22.5°, 67.5°), corresponding to Alice's measurement along the Z or X direction, while HWP1 and HWP5, representing measurements of Bob1 and Bob2, are set at (−11.25°, 33.75°) or (11.25°, 56.25°), corresponding to the ðÀZ þ XÞ= ffiffi ffi 2 p or ÀðZ þ XÞ= ffiffi ffi 2 p direction, respectively. For instance, if HWP1, HWP4 and HWP5 are rotated at −11.25°a nd HWP6 is fixed at 0°, the three-variable joint conditional probability P½a

DISCUSSION
In conclusion, we have observed double violations of the CHSH-Bell inequality for the entangled state of photon pairs by using a strength continuous-tunable optimal weak measurement. Our experimental results verify the non-locality distribution among multiple observers and shed new light on our understanding of the fascinating properties of non-locality and quantum measurement. The weak measurement technique used herein can find significant applications in unbounded randomness certification, 19,20 which is a valuable resource applied from quantum cryptography 37,38 and quantum gambling 39,40 to quantum simulation. 41 Here, the S value of the correlation between Alice and Bob2 is determined by the quality factor of Bob1's weak measurement, implying that Bob1 can control the non-local correlation of Alice and Bob2 by manipulating the strength of his measurement. This result provides tremendous motivation for the further quantum steering research. 21,22 Note added. After we have posted this work online we noticed that double Bell inequality violations was also observed using similar method in ref. 42 though they did not realize optimal weak measurement.

Weak measurement on a spin-1/2 particles
Consider a typical measurement of spin observable σ z of a spin-1/2 particle with eigenstates satisfy σ z |↑〉 = |↑〉 and σ z |↓〉 = −|↓〉. The initial pointer state |ϕ〉 is entangled with spin system after measurement interaction Uðϱ jϕihϕjÞÛ y ¼ π " ϱπ " jϕ " ihϕ " j þ π # ϱπ # jϕ # ihϕ # j þπ " ϱπ # jϕ " ihϕ # j þ π # ϱπ " jϕ # ihϕ " j; where ϱ represents the initial state of spin system, π ↑ ≡ |↑〉〈↑|, π ↓ ≡ |↓〉〈↓| are the projectors on the eigenstates of σ z and |ϕ ↑ 〉, |ϕ ↓ 〉 are the evolved pointer states corresponding to |↑〉, |↓〉, respectively. The state of spin system, after tracing out pointer, becomes ρ ¼ π " ϱπ " þ π # ϱπ # þ π " ϱπ # hϕ # jϕ " i þ π # ϱπ " hϕ " jϕ # i: To quantify the disturbance of measurement to the spin system, a quantity F called quality factor of measurement can be defined 18 F hϕ " jϕ # i: Usually, F is a complex value. Without loss of generality, here we take it as a real number for the simplicity of discussion. The Eq. (9) Fig. 3 Measurement setup. Polarization-entangled pairs of photons are produced by pumping a type-II apodized periodically poled potassium titanyl phosphate (PPKTP) crystal placed in the middle of a Sagnac-loop interferometer with dimensions of 1 mm × 2 mm × 20 mm and with end faces with anti-reflective coating at wavelengths of 399 nm and 798 nm. The photon emitted to Alice is measured via a combination of HWP6 and PBS. The green area shows the weak disturbance measurement setup of Bob1. During the experiment, HWP2, HWP3 are rotated by θ/2, π/4 − θ/2 according to the experimental requirement. HWP1 is used for Bob1's measurement, and HWP4 is rotated by the same angle as HWP1 to transform the photons polarization state back to the measurement basis after the photon passes through two beam displacers (BDs). The photon passing through HWP4 is then sent to Bob2 for a strong projective measurement with HWP5 and PBS. In the final stage, two-photon coincidences at 6 s are recorded by avalanche photodiode single-photon detectors and a coincidence counter (ID800) reformulated as ρ ¼ Fϱ þ ð1 À FÞðπ " ϱπ " þ π # ϱπ # Þ (11) as that ρ = π ↑ ρπ ↑ + π ↓ ρπ ↓ + π ↑ ρπ ↓ + π ↓ ρπ ↑ . If F = 0, the spin-up state |↑〉 and spin-down state |↓〉 can be distinguished definitely through the orthogonal pointer states |ϕ ↑ 〉 and | ϕ ↓ 〉. The state of spin system ρ is thus reduced to a state of completely decohered in the eigenbasis of σ z and the measurement in this case is called a strong measurement in which we can obtain the maximum information about the system. There is no measurement at all if F = 1, i.e., the pointer state is the same for spin up or spin down. The measurement is called weak when F ∈ (0, 1) in which we can obtain partial information of the spin states with partial disturbance on it. It is obvious from Eq. (11) that a weak measurement can be considered as the combination of a strong measurement and none of measurement operationally. The quality factor F thus reflects the strength of measurement and the disturbance of measurement.
Relation between weak measurement and Bell non-locality The connection between weak measurement and non-locality can be shown in the new Bell scenario that one pair of entangled spin- 1 2 particles are delivered to Alice, Bob1 and Bob2, here Bob1 and Bob2 access to the same particle shown in Fig. 1 of main text. Contrary to Bob2 who performs the strong measurement, Bob1 performs the weak measurement before Bob2. After the measurement of Bob1, the particle will be sent to Bob2 who has no idea of Bob1's existence. Suppose that the entangled pair is in the singlet state jΨi ¼ 1 ffiffi ffi 2 p ðj "ij #i À j #ij "iÞ: Similar to the standard Bell scenario, Alice, Bob1 and Bob2 each receives a binary input x, y 1 , y 2 ∈ {0, 1} and accordingly performs measurement of their spin along the corresponding directionλ x ;μ y1 ;ν y2 respectively. The outcomes of their measurement are labelled by a, b 1 , b 2 ∈ {+1, −1}. To study correlations between Alice and Bobs, we need to calculate the conditional probability distributions P(ab 1 b 2 |xy 1 y 2 ) that can be simplified by using no-signalling condition Pðab 1 b 2 jxy 1 y 2 Þ ¼ PðajxÞPðb 1 jxy 1 aÞPðb 2 jxy 1 y 2 ab 1 Þ: The probability of obtaining outcome a conditioned on x for Alice can be easily shown to be PðajxÞ ¼ 1 2 as that any strong measurement on onehalf of singlet state gives outcomes with equal probability. After the measurement of Alice, the spin state of another particle sent to Bob1 will collapse into the state that in an opposite spin direction with respect to Alice's post-measurement state where π Àaλx represents the spin projector along the direction Àaλ x and I;σ are identity operator and Pauli operator, respectively. The measurement of Bob1 is weak, the probability Pðb 1 jxy 1 aÞ is determined by Eq. (14) and where Tr π b1μ y 1 ρ jxa ¼ π Àμ y 1 ρ jxa π Àμ y 1 (22) with its norm trace Tr ρ jxy1ab1 ¼ P b 1 jxy 1 a ð Þand F is quality factor of the weak measurement. The probability of obtaining outcome b 2 for Bob2's strong measurement is Pðb1jxy1aÞ ρ jxy1ab1 is a normalized state. Now we can calculate conditional probabilities P(ab 1 b 2 |xy 1 y 2 ) according to Eq. (12) 1Àab2ðλx Áμ y 1 Þðμ y 1 Áνy 2 Þ 2 À b1G 4 aλx Áμ y 1 Àb2μ y 1 Áνy 2 2 : As probability lies between 0 and 1, ifλ 0 ¼Z;μ 0 ¼ ÀX andν 0 1 Zsinθ ÀXcosθ are chosen, we obtain along with outcomes a = b 2 = 1 and b 1 = −1. This is the expression of a tangent to the unit circle F 2 + G 2 = 1 and obviously the optimal pointer saturates this constraint.
M-J Hu et al.
The non-local correlation of Alice and Bobs's can be shown by calculating S value defined as S AliceÀBobn ¼ C n ð0;0Þ þ C n ð0;1Þ þ C n ð1;0Þ À C n ð1;1Þ ; (26) where n ∈ {1, 2} and C n ðx;ynÞ defines correlation of Alice and Bob's measurement outcomes C n ðx;ynÞ ¼ Tr ρ n σλ x σμ yn ¼ X a;bn ab n Pðab n jxy n Þ (27) with ρ n is the state of the spin-1 2 entangled pair possessed by Alice and Bobs, σλ x ; σμ yn represent the spin observables corresponding to directions λ x andμ yn , respectively.
Since that Pðab 1 jxy 1 Þ ¼ PðajxÞPðb 1 jxy 1 aÞ ¼ 1ÀGab1λx Áμ y 1 4 , the correlation C ðx;y1Þ of Alice and Bob1 is C ðx;y1Þ ¼ ÀGλ x Áμ y1 and thus we have Similarly Pðab 2 jxy 2 Þ ¼ P b1;y1 and The S value of Alice and Bob2 is calculated as In the case that quantum bound 2 ffiffi ffi 2 p is obtained, i.e., Alice measures in the directionsZ orX according to her inputs 0 or 1, whereas Bob1 and Bob2 measure in the directions ÀðZþXÞ ffiffi 2 p or ÀZþX ffiffi 2 p for their respective inputs 0 or 1, we obtain

Demonstration of entangled photons source in experiment
In our experiment, high-quality polarization-entangled photon source is produced by pumping a type-II apodized periodically poled potassium titanyl phosphate (PPKTP) crystal inside a Sagnac-loop interferometer. The PPKTP has dimensions of 1 mm × 2 mm × 20 mm and the end faces are anti-reflective coated at wavelengths of 399 nm and 798 nm. The temperature of the crystal is controlled by using a home-made temperature controller with stability of ±2 mK. The ultraviolet (UV) pump beam is generated from a commercial Moglabs ECD004 laser. One UV quarter wave plate (QWP) and one UV half wave plate (HWP) are placed at the input port of the interferometer for controlling the power and relative phase of pump beam inside the Sagnac-loop interferometer. Polarization orthogonal pump beams are separated by a dual wavelength polarized beam splitter (DPBS). The vertical polarized pump beam is rotated to horizontal polarization by using a dual wavelength HWP before interact with the PPKTP crystal for spontaneous parametric down-conversion (SPDC). Orthogonal polarized photon pairs are generated in two counter propagating directions combined at the DPBS. The photon emitted to Alice is first separated from the pump beam by using a dichromatic mirror (DM) and then measured projectively via combination of HWP and PBS by Alice.
The photon sent to Bob, first passes through the weak measurement setup of Bob1 and is subsequently sent to Bob2 for projective measurement. The state of photon pair output from the interferometer can be expressed as jψi ¼ 1 ffiffi ffi 2 p ðjHijVi þ e iϑ jVijHiÞ; (33) where the relative phase ϑ is determined by the relative position of QWP and HWP at the input port. In our experiment, the phase ϑ is tuned to π such that the singlet state is produced The quality of our source is characterized by using a two-photon polarization interference shown in Fig. 5. The 399 nm wavelength pump beam's power is fixed at 4.5 mW and the coincidence windows is set in 2 ns. The single counts in 6 s is about 375,000 and 275,000 and the maximum coincidence is about 19,000. The raw visibilities in 0°/90°and 45°/−45°are (99.70 ± 0.06) percent and (99.32 ± 0.08) percent, respectively. Therefore high visibilities guarantee the large violation of Bell-CHSH inequality.

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon request.

ADDITIONAL INFORMATION
Competing interests: The authors declare no competing interests.
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