Experimental sharing of Bell nonlocality with projective measurements

In the standard Bell experiment, two parties perform local projective measurements on a shared pair of entangled qubits to generate nonlocal correlations. However, these measurements completely destroy the entanglement, rendering the post-measurement state unable for subsequent use. For a long time, it was believed that only unsharp measurements can be used to share quantum correlations. Remarkably, recent research has shown that classical randomness assisted projective measurements are sufficient for sharing nonlocality (Steffinlongo and Tavakoli 2022 Phys. Rev. Lett. 129 230402). Here, by stochastically combining no more than two different projective measurement strategies, we report an experimental observation of double Clauser–Horne–Shimony–Holt inequality violations with two measurements in a sequence made on each pair of maximally and partially entangled polarization photons. Our results reveal that the double violation achieved by partially entangled states can be 11 standard deviations larger than that achieved by maximally entangled ones. Our scheme eliminates the requirement for entanglement assistance in previous unsharp-measurement-based sharing schemes, making it experimentally easier. Our work provides possibilities for sharing other types of quantum correlations in various physical systems with projective measurements.

Usually, to share quantum correlations between multiple pairs of parties simultaneously, each sequential party except the last one should perform unsharp measurements.If projective (sharp) measurements are performed, the state of the system collapses to one of the eigenstates of the measured observable, which means that the disturbance caused by projective measurements is maximal and any subsequent measurement on the same system will not provide any additional information about its original state.It seems that the use of projective measurement can not provide aforementioned quantum correlation sharing.However, this is not always true.Recently, Steffinlongo and Tavakoli proposed an approach to overcome the inabilities of projective measurements by leveraging classical randomness [59].Counterintuitively, they also found that partially entangled states can exhibit stronger shareability compared to maximally entangled ones.
Here, we go beyond unsharp measurement and report an experimental demonstration of sharing Bell nonlocality via randomly combining no more than two different projective measurement strategies.By witnessing the violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality [60], we find that Bell nonlocality can be shared among three independent parties for both maximally and partially entangled two-qubit states.And partially entangled states can achieve a stronger double violation than maximally entangled states by 11 standard deviations.In addition to Bell nonlocality, our work also suggests that the role of projective measurements should be re-examined in all other types of quantum correlations, which is fundamental in quantum information theory.Also, we provide a simpler approach for the physical implementation of various relevant quantum information protocols.

Scenario and theory
We consider a scenario involving three parties, namely Alice, Bob, and Charlie, who aim to share the Bell nonlocality of a two-qubit entangled state ρ using only projective measurements (see figure 1).The Bell nonlocality is verified by violating the CHSH inequality [60].Clearly, each party has two binary-outcome measurements, which can be categorized into three cases of projective measurement strategies: both measurements are basis projections (λ = 1); both are identity measurements (λ = 2); one is basis projection and the other is identity measurement (λ = 3) [59].Before the sharing experiment begins, all parties are allowed to share correlated strings of classical data λ ∈ {1, 2, 3}, subject to some probability distribution p(λ).Each party selects their measurement strategy based on the value of λ they receive at that particular moment.And their respective binary-outcome measurements are determined by the private inputs {x, y, z}, which are statistically uniformly distributed.
More specifically, to begin with, a two-qubit state ρ is shared between Alice and Bob.
Case 1: basis projection (λ = 1).Alice measures A 1 0 = σ 1 and A 1 1 = σ 3 .Bob measures B 1 0 = cosϕσ 1 + sinϕ σ 3 and B 1 1 = cosϕσ 1 − sinϕ σ 3 , and then applies the corresponding unitary operations and I is the identity matrix.This gives S ).Clearly, case 1 only enables the CHSH inequality violation between Alice and Bob, and both case 2 and case 3 only enable the violation between Alice and Charlie.Thus, we need to stochastically combine these cases to obtain the double violations.
Here, we focus on investigating the shareability of the Bell nonlocality of a two-qubit state Since the measurement strategy is determined by the value of λ shared among Alice, Bob and Charlie, these parties always perform their respective measurements in the same measurement case.Suppose that the measurement strategy is a stochastic combination of case s i and case s j , and the corresponding probabilities are p(λ = s i ) and p(λ = s j ), then CHSH parameters S AB and S AC can be expressed as where There are three types of stochastically combined measurement strategies: a combination of case 1 and case 2, a combination of case 1 and case 3, and a combination of case 2 and case 3.For a given measurement strategy, the violation of the CHSH inequality S AB ⩽ 2 (S AC ⩽ 2) demonstrates the existence of Bell nonlocality between Alice and Bob (Charlie).Interestingly, we find that for a suitable choice of φ, partially entangled states can achieve larger sequential violations than the maximally entangled states.correspond to the double CHSH inequality violation regions for the partially and maximally entangled states, respectively.Clearly, the double violation regions for partially entangled states are larger than those for maximally entangled states.
In the following, we will derive the optimal measurement settings for aforementioned three stochastically combined measurement strategies by analyzing the optimal trade-offs between S AB and S AC .Note that in all three deterministic cases, the trade-offs between S λ AC and S λ AB are concave.In the case of shared randomness, the optimal trade-off between S AC and S AB can be expressed as and S 0 is a real value.To keep both CHSH parameters S AB and S AC as large as possible above the optimal classical bound, the best way is to set measurement settings to maximize the minimum value of {S AC , S AB }.It is easy to find that S λ AC decreases as S λ AB increases.Similarly, S AC decreases as S AB increases.And the maximum value The optimal trade-offs between S λ AC and S λ AB for the maximally entangled state φ = 45 • when the measurement strategy involves case 1 (λ = 1) and case 2 (λ = 2).(e).The optimal trade-offs between S λ AC and S λ AB for the maximally entangled state φ = 45 • when the measurement strategy involves case 1 (λ = 1) and case 2 (λ = 3).(f).The optimal trade-offs between S λ AC and S λ AB for the maximally entangled state φ = 45 • when the measurement strategy involves case 2 (λ = 2) and case 3 (λ = 3).The green lines represent the optimal trade-offs of the corresponding stochastically combined measurement strategies.The purple circles, red triangles and blue squares represent the values of {S λ AB , S λ AC } which are obtained with the optimal measurement settings for case 1, case 2 and case 3, respectively.The dashed orange lines are the classical bounds of the CHSH inequality.
of the minimum value of {S AC , S AB } is obtained at S AC = S AB .This is the main idea for us to obtain the optimal measurement settings of the stochastically combined projective strategy.Substituting k = For an arbitrary combination strategy {s 1 , s 2 }, one can obtain the optimal state and measurement parameters {φ, ϕ, χ, θ} by finding the maximum value of S 0 /(1 − k), where χ = arctan(csc(2φ)).And for a particular state φ, the optimal measurement parameters {ϕ, χ, θ} can also be obtained by finding the maximum value of S 0 /(1 − k).The results for a maximally entangled qubit state φ = 45 • are shown in figures 2(d)-(f).With the optimal measurement settings, we can obtain the results of {S λ AB , S λ AC }, which are marked as purple circles, red triangles and blue squares for case 1, case 2 and case 3, respectively.Theoretically, AC } are the tangent points of the corresponding tangents [59].The optimal trade-offs between S AB and S AC can be obtained by stochastically combining the corresponding optimal measurement settings, which are represented as green lines.Clearly, both S λ AB and S λ AC are above the classical bound whether the measurement strategy is the combination of case 1 and case 2 or the combination of case 1 and case 3.However, when case 2 and case 3 are stochastically combined, only S λ AC can beat the classical bound.By comparing the trade-offs of the above mentioned three deterministic projective measurement strategies and three stochastically combined projective measurement strategies, we can further obtain the boundary of the attainable CHSH parameters {S AB , S AC }.The optimal trade-off between S AC and S AB over the entire range of S AB (0 ⩽ S AB ⩽ 2 √ 2) is divided into four parts.The first one is a combination of case 2 and case 3, the second one is a deterministic case 3, the third one is a combination of case 1 and case 3, and the forth one is a deterministic case 1.The results for states φ = 41.48 • (red curve) and φ = 45 • (blue curve) are shown in figure 2(c).The inset shows an enlarged view of the double violation results.It is clear that the partially entangled state can outperform the maximally entangled state.

Experimental setup and results
We focus on investigating the Bell nonlocality sharing scenario where no more than two different projective measurement strategies are combined.Figure 3 shows our experimental setup.A 405 nm CW laser is used to pump periodically poled potassium titanyl phosphate (PPKTP) crystal in clockwise and counter-clockwise directions to generate a two-qubit polarization-entangled photon state . H and V represent horizontal and vertical polarizations, respectively.The parameter φ is flexibly changed by the half-wave plate (PH2).The pump light is reflected by a dichroic mirror (DM).Two interference filters (IFs) are used to filter the down-conversion photons.One of the two photons is directly sent to Alice, who uses a half-wave plate (AH1) and a polarization beam splitter (PBS) to perform projective polarization measurement.The other photon is sent to an unbalanced interferometer and then subsequently sent to Bob and Charlie.In each path, Bob and Charlie carry out a particular measurement strategy.Bob uses linear polarizers (BPs), and Charlie employs half-wave plates (CHs) and PBSs to perform the associated projective measurement.The unitary operation setup, comprised of a quarter-wave plate (UQ), a half-wave plate (UH), and a quarter-wave plate (UQ) on both paths, allows Bob to implement the desired unitary operation.The relative combining probability between these two paths can be easily controlled by rotating the variable neutral density (ND) filters (VFs), see appendix A for more details.As in the previous unsharp-measurement-based schemes [18,19], all outputs of Bob and Charlie are read from the same detector D2, for simplicity.Specifically, Bob's (Charlie's) outputs are obtained by removing(inserting) the setup of Bob's unitary operation and Charlie's measurement.In addition, the identity measurements of Bob can be realized by removing the linear polarizers (BPs) or by respectively measuring the qubit on Bob's normalized conditional state and its corresponding orthogonal state.
We first investigate the sharing of Bell nonlocality for a maximally entangled state, i.e. φ = 45  4(a).The purple circles, red triangle, and blue squares correspond to case 1 (λ = 1), case 2 (λ = 2), and case 3 (λ = 3), respectively.The theoretical optimal trade-offs are represented as solid curves and compared with the experimental data.See appendix B for more details.In case 1 and case 3, with the increase of S λ AB , S λ AC first increases and then decreases.And case 1 enables Bell nonlocality shared between Alice and Bob but not between Alice and Charlie, while the opposite is true for case 2 and case 3. To obtain the optimal double violations, we stochastically combine case 1 and case 2 with the optimal measurement setting ϕ = 75 • and χ = 45 • .The operations in path 1 and path 2 correspond to the measurements in case 1 and case 2, respectively.By rotating VF1 and VF2, we can further change the probability p of case 1.The CHSH parameters S AB (purple circles) and S AC (blue squares) as a function of p are shown in figure 4(b).Clearly, as p increases, S AB increases and S AC decreases.The double violations of 3)/3.Experimentally, we get S AB = 2.0451 ± 0.0015 and S AC = 2.0676 ± 0.0014 when p = 0.83207 ± 0.0015, both of which are more than 30 standard deviations above the classical bound.We also study the Bell nonlocality sharing by stochastically combining case 1 and case 3. The optimal measurements are ϕ = 71.57• and θ = 18.43 • .Figure 4(c) presents S AB (purple circles) and S AC (blue squares) as a function of the probability p of case 1.The maximum values of the two CHSH parameters S AB = 2.1056 ± 0.0013 and S AC = 2.0927 ± 0.0013 are obtained at p = 0.3256 ± 0.0012, both more than 71 standard deviations above the classical bound.It is clear that double violations of the CHSH inequality are achieved, and both violations are stronger than those obtained by combining case 1 and case 2. What is more, the double violations enabled range of p is extended, exhibiting better performance.We further combine the stochastically combined strategy of case 1 and case 3 with other three types of measurement strategies, i.e. a combination of cases 2 and 3 (green curve), deterministic case 3 (blue curve), a combination of cases 1 and 3 (red curve) and deterministic case 1 (purple curve), to obtain the boundary of the attainable CHSH parameters S AB and S AC .This analysis is detailed in appendices A and B. The optimal trade-off between S AC and S AB over the entire range of S AB is shown in figure 4(d).Clearly, experimental results are in a good agreement with theoretical optimal trade-offs.
We further studied the shareability of Bell nonlocality with partially entangled states.By varing the probability p of case 1, we also observe the double violations of CHSH inequality whether the measurement strategy is a stochastic combination of case 1 and case 2 or a combination of case 1 with case 3. The corresponding CHSH parameters S AB and S AC as a function of p are shown in figure B2.And the optimal trade-offs between S AC and S AB satisfying the double violations of CHSH inequality for the states φ = 34.08 • and φ = 41.48 • are presented in figures 5(a) and (b), respectively.Red solid lines and red circles correspond to theoretical predictions and experimental results.These results are compared with those for the maximally entangled state, whose theoretical results are denoted as the blue solid lines in figures 5(a) and (b).We find that, for a suitable choice of φ, the partially entangled states can achieve stronger double violations whether stochastically combining case 1 with case 2 or combining case 1 with case 3. The corresponding S AC shown in the dashed green boxes are about 11 and 5 standard deviations higher than those achievable with the maximally entangled state in the aforementioned two cases, respectively.Moreover, by simply applying the deterministic measurement strategy, partially entangled state can also outperform the maximally entangled state.The results for states φ = {45 • , 39.23 • , 34.08 • , 28.32 • , 21.77 • } with deterministic measurement case 2 are presented in figure 5(c).All experimental results exceed the bound of the maximally entangled state.And the difference of S AC reaches the maximum when φ = 34.08 • .The corresponding state fidelities are further shown in figure 5(d), whose average value is about 0.9853 ± 0.0012.

Conclusion and discussion
In this study, we have experimentally demonstrated the sharing of Bell nonlocality for two-qubit entangled states using projective measurements and shared classical randomness.We verified shareability by witnessing violations of the CHSH inequality.For the maximally entangled state, we found that when stochastically combining basis projection and identity measurement strategies, double violations of CHSH inequality can be obtained.And these violations become even stronger when identity measurement strategy is replaced with mixed strategy.In contrast to the unsharp-measurement-based sharing schemes, our results showed that some partially entangled states can achieve double violations 11 standard deviations higher than those achieved by maximally entangled states.Additionally, partially entangled states with deterministic identity measurement strategies can outperform maximally entangled states with optimal stochastically combined measurement strategies.
Similar to previous schemes based on unsharp measurement, all outputs of Bob and Charlie are read from Charlie's detector.However, this introduces an additional experimental assumption: Charlie is required to inform Bob of whether his detector kicks after each round of Bob's measurement.This requirement can be circumvented if Bob performs a corresponding measurement (which demolishes the photon) and subsequently prepares a new photon in the post-measurement state, relaying it to Charlie.Unlike unsharp measurement-based schemes, post-measurement states are usually mixed, while projective measurement-based schemes yield pure states, making them easier to reproduce in experiments.
Our work represents a significant step towards the experimental realization of Bell nonlocality sharing beyond the conventional unsharp measurements, and challenges the common notion that projective measurement prevent Bell nonlocality sharing.It has important significance both theoretically and experimentally.
On the theoretical side, our work inspires the notion that quantum information protocols involving quantum correlations, such as entanglement, steering, coherence, and contextuality, which were previously thought to be achievable only with unsharp measurements, may also be implemented using projective measurements.These protocols have unique properties and advantages that can be used to achieve various quantum information tasks.It is important to investigate how to construct the projective measurement strategy to achieve the corresponding protocols, and further increase the number of correlation-reuse parties while also flexibly manipulating the direction of steering.Additionally, existing works only analyze the Bell nonlocality when single-sided sequence projective measurements are performed, and it is important to investigate what happens if both sides perform sequence projective measurements.
On the experimental side, our sharing scheme only requires local projective measurements on qubits, which can be achieved using polarizers and quarter-wave plates in optical systems.This advantage will become even more pronounced for multi-qubit and higher-dimensional systems.Furthermore, our proposal is readily implementable in other physical systems, including ion traps, superconductors, diamond NV centers, and cold atoms, thus providing possibilities for experimental sharing of quantum correlations in physical systems beyond optics.

changes of S λ=1
AB and S λ=1 AC as a function of ϕ and the changes of S λ=3 AB and S λ=3 AC as a function of θ are presented in figures B1(a) and (b), respectively.However, when λ = 2, the value of χ is fixed and equals to 45 • , the experimental results for S λ=2 AB and S λ=2 AC are 0.0171 ± 0.0013 and 2.8081 ± 0.0012, respectively.Clearly, case 1 (λ = 1) only enables the CHSH inequality violation between Alice and Bob, and both case 2 (λ = 2) and case 3 (λ = 3) only enable the violation between Alice and Charlie.
For a given stochastically combined measurement strategy, using the optimal measurement settings introduced in section 2, we can obtain the relationship between the CHSH parameter and the combination probability.Figure B2(a) presents the CHSH parameters S AB and S AC as a function of probability p(λ = 1) (abbreviated as p) when the initial state is φ = 34.08 • and the measurement strategy is a combination of case 1 (λ = 1) and case 2 (λ = 2).Figure B2(b) presents the CHSH parameters S AB and S AC as a function of probability p(λ = 1) (abbreviated as p) when the initial state is φ = 41.48 • and the measurement strategy is a combination of case 1 and case 3 (λ = 3).Figure B2(c) presents the CHSH parameters S AB and S AC as a function of probability p(λ = 2) (abbreviated as p) when the initial state is φ = 45 • and the measurement strategy is a combination of case 2 and case 3. Experimentally measured CHSH parameters are shown as symbols, where the purple circles and blue squares correspond to S AB and S AC , respectively.Theoretical predictions are also represented as curves, and compared with the experimental data.We find that, whether stochastically combining case 1 with case 2 or combining case 1 with case 3, for a suitable choice of φ, the partially entangled states can also achieve the double violations of CHSH inequality.However, when case 2 and case 3 are stochastically combined, only S AC can beat the classical bound even if the initial state is maximally entangled.

Figure 1 .
Figure 1.Sequential projective measurement scenario.A two-qubit entangled state ρ is distributed among three parties, Alice, Bob, and Charlie, with Bob and Charlie having sequentially access to the same qubit.Alice and Charlie perform projective measurements on their respective qubits, while Bob implements projective instruments and unitary operations on his qubit.Each party has three distinct projective measurement strategies, determined by shared input λ with a probability distribution of p(λ).Within each measurement strategy, each party has two binary-outcome measurements, determined by their respective private inputs {x, y, z}.The task is to demonstrate the existence of Bell nonlocality between Alice and Bob as well as between Alice and Charlie simultaneously.

Figure 2 .
Figure 2. (a).The double violation region for the partially entangled state φ = 34.08 • (red, PE) and the maximally entangled state φ = 45 • (blue, ME) when the measurement strategy is a combination of case 1 and case 2. (b).The double violation region for the partially entangled state φ = 41.48 • (red, PE) and the maximally entangled state φ = 45 • (blue, ME) when the measurement strategy is a combination of case 1 and case 3. (c).The optimal trade-off between SAB and SAC for the partially entangled state φ = 41.48 • (red curve, PE) and the maximally entangled state φ = 45 • (blue curve, ME) over the entire range of SAB.The inset shows an enlarged view of the double violation results.(d).The optimal trade-offs between S λAC and S λ AB for the maximally entangled state φ = 45 • when the measurement strategy involves case 1 (λ = 1) and case 2 (λ = 2).(e).The optimal trade-offs between S λ AC and S λ AB for the maximally entangled state φ = 45 • when the measurement strategy involves case 1 (λ = 1) and case 2 (λ = 3).(f).The optimal trade-offs between S λ AC and S λ AB for the maximally entangled state φ = 45 • when the measurement strategy involves case 2 (λ = 2) and case 3 (λ = 3).The green lines represent the optimal trade-offs of the corresponding stochastically combined measurement strategies.The purple circles, red triangles and blue squares represent the values of {S λ AB , S λ AC } which are obtained with the optimal measurement settings for case 1, case 2 and case 3, respectively.The dashed orange lines are the classical bounds of the CHSH inequality.

1
ABand S AC = S AB into S AC = k • S AB + S 0 , we get the objective function S AC = S AB = S 0 /(1 − k).

Figure 3 .
Figure 3. Experimental setup.A bright source of polarization-entangled photon pairs in the state |ψφ⟩ = cosφ |HH⟩ + sinφ |VV⟩ (φ ∈ [0 • , 45• ]) is generated by pumping a type-II cut PPKTP crystal located in a Sagnac interferometer with a 405 nm CW laser.These two photons are filtered by interference filters.One photon is sent to Alice, and the other is subsequently sent to Bob and Charlie via path 1 and path 2. These paths correspond to different measurement strategies, and their relative probabilities can be flexibly changed by rotating variable ND filters.Since Alice, Bob, and Charlie only need to measure linear polarizations, their setup can be simplified to a polarizer or a composition of a half-wave plate and a PBS.Bob's unitary operation can be realized by a composition of two quarter-wave plates and a half-wave plate.The photons are detected by single photon detectors (D1 and D2) and the signals are sent for coincidence.

Figure 4 .
Figure 4.The Bell nonlocality shareability for the maximally entangled state.(a).The optimal trade-off between S λ AC and S λ AB for case 1 (λ = 1, purple circles), case 2 (λ = 2, red triangle), and case 3 (λ = 3, blue squares) with various measurement settings {ϕ, χ, θ}.Only one point is obtained for the case 2 with identity measurement.(b).The CHSH parameters SAB (purple circles) and SAC (blue squares) as a function of probability p when the measurement strategy is a combination of case 1 and case 2. (c).The CHSH parameters SAB (purple circles) and SAC (blue squares) as a function of probability p when the measurement strategy is a combination of case 1 and case 3. Theoretical predictions in (a)-(c) are represented as solid curves with the corresponding colors.(d).The optimal trade-off between SAC and SAB.Theoretical predictions and experimental results are represented as curves and purple circles, respectively.The dashed orange lines are the classical bounds of the CHSH inequality.The insets in (b) and (d) show an enlarged view of the double violation results.Error bars are due to the Poissonian counting statistics.

Figure 5 .
Figure 5.The optimal trade-offs between SAC and SAB for various partially entangled states when the measurement strategy is: (a). a combination of case 1 (λ = 1) and case 2 (λ = 2); (b). a combination of case 1 and case 3 (λ = 3); (c).a determined case 2. Experimental results are shown as red circles.Theoretical predictions for partially (PE) and maximally (ME) entangled states are represented as red and blue curves, respectively.(d).The fidelity of the experimental states shown in (c).Error bars are due to the Poissonian counting statistics.

Figure B1 .
Figure B1.(a).The CHSH parameters S λ=1 AB (red circles) and S λ=1 AC (blue squares) as a function of measurement parameter ϕ when the initial state is φ = 45 • .(b).The CHSH parameters S λ=3 AB (red circles) and S λ=3 AC (blue squares) as a function of measurement parameter θ when the initial state is φ = 45 • .Theoretical predictions in (a) and (b) are represented as solid lines with the corresponding colors.The dashed orange lines are the classical bounds of the CHSH inequality.Error bars are due to the Poissonian counting statistics.

Figure B2 .
Figure B2.(a).The CHSH parameters SAB (purple circles) and SAC (blue squares) as a function of probability p(λ = 1) (abbreviated as p) when the initial state is φ = 34.08 • and the measurement strategy is a combination of case 1 (λ = 1) and case 2 (λ = 2).(b).The CHSH parameters SAB (purple circles) and SAC (blue squares) as a function of probability p(λ = 1) (abbreviated as p) when the initial state is φ = 41.48 • and the measurement strategy is a combination of case 1 and case 3 (λ = 3).(c).The CHSH parameters SAB (purple circles) and SAC (blue squares) as a function of probability p(λ = 2) (abbreviated as p) when the initial state is φ = 45 • and the measurement strategy is a combination of case 2 and case 3. Theoretical predictions in (a)-(c) are represented as solid lines with the corresponding colors.The dashed orange lines are the classical bounds of the CHSH inequality.Error bars are due to the Poissonian counting statistics.