Less entanglement exhibiting more nonlocality with noisy measurements

The Clauser–Horne–Shimony–Holt (CHSH) inequality test is widely used as a mean of invalidating the local deterministic theories. Most attempts to experimentally test nonlocality have presumed unphysical idealizations that do not hold in real experiments, namely, noiseless measurements. We demonstrate an experimental violation of the CHSH inequality that is free of idealization and rules out local models with high confidence. We show that the CHSH inequality can always be violated for any nonzero noise parameter of the measurement. Intriguingly, less entanglement exhibits more nonlocality in the CHSH test with noisy measurements. Furthermore, we theoretically propose and experimentally demonstrate how the CHSH test with noisy measurements can be used to detect weak entanglement on two-qubit states. Our results offer a deeper insight into the relation between entanglement and nonlocality.


INTRODUCTION
Quantum systems exhibit a wide range of non-classical and counter-intuitive phenomena, such as quantum entanglement 1 and Bell nonlocality 2 . Bell's inequality fundamentally provides a wealth of understanding in what differentiates the classical from the quantum world. Bell's insight that nonlocal correlations between quantum systems cannot be explained classically can be verified experimentally and has numerous applications in modern quantum information [3][4][5][6][7][8] . A great effort has also been devoted to understanding the relation between entanglement and nonlocality [9][10][11][12][13][14][15][16][17][18][19][20][21][22] . A fundamental problem with most proposals [13][14][15] for testing nonlocality and experiments [23][24][25][26][27] performed to date, is that measurements are assumed to have a deterministic response in the nonlocality model. It has been shown that this can only be justified under the idealization, namely, noiseless measurements 28 . For Bell's notion of local causality, the theoretical work by Clauser et al. 29 is critical to enabling an experimental test without unphysical idealizations, i.e., without the perfect anticorrelation presumed in Bell's original proof. Furthermore, Dilley and Chitambar 30 show how to devise the test that is free of the other idealization that measurements are noiseless, which is never satisfied precisely by any real experiment. They consider a scenario in which one party has inefficient detectors and can only perform noisy measurements and show that the Clauser-Horne-Shimony-Holt (CHSH) inequality can always be violated for measurements with any nonzero detection efficiency.
In this work, we perform an experimental violation of the CHSH inequality under a scenario when one party performs noisy measurements, which is known as an asymmetric Bell experiment. It should be mentioned that in our experiments the loopholes of a Bell test are not closed which requires extremely high experimental costs and rigorous technology [31][32][33][34] . First, we consider a range of non-maximal two-qubit entangled states and show that the CHSH inequality can always be violated for any nonzero noise parameter of the measurement. Surprisingly, less entanglement exhibits more nonlocality in the CHSH test with noisy measurements. Furthermore, as an application of the CHSH test with noisy measurements, we show how it is possible to also detect the presence of weak entanglement of a two-qubit system. Our results offer a thorough understanding of the relation between entanglement and nonlocality.

Theoretical scenarios
First, consider a hybrid scenario in which Alice and Bob share a two-qubit state. Alice's detector has perfect efficiency and she can perform an ideal measurement, while Bob can only perform a noisy measurement which has a conclusive detection only a fraction η of the time. Alice randomly chooses to measure her qubit in the directionâ x (x = 0, 1) and the outcome a is denoted by either 0 or 1. The projector for Alice's measurement is whereσ ¼ σ xx þ σ yŷ þ σ zẑ is a vector of Pauli matrices. In a realistic setup, it might not be possible to obtain a conclusive measurement outcome. We consider a simple scenario, where an inconclusive outcome can only arise on Bob's measurements, occurring with frequency (1 − η) for both of his measurement choices. Thus, Bob performs a positive operator-valued measure (POVM) with two outcomes whereb y (y = 0, 1) is the direction of Bob's measurement. The term ð1 À ηÞ1 corresponds to the inconclusive outcome. The CHSH inequality with detection efficiency η on Bob's measurement is where Eðx; yÞ :¼ is the expected value of the nonlocal function f(a, b) = (−1) a⊕b computed from the measurement outcomes, and the Bell operator for this scenario is 1jy . Without loss of generality, we choose the directions of Alice and Bob aŝ The Bell operator then becomes where and B CHSH is the standard Bell operator with η = 1. It is proven in ref. 30 that in a two-qubit CHSH test with detection efficiency η for one party and a perfect efficiency for the other, there exist measurement directions fâ 0 ;â 1 ;b 0 ;b 1 g and a corresponding entangled state ρ such that the CHSH inequality is violated, i.e., Tr ρB 1 ð Þ > 2 if η > 1/2. To analyze the entanglement in the state violating the CHSH inequality, if Tr ρB 1 ð Þ > 2ð1 þ κÞ for any κ = η − 1/2 ≥ 0, the squared concurrence of the state ρ is then For any pure state which satisfies C 2 (ρ) = κ(2 + κ), there exist suitable measurement directions to obtain the value of the CHSH inequality Tr ðρB 1 Þ ¼ 2ð1 þ κÞ. For η = 1, as the states are more entangled the violation of the CHSH inequality becomes larger. Surprisingly, for 1/2 < η < 1, more nonlocality with less entanglement is shown as the detection efficiency decreases 30 .
Comparing these two scenarios, the first one guarantees the existence of some measurement directions that generate nonlocal correlations for η > 1/2 and places an upper bound on the entanglement needed to violate the CHSH inequality. In contrast, the second scenario says that any measurement direction will do and also has implications for the relationship between nonlocality and measurement incompatibility. For any two non-commuting observables on Alice's side, nonlocality can always be demonstrated using incompatible POVMs on Bob's side in which one of them is any standard observable and the other is any noncommuting "coarse-grained" observable, i.e., having the form of Eq. (2).

Experimental violation of the CHSH inequality with noisy measurements
We experimentally test the CHSH inequality without unphysical idealization-noiseless measurements. The experimental setup illustrated in Fig. 1, consists of an entangled photon source, entangled state preparation, and polarization measurement 35 . By using two adjacent nonlinear crystals (β-barium borate, BBO) pumped by a 404-nm laser diode to produce spontaneous parametric down-conversion and a half-wave plate (HWP) with the setting angle θ ¼ 1 2 arcsin a, we can produce photon pairs in the family of entangled states where H and V represent horizontal and vertical polarizations, respectively. Fidelities of the states are about 97%. Concurrence of the states are obtained from the reconstructed density matrices via fully state tomography 36 . The polarization of each photon is analysed on an arbitrary basis, by means of a quarter-wave plate (QWP), HWP, and a polarizing beam splitter (PBS) in each arm. Making 16 measurements of the polarization correlations in various bases allows tomographic reconstruction of the density matrix of the two-photon states. Photons are detected in coincidence using silicon avalanche photodiodes (APDs), thereby projecting out the large vacuum state from the possibility of the pump photon not downconverting, and selecting the twophoton contribution to quantum states. Photon counts are taken Fig. 1 Experimental setup. Desired quantum states are generated via type-I spontaneous parametric down-conversion using two joint β-Barium Borate (BBO) crystals. In the measurement stage, a halfwave plate (HWP) at θ A and a polarizing beam splitter (PBS) on Alice's side perform the perfect projective measurement. For Bob's side, a POVM is performed by an HWP at θ B and a Mach-Zehnder interferometer with two beam displacers (BDs) and two HWPs at θ η and 45 ∘ inserted in each arm of the interferometer, respectively. For the second scenario, BDs and two HWPs are removed from the setup and an HWP at θ B is used to perform a projective measurement instead of a POVM. Here the noisy parameter of POVM η is adjusted by tuning the setting angle of the HWP as θ η ¼ arccos to be fair samples of the true probabilities for obtaining each outcome for every preparation-measurement pair.
To test the CHSH inequality, one of the photons is sent to Bob for his noisy measurement and the other is for Alice's perfect measurement. For Alice's side, the measurement of her observable O A x is a standard polarization measurement using an HWP at θ A and a PBS. The HWP is used to map the eigenstate of the observable corresponding to the eigenvalue 1 into H j i and the PBS is for projective measurement of the observable σ z -one of the standard Pauli operators. Two outcomes are read by APDs (D A0 and D A1 ).
In the first scenario, Bob performs a POVM fΠ B 0jy ;Π B 1jy g instead of projective measurement [37][38][39][40][41][42][43] . He needs two steps to implement the two-outcome measurements. First, a projector (the first term ofΠ B 0jy orΠ B 1jy ) is realized by an HWP at θ B . Then partially projecting polarizing elements involving two birefringent calcite beam displacers (BDs) and two HWPs to produce the required projectors with the appropriate weights, which are encoded into the angle of one of the HWPs θ η ¼ arccos ffiffiffi η p =2 (η is the efficiency of the noisy measurement which is simulated here via the optical implementation of the POVM). Two outcomes of Bob's measurement are read by APDs (D B0 and D B1 ). The conditional probability p (a, b|x, y), which denotes the probability obtained in the case Alice chooses to measure her qubit in the directionâ x and the outcome is a, while Bob chooses to measure his qubit in the directionb y with inefficient detectors and the outcome is b, is obtained by the coincidence counts N(a, b|x, y) between APDs (D Aa ,D Bb ) normalizing by the total photon counts. For example, p(0, 0|1, 1) = N(0, 0|1, 1)/∑ a,b N(a, b|1,1). For the second scenario, Bob performs a projective measurement (1) for y = 0 and a POVM (2) for y = 1 via the above setups, respectively. All the measurement events on Bob's side need to be heralded by Alice's detectors.
In Fig. 2a, the solid line signifies the upper bound on the amount of entanglement needed to violate the CHSH inequality for a given η. The states with the squared concurrences at or above the solid line (the shadow area) do not violate the CHSH inequality. For η > 1/2, there always exists a state maximally violating the CHSH inequality, and the squared concurrence of this state is given by the dashed line obtained theoretically. For η 1= ffiffi ffi 2 p , no maximally entangled states (C 2 = 1) can violate the CHSH inequality.
In Fig. 2b, we show the value of the CHSH violation versus the squared concurrence for the first scenario. For η = 1, the value of the CHSH violation increases with the squared concurrence linearly and the maximal entangled state Φ þ ¼ ð 00 2:7629 ± 0:0337 by 22 standard deviations. Surprisingly, for 1/2 < η < 1, as the detection efficiency η decreases, more nonlocality with less entanglement is observed, which agrees with the theoretical prediction in ref. 30 . For example, for η = 0.8, the state 0:7603 00 j i þ 0:6495 11 j i with squared concurrence 0.8568 ± 0.0048 violates the CHSH inequality with Tr ðρB 1 Þ ¼ 2:2911 ± 0:0323 by nine standard deviations. Whereas, the maximally entangled state violates the CHSH inequality with 2.1938 ± 0.0316 by only six standard deviations.
Experimental results of the second scenario are shown in Fig. 2c. For the POVM of Bob's side, we choose 5 different settings η = 0.2, 0.4, 0.5, 0.6, 1. For each η, five different two-qubit states ϕ j i(10) whose squared concurrences are 0.1, 0.4, 0.6, 0.7, 0.8, 0.92, 1, respectively, are chosen. For each η and ϕ j i, we optimize the directions of measurement for both sides fâ 0 ;â 1 ;b 0 ;b 1 g to obtain the maximal violation of the CHSH inequality. Furthermore, we also choose the states which violate the CHSH inequality maximally for each η. Similar to the first scenario, except for the case of η = 1, as the detection efficiency η decreases, more nonlocality with less entanglement is observed. Take η = 0.6 as an example, the state 0:8018 00 j i þ 0:5976 11 j i with squared concurrence of 0.8568 ± 0.0167 violates the CHSH inequality with Tr ðρB 1 Þ ¼ 2:3257 ± 0:0330 by ten standard deviations. While the maximally entangled state violates the CHSH inequality with 2.2879 ± 0.0334 by only nine standard deviations. In addition, different from that in the first scenario, the CHSH inequality can always be violated for an arbitrary small η > 0 in the second scenario. Fig. 2 Experimental results for the first and second scenarios. a Squared concurrence of the state C 2 (ρ) versus the detection efficiency of Bob's noisy measurement η. For a given η, a state with the squared concurrence at or above the solid curve (the shaded area) will never violate the CHSH inequality. For η > 1/2, there is always a state violating the CHSH inequality maximally and the squared concurrence of the state is given by the dashed curve. For η 1= ffiffi ffi 2 p , no maximally entangled states (C 2 = 1) can violate the CHSH inequality. In our experiment, more values of η in 1=2; 1= ffiffi ffi 2 p À Ã are chosen to manifest the behavior around η ¼ 1= ffiffi ffi 2 p . Black triangles denote the squared concurrence of the states which do not violate the CHSH inequality for a given η. Red dots denote C 2 of the states which violate the CHSH inequality. Blue squares denote C 2 of the states which violate the CHSH inequality maximally. b Violations of the CHSH inequality versus the squared concurrences of the states for various detection efficiency η. For η < 1, more nonlocality with less entanglement is observed. c Violations of the CHSH inequality versus the squared concurrences of the states for the second scenario, where only one of Bob's measurement choices is inefficient. The CHSH inequality can be violated for an arbitrary η > 0. To show the difference between the first and second scenarios, we choose more values of η in 0; 1=2 ½ . Error bars indicate the statistical uncertainty which is obtained based on assuming Poissonian statistics.
Detecting weak entanglement Furthermore, we theoretically propose and experimentally demonstrate how the CHSH test with noisy measurements can be used to detect the weak entanglement of a two-qubit system. Consider a two-qubit state whereφ ;1 g being the Schmidt basis of ϕ 0 j i and λ ± being the Schmidt coefficients. ϕ 0 j i is the eigenstate of B 1 with the directions of the measurements θ A = 0 and θ B ¼ arcsin½ð1 À τÞ 2 =ð1 þ τÞ 2 . ρ is weakly entangled with a monotonically decreasing squared concurrence C 2 (ρ) as parameter t varies among the interval and has an expected value Tr ρB 1 ð Þ > 2 if While, for values of t outside of the interval Eq. (12), ρ may exist stronger entanglement but with no violation of the CHSH equation. Figure 4 gives an intuitive presentation of such a situation, which is discussed in detail in the Method part. Such a property can be used to detect only nonlocal states with weak entanglement.
For experimental demonstration, we fix the detection efficiency of Bob's measurement η = 3/4. We choose total six values of t = 0, 0.02, 0.038, 0.3, 0.6, 0.83 and generate six states with various parameters t and fixed s = 1/50 and τ = 1/2 44 . In the measurement stage, the setup is as same as that for the pure state testing the CHSH inequality B 1 .
In Fig. 3b, weak entanglement is achieved for small t. The violation of the CHSH inequality is even larger than that obtained by the state with more entanglement (with larger t), which can be seen by comparing Fig. 4a, b. For t 2 0; 0:038 ½ , Tr ρB 1 ð Þ is larger than 2. The CHSH inequality is then violated for these states, which indicates Bell nonlocal states. For example, for t = 0, the state has only weak entanglement with the squared concurrence C 2 (ρ) = 0.5617 ± 0.0190. However, a violation of the CHSH inequality is still observed for Tr ρB 1 ð Þ¼ 2:0281 ± 0:0301. (Ideally, for t = 0 our method still allows to demonstrate nonlocality by violating the CHSH inequality (2.0704). However, the experimental result 2.0281 ± 0.0301 shows that if the error bar is considered the method is not always valid.). Actually, due to the squared concurrences in Fig. 3a, the states with t 2 0; 0:791 ½ Þ are entangled. However, a violation of the CHSH inequality is only observed as Tr ρB 1 ð Þ between 2.070 and 2 (theoretical predictions) for t 2 0; 0:038 ½ . The interval of separability will be pushed left on the graph for smaller values of s and τ; causing a violation to occur for only small values of parameter t.
The states belonging to the gap between the Bell nonlocal states with t 2 0; 0:038 ½ and the separable states with t 2 0:791; 0:865 ½ are so-called entangled Bell local states, which can be detected via entanglement witnesses 45 . An entanglement witness operator W acting on a bipartite system can be defined via Peres criterion as For separable states, Tr(ρW) ≥ 0, while for entangled states, Tr(ρW) < 0. In Fig. 3c, it is shown that Tr(ρW) < 0 for t 2 0; 0:791 ½ Þ , which indicates both Bell nonlocal states and entangled Bell local states can be detected by the entanglement witness.

DISCUSSION
Entanglement is a strong nonlocal correlation and an important resource in the development of technologies and protocols exploiting the properties of quantum systems 46,47 . Nonlocality inequalities set a bound on the possible strength of nonlocal correlations. Quantum mechanics predicts the existence of entangled states which violate a nonlocality inequality. Previous attempts to experimentally test nonlocality have all presumed unphysical idealizations that do not hold in real experiments, namely, noiseless measurements. In this work, we perform an experimental violation of the CHSH test that is free of the idealizations and rule out local models with high confidence in two scenarios. For the first scenario, we experimentally demonstrate that the CHSH inequality can always be violated for any efficiency η > 1/2 which was originally pointed out by 48,49 . We furthermore test the upper bound on the amount of entanglement needed to violate the CHSH inequality for a given η. Besides, we experimentally study a scenario, which was not previously considered in the literature. We show that the CHSH inequality can always be violated for any nonzero noise parameter of the measurement. Less entanglement exhibits more nonlocality in both of the scenarios. We also provide an application of testing the CHSH inequality with a noisy measurement; that is, the detection of weak entanglement of a two-qubit state. This approach has possible applications for scientific fields wherein quantum effects are important and for developing quantum technologies. For instance, we particularly expect to further develop it for the construction of entanglement witness that only detects nonlocal states with low entanglement.

Detailed explanation of the experimental results
Experimental results show that for η 1= ffiffi ffi 2 p , no maximally entangled states (C 2 = 1) can violate the CHSH inequality. It is straightforward to show why this is true. Note that we can write the expected value Tr ðρB 1 Þ as η Á 2 ðĉ 0 ; T ρb0 Þ cos ðθÞ þ ðĉ 1 ; T ρb1 Þ sin ðθÞ h i þ ð1 À ηÞ Á 2ðĉ 0 cos ðθÞ þĉ 1 sin ðθÞ; r ! Þ for measurements fĉ 0 ;ĉ 1 ;b 0 ;b 1 g, Alice's local Bloch vector r ! , and the correlation matrix T ρ . Since a maximally entangled state has local maximally mixed subsystems, the local Bloch vectors must both be zero for Alice and Bob. So the second term vanishes and we are left with the original term Horodecki et al. obtained in ref. 50 with a factor of η. Therefore, the maximum expected value has to be 2η ffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ 1 þ λ 2 p , where λ 1 , λ 2 are the largest eigenvalues of the symmetric matrix T T ρ T ρ . Due to the positivity of maximally entangled two-qubit states 51 , both eigenvalues must be unit which gives the optimal value of 2η ffiffi ffi 2 p . So maximally entangled states can only violate the CHSH inequality when η > 1= ffiffi ffi 2 p as suggested by the experimental results.

Details on the proposal of detecting weak entanglement
We theoretically propose and experimentally demonstrate how CHSH tests with noisy measurements can be used to detect weak entanglement on two-qubit states. Consider a two-qubit state ρ in Eq. (11) of the main text. It has an expected value Tr ρB 1 ð Þ > 2 if whereφ ;1 g being the Schmidt basis of ϕ 0 j i and λ ± being the Schmidt with f 0 j i; 1 j ig being the computational basis, is the eigenstate of B 1 corresponding to the eigenvalue 2 ffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 þ τ 2 p (τ = 2η − 1) with the directions of the measurements θ A = 0 and θ B ¼ arcsin½ð1 À τÞ 2 =ð1 þ τÞ 2 . The . By doing the singular value decomposition of the matrix form of ϕ 0 j i, given by the isomorphism between four vectors and 2 × 2 matrices, we are able to obtain the transformationφ ¼ 1 V ϕ 0 j i by the use of a local rotation V which is defined in the computational basis as The concurrence of the stateφ can be calculated as For experimental demonstration, we fix the detection efficiency of Bob's measurement at η = 3/4. For simplification, we generate the state in the computational basis with ϕ j i ¼ λ þ 00 j iþλ À 11 j i and Φ À j i ¼ ð 00 j iÀ 11 j iÞ= ffiffi ffi 2 p and the desired state ρ can be obtained by a change of basis, i.e., ρ ¼ 1 Vρ 0 1 V. We choose a total of six values of t = 0, 0.02, 0.038, 0.3, 0.6, 0.83 to generate six different states and fixed s = 1/50 and τ = 1/2.
In the measurement stage, for Alice's side, the setup is the same as that for the pure state testing of the CHSH inequality. For Bob's side, an extra unitary transformation V yΠ B bjy V is applied to perform a change of basis 52,53 . This unitary combines with another rotation that maps the eigenstate of the observable O B y , corresponding to the eigenvalue 1, into H j i and is realized by the HWP at θ B . The rest is the same as that for the pure state testing of the CHSH inequality associated with the Bell operator B 1 .
For the entanglement witness, in our experiment, as we actually generate the state ρ 0 , we measure Tr ðρ 0 W 0 Þ which is equivalent to Tr(ρW),

Experimental implementation of POVMs
To quantify the quality of the experimental realization of the POVMs, we define a modified two-norm distance 38 between the experimentally reconstructed matrix G exp and the ideal one G th as The value of the distance ranges between 0 for a perfect match and ffiffi ffi 2 p for a complete mismatch. To reconstruct the matrix form of G exp , we perform the measurement tomography 54 . More specifically, single photons are prepared in four testing states , and are detected by APDs in coincidence with the trigger photons after passing through the optical setup for realizing a certain POVM element. Photon counts give the measured probabilities. From these, we can obtain the matrix forms of all the elements of the POVMs via maximum-likelihood estimation. In our experiment, all the distances turn out to be smaller than 0.03, which validates the experimental realizations of the POVMs.

Experimental generation of mixed states
To detect weak entanglement, we generate mixed state ρ 0 (18) and fix the parameters as s = 1/50, κ = 1/2, and η = 3/4. The mixed state is then ρ 0 ¼ 1 1þt ð 25 27 ϕ j i ϕ h j þ 1 54 1 þ t Φ À j i Φ À h jÞ, where ϕ j i ¼ 0:851 00 j iþ 0:526 11 j i. We generate the two-photon pure entangled states and introduce quantum noise in a controlled way on one of the state subsystems 55,56 . As illustrated in Fig. 1 of the main text, by tuning the setting angle θ 1 of the HWP, we can generate two-photon pure entangled states with real coefficients. For example, with θ 1 ¼ 1 2 arccosð0:526Þ, ϕ j i is generated via type-I spontaneous parametric down-conversion using two BBO crystals. By introducing quantum noise through a depolarizing channel ε(ρ) = (1 − 3p/4)ρ + p(σ x ρσ x + σ y ρσ y + σ z ρσ z )/4, we can also generate a Werner state ρ W ¼ ð1 À pÞ Φ À j i Φ À h jþ p1=4. The depolarizing channel is introduced in a controlled way by employing two liquid crystal retarders (LCs) in the path of photon A. The LCs act as phase retarders, with the relative phase between the ordinary and extraordinary radiation components depending on the applied voltage. Precisely, V π and V 1 correspond to the case of LCs operating as HWP and as the identity operator, respectively. The two LCs' optical axes are set at 0 ∘ and 45 ∘ with respect to the V polarization. Then, when the voltage is applied, one of the LC acts as a σ z on the single-qubit and the other as σ x . The simultaneous application of V π on both LCs corresponds to the σ y operation. By controlling the activation time and the period of the LCs activation cycle, we can generate the Werner state ρ W with an arbitrary coefficient. Thus, we generate the pure entangled state ϕ j i and the Werner state ρ W respectively, and then by choosing different exposure time for each state, i.e., the time for coincidence measurement for each state, we can generate the mixed state ρ 0 . The ratio between exposure times for ϕ j i and ρ W is 25 27 : ð 2 27 þ tÞ.

DATA AVAILABILITY
Any related experimental background information not mentioned in the text and other findings of this study are available from the corresponding author upon reasonable request.