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,24,25,26 state tomography27,28 and in solving quantum paradoxes29 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.

Consider a von Neumann-type measurement30 on a spin-1/2 particle that is in the superposition state |ψ〉 = α|↑〉 + β|↓〉 with |α|2 + |β|2 = 1, where |↑〉 (|↓〉) denotes the spin up (down) state. After the measurement, the spin state is entangled with the pointer’s state, i.e., |ψ〉 ⊗ |ϕ〉 → α|↑〉 ⊗ |ϕ ↑ 〉 + β|↓〉 ⊗ |ϕ ↓ 〉, where |ϕ〉 is the initial state of the pointer and |ϕ ↑ 〉 (|ϕ ↓ ) indicates the measurement result of spin up (down). By tracing out the state of the pointer, the spin state becomes

$$\rho = F\rho _0 + (1 - F)(\pi _ \uparrow \rho _0\pi _ \uparrow + \pi _ \downarrow \rho _0\pi _ \downarrow ),$$ (1)

where ρ 0 = |ψ〉〈ψ|, π ↑ = |↑〉〈↑|, π ↓ = |↓〉〈↓| and F = 〈ϕ ↓ |ϕ ↑ 〉. The quantity F is called the measurement quality factor because it measures the disturbance of the measurement.18 If F = 0, the spin state is reduced to a completely decoherent state in the measurement eigenbasis, representing a strong measurement; otherwise, if F = 1, there is no measurement at all. For other cases, i.e., F∈ (0, 1), it refers to the measurement with intermediate strength called weak measurement.

Another important quantity associated with weak measurement is the information gain G that is determined by the precision of the measurement.18 In the case of strong measurement, the probability of obtaining the outcome +1 (−1) that corresponds to spin eigenstate |↑〉 (|↓〉) can be calculated by the Born rule P(+1) = Tr(π ↑ ρ 0 ) (P(−1) = Tr(π ↓ ρ 0 )). However, the non-orthogonality of the pointer states 〈ϕ ↑ |ϕ ↓ 〉 ≠ 0 in weak measurement results in ambiguous outcomes. An observer who performs a weak measurement must choose a complete orthogonal set of pointer states {|ϕ +1 〉, |ϕ −1 〉} as reading states to define the outcomes {+1, −1} corresponding to the spin eigenstates {|↑〉, |↓〉}. The probabilities of the outcome ±1 in weak measurement then become P(±1) = Tr(π ↑ ρ 0 )|〈ϕ ±1 |ϕ ↑ 〉|2 + Tr(π ↓ ρ 0 )|〈ϕ ±1 |ϕ ↓ 〉|2. Here, |〈ϕ +1 |ϕ ↑ 〉|2 and |〈ϕ −1 |ϕ ↓ 〉|2 correspond to the probabilities of obtaining the correct outcomes while |〈ϕ −1 |ϕ ↑ 〉|2 and |〈ϕ +1 |ϕ ↓ 〉|2 correspond to the probabilities of the wrong outcomes. For simplicity, we consider the case of symmetric ambiguousness in which |〈ϕ +1 |ϕ ↑ 〉|2 = |〈ϕ −1 |ϕ ↓ 〉|2 and |〈ϕ −1 |ϕ ↑ 〉|2 = |〈ϕ +1 |ϕ ↓ 〉|2, thus, the probabilities of the outcomes can be reformulated as

$$P( \pm 1) = G \cdot \frac{1}{2}[1 \pm Tr(\sigma \rho _0)] + (1 - G) \cdot \frac{1}{2},$$ (2)

where σ = π ↑ − π ↓ defines the spin observable and G = 1−|〈ϕ −1 |ϕ ↑ 〉|2−|〈ϕ +1 |ϕ ↓ 〉|2 represents the precision of the measurement (see more details in Methods). The quality factor F and the precision G are determined solely by the pointer states and satisfy the trade-off relation F2 + G2 ≤ 1.18 A weak measurement is optimal if F2 + G2 = 1 is satisfied.

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_{(x,y)} = \mathop {\sum}

olimits_{a,b} abP(ab|xy)\). The CHSH-Bell test is focused on the so-called S value defined by the combination of correlations

$$S = \left| {C_{(0,0)} + C_{(0,1)} + C_{(1,0)} - C_{(1,1)}} \right|.$$ (3)

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 Full size image

Whereas S ≤ 2 in any local hidden variable theory,4 quantum theory gives a more relaxed bound of \(2\sqrt 2\).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 non-locality 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 \(2\sqrt 2\) of the CHSH-Bell inequality can be attained, the calculation gives (see more details in Methods)

$$S_{A - B1} = 2\sqrt 2 G,S_{A - B2} = \sqrt 2 (1 + F).$$ (4)

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., F2 + G2 < 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) formalism33 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

$$|{\mathrm{\Psi }}_{ \pm 1}\rangle _s = \alpha \langle \phi _{ \pm 1}|\phi _ \uparrow \rangle | \uparrow \rangle + \beta \langle \phi _{ \pm 1}|\phi _ \downarrow \rangle | \downarrow \rangle$$ (5)

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 operators34

$$M_{ \pm 1} = \langle \phi _{ \pm 1}|\phi _ \uparrow \rangle | \uparrow \rangle \langle \uparrow | + \langle \phi _{ \pm 1}|\phi _ \downarrow \rangle | \downarrow \rangle \langle \downarrow |$$ (6)

corresponding to outcome ±1.

In our realization of weak measurement of Bob1 with photonic elements as shown in Fig. 2a, the measured photons are in polarization state and the path degree of freedom of photons is 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.

Fig. 2 Optimal weak measurement realized in a photonic system. a HWP2 and HWP3 are rotated at θ/2 and π/4 − θ/2 degree determining the strength of measurement F = sin2θ. Photons with vertical polarization state |V〉 transmit calcite beam displacer (BD) without change of its path while photons with horizontal polarization state |H〉 suffer a shift away from its original path. HWP1, HWP4 and HWP5 are rotated at the same degree φ/2 to realize weak measurement of polarization observable σ φ = |φ〉〈φ| − |φ⊥〉〈φ⊥|. The measurement outcome +1(−1) is encoded in path 0(1) separately. b The setup, used in actual experiment, realizes same optimal weak measurement as shown above. The only difference is that specific outcome +1(−1) can be selected by rotating HWP1 and HWP4. In the measurement of observation σ φ with HWP1 and HWP4 rotated at φ/2 degree, outcome +1 is obtained when photons comes out of the setup and outcome −1 is obtained when HWP1 and HWP4 rotated at φ/2 + π/4. Note that measurement outcome values are extracted in the final coincidence detection Full size image

The key part of our setup is the realization of weak measurement of observable σ H and this is achieved by interference between calcite beam displacers (BDs) (Fig. 2). Consider photons with polarization state |Φ〉 = α|H〉 + β|V〉 to be measured, after interaction, the composite state of photons becomes |ψ〉 = α|H〉|ϕ H 〉 + β|V〉|ϕ V 〉 with |ϕ H 〉 (|ϕ V 〉) is the corresponding pointer state. The reading states {|ϕ +1 〉, |ϕ −1 〉} in our realization are chosen as states of two separated paths 0 and 1 (Fig. 2a) denoted by |0〉 and |1〉. By rotating HWP2 and HWP3 between BDs at θ/2 and π/4 − θ/2 degrees respectively, the pointer states become

$$\begin{array}{*{20}{l}} {|\phi _H\rangle } \hfill & = \hfill & {{\mathrm{cos}}\theta |0\rangle + {\mathrm{sin}}\theta |1\rangle ,} \hfill \\ {|\phi _V\rangle } \hfill & = \hfill & {{\mathrm{sin}}\theta |0\rangle + {\mathrm{cos}}\theta |1\rangle } \hfill \end{array}$$ (7)

with 0 ≤ θ ≤ π/2. The quality factor and information gain in our case are F = 〈ϕ H |ϕ V 〉 = sin2θ and G = 1 − |〈1|ϕ H 〉|2 − |〈0|ϕ V 〉|2 = cos2θ. The condition of optimal weak measurement F2 + G2 = 1 is satisfied.

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 \((|H\rangle |V\rangle - |V\rangle |H\rangle )/\sqrt 2\) 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 \(\{ (|H\rangle \pm |V\rangle )/\sqrt 2 \}\), achieved by rotating the polarization analyzers for two photons.

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) Full size image

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 \(2\sqrt 2\) can be approached, Alice chooses measurement along direction Z or X, while Bobs choose measurement along \(( - Z + X)/\sqrt 2\) or \(- (Z + X)/\sqrt 2\) 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 \(( - Z + X)/\sqrt 2\) or \(- (Z + X)/\sqrt 2\) direction, respectively. For instance, if HWP1, HWP4 and HWP5 are rotated at −11.25° and HWP6 is fixed at 0°, the three-variable joint conditional probability \(P[a = 1,b_1 = 1,b_2 = 1|x = Z,y_1 = ( - Z + X)/\sqrt 2 ,y_2 = ( - Z + X)/\sqrt 2 ]\) is obtained by the final coincidence detection. The other joint conditional probabilities can be detected via similar various combination of HWP1, HWP4, HWP5 and HWP6.

Five different angles θ = {4°, 16.4°, 18.4°, 20.5°, 28°} are chosen from which the values of θ = {16.4°, 18.4°, 20.5°} are located in the region where double violations are predicted to be observed. In particular, the balanced double violations S A−B1 = S A−B2 = 2.26 are presented under optimal weak measurement when F = 0.6, corresponding to θ = 18.4°. Our final results are shown in Fig. 4, where double violations are clearly displayed at θ = {16.4°, 18.4°, 20.5°} with ~10 standard deviations. Specifically, when θ = 18.4° we obtain S A−B1 = 2.20 ± 0.02 and S A−B2 = 2.17 ± 0.02. Considering the possible statistical error, systematic error and imperfection of our apparatus, these experimental results fit well within the theoretical predictions.