1. Introduction

In 1935, Einstein, Podolsky, and Rosen (EPR) posed a famous paradox with the purpose of demonstrating the incompleteness of quantum mechanics [1]. In 1964, Bell formulated Bell’s inequality to resolve the EPR paradox, which stated that no hidden variable theory can reproduce all of the predictions of quantum mechanics, or, alternatively, that quantum correlation is beyond classical correlation [2]. Hitherto, all experiments with different physical systems violated Bell-type inequalities, confirming the predictions of quantum mechanics [3–12]. Compared with considering the local hidden variable in Bell’s theorem, in 2003, Leggett [13] proposed an incompatible theorem between the nonlocal hidden variable theory and quantum mechanics, that is Leggett inequality. Which aims to test a family of realistic theories involving non-local hidden variables. Similar to Bell’s inequality, Leggett inequality can also be violated in quantum mechanics for certain measurement parameter regions, which has been demonstrated in photonic systems [14–19], solid-state spin systems [20], and even in high-energy physics [21].

We note that, for demonstrating Leggett’s theorem, the maximally entangled states are usually required. As we know, by selecting suitable measurement bases, the violation of Bell inequality can always occur with arbitrary two-photon entangled pure state. Due to the fact that the restrictive condition of Leggett inequality is different from that of Bell’s inequality. That is the nonlocal hidden-variable for Leggett inequality and the local hidden-variable for Bell’s inequality. Accordingly, some interesting questions arise naturally: is it still possible that the violation of Leggett inequality can occur with arbitrary two-photon pure state? In this article, we leverage nonmaximally orbital angular momentum (OAM) entangled photon pairs to demonstrate the violation of the Leggett inequality. In theory, inspired by Seiler’s idea of geometrizing the Clauser-Horne-Shimony-Holt (CHSH) inequality [23], we construct a geometrical interpretation of Leggett’s parameter and demonstrate that, for nonmaximally entangled states, the problem of maximizing the correlation measure in the Leggett inequality can be reduced as maximizing the sum of an ellipse’s diameter and a semidiameter. In addition, unlike the violation of CHSH inequality can occur with any two-qubit entangled pure state, the violation of the Leggett inequality only occurs under concurrence $2\sqrt 2 - 2 < \mathcal {C} \leqslant 1$. In experiment, we demonstrate the violation of the Leggett inequality with nonmaximal biphoton orbital angular momentum (OAM) entanglement states. Our theoretical predictions and experimental observations reveal that in order to observe the quantum correlation, the requirement of entanglement must be stressed once locality constraint are relaxed.

2. Theoretical scheme

Let us first revisit the Leggett inequality presented by Branciard and co-workers [17]. Consider a general nonlocal model with a hidden variable in which two observers, Alice and Bob, perform measurements ${\mathbf {a}}$ and ${\mathbf {b}}$ on their qubits, respectively. Their outcomes are indicated by $\alpha$ and $\beta$ $\left ( {\alpha,\beta = \pm 1} \right )$. The conditional probability distribution $P\left ( {\alpha,\beta \left | {{\mathbf {a}},{\mathbf {b}}} \right.} \right )$ can be decomposed into a statistical mixture of correlations predetermined by the hidden variable $\lambda$,

In light of ${{\mathbf {b}}_i} - {\mathbf {b}}^\prime _i = 2\sin \left ( {\varphi /2} \right ){{\mathbf {e}}_i}$, we can also obtain ${{\mathbf {b}}_i} + {\mathbf {b}}^\prime _i = 2\cos \left ( {\varphi /2} \right ){{\mathbf {e}}^\prime _i}$, with ${{\mathbf {e}}^\prime _i} = {{\mathbf {e}}_j}$ for any $i,j = 1,2,3{\text { }}\left ( {i \ne j} \right )$. Within the properties of trigonometric functions, $\mathcal {I}$ can be rewritten as,

Recent years have also witnessed a growing interest in the geometrically characterizing the quantum system [25]. Such as revealing the geometry of multidimensional quantum systems [26,27] and probing the geometry of correlation matrices with randomized measurements [28]. Particularly, we note that, there are several attempts to connect the Bell’s inequality with the structure of a parallelogram [23,29]. Inspiringly, here we construct a geometrical interpretation of the Leggett inequality. As shown in Fig. 1(a), the vectors ${{\mathbf {a}}_i}{\text { }}\left ( {i = 1,2,3} \right )$ end on the Bloch sphere of subsystem A, while the vectors ${{\mathbf {b}}_i},{{\mathbf {b}}^\prime _i},{{\mathbf {e}}^\prime _i}{\text { }}\left ( {i = 1,2,3} \right )$ terminate on the Bloch sphere of subsystem B. Two Bloch sphere are connected by the correlation matrix $K$. The Leggett inequality contains six possible pairwise combinations of one vector ${{\mathbf {a}}_i}$ from subsystem A and one vector ${{\mathbf {b}}_i}$ or ${{\mathbf {b}}^\prime _i}$ from subsystem B. According to Eq. (6), the violation of the Leggett inequality can be characterized by $L$, which comprises three possible pairwise combinations of one vector ${{\mathbf {a}}_i}$ from subsystem A and one vector ${{\mathbf {e}}^\prime _i}$ from subsystem B.

 

Fig. 1. Geometric interpretation of the Leggett inequality. The relationship between the vectors on the two Bloch spheres is established by (a) the correlation matrix $K$ and (b) the identity matrix $\mathbf {1}$. The vectors ${{\mathbf {a}}_i}$ (blue) as well as ${{\mathbf {b}}_i},{{\mathbf {b}}^\prime _i}$ (red) are unit vectors on the Bloch spheres of the subsystems A and B, respectively, representing the desired measurement in the case of concurrence $0.5 \leqslant \mathcal {C} \leqslant 1$. The unit vectors ${{\mathbf {e}}^\prime _i}$ (orange) describe the directions of ${{\mathbf {b}}_i} + {{\mathbf {b}}^\prime _i}$. Due to the matrix $K$, the new vectors ${{\mathbf {e}}^\prime _{Ki}}$ point to the surface of a prolate spheroid.

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For simplicity, we defined new vectors ${{\mathbf {e}}^\prime _{Ki}} \equiv K{{\mathbf {e}}^\prime _i}$, which are no longer unit vectors but instead reside on an ellipsoid with semiaxes of length $\mathcal {C}$ and $1$, as illustrated in Fig. 1(b). In this context, the correlation matrix $K$ is eliminated, and the relationship between the Bloch sphere of subsystem A and the prolate spheroid of subsystem B is established by the identity matrix $\mathbf {1}$. Based on these new vectors, $L$ can be rewritten as $L = \sum \nolimits _{i = 1}^3 {{{\mathbf {a}}_i} \cdot {{\mathbf {e}}^\prime _{Ki}}} = \sum \nolimits _{i = 1}^3 {\left | {{{\mathbf {a}}_i}} \right |\left | {{{\mathbf {e}}^\prime _{Ki}}} \right |\cos {\omega _i}}$, where ${\omega _i}$ are the angles between ${{\mathbf {a}}_i}$ and ${{\mathbf {e}}^\prime _{Ki}}$. Then we can easily maximize the vectors ${{\mathbf {a}}_i}$ by setting ${\omega _1} = {\omega _2} = {\omega _3} = 0$, i.e., ${{\mathbf {a}}_1} = {{\mathbf {e}}^\prime _{K1}}/\left | {{\mathbf {e}}^\prime _{K1}} \right |$, ${{\mathbf {a}}_2} = {{\mathbf {e}}^\prime _{K2}}/\left | {{\mathbf {e}}^\prime _{K2}} \right |$, ${{\mathbf {a}}_3} = {{\mathbf {e}}^\prime _{K3}}/\left | {{\mathbf {e}}^\prime _{K3}} \right |$, with which $L$ can be optimized as:

Here, we set a generally orthogonal basis as ${{\mathbf {e}}_1} = \left ( {\cos \theta \cos \phi,\cos \theta \sin \phi, - \sin \theta } \right )$, ${{\mathbf {e}}_2} = \left ( { - \sin \phi,\cos \phi,0} \right )$, ${{\mathbf {e}}_3} = \left ( {\sin \theta \cos \phi,\sin \theta \sin \phi,\cos \theta } \right )$, leading to three possible outcomes of $\left | {{{\mathbf {e}}^\prime _{Ki}}} \right |$, namely $\left | {{{\mathbf {e}}_{K1}}} \right | \equiv \left | {K{{\mathbf {e}}_1}} \right | = \sqrt {{\mathcal {C}^2}{{\cos }^2}\theta + {{\sin }^2}\theta }$, $\left | {{{\mathbf {e}}_{K2}}} \right | \equiv \left | {K{{\mathbf {e}}_2}} \right | = \mathcal {C}$, $\left | {{{\mathbf {e}}_{K3}}} \right | \equiv \left | {K{{\mathbf {e}}_3}} \right | = \sqrt {{\mathcal {C}^2}{{\sin }^2}\theta + {{\cos }^2}\theta }$. For the sake of simplicity, we restrict $\theta$ at $0 \leqslant \theta \leqslant \pi /4$, which ensures the condition that $\left | {{{\mathbf {e}}_{K3}}} \right | \geqslant \left | {{{\mathbf {e}}_{K1}}} \right | \geqslant \left | {{{\mathbf {e}}_{K2}}} \right |$. It’s worth noting that this constraint can always be fulfilled, as for $\theta > \pi /4$, we can simply exchange the values of $\left | {{{\mathbf {e}}_{K3}}} \right |$ and $\left | {{{\mathbf {e}}_{K1}}} \right |$. Thereby, by selecting ${{\mathbf {e}}^\prime _{K1}} = {{\mathbf {e}}^\prime _{K2}} = {{\mathbf {e}}_{K3}}$ and ${{\mathbf {e}}^\prime _{K3}} = {{\mathbf {e}}_{K1}}$, we can obtain the optimal ${L_m}$,

Since the vector ${{\mathbf {e}}_{K2}}$ resides in the XOY plane, ${{\mathbf {e}}_{K3}}$ and ${{\mathbf {e}}_{K1}}$ naturally span a two-dimensional plane $P$ in $\xi OZ$ Plane, as depicted in the left-hand side of Fig. 2. This plane intersects with the ellipsoid, and thus forms an ellipse characterized by a semiminor axis of length $\mathcal {C}$ and a semimajor axis of length 1, as depicted on the right-hand side of Fig. 2. In this ellipse, the vectors ${{\mathbf {K}}_3}{{\mathbf {K}}^\prime _3}$ and ${\mathbf {O}}{{\mathbf {K}}_1}$ possess the lengths $2\left | {{{\mathbf {e}}_{K3}}} \right |$ and $\left | {{{\mathbf {e}}_{K1}}} \right |$, respectively. Thus, the optimization problem can be further refined as the pursuit of the maximum length of $\left | {{{\mathbf {K}}_3}{{\mathbf {K}}^\prime _3}} \right | + \left | {{\mathbf {O}}{{\mathbf {K}}_1}} \right |$, while adhering to the constraint that the eccentric angles of points $K_3$ and $K_1$ sum to $\pi /2$. Mathematically, this problem can be solved by setting the gradient of ${L_m}$ equal to zero, i.e., $\partial {L_m}\left ( \theta \right )/\partial \theta = 0$. After some algebra, we derive that the maximum value is $2+\mathcal {C}$, which is achieved at $\theta = 0$ when $0.5 \leqslant \mathcal {C} \leqslant 1$. In the scenario where $0 \leqslant \mathcal {C} < 0.5$, the maximal value is found to be $\sqrt {5\left ( {1 + {\mathcal {C}^2}} \right )}$, and this maximum occurs when $\theta = \arctan \left ( {\sqrt {\left ( {4{\mathcal {C}^2} - 1} \right )/\left ( {{\mathcal {C}^2} - 4} \right )} } \right )$. As a result, the maximum value of ${\mathcal {I}}$ is

 

Fig. 2. Geometrical interpretation of ${L_m}$ as the sum of an ellipse’s diameter and semidiameter. We consider subsystem $B$ of Fig. 1(b) represented on the left by the prolate spheroid. The vectors ${{\mathbf {e}}_{K1}}$ and ${{\mathbf {e}}_{K3}}$ span a plane $P$ which cuts the spheroid in an ellipse with a semiminor axis of length $\mathcal {C}$ and a semimajor axis of length 1. The vectors ${{\mathbf {e}}_{K1}}$ and $2{{\mathbf {e}}_{K3}}$ touch the ellipse in the points ${K_1}$, ${K_3}$ and ${K'_3}$, respectively. ${\theta _{{E_1}}}$ and ${\theta _{{E_3}}}$ are eccentric anomalies of points ${K_1}$ and ${K_3}$, respectively, satisfying ${\theta _{{E_1}}} + {\theta _{{E_3}}} = \pi /2$.

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Fig. 3. Theoretical maximum violation of the Leggett inequality for nonmaximally entangled states: (a) pure state $\left | \Psi \right \rangle$, (b) mixed states $\hat \rho _M$ and $\hat \rho _w$. The dashed red line indicates the bound of the Leggett inequality, while the solid blue line, solid green line, and solid violet line denote the quantum mechanical prediction of pure state $\left | \Psi \right \rangle$, mixed state $\hat \rho _M$ with $\mathcal {C} = \mathcal {D} = 1$, and mixed state $\hat \rho _w$, respectively (more details see Appendix A).

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3. Experimental setup and results

Compared with the spin angular momentum of light, the OAM of light exhibits an inherent capacity for high-dimensional quantum information processing [30–32]. The nonlocal feature of OAM entanglement was verified by using the generalized Bell inequalities [33] and Hardy’s paradox [34]. Here, we adopt two-dimensional OAM entanglement states to prove the aforementioned theory. Our experimental setup is shown in Fig. 4. The 140 fs pulsed ultraviolet Gaussian pump beam centered at 355 nm is guided to pump a 3-mm-thick BBO crystal, with type-I and collinear phase matching condition. The frequency-degenerated photon pairs at 710 nm are generated via spontaneous parametric down-conversion (SPDC), and then separated by a beam splitter (BS) to illuminate two spatial light modulators (SLMs) respectively. Note that, here we use a 4f imaging system constituted by lens 1 (L1, $f1$ = 200 mm) and lens 2 (L2, $f2$ = 400 mm) to image the output face of the BBO crystal onto the SLMs, which is routinely adopted for detecting OAM entanglement. Two additional lenses L3 and L4 ($f3$ = 500 mm and $f4$ = 2 mm) are used to re-image two SLMs onto the face of single-mode fiber (SMF). Here, the SLM and SMF serve as the OAM mode filter [35] to select the desired OAM modes via loading the corresponding computational hologram grating (see the insets of Fig. 4). Subsequently, the filtered photons incident into the single photon counting model (SPCM) connected with a coincidence circuit to acquire the coincidence counts.

 

Fig. 4. Experimental setup for implementing Leggett inequalities with nonmaximally entangled OAM states. The top and bottom insets are examples of phase and intensity of measurement modes and the desired holograms projected on both SLMs.

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The generated OAM entangled state via SPDC can be written as ${\left | \psi \right \rangle _{{\text {SPDC}}}} = \sum \nolimits _\ell {{c_\ell }} {\left | \ell \right \rangle _A}{\left | { - \ell } \right \rangle _B}$, where ${{c_\ell }}$ indicates the probability amplitude of finding one signal photon (A) with $\ell \hbar$ OAM and its partner idler photon (B) with $- \ell \hbar$ OAM [36]. By restricting our attention to the two-dimensional subspace spanned by OAM modes of $\ell = + 1, - 1$, we prepare the maximally entangled state ${\left | \psi \right \rangle _{AB}} = \sqrt {\frac {1}{2}} \left ( {{{\left | { + 1} \right \rangle }_A}{{\left | { - 1} \right \rangle }_B} + {{\left | { - 1} \right \rangle }_A}{{\left | { + 1} \right \rangle }_B}} \right ).$ In order to prepare nonmaximally entangled states for examining the aforementioned theory, we employ the Procrustean method of entanglement concentration [37] to modify the initial state into the desired one,

Here, $\mathcal {C}$ represents the concurrence of OAM entanglement. Without loss of generality, we select the states ${\left | \psi \right \rangle _{AB}}$ with $\mathcal {C}$ = 1, 0.95, 0.9, and 0.85 to experimentally observe the violation of the Leggett inequalities. Specifically, we adjust the contrast of the blazed phase grating to modulate the diffraction efficiency for ${\left | { - 1} \right \rangle _A}$ mode and ${\left | { + 1} \right \rangle _B}$ mode. Consequently, we compensate the weight amplitudes of each OAM mode between the target pure states and the experimental states [33,34]. Under this subspace, we can devise a Bloch sphere comparable to the Poincaré sphere for polarization [19], as depicted in Fig. 5. On our Bloch sphere, The vectors $\mathbf {a}$ and $\mathbf {b}$ correspond to the states

 

Fig. 5. The OAM Bloch sphere of Alice’s side or Bob’s side. On the Bloch sphere, the states at the poles are $\left | { \pm 1} \right \rangle$, and all other states on the Bloch sphere are complex superpositions of $\left | { \pm 1} \right \rangle$. Each point (green dot) on this sphere represents a state determined by equation (11). The insets are examples of intensity of measurement modes.

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Fig. 6. Experimental violation of Leggett inequalities for pure two-qubit states with $\varphi$ range from $0^\circ$ to $90^\circ$. The gray points with error bars represent the experimental $\mathcal {I}$, while the blue solid lines depict the theoretically predicted violation based on nonmaximally entangled states. Additionally, the red dashed lines represent the Local Hidden Variables limit in comparison with the experimental results. The experimental data closely match the prediction of quantum mechanics and that the inequality is violated over a wide range of angles.

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4. Conclusion

In summary, we have investigated the quantumness of nonmaximal OAM entanglement systems within the framework of Leggett’s nonlocal model. Theoretically, we construct a geometrical interpretation of Leggett’s model and show that the problem of maximizing the correlation measure in the Leggett inequality can be reduced to maximizing the total of an ellipse’s diameter and semi-diameter for arbitrary entangled states. Consequently, we find the boundaries of quantum correlation and the limitations imposed by non-local hidden variables and demonstrate that the falsification of Leggett’s model requires a more robust entanglement than that of Bell’s theory. Experimentally, we have demonstrated the violation of the Leggett inequalities using two-photon entangled states residing in two-dimensional OAM subspaces with flexibly adjustable concurrence. The experimental results are in conflict with Leggett’s model, thus excluding a particular class of non-local hidden variable theories in the framework of nonmaximal entanglement. As the practical state in realistic environments is usually nonmaximal entanglement, our findings may contribute to the practical implementation of quantum information science protocols, such as quantum key distribution [38] and quasi-deterministic secure quantum communication [39].

Appendix A: the violation of Leggett inequality by mixed states

In this Appendix, we investigate the violation of Leggett inequality by two-qubit mixed states ${\hat \rho _{AB}}$. The correlation matrix is reformulated as $K \equiv {\text {tr}}\left ( {{{\hat \rho }_{AB}} \cdot {{{\mathbf {\hat \sigma }}}_A} \cdot {{{\mathbf {\hat \sigma }}}_B}} \right )$. According to Ref. [23], any mixed state can also transform the Bloch sphere of subsystem B into an ellipsoid, generally exhibiting less symmetry than the prolate spheroid associated with pure states. The optimization strategy remains consistent with that of pure states, discussed in Sec. 2. The crucial distinction lies in the correlation matrix, transitioning from $K = {\text {diag}}\left ( {\mathcal {C},\mathcal {C}, - 1} \right )$ to $K = {\text {diag}}\left ( {a,b,c} \right )$, facilitating the transformation of the Bloch sphere of subsystem B into an ellipsoid with three distinct semi-axes $a$, $b$, and $c$. The orientation of the ellipsoid with respect to our coordinate system and the lengths $a$, $b$, and $c$ depend on the state. On this ellipsoid with a generally orthogonal basis $\{{{\mathbf {e}}_1},{{\mathbf {e}}_2},{{\mathbf {e}}_3}\}$, we also obtain three possible outcomes of $\left | {{{\mathbf {e}}^\prime _{Ki}}} \right |$: $\left | {{{\mathbf {e}}_{K1}}} \right | = \sqrt {{a^2}{{\cos }^2}\theta {{\cos }^2}\phi,{\text { }}{b^2}{{\cos }^2}\theta {{\sin }^2}\phi,{\text { }}{c^2}{{\sin }^2}\theta }$, $\left | {{{\mathbf {e}}_{K2}}} \right | = \sqrt {a^2{{\sin }^2}\phi + b^2{{\cos }^2}\phi }$ , $\left | {{{\mathbf {e}}_{K3}}} \right | = \sqrt {{a^2}{{\sin }^2}\theta {{\cos }^2}\phi,{\text { }}{b^2}{{\sin }^2}\theta {{\sin }^2}\phi,{\text { }}{c^2}{{\cos }^2}\theta }$. For a fix set of $\theta$ and $\phi$, $\mathcal {I}^{mix}\left ( {\theta,\phi } \right ) = 2\sqrt {L^{mix}\left ( {\theta,\phi } \right ) + 1}$, where ${L^{mix}}\left ( {\theta,\phi } \right ) = 2\max \left ( {\left | {{{\mathbf {e}}_{K1}}} \right |,\left | {{{\mathbf {e}}_{K2}}} \right |,\left | {{{\mathbf {e}}_{K3}}} \right |} \right ) + {\text {secondmax}} \left ( {\left | {{{\mathbf {e}}_{K1}}} \right |,\left | {{{\mathbf {e}}_{K2}}} \right |,\left | {{{\mathbf {e}}_{K3}}} \right |} \right )$. Consequently, the problem of maximizing $\mathcal {I}^{mix}$ can also be reduced to maximizing the total of an ellipse’s diameter and semi-diameter. Hereafter we denote by $\mathcal {I}_m^{mix}$ the optimal value of Leggett inequality by ranging over $\theta$ and $\phi$ from 0 to $2\pi$ with a given mixed state. In the following, we present two examples of entangled two-qubit mixed states to demonstrate that at a similar level of concurrence, Leggett inequalities are not violated.

Firstly, we consider the mixed state ${\hat \rho _M} = p\left | \Psi \right \rangle \left \langle \Psi \right | + \left ( {1 - p} \right )\left | \Phi \right \rangle \left \langle \Phi \right |$ comprising the pure state $\left | \Psi \right \rangle$ in Eq. (3) with concurrence $\mathcal {C}$ and probability $p$, and $\left | \Phi \right \rangle = \sqrt {\frac {{1 + \sqrt {1 - {\mathcal {D}^2}} }}{2}} {\left | 0 \right \rangle _A}{\left | 0 \right \rangle _B} + \sqrt {\frac {{1 - \sqrt {1 - {\mathcal {D}^2}} }}{2}} {\left | 1 \right \rangle _A}{\left | 1 \right \rangle _B}$ with concurrence $\mathcal {D}$ and probability $1-p$. The correlation matrix here can be rewritten as $K = {\text {diag}}\left ( {p\mathcal {C} + \left ( {1 - p} \right )\mathcal {D}{\text {,}}p\mathcal {C} - \left ( {1 - p} \right )\mathcal {D}{\text {,}}1 - 2p} \right )$. Based on the numeric strategy mentioned above with mixed state ${\hat \rho _M}$, we calculate $\mathcal {I}_m^{M}$ with a fix set of $\left \{ {p,\mathcal {C},\mathcal {D}} \right \}$. For instance, assuming $\mathcal {C} = \mathcal {D} = 1$, the correlation matrix becomes $K = {\text {diag}}\left ( {1,2p - 1, - \left ( {2p - 1} \right )} \right )$, akin to that of pure states. Then we directly obtained

(12)$${\mathcal{I}_m} = \left\{ {\begin{array}{cl} {2\sqrt {{{\left( {2q + 1} \right)}^2} + 1}, \quad\qquad\qquad {\text{if 0.75}} \leqslant \mathcal{C} \leqslant {\text{1}}}, \\ {2\sqrt {5\left( {1 + {{\left( {2q - 1} \right)}^2}} \right) + 1}, \qquad\quad{\text{if 0.25}} \leqslant \mathcal{C} < {\text{0.75}}}, \\ {2\sqrt {{{\left( {3 - 2q} \right)}^2} + 1}, \quad\qquad\qquad{\text{if 0}}\leqslant \mathcal{C} < {\text{0.25}}}. \end{array}} \right.$$

Consequently, the violation of the Leggett inequality is constrained to the interval $\sqrt 2 - 0.5 < p \leqslant 1$ and $0 \leqslant p < 1.5 - \sqrt 2$. In this case, Bell’s inequality can not be violated only at $p=0.5$. Another scenario arises when $p = \mathcal {C} = \mathcal {D}$, resulting in the correlation matrix $K = {\text {diag}}\left ( {p,2{p^2} - p,1 - 2p} \right )$. Applying the aforementioned numeric strategy reveals that the Leggett inequality violation occurs only within the interval $0.9571 < p \leqslant 1$. In this case, Bell’s inequality can be violated within the interval $0.8 < p \leqslant 1$.

Secondly, we consider the famous Werner state ${\hat \rho _w} = p\left | {{\psi ^ - }} \right \rangle \left \langle {{\psi ^ - }} \right | + \frac {{\left ( {1 - p} \right )}}{4}{\mathbf {\hat 1}}$. This state represents a mixture of the maximally entangled singlet state $\left | {{\psi ^ - }} \right \rangle$ with probability $p$ and the completely mixed state $\mathbf {\hat 1}/4$ in four dimensions with probability $1-p$. The correlation matrix is $K = {\text {diag}}\left ( { - p{\text {,}} - p{\text {,}} - p} \right )$, transforming the Bloch sphere of subsystem B into a smaller sphere with radius $p$. Consequently, the optimal measurement strategies align with those of the maximally entangled pure state. From Eq. (9), we derive $\mathcal {I}_m^w = 2\sqrt {9{p^2} + 1}$, indicating that the violation of the Leggett inequality is confined to the interval $2\sqrt 2 /3 < p \leqslant 1$. In this case, Bell’s inequality can be violated within the interval $\sqrt 2 /2 < p \leqslant 1$.

Appendix B: desired measurement settings of the Leggett inequality for pure states

In this Appendix, we obtain the optimal measurement settings to achieve the maximal value of Leggett’s parameter. In the case of $0.5 \leqslant \mathcal {C} \leqslant 1$, the maximum violation occur at $\theta = 0$ and the optimal basis can be rewritten as ${{\mathbf {e}}_1} = \left ( {\cos \phi,\sin \phi,0} \right )$, ${{\mathbf {e}}_2} = \left ( { - \sin \phi,\cos \phi,0} \right )$, ${\text { }}{{\mathbf {e}}_3} = \left ( {0,0,1} \right )$. For simplicity, we choose $\phi = 0$, i.e., ${{\mathbf {e}}_1} = {\mathbf {x}},{{\mathbf {e}}_2} = {\mathbf {y}},{{\mathbf {e}}_3} = {\mathbf {z}}$ , to construct the desired measurements. Combining with ${{\mathbf {e}}^\prime _1} = {{\mathbf {e}}^\prime _2} = {{\mathbf {e}}_3},{{\mathbf {e}}^\prime _3} = {{\mathbf {e}}_1}$ , we obtain the optimal measurement setting (see Fig. 1(a)):

(13)$$\begin{array}{ll} {{{\mathbf{a}}_1} = {\mathbf{z}},} & {{{{{\mathbf{b}}_1}} \mathord{\left/ {\vphantom {{{{\mathbf{b}}_1}} {{\mathbf{b}}^\prime_1}}} \right. } {{\mathbf{b}}^\prime_1}} = \left( { \pm \sin \frac{\varphi }{2},0,\cos \frac{\varphi }{2}} \right),} \\ {{{\mathbf{a}}_2} = {\mathbf{z}},} & {{{{{\mathbf{b}}_2}} \mathord{\left/ {\vphantom {{{{\mathbf{b}}_2}} {{\mathbf{b}}^\prime_2}}} \right. } {{\mathbf{b}}^\prime_2}} = \left( {0, \pm \sin \frac{\varphi }{2},\cos \frac{\varphi }{2}} \right),} \\ {{{\mathbf{a}}_3} = {\mathbf{x}},} & {{{{{\mathbf{b}}_3}} \mathord{\left/ {\vphantom {{{{\mathbf{b}}_3}} {{\mathbf{b}}^\prime_3}}} \right. } {{\mathbf{b}}^\prime_3}} = \left( {\cos \frac{\varphi }{2},0, \pm \sin \frac{\varphi }{2}} \right).} \end{array}$$

In the case of $0 \leqslant \mathcal {C} < 0.5$, the maximum violation occur at $\theta = \arctan \left ( \delta \right )$ and the optimal basis can be rewritten as ${{\mathbf {e}}_1} = \left ( {\cos \phi, \sin \phi, - \delta } \right )/\sqrt {1 + {\delta ^2}}$, ${{\mathbf {e}}_2} = \left ( { - \sin \phi,\cos \phi,0} \right )$, ${{\mathbf {e}}_3} = \left ( {\delta \cos \phi,\delta \sin \phi,1} \right )/\sqrt {1 + {\delta ^2}}$, where $\delta = \left ( {\sqrt {\left ( {4{\mathcal {C}^2} - 1} \right )/\left ( {{\mathcal {C}^2} - 4} \right )} } \right )$. For simplicity, we choose $\phi = \pi /2$, i.e., ${{\mathbf {e}}_1} = \left ( {0,{\text { }}1,{\text { }} - \delta } \right )/\sqrt {1 + {\delta ^2}}$, ${{\mathbf {e}}_2} = - {\mathbf {x}}$, ${{\mathbf {e}}_3} = \left ( {0,{\text { }}\delta,{\text { }}1} \right )/\sqrt {1 + {\delta ^2}}$, to construct the desired measurements. Combining with ${{\mathbf {e}}^\prime _1} = {{\mathbf {e}}^\prime _2} = {{\mathbf {e}}_3}$, ${{\mathbf {e}}^\prime _3} = {{\mathbf {e}}_1}$, we obtain the optimal measurement setting:

(14)$$\begin{array}{lc} {{{\mathbf{a}}_1} = \left( {0,\frac{{ - \mathcal{C}\delta }}{{\sqrt {{\mathcal{C}^2}{\delta ^2} + 1} }},\frac{1}{{\sqrt {{\mathcal{C}^2}{\delta ^2} + 1} }}} \right),} & {{{{{\mathbf{b}}_1}} \mathord{\left/ {\vphantom {{{{\mathbf{b}}_1}} {{\mathbf{b}}^\prime_1}}} \right. } {{\mathbf{b}}^\prime_1}} = \left( {0,\sin \left( {\gamma \pm \frac{\varphi }{2}} \right),\cos \left( {\gamma \pm \frac{\varphi }{2}} \right)} \right),} \\ {{{\mathbf{a}}_2} = \left( {0,\frac{{ - \mathcal{C}\delta }}{{\sqrt {{\mathcal{C}^2}{\delta ^2} + 1} }},\frac{1}{{\sqrt {{\mathcal{C}^2}{\delta ^2} + 1} }}} \right),} & {{{{{\mathbf{b}}_2}} \mathord{\left/ {\vphantom {{{{\mathbf{b}}_2}} {{\mathbf{b}}^\prime_2}}} \right. } {{\mathbf{b}}^\prime_2}} = \left( { \mp \sin \frac{\varphi }{2},\sin \gamma \cos \frac{\varphi }{2},\cos \gamma \cos \frac{\varphi }{2}} \right),} \\ {{{\mathbf{a}}_3} ={-} \left( {0,\frac{\mathcal{C}}{{\sqrt {{\mathcal{C}^2} + {\delta ^2}} }},\frac{\delta }{{\sqrt {{\mathcal{C}^2} + {\delta ^2}} }}} \right),} & {{{{{\mathbf{b}}_3}} \mathord{\left/ {\vphantom {{{{\mathbf{b}}_3}} {{\mathbf{b}}^\prime_3}}} \right. } {{\mathbf{b}}^\prime_3}} = \left( {0,\cos \left( {\frac{\varphi }{2} \mp \gamma } \right), \pm \sin \left( {\frac{\varphi }{2} \mp \gamma } \right)} \right),} \end{array}$$

where $\gamma = \arccos \left ( {1/\sqrt {1 + {\delta ^2}} } \right )$.

Appendix C: extended experimental results

In this Appendix, we supplement two groups of experiment results. First, we vary the angle $\varphi$ range from $0^\circ$ to $90^\circ$ and compare the result to the maximal value of $\mathcal {I}$ permitted by Leggett’s mode while fixing concurrence $\mathcal {C}$=0.9, 0.6 and 0.3. The experimental observations show a good agreement with the quantum-mechanical predictions, as shown in Fig. 7.

 

Fig. 7. Experimental results for the observed $\mathcal {I}$ for pure two-qubit states with concurrence $\mathcal {C}$=0.9, 0.6, 0.3, respectively. The red dashed line is the bound arising from Leggett’s model, while the cubes are experimental results. The measurement results are in reasonable agreement with the quantum-mechanical predictions (blue solid line).

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Second, we fix $\varphi = 0$ and vary $\mathcal {C}$ from 0 to 1 to observe $\mathcal {I}\left ( {\varphi =0} \right ) = 2L$. For this, we take the desired OAM measurement states in Appendix B. For $\mathcal {C} \geqslant 0.2$, the experimental observations show a good agreement with the quantum-mechanical predictions, as shown in Fig. 8. However, due to the imperfect method of entanglement concentration, very low entangled states ($\mathcal {C}< 0.2$) cannot be precisely generated, thereby leading to the deviation between the experimental results and theoretical predictions.

 

Fig. 8. Experimental results for the observed $\mathcal {I}$ for pure two-qubit state at $\varphi = 0$ with concurrence $\mathcal {C}$ range from 0 to 1. The measurement results (gray) are in reasonable agreement with the quantum-mechanical predictions (blue solid line).

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Funding

National Natural Science Foundation of China (12034016, 12205107, 61975169); Scientific Research Funds of Huaqiao University (605-50Y21054); Natural Science Foundation of Xiamen City (3502Z20227033); Natural Science Foundation of Fujian Province (2021J02002); Distinguished Young Scientists (2015J06002); Program for New Century Excellent Talents in University (NCET-13-0495).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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