Generation and measurement of arbitrary four-dimensional spatial entanglement between photons in multicore fibers

Hee Jung Lee; Hee Su Park

doi:doi:10.1364/PRJ.7.000019

Photonics Research, 2019, 7 (1): 01000019, Published Online: Feb. 21, 2019

Generation and measurement of arbitrary four-dimensional spatial entanglement between photons in multicore fibers

论文大纲

Hee Jung Lee ^1,2Hee Su Park ^1,*

Author Affiliations

¹ Korea Research Institute of Standards and Science (KRISS), Daejeon 34113, South Korea

² Current address: Agency for Defense Development, Daejeon 34186, South Korea

Abstract

High-dimensional entanglement is a valuable resource for secure and efficient quantum information processing. A major challenge for practical use of multidimensional quantum systems is the establishment of controls over arbitrary superposition states in realistic conditions. This work demonstrates spatially entangled photon pairs propagating through two separate four-core optical fibers with the amplitudes and phases of the superposition being independently controllable. Using quantum state analyzers that can detect arbitrary multicore superposition states, Bell-type CGLMP inequalities in two, three, and four dimensions are directly tested. Enhanced violation of the inequality by slight nonmaximality of entanglement is also demonstrated.

1. INTRODUCTION

Quantum entanglement is a key feature of quantum mechanics that is essential for quantum information processing applications. In recent years, entanglement in more than two dimensions has attracted interest owing to a larger information capacity and better resilience to errors than those of a simplest qubit entanglement ^{[1–5" target="_self" style="display: inline;">–5]}. A stronger violation of Bell inequalities has also been demonstrated in higher dimensions ^[6,7].

Experimental realizations of high-dimensional entanglement with photons have relied on systems exploiting optical paths ^{[8–12" target="_self" style="display: inline;">–12]}, orbital angular momenta ^{[13–16" target="_self" style="display: inline;">–16]}, temporal modes ^[5,17,18], and frequency modes ^[19,20]. While remarkable progress has been made, preparation of arbitrary entangled states whose component coefficients are dynamically controllable remains challenging. Recent studies have demonstrated arbitrary entanglement control in up to 10 and 15 dimensions based on temporal modes ^[20] and spatial modes ^[12], respectively, on integrated-optics platforms.

Spatial-mode encoding has an advantage in that arbitrary unitary operations are relatively straightforward compared with time-bin or frequency encoding ^{[2123" target="_self" style="display: inline;">–23]}. An important technical issue to utilize spatial qudits of photons is that all the spatial modes have to share the optical components to avoid phase instability owing to ambient noise. Multicore fibers (MCFs) have recently been proposed as such common media to carry high-dimensional quantum information based on multiple single-mode cores inside a common cladding ^[11,24,25]. We have previously shown that spatially entangled photon pairs produced by spontaneous parametric downconversion (SPDC) can be transmitted through two separate MCFs ^[11]. Quantum state tomography (QST) based on projection measurements to one- and two-core states verified the entanglement of the output photons after transmission. This work expands this line of research to demonstrate novel techniques to generate arbitrarily controllable high-dimensional spatial entanglement between MCFs and to directly verify the nonclassicality of the photon pairs through measurement of the correlations between N-dimensional $(N > 2)$ superposition states.

2. GENERATION OF ARBITRARY ENTANGLEMENT BETWEEN MCFs

Energy and momentum conservation during SPDC relates the spatial/temporal profile of a pump beam with the quantum state of downconverted photon pairs. For example, spatial pump beam shaping can redistribute correlations between orbital angular momentum states ^[26] or define the dimensionality and spatial parity correlations of the Hermite–Gaussian modes ^[27]. To generate arbitrarily controllable entanglement along MCFs, a multispot pump beam with independent control over the amplitudes and phases of all spots is generated inside an SPDC crystal with a spatial light modulator (SLM), as shown in Fig. 1(a). Photon pairs are produced by noncollinear degenerate type-0 SPDC in a periodically poled lithium niobate (PPLN) crystal (poling period of 19.5 μm, thickness of 1 mm), pumped by a continuous-wave diode laser (wavelength of 775 nm, nominal linewidth $< 1 MHz$ , optical power of 30 mW). The SLM, a reflection-mode multipixel phase shifter (Hamamatsu X10468-02, $792 pixel \times 600 pixel$ , pixel pitch 20 μm), applies a pattern with subsections whose number leads to the number of pump beam spots. The position and phase of each spot are determined by the direction and phase of the grating-like linear phase gradient pattern of each subsection, respectively, as shown in the inset of Fig. 1(a). The brightness and phase of each spot eventually determine the complex coefficient of the corresponding component of the final entangled state. The brightness is controlled by varying the arc angles of the pump beam slices to adjust the relative focused beam powers. The phase is controlled by shifting the grating pattern to the phase gradient direction. Because all the spots share the optical components, the relative phases between the spots are inherently stable, which is critical for maintaining coherence and is not straightforward for multisource schemes. The maximum number of spots with this scheme can be estimated to be the cross-section area of the SPDC crystal divided by the minimum beam spot area to guarantee a Rayleigh range longer than the crystal length.

Schematic of the experimental setup. (a) Arrangement of the optical components used to generate and measure the entangled photon pairs between two multicore fibers. (b) Imaging configuration for the pump beam and photons coupled to the fiber cores (L=250 mm, f1=50 mm, f2=75 mm, and f3=15 mm). Inset: phase patterns on SLM0. SLM, spatial light modulator; L, lens; MCF, multicore fiber; Q, quarter-wave plate; H, half-wave plate; PBS, polarizing beam splitter; IF, interference filter; FC, fiber coupler; SMF, single-mode fiber.

Fig. 1. Schematic of the experimental setup. (a) Arrangement of the optical components used to generate and measure the entangled photon pairs between two multicore fibers. (b) Imaging configuration for the pump beam and photons coupled to the fiber cores ( $L = 250 mm$ , $f_{1} = 50 mm$ , $f_{2} = 75 mm$ , and $f_{3} = 15 mm$ ). Inset: phase patterns on SLM0. SLM, spatial light modulator; L, lens; MCF, multicore fiber; Q, quarter-wave plate; H, half-wave plate; PBS, polarizing beam splitter; IF, interference filter; FC, fiber coupler; SMF, single-mode fiber.

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The end faces of two MCFs (Fibercore SM-4C1550, length of 50 cm, NA of 0.14–0.17, mode field diameter of 7.4–8.5 μm, and single-mode cores at the vertices of a centered $37 μm \times 37 μm$ square) are imaged onto the PPLN crystal to capture the downconverted photons with a center wavelength of 1550 nm. The four core images coincide with the pump beam spots that are aligned with the vertices of a centered square of $185 μm \times 185 μm$ at the crystal. Figure 1(b) shows the lens configuration to ensure parallel propagation of all the beams inside the crystal. The pump laser incident on SLM0 has a $1 / e^{2}$ diameter of 1.3 mm. The distance $L$ (250 mm) is chosen such that the centroids of the four sliced Gaussian beams from SLM0 overlap at the back focal plane of $L_{p}$ considering the beam propagation angles. Each optical path of the downconverted photons includes a $4 f$ configuration ( $L_{2}$ and $L_{3}$ ) that relays images between the focal planes of $L_{1}$ and $L_{f}$ , as shown in Fig. 1(b). In the real implementation, the distance between $L_{1}$ and $L_{2}$ was 130 mm larger than the design length (150 mm); therefore, the four core images propagate with a slight tilt angle (3.0 mrad) toward the optic axis inside the PPLN crystal.

The length of the MCF is practically limited by decoherence between the core modes due to the intercore differential group delay (DGD). We note that the coherence of this entangled state based on a continuous-wave-pumped SPDC can be maintained even when the cores are different only if the two MCFs are identical, and the same cores of the two MCFs are aligned to coincide within the images at the crystal ^[11]. However, the current MCF shows inhomogeneity along the fiber length, which hinders complete entanglement protection by geometric alignment or DGD compensation by core permutation through sequential offset splicings as in Ref. ^[28]. Therefore, the accumulated inhomogeneous DGD has to be kept smaller than the coherence length of photons. Development of entangled photon pair sources with a narrower bandwidth can increase the usable fiber length, possibly up to several hundreds of meters, incorporating an active phase stabilization setup, as shown in the QKD experiments using a faint laser source ^[25].

3. MEASUREMENT SETUP

After transmission through the MCFs in Fig. 1(a), the output polarizations of the photons differ between the cores because of a differential birefringence by fiber curvatures and intrinsic stresses. A quarter-wave plate and a half-wave plate convert the average polarization calculated on the Poincaré sphere into horizontal polarization, and a polarizing beam splitter on each photon path filters out vertical-polarization components. SLM1 and SLM2 project superposition states over multiple cores, whose number is also given by the number of subsections, into single-mode fibers. Fiber-coupled single-photon counters (id Quantique id220, efficiency 20%) and a time-to-amplitude converter module with a coincidence window of 3.5 ns record the coincidence counts of the photon pairs. The wavelength bandwidth of the photons is set by interference filters with a bandwidth of 3 nm (1.8 nm) for photon 1 (photon 2). The asymmetric bandwidth pair is used to suppress fluctuation of the coincidence counts due to drift of the pump laser wavelength.

4. CONTROL OVER THE AMPLITUDE AND PHASE OF SUPERPOSED COMPONENTS

The post-selected two-photon state coupled to the two MCFs is given as where ${| k ⟩}_{i}$ denotes the photon state propagating through core $k$ of $MCF i$ , and $C_{k}$ represents complex constants controlled by the SLM0 pattern. The controllability over the amplitudes of the coefficients is first verified. By changing the number of subsections, two-, three-, and four-dimensional entanglements are produced and measured by QST based on 256 projection measurements and maximum likelihood estimation (see Appendix A for the raw data). The subsections have the same size; therefore, the amplitudes of $C_{k}$ are equal unless the multiple cores show different birefringences or coupling efficiencies. The results are shown in Figs. 2(a)–2(c). The excited core modes are denoted as yellow circles in the insets of Fig. 2. Each projection measurement was performed for 60 s, and the maximum counts were approximately 1000–2000. For comparison purposes, phase offsets of 0.1–0.7 rad have been applied to the ${| k ⟩}_{2}$ ( $k = 1, 2, 3$ ) components of the reconstructed density matrix after the measurements. The fidelities $Tr {{\hat{ρ}}^{1 / 2} | β ⟩ ⟨ β | {\hat{ρ}}^{1 / 2}}$ of the re-phased density matrix $\hat{ρ}$ with the ideal maximally entangled state $| β ⟩ = \sum_{k = 0}^{N - 1} | k ⟩ | k ⟩ / \sqrt{N}$ ( $N = 2, 3, 4$ ) were $0.88 \pm 0.01$ , $0.85 \pm 0.01$ , and $0.87 \pm 0.01$ , respectively. The uncertainties denote the standard deviations due to the Poisson distribution of counts, $\pm \sqrt{counts}$ .

Fig. 2. Reconstructed density matrices by quantum state tomography: (a) $d = 2$ , (b) $d = 3$ , and (c) $d = 4$ .

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The phase controllability of the coefficients of the terms in Eq. (1) is elaborated by measuring the correlations between two-core superposition states, while varying the phases of the pump beam spots. Photon 1 in MCF1 is projected to $({| j ⟩}_{1} + {| k ⟩}_{1}) / \sqrt{2}$ , and photon 2 in MCF2 is projected to $({| j ⟩}_{2} + {| k ⟩}_{2}) / \sqrt{2}$ or $({| j ⟩}_{2} + i {| k ⟩}_{2}) / \sqrt{2}$ , where $(j, k) = (0, 1), (1, 2), (2, 3), (3, 0)$ . A phase factor $e^{i ϕ}$ is added only to $C_{k}$ by adjusting the phase of the grating-like pattern leading to the ${| k ⟩}_{1} {| k ⟩}_{2}$ component on SLM0, and coincidence counts are measured as shown in Fig. 3. The results clearly show interference fringes with an average visibility of $0.91 \pm 0.02$ . The phase references ( $ϕ = 0$ ) have been chosen to maximize all the coincidence counts with projection to $({| j ⟩}_{1} + {| k ⟩}_{1}) ({| j ⟩}_{2} + {| k ⟩}_{2}) / 2$ . These measurements guarantee in-phase superposition, i.e., $\arg (C_{k}) = 0$ for all $k$ , that is required for direct application of the CGLMP inequality formula ^[6]. The core dependence of the visibility is thought to be mainly caused by the DGD fluctuation between the two MCF sections ^[28]. The phases showed no significant temperature dependence; however, fiber bending affected the interference fringes in Fig. 3. Analysis of the differential phase between the cores caused by fiber curvature can be found in Ref. ^[29]. The experiments were performed without mechanical isolation on an optical table.

Quantum correlations between the two-core superposition states, while changing the relative phase of the pump beam spots. Photon 1 in MCF1 is projected onto the state (|j⟩+|k⟩)/2. Photon 2 in MCF2 is projected onto (|j⟩+|k⟩)/2 (red circles) or (|j⟩+i|k⟩)/2 (black squares). The phase ϕ of the two-photon state component |k⟩|k⟩ is scanned from 0 to 2π by a relative phase of the relevant subsection pattern on SLM0. The coincidence counting period was 60 s. (a) j=0, k=1. (b) j=1, k=2. (c) j=2, k=3. (d) j=3, k=0.

Fig. 3. Quantum correlations between the two-core superposition states, while changing the relative phase of the pump beam spots. Photon 1 in MCF1 is projected onto the state $(| j ⟩ + | k ⟩) / \sqrt{2}$ . Photon 2 in MCF2 is projected onto $(| j ⟩ + | k ⟩) / \sqrt{2}$ (red circles) or $(| j ⟩ + i | k ⟩) / \sqrt{2}$ (black squares). The phase $ϕ$ of the two-photon state component $| k ⟩ | k ⟩$ is scanned from 0 to $2 π$ by a relative phase of the relevant subsection pattern on SLM0. The coincidence counting period was 60 s. (a) $j = 0$ , $k = 1$ . (b) $j = 1$ , $k = 2$ . (c) $j = 2$ , $k = 3$ . (d) $j = 3$ , $k = 0$ .

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5. DIRECT TEST OF THE GENERALIZED BELL INEQUALITIES

The entanglement is directly measured by testing a generalized Bell-type (CGLMP) inequality ^[6]. The Bell expression $S_{d}$ is given by a combination of joint measurement probabilities as where $d$ is the dimension of single-photon states. $A_{a}$ and $B_{b}$ ( $a, b = 0, 1$ ) denote measurements of photon 1 and photon 2, respectively, having $d$ possible outcomes: $0, 1, \dots, d - 1$ . $P (A_{a} = B_{b} + k)$ is the joint probability for the outcome of $A_{a}$ to be equal to the outcome of $B_{b}$ plus $k$ modulo $d$ . Measurements $A_{a}$ and $B_{b}$ are, respectively, composed of projections to basis states in Fourier domains ${| v ⟩}_{a}^{A}$ and ${| w ⟩}_{b}^{B}$ ( $v, w = 0, \dots, d - 1$ ): where $α_{0} = 0$ , $α_{1} = 1 / 2$ , $β_{0} = 1 / 4$ , and $β_{1} = - 1 / 4$ ^[6,14]. The maximum value for $S_{d}$ based on local variable theories (LVT) is 2, while the maximally entangled states of the form $(1 / \sqrt{d}) \sum_{k = 0}^{d - 1} {| k ⟩}_{1} {| k ⟩}_{2}$ beat this limit. The number of projection measurements is 16, 36, and 64 for $S_{2}$ , $S_{3}$ , and $S_{4}$ , respectively, in contrast with 256 for QST.

Patterns of $d$ segments are sequentially loaded onto SLM1 and SLM2 with only changing phase combinations. The raw data of the coincidence counts for the $d = 2$ , $d = 3$ , and $d = 4$ cases are provided in Appendix A. Figure 4 compares the experimental results ( $S_{2} = 2.77 \pm 0.06$ , $S_{3} = 2.45 \pm 0.04$ , $S_{4} = 2.41 \pm 0.04$ ) with the local variable theory limit and the theoretical values for the maximally entangled states. All the $S_{d}$ values violate the inequality $S_{d} \leq 2$ by $> 10$ standard deviations. The unit measurement periods for each projection were 60, 100, and 100 s for the three cases, respectively.

Measured Bell-type parameter Sd (squares), compared with the limit by the local variable theories (triangles) and the theoretical values for the maximally entangled states (circles).

Fig. 4. Measured Bell-type parameter $S_{d}$ (squares), compared with the limit by the local variable theories (triangles) and the theoretical values for the maximally entangled states (circles).

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$S_{d}$ can also be calculated from the reconstructed density matrices in Fig. 2 as $S_{2}^{(QST)} = 2.39 \pm 0.08$ , $S_{3}^{(QST)} = 2.43 \pm 0.07$ , and $S_{4}^{(QST)} = 2.45 \pm 0.03$ . The direct estimation had a slightly lower level of uncertainty than the QST if the standard deviations normalized by the square root of the total measurement time were compared. The total measurement time was $60 s \times 256 = 15,360 s$ for the QST and $60 s \times 16 = 960 s$ , $100 s \times 36 = 3600 s$ , and $100 s \times 64 = 6400 s$ for the direct estimation of $S_{2}$ , $S_{3}$ , and $S_{4}$ , respectively. In terms of the number of coincidence counts for a fixed time, the direct estimation of $S_{d}$ detects fewer counts of $2.56 / d^{2}$ on average because the coupling efficiency of the photons in an $N$ -core superposition state to the output single-mode fiber scales as $1 / N$ ^[11], and the QST is composed of coincidences between four single-core and 12 two-core mode projections. The discrepancy between the experimental results and the theoretical maxima is attributed mainly to the above-mentioned dephasing effect between the core modes because unwanted coincidence counts between different cores (i.e., $| i ⟩ | j ⟩$ , $i \neq j$ ) were, on average, less than 1% of one of the major components.

It has been shown theoretically ^{[3032" target="_self" style="display: inline;">–32]} and experimentally ^[33,34] that the high-dimensional Bell-type inequality is maximally violated by slightly nonuniform entangled states. The following four-dimensional entangled state $| Ψ ⟩$ is considered: where $γ$ is a constant. An eigenstate of the Bell-type operator evaluating $S_{4}$ has a $γ$ value of 0.739 and maximizes $S_{4}$ ^[32]. By adjusting the relative areas of the subsections on SLM0 in Fig. 1(a), states with $γ$ values of 1.35, 1, 0.74, and 0.32 were generated, and their $S_{4}$ values were measured as shown in Fig. 5. The optimized entangled state in Fig. 5(c) shows a stronger violation of the inequality by three standard deviations than the maximally entangled state in Fig. 5(b).

Relative amounts of the core-mode components and the Bell parameter S4 in entangled states (|0⟩|0⟩+γ|1⟩|1⟩+γ|2⟩|2⟩+|3⟩|3⟩)/2(1+γ2). Dashed lines denote the design values. (a) γ≈1.35. (b) γ≈1.00. (c) γ≈0.74. (d) γ≈0.32.

Fig. 5. Relative amounts of the core-mode components and the Bell parameter $S_{4}$ in entangled states $(| 0 ⟩ | 0 ⟩ + γ | 1 ⟩ | 1 ⟩ + γ | 2 ⟩ | 2 ⟩ + | 3 ⟩ | 3 ⟩) / \sqrt{2 (1 + γ^{2})}$ . Dashed lines denote the design values. (a) $γ \approx 1.35$ . (b) $γ \approx 1.00$ . (c) $γ \approx 0.74$ . (d) $γ \approx 0.32$ .

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The results in Fig. 5 verify the simultaneous controllability over the phases and amplitudes of the superposed components. All phases of the four components are adjusted to zero based on the measurements shown in Fig. 3 during the CGLMP inequality measurements because the applied inequality formula detects entangled states with in-phase coherent superposition of the components ^[6]. The phase references are experimentally given by the measurement setup for projections to the Fourier basis states in Eq. (3). Fixing the pump phases at different $ϕ$ values in Fig. 3 simply re-phases the coefficients $C_{k}$ in Eq. (1) with an uncertainty of 0.03 rad, which corresponds to the phase resolution of the SLMs. The control accuracy of the amplitudes $| C_{k} |$ was estimated to be 0.03 from the difference between the design amplitudes and the measured results in Fig. 5. The magnitude of unwanted peaks in Fig. 2 [for example, $| 2 ⟩ | 2 ⟩ ⟨ 2 | ⟨ 2 |$ and $| 3 ⟩ | 3 ⟩ ⟨ 3 | ⟨ 3 |$ components in Fig. 2(a) and $| i ⟩ | j ⟩ ⟨ j | ⟨ i |$ components with $i \neq j$ ] manifests the crosstalk between the components, which was $< 0.6 %$ .

The entanglement of an arbitrary rank, as shown in Fig. 2, can be transformed to arbitrary states containing 16 components $| i ⟩ | j ⟩$ ( $i, j = 0, 1, 2, 3$ ) by incorporating mode converters in each photon path. Such mode converters can be realized by interferometric structures (possibly utilizing additional SLMs) with input and output ports being coupled to MCF cores, respectively. Additionally, an interesting proposal has recently been made to switch core modes of an MCF in-line, where acoustically induced gratings realize two-step mode conversion between the core modes via a common cladding mode ^[35]. Incorporation of such spatial-mode conversion techniques together with high-dimensional quantum interference ^[36] will lead to a greater variety of multiphoton multidimensional states.

6. CONCLUSIONS

In conclusion, a direct test of the generalized Bell-type inequality has been performed by spatially entangled photon pairs in up to four dimensions. The amplitudes and phases of the superposed photon-pair states propagating through two MCFs could be controlled by a pump beam-shaping technique utilizing an SLM and accompanying optical imaging configuration. Enhanced violation of the inequality by a four-dimensional nonmaximally entangled state compared with the maximally entangled state has been verified based on controlled entanglement generation and direct measurement of multimode superpositions. We believe that these results provide valuable guidelines to the design of high-dimensional quantum communications and quantum interfaces utilizing a space-division multiplexing framework.

7 Acknowledgment

Acknowledgment. The authors thank Eunjoo Lee for help with the experimental setup.

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1. INTRODUCTION

2. GENERATION OF ARBITRARY ENTANGLEMENT BETWEEN MCFs

3. MEASUREMENT SETUP

4. CONTROL OVER THE AMPLITUDE AND PHASE OF SUPERPOSED COMPONENTS

5. DIRECT TEST OF THE GENERALIZED BELL INEQUALITIES

6. CONCLUSIONS

7 Acknowledgment

Hee Jung Lee, Hee Su Park. Generation and measurement of arbitrary four-dimensional spatial entanglement between photons in multicore fibers[J]. Photonics Research, 2019, 7(1): 01000019.

Generation and measurement of arbitrary four-dimensional spatial entanglement between photons in multicore fibers

1. INTRODUCTION

2. GENERATION OF ARBITRARY ENTANGLEMENT BETWEEN MCFs

3. MEASUREMENT SETUP

4. CONTROL OVER THE AMPLITUDE AND PHASE OF SUPERPOSED COMPONENTS

Fig. 2. Reconstructed density matrices by quantum state tomography: (a) $d = 2$ , (b) $d = 3$ , and (c) $d = 4$ .

5. DIRECT TEST OF THE GENERALIZED BELL INEQUALITIES

Fig. 4. Measured Bell-type parameter $S_{d}$ (squares), compared with the limit by the local variable theories (triangles) and the theoretical values for the maximally entangled states (circles).

6. CONCLUSIONS

7 Acknowledgment

Article Outline

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Generation and measurement of arbitrary four-dimensional spatial entanglement between photons in multicore fibers

1. INTRODUCTION

2. GENERATION OF ARBITRARY ENTANGLEMENT BETWEEN MCFs

3. MEASUREMENT SETUP

4. CONTROL OVER THE AMPLITUDE AND PHASE OF SUPERPOSED COMPONENTS

Fig. 2. Reconstructed density matrices by quantum state tomography: (a) d=2, (b) d=3, and (c) d=4.

5. DIRECT TEST OF THE GENERALIZED BELL INEQUALITIES

Fig. 4. Measured Bell-type parameter Sd (squares), compared with the limit by the local variable theories (triangles) and the theoretical values for the maximally entangled states (circles).

Fig. 5. Relative amounts of the core-mode components and the Bell parameter S4 in entangled states (|0⟩|0⟩+γ|1⟩|1⟩+γ|2⟩|2⟩+|3⟩|3⟩)/2(1+γ2). Dashed lines denote the design values. (a) γ≈1.35. (b) γ≈1.00. (c) γ≈0.74. (d) γ≈0.32.

6. CONCLUSIONS

7 Acknowledgment

Article Outline

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Fig. 2. Reconstructed density matrices by quantum state tomography: (a) $d = 2$ , (b) $d = 3$ , and (c) $d = 4$ .

Fig. 4. Measured Bell-type parameter $S_{d}$ (squares), compared with the limit by the local variable theories (triangles) and the theoretical values for the maximally entangled states (circles).