Here we report the first demonstration of entanglement distribution over a record distance of 200 km which is of sufficient fidelity to realize secure communication. In contrast to previous entanglement distribution schemes, we use detection elements based on practical avalanche photodiodes (APDs) operating in a self-differencing mode. These APDs are low-cost, compact and easy to operate requiring only electrical cooling to achieve high single photon detection efficiency. The self-differencing APDs in combination with a reliable parametric down-conversion source demonstrate that entanglement distribution over ultra-long distances has become both possible and practical. Consequently the outlook is extremely promising for real world entanglement-based communication between distantly separated parties.
© 2009 OSA
Quantum states that cannot be separated into product states of their isolated systems are said to be entangled. These entangled states are well known to exhibit non-local effects  and Bell inequalities have been derived to describe the degree of entanglement not explained by some hidden variable theory . Over the past two decades, such states have been under intensive investigation for applications in quantum information [3, 4]. In particular, entangled states are useful in quantum communication [5–7] and several entangled based quantum key distribution (QKD) protocols have been put forward [7–10].
For realizing entanglement based quantum communication, distribution of entanglement is a necessary and important precursor. Many demonstrations of entanglement distribution employ photonic based entangled states to encode quantum information [10–14]. This is advantageous from a practical point of view. Firstly, the photons’ bosonic character makes them particularly resistant to environmental decoherence. Secondly, photonic based entanglement schemes in the telecom wavelength band can make use of the massive worldwide telecommunication infrastructure. Depending on the classical traffic load constraints, utilizing installed telecom dark optical fiber links offers both a practical and cost-effective method of realizing global entangled quantum communication.
Coupled with such telecom optical fiber links, it is highly desirable a telecom based entanglement distribution system should also be functional and constructed from low cost components. Single photon detectors form vital components of any entanglement distribution system and should also be as practical as possible. Inexpensive single photon detectors such as InGaAs APDs are widely used in quantum key distribution (QKD) systems and are preferred over any other detector in commercial QKD systems. However, their suitability for telecom-based entanglement distribution is questionable. Work to date with InGaAs APDs has shown a maximum range (that also violates Bell’s inequality) of 82km [13–15]. More seriously for quantum communication, high coincidence rates are simply not possible; and as distribution distances are extended to 100km coincidence rates drop to as low as ~0.01 Hz [15, 16].
Extending telecom entanglement distribution distances as well as increasing coincidence rates has therefore prompted work with new types of detector technology. For example, entanglement distribution with upconversion single photon detectors  or superconducting single photon detectors (SSPDs)  show marked improvements over conventional InGaAs APDs. Upconversion detectors display modest dark count rates of the order of a few kHz while SPPD detectors stand out as possessing very low dark count rates of around ~100 Hz or less. However the key distribution distance of entanglement-based QKD using SSPD or upconversion detectors is not limited by the dark count rate but rather by extremely small photon coincidence rates. This is due to the fact that both upconversion and SSPD detectors suffer from low single photon efficiencies ~1%. While not so problematic for point to point QKD where the secure key rate scales with detector efficiency [19, 20], it is a serious problem for entangled QKD, where instead the secure key rate scales with the square of the detector efficiency. Moreover, ~1% photon detection efficiencies lead to extraordinarily long and impractical experimental integration times. This major drawback becomes more evident as the entanglement is distributed over longer distances.
In this article we show, despite current opinion, InGaAs APDs can be used effectively in high speed and long distance entanglement distribution. When operated at 1 GHz in a self-differencing mode  they display high single photon efficiencies ~10% which in turn give rise to high photon coincidence rates, much reducing experimental integration times. These positive characteristics prove InGaAs APDs are suitable for an array of applications in quantum information [22–24]. This is particularly apparent in the telecom band window where until recently high speed, high efficiency single photon detectors have been lacking. These self-differencing single photon APDs have allowed us to perform entanglement distribution for the first time over a record distance of 200 km while simultaneously violating Bell’s inequality. The results have been obtained without subtracting any accidental coincidences. Furthermore, the coincidence rates achieved are almost three orders of magnitude greater than obtained previously with conventionally operated InGaAs APDs [15, 16]. The coincidence rates are also two orders of magnitude greater than achieved with upconversion or SSPD detectors. Therefore, self-differencing single photon APDs are currently the detectors of choice for entanglement distribution and entanglement QKD over optical fiber. They operate at close to ambient temperatures and therefore only require simple electrical cooling. With additional advantages such as low cost and compactness, they will emerge as a key component for real-world deployment.
2. Time-bin encoded entanglement distribution
The source of entangled photons used in this article is based on time-bin encoded photons  and they offer a robust approach to encoding quantum information. Telecom band time bin entangled photons are generated by employing a well established technique based on parametric down-conversion (PDC) in periodically poled Lithium Niobate waveguides (PPLN) . Figure 1 depicts the experimental arrangement. A continuous wave 1551.1 nm light from an external cavity laser is modulated into a sequence of 41 ps pulses at a repetition rate of 1 GHz. The pulses are amplified by an erbium doped fiber amplifier (EDFA) before being passed through a fiber Bragg grating (FBG) to filter out any residual amplified spontaneous emission (filter 1). The resulting pulses are transmitted through a polarization controller and serve as pump pulses for second harmonic generation (SHG) in the first periodically poled Lithium Niobate waveguide (PPLN SHG). The 775 nm pulses from the SHG process are then transmitted though a 1550 nm band blocking filter (filter 2) to suppress stray pump light and then launched through a second polarization controller. The pulses act as pump pulses for parametric downconversion (PDC) in the second PPLN waveguide. A subsequent filter (filter 3) blocks residual 775 nm radiation. The 775 nm pulses create non-degenerate pairs of photons via PDC with wavelengths of 1547 nm and 1555 nm for the signal and idler respectively. These wavelengths are within the telecom band for efficient (low loss) optical fiber transmission.
The wavelengths of the signal and idler pulses, being non-degenerate, can be spatially separated using a dielectric band pass filter, bandwidth 100 GHz (filter 4). The losses from the exit point of the PPLN PDC waveguide through the filters 3 & 4 is approximately 4.0 dB each for both signal and idler channels.
Time bin entanglement of the photon pairs arises from the long coherence time of the pump photons. To see this, we note the pump laser linewidth is approximately 100 kHz which translates to a laser coherence time, tcoh of the order of tcoh~10 µs. This coherence time extends over many clock cycles (time slots) of the optical pulse repetition rate, R=1 GHz. For very low mean photon pair number, µc, the total state wavefunction |ψ> of each photon pair can be written as:
Here x refers to the time slot which can extend from x=→N=R×tcoh. The amplitude terms ax and relative phase terms φx are fixed by the laser and can be assumed to be constant; at least over tcoh for a well stabilized laser. As it is not possible to affirm which time slot the photon pair was created in, the state wavefunction |ψ> can be viewed as high dimensional time bin entanglement  between each pair of states |x>s|x>i, with the dimension fixed by N.
After spatial separation the signal and idler photons are then launched into individual fibers. Each fiber is connected to a series of dispersion shifted fiber spools (ITU-T G.653, Fujikura), featuring a loss characteristic of 0.21 dB/km. The fiber reels used had discrete spans of 0, 75 and 100 km for both signal and idler photons. After distributing the photons through the fiber reels, we analyzed the degree of entanglement correlation between the signal and idler photon pairs. The signature of entanglement correlation is two-photon interference between signal and idler photon pairs. To observe two-photon interference, 1-bit delay planar light wave circuits (PLCs) are employed, which display excellent long term stability . The action of the 1-bit delay PLCs for a single PLC output port is equivalent to (unnormalized) transformation |x>s,i to |x>s,i+exp(iφ s,i)|x+1>s,i. Here, θ s,i is the phase of the signal (idler) PLCs. The path difference (phase θ) of the PLCs can be tuned by adjusting the PLC substrate temperature. Under these conditions, single photon interference cannot occur. This is because the PLC path differences of 1 ns is much longer than the coherence time of the photon pairs, ~10 ps. In fact the coherence time of the signal and idler photons is determined by the combination of the dielectric filter and PPLN bandwidths.
The contributions to two-photon interference on a single output port of each PLC are given by the (unnormalized) correlations:
Here y refers to the state of the final time slots. By measuring coincidence time slots we select terms when x=y and x+1=y in Eq. (2). This post-selection process corresponds to measuring photons travelling down the same paths of the PLC interferometers. As it is not possible to distinguish between the two processes of photons travelling down (1) both the short arms or (2) both long arms of the PLC interferometers we obtain two-photon interference. Adjusting the PLCs phases θ s,i varies the magnitude of this interference, so the visibility of the two-photon interference can be measured. To separate any classical correlations from quantum correlations, it is usually necessary to measure two-photon interference patterns in at least two non-orthogonal bases. In this way, the Bell parameter S of the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality [33, 34] can be employed to quantify the degree of entanglement in the experiment.
3. Self-differencing APDs
InGaAs APDs are used to detect the signal (idler) photons after the PLCs, Fig. 1. A DC voltage of about 52V was applied to each APD which was slightly below the APD breakdown voltage. A 1 GHz square wave with a peak to peak voltage of about 7V was superimposed upon the DC bias. This brought the APD above and below breakdown voltage with an alternating gate width of 500 ps. The square wave was synchronized to the time slot frequency of the pump laser, R so the photon arrival time coincided with an active APD gate. Cooling of the APDs was achieved by using small, low power ~20 W Peltier cooler elements. The temperature of the APDs was set at -30 degrees C for all measurements.
Photons impinging on the APDs create avalanches which are greatly obscured by the APD large capacitive response. To isolate the small avalanches a self-differencing technique is employed . The APD electrical signal is divided into two signals. One signal is launched through a 1 ns electrical delay, while the other signal is unaffected. The two signals are then united with a subtractive combiner. This results in removal of the periodic capacitive response and the small avalanche remaining is then clearly resolvable. After amplification, fast discriminators were used to convert the avalanches to fast NIM signals, suitable for triggering a Time Acquisition Card (TAC), operating in start-stop histogram mode. The efficiencies of both APDs were set at 10% which corresponded to dark count rates of 4500 Hz. The APDs were also carefully prepared to minimize afterpulsing which was below 2% for both detectors.
The APDs working in self-differencing mode displays exemplary timing characteristics. The timing jitter is Gaussian broadened and measured to be 60 ps FWHM . However in the experiment the jitter was dominated mainly by the electronic jitter of the fast discriminators. Electronic jitter broadened the APD response to <200 ps FWHM, which was still small enough to distinguish the true coincidence counts from mis-matched counts (see below). Moreover, with the use of low jitter discriminators, time-bin entanglement distribution with self-differencing APDs up to clock rates of a few GHz should be possible .
4. Experimental entanglement distribution
Before performing long distance entanglement distribution, we first measured the quality of entanglement distribution with the APDs directly connected to the PLC outputs (0 km). For this measurement we set the pump power to give a mean photon pair number of µc=0.002. The mean photon number µc is calculated directly from the average count rates of the APDs, c in Hz. The calculation takes into account the 10% efficiency of the APD (η=0.1), the loss of any fiber transmission (α), the 2.0 dB loss of the PLC (ξ=0.63) and the 4.0 dB fiber coupling loss of the PDC PPLN waveguide (ω=0.4). The coupling loss of the PPLN waveguide also includes the loss of the dielectic band pass filter. Thus the count rate can be expressed as c=R(αηξωµc/+d), where d is the dark count probability of the detector and hence µc can be easily extracted. Note that the first term in the expression for c is divided by 2 reflecting the fact that half of the photons are lost through the unused port of the PLC interferometer.
Figure 2(a) shows the histogram of coincidence counts for the two APDs with the phase settings of the two PLCs set to give constructive interference. A clear central coincidence peak is observed with a FWHM of ~200 ps. The two side lobes due to non-coincidence counts are temporally well distinguished from the main coincidence peak. When the PLCs have their phases adjusted to give destructive interference, the central coincidence peak is greatly suppressed to just above the dark count level. By applying a 700 ps time gate around the coincidence peak we were able to unambiguously select the coincidence counts, Figs. 2(c) & 2(d).
We held one PLC at a fixed temperature (phase) while adjusting the temperature of the other PLC in steps of 0.05 degrees C. The average coincidence count rate was >200 counts per second. Two and half interference fringes were observed giving an average visibility of 97.5 ±1.6%. For the corresponding non-orthogonal basis , we also observed two and half interference fringes yielding an average visibility of 98.6 ± 2.5%. Visibilities at this distance are limited by the PLC’s extinction ratio ~20 dB, translating to a maximum theoretical visibility of 99%. Therefore these two-photon interference results indicate high quality and stable performance of the entangled photon source as the visibilities are very close to the theoretical maximum.
Next we studied entanglement distribution at non-zero distances. Please note the PDC entangled photon pairs are not completely pure states as described by Eq. (1). Increasing the pump power, increases the probability of detecting photons from uncorrelated photon pairs leading to a reduction in the overall entangled photon pair visibility [27,32]. On the other hand, as the photon pair transmission distance increases, the contribution from the detector dark counts reduces the visibility. In other words there exists a trade-off between pump power and the dark count rate of the APDs. Consequently optimization of the mean photon number µc was necessary. The theory is based on a visibility calculation pertinent for PDC photon pairs :
Here α s,i is the total transmittance loss for either the signal or idler channel and d s,i is the dark count probability of either the signal or idler detector. To obtain the maximum visibility at each distance µc was scanned numerically. Once the optimal value of was found, µc was set experimentally by adjusting the pump power to give the corresponding maximum visibility for each fiber distance.
We inserted 2×75 km reels for the signal and idler channels. The combined overall loss of both channels corresponds to 31.5 dB. The pump power was increased so µc=0.016, resulting in a coincidence count rate of >1 counts per second. As in the previous experiment, we kept one PLC at a fixed temperature while the phase of the other PLC was varied. Figure 2(c) shows the experimental results. We observe two and half high quality fringes (squares). A fit to the data yields an average visibility of 88.5±5.5%. Switching to the non-orthogonal basis (circles, Fig. 2(c)), we also observe high quality interference fringes. A fit to this data yields a similar average visibility of 88.4±4.1%. These results both violate the Bell’s inequality of 70.7% for entanglement distribution by 3 (4) standard deviations respectively . This is the first demonstration of entanglement distribution violating Bell’s inequality at a distance of 150 km.
Finally, we inserted 2×100 km dispersion shifted fiber reels. The overall combined loss now is 42 dB≡200 km. In this experiment the PLC temperature resolution was reduced from 0.05 degrees C to 0.1 degrees C. This was necessary to reduce the overall time to collect the coincidence counts; the average coincidence count rate was now ~0.25 counts per second. The corresponding mean photon pair flux µc=0.03. Figure 2(d) shows that it is still possible to distribute entanglement at 200 km. We can observe two and half interference fringes for both non-orthogonal bases, with average visibilities of 78.9±4.1% and 80.5±8.0% . Importantly, these visibilities violate the Bell’s inequality; in this case by more than one standard deviation. It is important to emphasize the violation of Bell’s inequality is made possible without subtracting any accidental coincidences from dark counts. Consequently we have demonstrated successful entanglement distribution, for the first time, over 200 km of fiber.
To summarize the results, we examine the experimental visibility distance dependence with theory. To this end, experimental visibilities were also measured by introducing an attenuator in each fiber path equivalent to fiber distances of 100 km and 220 km. The results are plotted together with the visibilities for 0, 150 and 220 km in Fig. 3. Good agreement between experiment (points) and theory (solid line, based on Eq. (3)) are obtained. Bell’s inequality is clearly violated over the fiber distance 0→200 km. Also plotted is the distance dependence of µc (dotted line), derived from the optimization of Eq. (3).
It may be possible to extend the distance of entanglement distribution beyond 200 km by operating the APDs with lower dark count rates (and hence lower efficiencies). Of course under such conditions, the overall coincidence count rate will be reduced. It is important to note that also our setup can be straightforwardly modified to implement QKD based on entangled photons. To perform time-energy entangled QKD four single photon detectors are required. Nevertheless, as our detectors are low-cost and practical components, it is a simple matter to add two extra InGaAs detectors working in self-differential mode to realize an entangled based QKD system.
In conclusion we have demonstrated that InGaAs APDs are suitable for entanglement distribution over ultra-long distances of optical fiber. Combining a PPLN PDC entangled photon source and high speed self-differential APDs we were able to distribute entanglement over 200 km with high coincidence rates. These results show that entanglement distribution over ultra long distances is possible as well as practical for real world applications.
Both J. F. D and H. T. contributed to this work equally. The authors would like to thank Y. Sata and Y. Tokura for support during this research. In particular J.F.D. would like to thank Y. Tokura for supporting his stay at NTT Basic Research Laboratories, Japan. Financial support was provided by the CREST program of the Japan Science and Technology Agency (JST).
References and links
1. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47(10), 777–780 (1935). [CrossRef]
2. J. S. Bell, “On the Einstein-Podolsky-Rosen paradox,” Physics (Long Island City, N. Y.) 1, 195–200 (1964).
3. C. H. Bennett and D. P. DiVincenzo, “Quantum information and computation,” Nature 404(6775), 247–255 (2000). [CrossRef]
4. M. A. Nielson and I. L. Chuang, Quantum computation and quantum information, (Cambridge University Press, 2000).
5. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum Cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002). [CrossRef]
6. R. See Thew and N. Gisin, “Quantum communication,” Nat. Photon. 1, 165–171 (2007) for a review. [CrossRef]
7. A. Acín, N. Gisin, and L. Masanes, “From Bell’s theorem to secure quantum key distribution,” Phys. Rev. Lett. 97(12), 120405 (2006). [CrossRef]
8. A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991). [CrossRef]
9. C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without Bell’s Theorem,” Phys. Rev. Lett. 68(5), 557–559 (1992). [CrossRef]
10. W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time bell states,” Phys. Rev. Lett. 84(20), 4737–4740 (2000). [CrossRef]
11. R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Oemer, M. Fuerst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement based quantum communication over 144km,” Nat. Phys. 3(7), 481–486 (2007). [CrossRef]
12. A. Fedrizzi, R. Ursin, T. Herbst, M. Nespoli, R. Prevedel, T. Scheidl, F. Tiefenbacher, T. Jennewein, and A. Zeilinger, “High-fidelity transmission of entanglement over a high-loss freespace channel,” http://arxiv.org/abs/0902.2015.
13. M. Pfennigbauer, M. Aspelmeyer, W. Leeb, G. Baister, T. Dreischer, T. Jennewein, G. Neckamm, J. Perdigues, H. Weinfurter, and A. Zeilinger, “Satellite-based quantum communication terminal employing state-of-the-art technology,” J. Opt. Netw. 4(9), 549–560 (2005). [CrossRef]
14. H. Hubel, M. R. Vanner, T. Lederer, B. Blauensteiner, T. Lorünser, A. Poppe, and A. Zeilinger, “High-fidelity transmission of polarization encoded qubits from an entangled source over 100km of fiber,” Opt. Express 15(12), 7853–7862 (2007). [CrossRef]
15. H. C. Lim, A. Yoshizawa, H. Tsuchida, and K. Kikuchi, “Distribution of polarization-entangled photon pairs produced via spontaneous parametric down-conversion within a local-area fiber network: Theoretical model and experiment,” Opt. Express 16(19), 14512–14523 (2008). [CrossRef]
16. C. Liang, K. F. Lee, J. Chen, and P. Kumar, “Distribution of fiber-generated polarization entangled photon-pairs over 100 km of standard fiber in OC-192 WDM environment,” post deadline paper, Optical Fiber Communications Conference (OFC’2006), paper PDP35.
17. T. Honjo, H. Takesue, H. Kamada, Y. Nishida, O. Tadanaga, M. Asobe, and K. Inoue, “Long-distance distribution of time-bin entangled photon pairs over 100 km using frequency up-conversion detectors,” Opt. Express 15(21), 13957–13964 (2007). [CrossRef]
18. Q. Zhang, H. Takesue, S. W. Nam, C. Langrock, X. Xie, B. Baek, M. M. Fejer, and Y. Yamamoto, “Distribution of time-energy entanglement over 100 km fiber using superconducting single-photon detectors,” Opt. Express 16(8), 5776–5781 (2008). [CrossRef]
19. H. Takesue, S. W. Nam, Q. Zhang, R. H. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum key distribution over a 40-dB channel loss using superconducting single-photon detectors,” Nat. Phot. 1(6), 343–348 (2007). [CrossRef]
20. T. Honjo, S. W. Nam, H. Takesue, Q. Zhang, H. Kamada, Y. Nishida, O. Tadanaga, M. Asobe, B. Baek, R. Hadfield, S. Miki, M. Fujiwara, M. Sasaki, Z. Wang, K. Inoue, and Y. Yamamoto, “Long-distance entanglement-based quantum key distribution over optical fiber,” Opt. Express 16(23), 19118–19126 (2008). [CrossRef]
21. Z. L. Yuan, B. E. Kardynal, A. W. Sharpe, and A. J. Shields, “High speed single photon detection in the near infrared,” Appl. Phys. Lett. 91(4), 041114 (2007). [CrossRef]
22. A. R. Dixon, Z. L. Yuan, J. F. Dynes, A. W. Sharpe, and A. J. Shields, “Gigahertz decoy quantum key distribution with 1Mbit/s secure key rate,” Opt. Express 16(23), 18790–18979 (2008). [CrossRef]
23. J. F. Dynes, Z. L. Yuan, A. W. Sharpe, and A. J. Shields, “A high speed, post-processing free, random number generator,” Appl. Phys. Lett. 93, 031109 (2008). [CrossRef]
24. B. E. Kardynal, Z. L. Yuan, and A. J. Shields, “An avalanche-photodiode-based photon-number-resolving detector,” Nat. Photonics 2(7), 425–428 (2008). [CrossRef]
25. J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett. 62(19), 2205–2208 (1989). [CrossRef]
26. T. Honjo, H. Takesue, and K. Inoue, “Generation of energy-time entangled photon pairs in 1.5µm band with periodically poled lithium niobate waveguide,” Opt. Express 15(4), 1679–1683 (2007). [CrossRef]
27. H. de Riedmatten, I. Marcikic, V. Scarani, W. Tittel, H. Zbinden, and N. Gisin, “Tailoring photonic entanglement in high-dimensional Hilbert spaces,” Phys. Rev. A 69(5), 050304 (2004). [CrossRef]
28. H. Takesue and K. Inoue, “Generation of 1.5µm time-bin entanglement using spontaneous fiber four-wave mixing and planar-lightwave circuit interferometers,” Phys. Rev. A 72(4), 041804 (2005). [CrossRef]
29. Z. L. Yuan, A. R. Dixon, J. F. Dynes, A. W. Sharpe, and A. J. Shields, “Gigahertz quantum key distribution with InGaAs avalanche photodiodes,” Appl. Phys. Lett. 92(20), 201104 (2008). [CrossRef]
30. Q. Zhang, C. Langrock, H. Takesue, X. Xie, M. Fejer, and Y. Yamamoto, “Generation of 10-GHz clock sequential time-bin entanglement,” Opt. Express 16(5), 3293–3298 (2008). [CrossRef]
31. By shifting the phase of the fixed temperature PLC by π/2 the non-orthogonal basis was measured.
32. I. Marcikic, H. de Riedmatten, W. Tittel, V. Scarani, H. Zbinden, and N. Gisin, “Time-bin entangled qubits for quantum communication created by femtosecond pulses,” Phys. Rev. A 66(6), 062308 (2002). [CrossRef]
33. The observed correlation in the experiment, C is connected to the visibility V thus: C=V cos2 (f) where f is a function of signal and idler phases, f(θs,θi). We use the Clauser-Horne-Shimony-Holt Bell inequality  with associated Bell parameter S=|C(θs,θi)-C(θs,θ′i)+C(θ′s,θ′i)+(θ′s,θ′i)|≤2. Quantum-mechanically, this inequality can be violated with maximum violation of S=2√2. The violation of the Bell CHSH inequality in terms of visibility is hence V>1/√2.
34. J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden variable theories,” Phys. Rev. Lett. 23(15), 880–884 (1969). [CrossRef]
35. The factor of two variation in error bar values between the two basis measurements arises from statistical variations from one experimental run to another.