Higher amounts of loophole-free Bell violation using a heralded entangled source

Loophole-free Bell non-locality test plays a central role in device-independent quantum information processing tasks, such as device-independent quantum random number generation (QRNG) and quantum key distribution (QKD). Inspired by the scheme of heralded spontaneous parametric down conversion (SPDC) source proposed in (Śliwa C and Banaszek K 2003 Phys. Rev. A 67 030101), we present a loophole-free Bell test scheme that employs the heralded type of entangled photon pairs. Our proposal enables a much higher degree of Bell violation with realistic photonic devices over the one using the conventional SPDC source, thus allowing the implementation of device-independent QRNG and QKD with significant advantages. We anticipate that the scheme will enable variously subtle applications in device-independent quantum information processing tasks.


Introduction
Bell non-locality [1][2][3][4] draws one of the most important distinctions between quantum mechanics and classical mechanics in the sense that Bell nonlocal correlations can not be explained by the local-realistic theory [5]. Despite this fundamental interest, the Bell non-locality has also laid a significant foundation for many promising quantum information processing tasks. Indeed, in a model that reproduces Bell nonlocal correlations, the measurement outcomes cannot be fully determined by sharing local variables. Therefore, once Bell non-locality is certified in a quantum information processing task, the measurement outputs cannot be fully predicted by a third party no matter the mechanism underlying the devices, i.e. device independent security [6]. For instance, the Bell non-locality guarantees the security of device-independent quantum key distribution (DI-QKD) [6][7][8][9][10][11] and device-independent quantum random number generation (DI-QRNG) [12][13][14][15][16][17][18][19][20]. See [21,22] for a general discussion on the subjects of QKD and QRNG.
• The locality loophole. Once the local parties are not spacelike separated, there could be signals about the measurement sites from one party to influence the measurements at the other party. Specifically, spacelike separations of the event that determines the measurement setting choice at one party and the event that generates the measurement outcome at the other party can close the locality loophole [4].
• The freedom-of-choice loophole. For the case that the measurement settings are not independent from local hidden variables (LHV), the measurement results would be manipulated arbitrarily. The freedom-of-choice loophole can be closed when the measurement setting choices are generated by true random number generators, and the setting choices events are spacelike separated from the source pairs generation events [29].
• The fair-sampling loophole. Supposing the parties select only the detected events and discard the undetected events, fair-sampling assumption states that the subensemble of detected events can represent the total ensemble. However, if the detection efficiency is non-unit, the detection events could be controlled by LHV depending on the measurement settings choices of the parties [30,31]. The fair-sampling loophole-free (or detection loophole-free) Bell test can be realized by considering all the detected and undetected events, and using sufficiently high efficiency detectors [4].
There have been several experimental Bell tests reported to be loophole-free. For examples, the experiments reported in references [32,33] are the first two experiments which closed the detection loophole using photons, the experiments in references [34][35][36] have closed above three loopholes simultaneously, the experiment in [37] has closed the locality loophole and the fair-sampling loophole simultaneously using entangled atoms. It is noteworthy that DI-QRNG has also been realized [18][19][20]. However, until now the loophole-free Bell tests are still extremely limited even with state-of-the-art technologies. To push forward the application of deviceindependent quantum information processing, one must overcome a number of practical obstacles such as imperfections of the source of entanglement and the detectors. As Bell non-locality has been vital for many quantum information processing tasks, it is of great significance for achieving higher amounts of violation of Bell inequalities in Bell tests. Experimentally, the spontaneous parametric down-conversion (SPDC) [38] source has been widely adopted to generate entanglement photon pairs in quantum information processing. However, the maximum loophole-free CHSH value using the conventional entangled SPDC source is only 2.35 even with a unit efficiency detector theoretically [39], which is far from the maximal value of 2 2 allowed by Tsirelson's bound in quantum theory [40]. The main reason is that vacuum and multi-pair photons dominate the output mode of the conventional SPDC source which will be very harmful to the violation of CHSH inequality.
In this letter, we propose a loophole-free Bell test scheme using entangled photon pairs based on the heralded SPDC source proposed in [41]. Owing to its heralding mechanism [41][42][43], one may expect that higher amounts of violation of Bell inequalities can be achieved using the heralded SPDC source compared with that using the conventional SPDC source. It is demonstrated that the degree of violation for the CHSH inequality using the heralded entangled source can significantly surpass the one using the conventional SPDC source [39] for both non-maximally entangled output modes and maximally entangled output modes. In the ideal case according to our proposal, the maximal violation for CHSH inequality using the heralded entangled source is 2.828, which is approaching the Tsirelson's bound [40]. This result is significantly higher than 2.35 which is the maximal violation for CHSH inequality using multi-mode emission of the conventional SPDC source even with unit detector efficiency [39]. The Clauser-Horne (CH) inequality [3] violation using the heralded entangled source can be two orders of magnitude better than that is reported in [34] for typical experiment parameters. As one of the most promising applications of Bell non-locality, we also present a DI-QKD scheme using the heralded entangled source, and show that one can reduce the lower bound of detector efficiency for positive key rate from 95.3% to 91%, and obtain a 2.62 times higher key generation rate compared with that using the conventional SPDC source.

The heralded polarization-entangled source
The heralded entangled source consists of a conventional SPDC source and a linear optical part which is used to herald generations of entangled photon pairs (see appendix C for detail analysis). As shown in figure 1, the coincidence of four detectors D 1 , D 2 , D 3 and D 4 heralds a successful preparation of one pair of entanglement photons. The output mode of the conventional SPDC source has the form [41,44] å l y where λ n is the probability amplitude of the nth order of the output mode, y ñ | n is the wave function for n pairs of photons. As to the heralded SPDC source, the third order element y ñ | 3 is the lowest order that contributes to the coincidence of four detectors [41], and the corresponding output mode is an entangled state where {H, V} is an orthogonal polarizing basis, r depends on the squeezing parameters associated with the source (see appendix B). When r=1, it reduces to the maximally entangled state In fact, the components  n 4 will also contribute to the coincidence events of four detectors. The corresponding output modes of the coincidence events for  n 4 will reduce the fidelity of the entanglement pairs. For example, considering the next leading order y ñ | 4 , after the optimization for the probability of coincidence events caused by the component y ñ | 3 , the contribution of the next leading order term y ñ | 4 to the coincidence events is rather small as analyzed in [41]. In our simulation, we have considered the multiple pair events up to n=5, where the CHSH value contributed by the component of  n 6 is negligible, as compared with that contributed by the dark-count events of p 10 d 8 [45,46]. Thus, even though higher orders of output modes of the conventional SPDC source may also contribute to coincidence events, the heralded entangled photon pairs dominate the output state of the heralded entangled source. Based on the advantage of the heralded SPDC source, we explore its potential improvement on Bell tests and quantum information processing tasks.
3. Bell tests using heralded entangled source 3.1. Loophole-free CHSH test using heralded entangled source The CHSH inequality has been extensively employed in Bell tests. There are two separated players, Alice and Bob, sharing entanglement pairs. Alice randomly chooses a measurement x from measurement setting choices {α 0 =0, α 1 =1} to measure the particle in her side, similarly for Bob with measurement setting choices yä{β 0 =0, β 1 =1}. What they obtain are binary results {±1}, labeled by i and j for Alice and Bob, respectively. The trial is repeated for many times to accurately estimate the probability distribution ( | ) P ij xy (see appendix D for details). Alice and Bob then calculate the CHSH value [2] where xä{α 0 , α 1 }, yä{β 0 , β 1 }. The maximal CHSH values are 2 and 2 2 for LHV models and quantum theory [2,40], respectively. If S>2, Alice's and Bob's correlation are nonlocal. The schematic setup, as shown in figure 1, is composed of two parts: the heralded entangled source and the CHSH test. The experiment setup for CHSH test consists of a polarization modulator (PM), a polarization beam splitter (PBS) and two detectors in Alice's side and similarly for Bob. The PM is used to choose different measurement settings. The coincidence of four detectors D 1 , D 2 , D 3 and D 4 heralds the output of entangled photon pairs. However, because of the inefficiency of the detector, the dark-count and the higher order components of the conventional SPDC source, there are also vacuum state and multi-pair components in the output mode of the heralded entangled source. As for the CHSH test, there are four detection events at Alice's (Bob's) side: double clicks, ±1 click and no click. In order to close the fair-sampling loophole [28,33], one need to take all the detection events into consideration. One denotes the events that A H does not click and A V clicks as −1, other events are counted as +1 for Alice. Also, the events that B H does not click and B V clicks are counted as −1, others counted as +1 for Bob [39]. To compare with the CHSH test using the conventional SPDC source, one can calculate the CHSH value S when the dark-count rate equals p d =0 which is chosen according to [39]. At the same time, the transmission coefficient η trans of the BSs in the heralded entangled source approaches to 1. The simulation is cut off at n=5, which is reasonable, because it can be certified through simple calculation that the CHSH value contributed by the component of  n 6 from equation (1) is negligible compared with that contributed by the dark-count events when p 10 d 8 which is reasonably chosen according to [45,46]. As shown in figure 2, one can see that the CHSH values for non-maximally entangled output modes are higher than the one for maximally entangled output modes from both the heralded entangled source and the conventional SPDC source. This is because that non-maximally entangled states can tolerate more noises than maximally entangled states [23]. Without considering the system error, the maximal violation for CHSH inequality using the heralded entangled source is 2.828 when η trans =0.999 999, which is approaching 2 2 allowed by Tsirelson's bound in quantum theory [40]. This result is significantly higher than 2.35 which is the maximal violation for CHSH inequality using multi-mode emission of the conventional SPDC source even with detector efficiency η=1 [39]. The lower bound of detector efficiency for loophole-free CHSH inequality violation using the heralded entangled source approaches 67.3%, and the corresponding CHSH value is S=2.000 01. As shown in [23], the lower bound for a loophole-free Bell inequality violation is 2/3 when nonmaximally entangled state is adopted. It is shown in equation (C.7) that the fidelity of the output entangled state increases as the transmission coefficient increases. When h  1 trans , one can achieve a lower bound of detector efficiency h  2 3for CHSH inequality violation using the heralded entangled source.
To match the actual experiment setups, one can take reasonable transmission coefficients η trans and darkcount rate p d when estimating the CHSH value. Specifically, we provide simulation for the CHSH test with nonmaximally and maximally entangled output modes from the heralded entangled source at two different transmission coefficients. The first one is h = 0.99 trans 1 which is derived from [47]. According to equation (C.6) the preparation probability of entangled pairs approaches 0 when transmission coefficient approaches to 1. The second one is chosen at h = 0.9 trans 2 to make a tradeoff between the preparation probability and the amounts of violation of CHSH inequality. At the same time, the dark count rate is set to be p d =10 −6 which is derived according to [35].
As shown in figure 3, the CHSH value S at h = 0.99 trans 1 is higher than that at h = 0.9 trans 2 for both the maximally and non-maximally entangled output modes at most values of the detection efficiency η. That is because the fidelity of the output entangled state increases as the transmission coefficient increases according to equation (C.7) and entangled states with higher fidelity can achieve better CHSH inequality violation. For the non-maximally entangled output mode, when h = 0.99 trans 1 , the minimal detector efficiency for violating the CHSH inequality is η≈80.9%. But when h = 0.9 trans 2 , the minimal detector efficiency for violating the CHSH inequality is η≈80.3% which is lower. The reason for this counter-intuitive phenomenon is that the darkcount rate in case is big enough to break down the advantage in fidelity of entangled state induced by higher transmission coefficient which will cause lower preparation probability of the heralded entangled source according to equation (C.6). More specifically, as shown in table 1, one can see that the CHSH value S with at η=80.9% when the dark-count rate is turn down to p d =10 −7 . According to the detector efficiency η≈82.28% reported in a recent experiment [17], the detector within current technology is enough for loophole-free CHSH tests using the heralded entangled source.

Loophole-free CH test using heralded entangled source
Among recently reported experimental loophole-free Bell tests [32][33][34][35][36][37], the CH inequality [3,23] is employed extensively. In order to make a comparison with the reported results, we propose a CH test scheme using the heralded entangled source and experiment parameters comparable to that in [34]. The CH inequality is of the The experimental setup for the CH test is similar to the CHSH test in figure 1, except that there is only one detector in Alice's (Bob's) side. As for the CH test, there are only two detection events in Alice's (Bob's) side: click and no click. One counts events that Alice's (Bob's) detector clicks as + and does not click as 0 in the CH test. The trial is repeated enough times to accurately calculate the CH value J. The violation of the CH inequality means that Alice's and Bob's correlation are nonlocal. As illustrated in figure 4, the corresponding experiment parameters are set to be p d =10 −7 and η trans =0.95 without loss of generality. The red X mark denotes the result reported in [34] that J=7.27×10 −6 when η=77.39% (the detector efficiency is taken as a geometric average over Alice's and Bob's detector efficiencies in [34]). The CH value using the heralded entangled source is J=9.06×10 −4 which is two order of magnitude better than that reported in [34] at the same detector efficiency. As presented infigure 3, better violations can be achieved for the CHSH test by adjusting the transmission coefficients of the BSs in the heralded entangled source because one can achieve a higher fidelity of the entangled source by adjusting η trans according to equation (C.7). It should be noted that one can also obtain even better results by adjusting the transmission coefficients of the BSs in the heralded entangled source for the same reason. for heralded non-maximally entangled output and heralded maximally entangled output mode, respectively. Note. a The simulations are implemented with Wolfram Mathematica on a standard laptop computer by default accuracy. Due to the round-off of g,ḡ , α 0 , α 1 , β 0 and β 1 to four decimal places, there are differences between the presented values of S and the true optimal values which have also been presented in the round brackets.

Application: DI-QKD using heralded entangled source
As one of the most promising applications of Bell non-locality, the DI-QKD provides a way that Alice and Bob can conduct secure communication without trusting the devices [6,7,10,11]. The schematic setup for DI-QKD protocol is similar with that for CHSH test shown in figure 1, except that Alice's measurement setting choices is {α 0 , α 1 , α 2 }. The DI-QKD protocol runs as follows. Alice and Bob share entangled photon pairs from a third party, who can be untrustworthy. Alice randomly chooses a measurement x from measurement setting choices {α 0 , α 1 , α 2 } to measure the particle in her side, and similarly for Bob with measurement setting choices yä{β 0 , β 1 }, and the trial is repeated many times. After their measurements, Alice and Bob obtain two sequences of measurement results. They use the measurement results generated by xä{α 0 , α 1 } and yä{β 0 , β 1 } to estimate the loophole-free CHSH value S. The measurement setting α 2 is set to be the same with β 0 , and the measurement results generated by α 2 and β 0 are used to estimate the quantum bit error rate Q. The key generation rate R is bounded by [6,7]  -- is the binary entropy, a b = ¹ ( | ) Q P i j 2 0 is the probability that Alice and Bob obtain different measurement results when they choose measurement settings α 2 and β 0 , respectively.
Let us compare the DI-QKD key generation rate using the non-maximally entangled output mode from the heralded entangled source with that using the non-maximally entangled output mode from the conventional SPDC source with p d =0. The measurement setting choices {α 0 , α 1 , α 2 } and {β 0 , β 1 } are determined by optimizing the CHSH value S at different given parameters. As shown in figure 5, the blue line denotes key generation rate using non-maximally entangled photon pairs from the heralded entangled source when the transmission coefficient η trans approaches to 1. The yellow lower line denotes key generation rate with nonmaximally entangled photon pairs from the conventional SPDC source.
As a result, a lower bound of η L ≈91% is obtained with the heralded entangled source, and further calculation shows that this result is close to the one using non-maximally entangled state as equation (2). The lower bound is better than h » ¢ 95.3% L which is the lower bound using the conventional SPDC source. At the same time, the key generation rate using the heralded entangled source is significantly higher than that using the conventional SPDC source. When η=1, we realize a 2.62 times improvement in key generation rate compare with the DI-QKD setup with conventional SPDC source. The previous work shows that the detector efficiency should be  h 92.4% to conduct a DI-QKD protocol [7], we illustrate that it is possible to lower significantly the requirement of the efficiency of detectors in a DI-QKD protocol.

Discussions and conclusions
In conclusion, a new loophole-free Bell test scheme based on the heralded entangled source has been presented. We demonstrate that the CHSH test using the heralded entangled source can dramatically surpass the one using the conventional SPDC source, for both non-maximally entangled and maximally entangled output modes. The maximal violation of CHSH inequality using the heralded entangled source can almost approach 2 2 which is bounded by quantum theory. It is also shown that the CH inequality violation using the heralded entangled source can be two orders of magnitude better compared with that reported in [34] at typical experiment parameters. Being one of the most promising applications of Bell non-locality, we show that it is possible to lower significantly the requirement of the efficiency of detectors in a DI-QKD protocol. At the same time, the DI-QKD key generation rate using the heralded entangled source is much higher than the one using the conventional SPDC source with a more broad range. After considering the experimentally reasonable parameters, it turns out that the detector within current technology is feasible for loophole-free CHSH tests by implementing this scheme. With the merit of lower requirements than conventional SPDC source, we anticipant that our proposals may facilitate ongoing Bell test experiments and various device-independent quantum information tasks.
We consider the non-photon-number-resolving detectors with overall detection efficiency η 0 which includes the detector efficiency and internal transmission efficiency. The fiber channel loss is h = -x 10 l l 10 with a loss rate ξ measured in dB km −1 , and l represent the distance of the channel. Without loss of generality, the channel and detection can be modeled as a beam splitter (BS) with transmission rate η and a perfect non-photon-numberresolving single photon detector h h h = · ( ) .
A . 1 l 0 Now, combining the consideration of the dark-count rate p d , the no-click probability P NC and the click probability P C of a non-photon-number-resolving detector for incoming m photons is h = --= - where η is the transmission rate of the BS in the above model of the detection and channel. Without causing misunderstanding, we will directly call η the detection efficiency of a non-photon-number-resolving detector in the main text and other parts of the appendix. The probability of the coincidence of the four heralding non-photon-number-resolving events is The correlation between the coincidence probability P co and fidelity F are similar with that in [39]  To calculate the CHSH value, we need to calculate the probability distribution ( | ) P ij xy for i, jä{+1,−1}, xä{α 0 , α 1 } and yä{β 0 , β 1 } in equation (4). For the chosen measurement settings x and y, there are four detection events at Alice's (Bob's) side: double clicks, ±1 click and no click. In order to close the fair-sampling loophole, one need to take all the detection events into consideration. without loss of generality, one denotes the events that A H does not click and A V clicks as −1, other events are counted as +1 for Alice. Also, the events that B H does not click and B V clicks are counted as −1, others counted as +1 for Bob [39]. Thus, according to equations (A.2) and (D.2), the corresponding probability distributions ( | ) P ij xy are expressed as å --= Here, for the sake of simplicity and without loss of generality, we have set the detection efficiency η to be the same for all eight detectors. From equations (B.1), (D.2) and (D.3) to (D.6), one can optimize the CHSH value over the squeezing parameters g andḡ , measurement settings {α 0 , α 1 } and {β 0 , β 1 } for given transmission efficiency η trans , detection efficiencies of the detectors and the dark-count rate p d .