Guaranteed violation of a Bell inequality without aligned reference frames or calibrated devices
Peter Shadbolt, Tamas Vertesi, Yeong-Cherng Liang, Cyril Branciard, Nicolas Brunner, Jeremy L. O'Brien
GGuaranteed violation of a Bell inequality without aligned reference frames or calibrated devices
Peter Shadbolt, Tam´as V´ertesi, Yeong-Cherng Liang, Cyril Branciard, Nicolas Brunner, ∗ and Jeremy L. O’Brien Centre for Quantum Photonics, H. H. Wills Physics Laboratory & Department of Electrical and Electronic Engineering,University of Bristol, Merchant Venturers Building, Woodland Road, Bristol, BS8 1UB, UK Institute of Nuclear Research of the Hungarian Academy of Sciences, H-4001 Debrecen, P.O. Box 51, Hungary. Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland School of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072, Australia H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, United Kingdom (Dated: August 27, 2018)Bell tests—the experimental demonstration of a Bell inequality violation—are central to understanding thefoundations of quantum mechanics, underpin quantum technologies, and are a powerful diagnostic tool for tech-nological developments in these areas. To date, Bell tests have relied on careful calibration of the measurementdevices and alignment of a shared reference frame between the two parties—both technically demanding tasksin general. Surprisingly, we show that neither of these operations are necessary, violating Bell inequalities withnear certainty with (i) unaligned, but calibrated, measurement devices, and (ii) uncalibrated and unaligned de-vices. We demonstrate generic quantum nonlocality with randomly chosen local measurements on a singletstate of two photons implemented with reconfigurable integrated optical waveguide circuits based on voltage-controlled phase shifters. The observed results demonstrate the robustness of our schemes to imperfections andstatistical noise. This new approach is likely to have important applications in both fundamental science and inquantum technologies, including device independent quantum key distribution.
Nonlocality is arguably among the most striking aspectsof quantum mechanics, defying our intuition about space andtime in a dramatic way [1]. Although this feature was initiallyregarded as evidence of the incompleteness of the theory [2],there is today overwhelming experimental evidence that na-ture is indeed nonlocal [3]. Moreover, nonlocality plays a cen-tral role in quantum information science, where it proves to bea powerful resource, allowing, for instance, for the reductionof communication complexity [4] and for device-independentinformation processing [5–8].In a quantum Bell test, two (or more) parties perform localmeasurements on an entangled quantum state, Fig. 1(a). Af-ter accumulating enough data, both parties can compute theirjoint statistics and assess the presence of nonlocality by check-ing for the violation of a Bell inequality. Although entangle-ment is necessary for obtaining nonlocality it is not sufficient.First, there exist some mixed entangled states that can prov-ably not violate any Bell inequality since they admit a localmodel [9]. Second, even for sufficiently entangled states, oneneeds judiciously chosen measurement settings [10]. Thus al-though nonlocality reveals the presence of entanglement in adevice-independent way, that is, irrespectively of the detailedfunctioning of the measurement devices, one generally con-siders carefully calibrated and aligned measuring devices inorder to obtain a Bell inequality violation. This in generalamounts to having the distant parties share a common ref-erence frame and well calibrated devices. Although this as-sumption is typically made implicitly in theoretical works, es-tablishing a common reference frame, as well as aligning andcalibrating measurement devices in experimental situationsare never trivial issues. It is therefore an interesting and im-portant question whether such requirements can be dispensedwith.It was recently shown [11] that, for Bell tests performedin the absence of a shared reference frame, i.e., using ran- domly chosen measurement settings, the probability of ob-taining quantum nonlocality can be significant. For instance,considering the simple Clauser-Horne-Shimony-Holt (CHSH)scenario [12], randomly chosen measurements on the singletstate lead to a violation of the CHSH inequality with proba-bility of ∼ ; moreover this probability can be increasedto ∼ by considering unbiased measurement bases. Thegeneralization of these results to the multipartite case wereconsidered in Refs. [11, 13], as well as as schemes basedon decoherence-free subspaces [14]. Although these worksdemonstrate that nonlocality can be a relatively common fea-ture of entangled quantum states and random measurements,it is of fundamental interest and practical importance to estab-lish whether Bell inequality violation can be ubiquitous.Here we demonstrate that nonlocality is in fact a far moregeneric feature than previously thought, violating CHSH in-equalities without a shared frame of reference, and evenwith uncalibrated devices, with near-certainty. We first showthat whenever two parties perform three mutually unbiased(but randomly chosen) measurements on a maximally entan-gled qubit pair, they obtain a Bell inequality violation withcertainty—a scheme that requires no common reference framebetween the parties, but only a local calibration of each mea-suring device. We further show that when all measurementsare chosen at random ( i.e. , calibration of the devices is notnecessary anymore), although Bell violation is not obtainedwith certainty, the probability of obtaining nonlocality rapidlyincreases towards one as the number of different local mea-surements increases. We perform these random measurementson the singlet state of two photons using a reconfigurable in-tegrated waveguide circuit, based on voltage-controlled phaseshifters. The data confirm the near-unit probability of violat-ing an inequality as well as the robustness of the scheme to ex-perimental imperfections—in particular the non-unit visibilityof the entangled state—and statistical uncertainty. These new a r X i v : . [ qu a n t - ph ] N ov FIG. 1:
Bell violations with random measurements. (a) Schematicrepresentation of a Bell test:. (b) Schematic of the integrated waveg-uide chip used to implement the new schemes described here. Al-ice and Bob’s measurement circuits consist of waveguides to encodephotonic qubits, directional couplers that implement Hadamard-likeoperations, thermal phase shifters to implement arbitrary measure-ments and detectors. schemes exhibit a surprising robustness of the observation ofnonlocality that is likely to find important applications in di-agnostics of quantum devices ( e.g.. removing the need to cali-brate the reconfigurable circuits used here) and quantum infor-mation protocols, including device independent quantum keydistribution [5] and other protocols based on quantum nonlo-cality [6–8] and quantum steering [15].
Results
Bell test using random measurement triads.
Two distant par-ties, Alice and Bob, share a Bell state. Here we will focus onthe singlet state | Ψ − (cid:105) = 1 √ | (cid:105) A | (cid:105) B − | (cid:105) A | (cid:105) B ) , (1)though all our results can be adapted to hold for any two-qubitmaximally entangled state. Let us consider a Bell scenarioin which each party can perform 3 possible qubit measure-ments labeled by the Bloch vectors (cid:126)a x and (cid:126)b y ( x, y = 1 , , ,and where each measurement gives outcomes ± . After suf-ficiently many runs of the experiment, the average value ofthe product of the measurement outcomes, i.e. the correla-tors E xy = − (cid:126)a x · (cid:126)b y , can be estimated from the experimentaldata. In this scenario, it is known that all local measurementstatistics must satisfy the CHSH inequalities: | E xy + E xy (cid:48) + E x (cid:48) y − E x (cid:48) y (cid:48) | ≤ , (2)and their equivalent forms where the negative sign is permutedto the other terms and for different pairs x, x (cid:48) and y, y (cid:48) ; thereare, in total, 36 such inequalities.Interestingly, it turns out that whenever the measurementsettings are unbiased, i.e. (cid:126)a x · (cid:126)a x (cid:48) = δ x,x (cid:48) and (cid:126)b y · (cid:126)b y (cid:48) = δ y,y (cid:48) ,then at least one of the above CHSH inequalities must be vio-lated — except for the case where the orthogonal triads (from now on simply referred to as triads) are perfectly aligned, i.e.for each x , there is a y such that (cid:126)a x = ± (cid:126)b y . Therefore, ageneric random choice of unbiased measurement settings —where the probability that Alice and Bob’s settings are per-fectly aligned is zero (for instance if they share no commonreference frame) — will always lead to the violation of aCHSH inequality. Proof.
Assume that { (cid:126)a x } and { (cid:126)b y } are orthonormal bases.Since the correlators of the singlet state have the simple scalarproduct form E xy = − (cid:126)a x · (cid:126)b y , the matrix E = E E E E E E E E E (3)contains (in each column) the coordinates of the three vectors − (cid:126)b y , written in the basis { (cid:126)a x } .By possibly permuting rows and/or columns, and by possi-bly changing their signs (which corresponds to relabeling Al-ice and Bob’s settings and outcomes), we can assume, withoutloss of generality, that E , E > and that E > is thelargest element (in absolute value) in the matrix E . Noting that (cid:126)b = ± (cid:126)b × (cid:126)b and therefore | E | = | E E − E E | ,these assumptions actually imply E = E E − E E ≥ E , E , | E | , | E | and E E ≤ ; we will assume that E ≤ and E ≥ (one can multiply both the x = 2 rowand the y = 2 column by -1 if this is not the case).With these assumptions, ( E + E ) max[ − E , E ] ≥ E E − E E = E ≥ max[ − E , E ] , and by divid-ing by max[ − E , E ] > , we get E + E ≥ . One canshow in a similar way that − E + E ≥ . Adding theselast two inequalities, we obtain E + E − E + E ≥ . (4)Since E is an orthogonal matrix, one can check that equality isobtained above (which requires that both E + E = 1 and − E + E = 1 ) if and only if (cid:126)a = (cid:126)b , (cid:126)a = (cid:126)b and (cid:126)a = (cid:126)b .Therefore, if the two sets of mutually unbiased measurementsettings { (cid:126)a x } and { (cid:126)b y } are not aligned, then inequality (4)is strict: a CHSH inequality is violated. Numerical evidencesuggests that the above construction always gives the largestCHSH violation obtainable from the correlations (3). (cid:4) While the above result shows that a random choice of mea-surement triads will lead to nonlocality with certainty, we stillneed to know how these CHSH violations are distributed; thatis, whether the typical violations will be rather small or large.This is crucial especially for experimental implementations,since in practice, various sources of imperfections will reducethe strength of the observed correlations. Here we considertwo main sources of imperfections: limited visibility of theentangled state, and finite statistics.First, the preparation of a pure singlet state is impossibleexperimentally and it is thus desirable to understand the effectof depolarizing noise, giving rise to a reduced visibility V : | Ψ − (cid:105) → ρ V = V | Ψ − (cid:105)(cid:104) Ψ − | + (1 − V ) . (5) CHSH P r obab ili t y den s i t y V P r obab ili t y o f v i o l a t i on FIG. 2:
Bell tests using random measurement triads (theory).
Distribution of the (maximum) CHSH violations for uniformly ran-dom measurement triads on a singlet state. The inset shows the prob-ability of obtaining a CHSH violation as a function of the visibil-ity V of the Werner state; this probability is obtained by integratingthe distribution of CHSH violations (main graph) over the interval [2 /V, √ . This, in turn, results in the decrease of the strength of corre-lations by a factor V . In particular, when V ≤ / √ , thestate (5) ceases to violate the CHSH inequality. States ρ V areknown as Werner states [9].Second, in any experiment the correlations are estimatedfrom a finite set of data, resulting in an experimental uncer-tainty. To take into account this finite-size effect, we willconsider a shifted classical bound S ≥ of the CHSH expres-sion (2) such that an observed correlation is only considered togive a conclusive demonstration of nonlocality if CHSH > S .Thus, if the CHSH value is estimated experimentally up to aprecision of δ , then considering a shifted classical bound of S = 2 + δ ensures that only statistically significant Bell vio-lations are considered.We have estimated numerically the distribution of theCHSH violations (the maximum of the left-hand-side of (2)over all x, x (cid:48) , y, y (cid:48) ) for uniformly random measurement tri-ads on the singlet state (see Fig. 2). Interestingly, typical vi-olations are quite large; the average CHSH value is ∼ . ,while only ∼ . of the violations are below . . Thus thisphenomenon of generic nonlocality is very robust against theeffect of finite statistics and of limited visibility, even in thecase where both are combined. For instance, even after rais-ing the cutoff to S = 2 . and decreasing the singlet visibilityto V = 0 . , our numerical simulation shows that the proba-bility of violation is still greater than 98.2% (see Fig 2). Bell tests using completely random measurements.
Althoughperforming unbiased measurements does not require the spa-tially separated parties to share a common reference frame,it still requires each party to have good control of the localmeasurement device. Clearly, local alignment errors (that is,if the measurements are not exactly unbiased) will reduce theprobability of obtaining nonlocality. In practice the difficulty V P r obab ili t y o f v i o l a t i on ( % ) m=2m=3m=4m=5m=6m=7m=8 FIG. 3:
Bell tests using completely random measurements (the-ory).
Plot of the probability of Bell violation as a function of thevisibility V of the Werner state, for different numbers m of (com-pletely random) measurements per party. of correctly aligning the local measurement settings dependson the type of encoding that is used. For instance, using thepolarization of photons, it is rather simple to generate locallya measurement triad, using wave-plates. However, for othertypes of encoding, generating unbiased measurements mightbe much more complicated (see experimental part below).This leads us to investigate next the case where all measure-ment directions { (cid:126)a x ,(cid:126)b y } are chosen randomly and indepen-dently. For simplicity, we will focus here on the case whereall measurements are chosen according to a uniform distribu-tion on the Bloch sphere. Although this represents a particularchoice of distribution, we believe that most random distribu-tions that will naturally arise in an experiment will lead toqualitatively similar results, as indicated by our experimentalresults.We thus now consider a Bell test in which Alice and Bobshare a singlet, and each party can use m possible mea-surement settings, all chosen randomly and uniformly on theBloch sphere. We estimated numerically the probability ofgetting a Bell violation as a function of the visibility V [ofthe state (5)] for m = 2 , . . . , ; see Fig. 3. Note that for m ≥ , additional Bell inequalities appear [16]; we havechecked however, that ignoring these inequalities and consid-ering only CHSH leads to the same results up to a very goodapproximation. Fig. 3 clearly shows that the chance of find-ing a nonlocal correlation rapidly increases with the numberof settings m . Intuitively, this is because when choosing an in-creasing number of measurements at random, the probabilitythat at least one set of four measurements (2 for Alice and 2 forBob) violates the CHSH inequality increases rapidly. For ex-ample, with m = 3 settings, this probability is . but with m = 4 , it is already 96.2%, and for m = 5 it becomes 99.5%.Also, as with the case of unbiased measurements, the proba-bility of violation turns out to be highly robust against depo-larizing noise; for instance, for V = 0 . and m ≥ , there isstill at least 96.9% chance of finding a subset { (cid:126)a x , (cid:126)a x (cid:48) ,(cid:126)b y ,(cid:126)b y (cid:48) } p CH S H Iteration(a) V P r obab ili t y o f V i o l a t i on , % (b) FIG. 4:
Bell tests requiring no shared reference frame.
Here we perform Bell tests on a two-qubit Bell state, using randomly chosenmeasurement triads. Thus our experiment requires effectively no common reference frame between Alice and Bob. (a) 100 successive Belltests; in each iteration, both Alice and Bob use a randomly-chosen measurement triad. For each iteration, the maximal CHSH value is plotted(black points). In all iterations, we get a CHSH violation; the red line indicates the local bound (CHSH=2). The smallest CHSH value is ∼ . ,while the mean CHSH value (dashed line) is ∼ . . This leads to an estimate of the visibility of V = . . (cid:39) . , to be compared with . ± . obtained by maximum likelihood quantum state tomography [26]. This slight discrepancy is due to the fact that our entangledstate is not exactly of the form of a Werner state. Error bars, which are too small to draw, were estimated using a Monte Carlo technique,assuming Poissonian photon statistics. (b) The experiment of (a) is repeated for Bell states with reduced visibility, illustrating the robustnessof the scheme. Each point shows the probability of CHSH violation estimated using 100 trials. Uncertainty in probability is estimated as thestandard error. Visibility for each point is estimated by maximum-likelihood quantum state tomography, where the error bar is calculated usinga Monte Carlo approach, again assuming Poissonian statistics. Red points show data corrected for accidental coincidences (see Methods), thecorresponding uncorrected data is shown in blue. Again, since our state is not exactly of the Werner form, we get slightly higher probabilitiesof CHSH violation than expected. among our randomly chosen measurements that gives nonlo-cal correlations. Measurement devices.
We use the device shown in Fig. 1(b)to implement Alice and Bob’s random measurements on anentangled state of two photons. The path encoded singletstate is generated from two unentangled photons using an inte-grated waveguide implementation [17, 18] of a nondetermin-istic
CNOT gate [19–21] to enable deliberate introduction ofmixture. This state is then shared between Alice and Bobwho each have a Mach-Zehnder (MZ) interferometer, con-sisting of two directional couplers (equivalent to beamsplit-ters) and variable phase shifters ( A , and B , ) and sin-gle photon detectors. This enables Alice and Bob to inde-pendently make a projective measurement in any basis bysetting their phase shifters to the required values [22, 23].The effect of the first phase-shifter is a rotation around the Z axis of the Bloch sphere ( R Z ( φ ) = e − iφ σ Z / ); sinceeach directional coupler implements a Hadamard-like oper-ation ( H (cid:48) = e iπ/ e − iπσ Z / He − iπσ Z / , where H is the usualHadamard gate), the effect of the second phase shifter is arotation around the Y axis ( R Y ( φ ) = e − iφ σ Y / ). Over-all the MZ interferometer and phase shifters implement theunitary transformation U ( φ , φ ) = R Y ( φ ) R Z ( φ ) , whichenables projective measurement in any qubit basis when com-bined with a final measurement in the logical ( Z ) basis usingavalanche photodiode single photon detectors (APDs).Each thermal phase shifter is implemented as a resistive element, lithographically patterned onto the surface of thewaveguide cladding. Applying a voltage v to the heater hasthe effect of locally heating the waveguide, thereby inducinga small change in refractive index n ( dn/dT ≈ × − K )which manifests as a phase shift in the MZ interferometer.There is a nonlinear relationship between the voltage appliedand the resulting phase shift φ ( v ) , which is generally well ap-proximated by a quadratic relation of the form φ ( v ) = α + βv . (6)In general, each heater must be characterized individu-ally, i.e. by estimating the function φ ( v ) . This is achievedby measuring single-photon interference fringes from eachheater. The parameter α can take any value between 0 and π depending on the fabrication of the heater, while typically β ∼ . rad V [22, 23]. For any desired phase, the correctedvoltage can then be determined. This operation is necessaryboth for state tomography, and for implementing random mea-surement triads. In contrast, this calibration can be dispensedwith when implementing completely random measurements.Indeed this represents the central point of this experiment,which requires no a priori calibration of the devices. Thus,In this case we simply choose random voltages from a uni-form distribution, in the range [0 V , V ] , which is adequate toaddress phases in the range ≤ φ ≤ π — i.e. no a prioricalibration of Alice and Bob’s devices is necessary. Experimental violations with random measurement triads.
We first investigate the situation in which Alice and Bob bothuse 3 orthogonal measurements. We generate randomly cho-sen measurement triads using a pseudo-random number gen-erator. Having calibrated the phase/voltage relationship of thephase shifters, we then apply the corresponding voltages onthe chip. For each pair of measurement settings, the coin-cidence counts between all of the 4 pairs of APDs are thenmeasured for a fixed amount of time—the typical numberof simultaneous photon detection coincidences is ∼ kHz.From these data we compute the maximal CHSH value as de-tailed above. This entire procedure is then repeated 100 times.The results are presented in Fig. 4a, where accidental coinci-dences, arising primarily from photons originating from dif-ferent down-conversion events, which are measured through-out the experiment, have been subtracted from the data (theraw data, as well as more details on accidental events, can befound in the Methods). Remarkably, all 100 trials lead to aclear CHSH violation; the average CHSH value we observe is ∼ . , while the smallest measured value is ∼ . .We next investigate the effect of decreasing the visibility ofthe singlet state: By deliberately introducing a temporal de-lay between the two photons arriving at the CNOT gate, wecan increase the degree of distinguishability between the twophotons. Since the photonic
CNOT circuit relies on quantuminterference [24] a finite degree of distinguishability betweenthe photons results in this circuit implementing an incoher-ent mixture of the CNOT operation and the identity operation[25]. By gradually increasing the delay we can create states ρ V with decreasing visibilities. For each case, the protocoldescribed above is repeated, which allows us to estimate theaverage CHSH value (over 100 trials). For each case we alsoestimate the visibility via maximum likelihood quantum statetomography. Figure 4b clearly demonstrates the robustness ofour scheme, in good agreement with theoretical predictions: aconsiderable amount of mixture must be introduced in orderto get an average CHSH value below 2.Together these results show that large Bell violations canbe obtained without a shared reference frame even in the pres-ence of considerable mixture. Experimental violations with completely random measure-ments.
We now investigate the case where all measurementsare chosen at random. The procedure is similar to the first ex-periment, but we now apply voltages chosen randomly froma uniform distribution, and independently for each measure-ment setting. Thus our experiment requires no calibration ofthe measurement MZ interferometers ( i.e. the characterizationof the phase-voltage relation), which is generally a cumber-some task. By increasing the number of measurements per-formed by each party ( m = 2 , , , ), we obtain CHSH vi-olations with a rapidly increasing probability, see Fig. 5. For m = 5 , we find 95 out of 100 trials lead to a CHSH viola-tion. The visibility V of the state used for this experiment wasmeasured using state tomography to be . ± . , clearlydemonstrating that robust violation of Bell inequalities is pos-sible for completely random measurements. p CH S H Iteration m =2 p CH S H Iteration m =3 p CH S H Iteration m =4 p CH S H Iteration m =5 FIG. 5:
Experimental Bell tests using uncalibrated devices.
Weperform Bell tests on a two-qubit Bell state, using uncalibrated mea-surement interferometers, that is, using randomly-chosen voltages.For m = 2 , , , local measurement settings, we perform 100 tri-als (for each value of m ). As the number of measurement settings m increases, the probability of obtaining a Bell violation rapidly ap-proaches one. For m ≥ , the average CHSH value (dashed line) isabove the local bound of CHSH=2 (red line). Error bars, which aretoo small to draw, were estimated by a Monte Carlo technique, as-suming Poissonian statistics. Data has been corrected for accidentals(see Methods). It is interesting to note that the relation between the phaseand the applied voltage is typically quadratic, see Eq. (6).Thus, by choosing voltages from a uniform distribution, thecorresponding phase distribution is clearly biased. Our exper-imental results indicate that this bias has only a minor effecton the probability of obtaining nonlocality.
Discussion.
Bell tests provide one of the most important paths to gain-ing insight into the fundamental nature of quantum physics.The fact that they can be robustly realised without the needfor a shared reference frame or calibrated devices promises toprovide new fundamental insight. In the future it would be in-teresting to investigate these ideas in the context of other Belltests, for instance considering other entangled states or in themultipartite situation (see [27] for recent progress in this di-rection), as well as in the context of quantum reference frames[28, 29].The ability to violate Bell inequalities with a completelyuncalibrated device, as was demonstrated here, has importantapplication for the technological development of quantum in-formation science and technology: Bell violations provide anunambiguous signature of quantum operation and the abilityto perform such diagnostics without the need to first performcumbersome calibration of devices should enable a significantsaving in all physical platforms. These ideas could be particu-larly helpful for the characterization of entanglement sourceswithout the need for calibrated and aligned measurement de-vices.Finally Bell violations underpin many quantum informationprotocols, and therefore, the ability to realise them with dra-matically simplified device requirements holds considerablepromise for simplifying the protocols themselves. For exam-ple, device independent quantum key distribution [5] allowstwo parties to exchange a cryptographic key and, by checkingfor the violation of a Bell inequality, to guarantee its securitywithout having a detailed knowledge of the devices used in theprotocol. Such schemes, however, do typically require precisecontrol of the apparatus in order to obtain a sufficiently largeviolation. In other words, although a Bell inequality violationis an assessment of entanglement that is device-independent,one usually needs carefully calibrated devices to obtain such aviolation. The ability to violate Bell inequalities without theserequirements could dramatically simplify these communica-tion tasks. The implementation of protocols based on quantumsteering [15] may also be simplified by removing calibrationrequirements.
Note added.
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METHODS
Photon counting and accidentals.
In our experiments, wepostselect on successful operation of the linear-optical
CNOT gate by counting coincidence events, that is, by measuring therate of coincidental detection of photon pairs. Single photonsare first detected using silicon avalanche photodiodes (APDs).Coincidences are then counted using a Field-ProgrammableGate Array (FPGA) with a time window of ∼ ns. We referto these coincidence events as { t A , t B } .Accidental coincidences have two main contributions: first,from photons originating from different down-conversionevents arriving at the detectors within the time window; sec-ond, due to dark counts in the detectors. Here we directly mea-sure the (dynamic) rate of accidental coincidences in real time,for the full duration of all the experiments described here. Todo so, for each pair of detectors we measure a second coin-cidence count rate, namely { t A , t B } , with | t − t | = 30 ns.In order to do this, we first split (duplicate) the electrical TTLpulse from each detector into two BNC cables. An electri-cal delay of 30ns is introduced into one channel, and coinci-dences (i.e. at { t A , t B } ) are then counted directly. Finally weobtain the corrected coincidence counts by subtracting coinci-dence counts at { t A , t B } from the raw coincidence counts at { t A , t B } .All experimental results presented in the main text havebeen corrected for accidentals. Here we provide the raw data.Fig 6 presents the raw data for Fig. 4(a) while Fig. 7 presentsthe raw data for Fig. 5. Notably, in Fig. 6, corresponding tothe case of randomly chosen triads, all but one of the hundredtrials feature a CHSH violation. The average violation is now ∼ . . p CH S H Iteration
FIG. 6:
Raw data of experimental Bell tests requiring no sharedreference frame.
This figure shows the raw data, without correctingfor accidental coincidences, of Fig. 4a. Here the average CHSH valueis 2.30 (dashed line), leading to an estimate of the visibility of V = . . (cid:39) . , while the estimate from quantum state tomography is V = 0 . ± . . Again, this discrepancy is due to the fact thatour entangled state is not exactly of the form of a Werner state. Errorbars, which are too small to draw, were estimated using a Monte-Carlo technique, assuming Poissonian photon statistics. p CH S H Iteration m =2 p CH S H Iteration m =3 p CH S H Iteration m =4 p CH S H Iteration m =5 FIG. 7:
Raw data of experimental Bell tests using uncalibratedmeasurement interferometers (random voltages).
This figureshows the raw data, without correcting for accidental coincidences,of Fig. 5. Error bars are estimated by a Monte Carlo technique,assuming Poissonian statistics. The visibility V of the state usedfor this experiment was measured using state tomography to be . ± .003