Bell experiments can be used to generate private random numbers. An ideal
Bell experiment would involve measuring a state of two maximally entangled
qubits, but in practice any state produced is subject to noise. Here we
consider how the techniques presented in arXiv:1309.3894 and arXiv:1309.3930,
i.e. using an optimized Bell inequality, and taking advantage of the fact that
the device provider is not our adversary, can be used to improve the rate of
randomness generation in Bell-like tests performed on singlet states subject to
either white or dephasing noise.Comment: 4 pages, 2 figures; to appear in Proceedings of TQC 2014; published
  versio

Bancal, Jean-Daniel

Scarani, Valerio

English

arXiv

Bell experiments can be used to generate private random numbers. An ideal Bell experiment would involve measuring a state of two maximally entangled qubits, but in practice any state produced is subject to noise. Here we consider how the techniques presented in Refs [Bancal et al., New J. Phys., 2014] and [Nieto-Silleras, New J. Phys., 2014], i.e. using an optimized Bell inequality, and taking advantage of the fact that the device provider is not our adversary, can be used to improve the rate of randomness generation in Bell-like tests performed on singlet states subject to either white or dephasing noise

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More Randomness From Noisy Sources∗Jean-Daniel Bancal1 and Valerio Scarani1,21 Centre for Quantum Technologies, National University of Singapore3 Science Drive 2, Singapore 1175432 Department of Physics, National University of Singapore2 Science Drive 3, Singapore 117542AbstractBell experiments can be used to generate private random numbers. An ideal Bell experimentwould involve measuring a state of two maximally entangled qubits, but in practice any stateproduced is subject to noise. Here we consider how the techniques presented in [1] and [2], i. e.using an optimized Bell inequality, and taking advantage of the fact that the device provider isnot our adversary, can be used to improve the rate of randomness generation in Bell-like testsperformed on singlet states subject to either white or dephasing noise.1998 ACM Subject Classification J.2 Physical Sciences and EngineeringKeywords and phrases Randomness, Bell inequalities, Trusted provider assumptionDigital Object Identifier 10.4230/LIPIcs.TQC.2014.11 IntroductionIt is well known that the violation of a Bell inequality rules out the possibility for theoutcomes of a Bell-type experiment to be known in advance [3]. Therefore, these outcomesare certifiably unpredictable. Recent works have shown that the uncertainty present in theseoutcomes can be quantified, thus allowing one to lower bound the number of random bitsthat can be extracted from a given Bell-type experiment [4, 5].This possibility has given rise to a variety of randomness-related studies based on asimilarly varied set of working assumptions. For instance, many works considered the casein which the adversary (the actor for whom outcomes are to be certifiably unpredictable)is allowed to distribute the quantum state measured by the authorized parties, and keep apurification of this state. Under this assumption, it was shown that randomness expansionis possible: if the user holds a secret string of finite length, he can expand it into a longerone [6], or, in principle, even an infinite one [8, 7].Also, the outcomes observed by the user can be certified to contain some amount ofrandomness even when the adversary, in addition to distributing the state, holds partialinformation about the initial random string of the user [9]. This possibility, refered to asrandomness amplification, was proved recently for initial randomness issued from a genericmin-entropy source [10, 11] after a series of partial results [12, 13].These results show the full power of quantum certification in principle. However, whenit comes to realizing such protocols, a number of questions arise. For instance, in whichpractical situation would one wish to expand a random string if we already have accessto a source that can produce initial random strings? Also, in the context of randomness∗ This work was supported by the National Research Foundation (partly through the Academic ResearchFund Tier 3 MOE2012-T3-1-009) and the Ministry of Education, Singapore.T Q C© Jean-Daniel Bancal and Valerio Scarani;licensed under Creative Commons License CC-BY9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC’14).Editors: Steven T. Flammia and Aram W. Harrow; pp. 1–6Leibniz International Proceedings in InformaticsSchloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany2 More Randomness From Noisy Sourcesamplification, it is unclear in which meaningful situation the dependence would exist at allbut be bounded. If the adversary is allowed to tamper with the devices, for instance, or evento produce them, then he may have hidden some kind of emitter inside the boxes, in orderto retrieve all numbers produced by the boxes (which otherwise work as expected). Thissimple possibility would compromise any certification of randomness.For these reasons, in the design and assessment of practical realization of randomnessprotocols , it is very reasonable to work under the assumption that the adversary has noaccess to the devices used by the authorized partners. This trusted provider assumptionwas already introduced in the context of randomness protocols in [14], where it was shownthat it restricts the adversary to hold only classical side information (i. e. he cannot hold apurification of the quantum state). Note that this contrasts with the case of quantum keydistribution (QKD): practical QKD also requires the trusted provider assumption, for thesame reason as mentione above, however the adversary can still hold a purification in thiscase since the quantum state passes in his hands. Another consequence mentioned in [14]is that the initial string used by the user to choose settings for his Bell test need not beprivate, but can be fully known in advance by the adversary. One thus speaks of randomnessgeneration in this context.It was shown in [1] that additional randomness can be certified under the trusted providerassumption compared to that granted by randomness expansion protocols, by extractingrandomness from all the settings. Moreover, this same paper as well as [2] demonstratedthat Bell-like inequalities that certify more randomness than usual Bell inequalities (likee. g. CHSH) can be derived from knowledge of the full correlations. In this paper, weanalyse the advantage provided by these techniques when the quantum state measured bythe user is a singlet states mixed either with white or dephasing noise. White noise typicallydescribes the effect of many small errors in a setup whereas dephasing noise is the dominantnoise in SPDC-based sources when the pump power is low. The case of white noise wasalready partially studied in both [1] and [2]. The analysis given here gathers the informationpresented in both studies and provides a comparison with the dephasing noise case.Even though our analysis relies on the trusted provider assumption, it is worth notingthat some of the results obtained here could also apply to more general adversaries; we referto [15] for a concise review of adversarial classes relevant to randomness protocols.For the present paper, we assume that the source emits exactly one pair of particles perunit time and that these are detected with certainty. The case of finite detection efficiencywas studied in [1], in absence of noise; when the emission is not heralded, more effects comeinto play, see e. g. [16].Another assumption that we make here is that the devices used by the user are i.i.d. andthat he can use as many of them as he wants. We thus focus on the rate of randomnessgeneration, defined as the number of random bits generated in each use of the devices.2 Randomness analysisWe consider here a usual Bell-type experiment performed by a user [3]. At each round,the user chooses some inputs x, y for his two devices to use as measurement settings, andobserves their outcomes a, b. The i.i.d. behavior of the boxes follows the quantum conditionalprobability P (a, b|x, y) ∈ Q.In general, these correlation can admit a decomposition {qλ, Pλ} such thatP (ab|xy) =∑λqλPλ(ab|xy) (1)J.-D. Bancal and V. Scarani 3with qλ ≥ 0,∑λ qλ = 1, Pλ(ab|xy) ∈ Q. When this decomposition is not trivial, by knowingin each round which value of λ corresponds to the realization of the box, the adversary canhold a more precise decription of the box’s behavior for that run, as given by Pλ(ab|xy).Following [1, 2], we thus define the adversary’s guessing probability on the outcomesobserved by the user when using settings x, y and in presence of the decomposition {qλ, Pλ}asGx,y({qλ, Pλ}) =∑λqλmaxa,bPλ(ab|xy). (2)The average guessing probability when settings are chosen with probability p(x, y) is thenthe maximum ofG(P ) =∑xyp(x, y)∑λqλmaxa,bPλ(ab|xy) (3)over all decompositions (1) compatible with the correlations P (ab|xy).It was shown in [1] that this quantity can be upper bounded by considering an SDP(Semidefinite Program) relaxation of the set of quantum correlations [17]. In the followingsection, we thus use this program to evaluate the rate, as given by the min entropyHmin(P ) = − log2(G(P )), (4)at which random bits are generated in the experiment.Note that the particular case of this optimization where randomness is extracted froma fixed choice of settings (pxy = δx,x0δy,y0), or where the outcomes of different settings areallowed to by guessed with different decomposition, was also presented independently in [2].In the following we compare three quantities:1. The rate of randomness obtained from a fixed set of settings as certified by a CHSHviolation.2. The rate of randomness obtained from a fixed set of settings as certified by an optimizedBell-type expression.3. The rate of randomness obtained when using all settings with the same probability ascertified by an optimized Bell-type expression.Note that here we consider extracting randomness from the pair of outcomes (a, b) ratherthan from the outcome of a single party. A similar computation could be done by takingonly one party’s outcome into consideration, but would result in a lower rate. Also, for thefirst two quantities, the fixed set of settings is chosen as to maximize the rate of randomness.For all results presented next, the numerical computations were performed using therelaxation of the SDP hierarchy at local level 2 [18].2.1 White noiseFirst, let us consider the case in which the measured state isρ(V ) = V |φ+〉〈φ+|+ (1− V )1/4, (5)for some visibility V . The settings which provide the largest violation 2√2V of the CHSHinequalityS = 〈A0B0〉+ 〈A0B1〉+ 〈A1B0〉 − 〈A1B1〉 ≤ 2, (6)TQC’144 More Randomness From Noisy Sources0.7 0.75 0.8 0.85 0.9 0.95 100.20.40.60.811.2VRandomness [bits/run]0.7 0.8 0.9 1123VratioFigure 1 Rate of private randomness generation certified by the measure 1, 2 and 3 for a singletstate mixed with white noise. The inset presents the ratio of the curves to the lowest one.where Ax, By are Alice’s and Bob’s observables, are the same for all V :A0 = σz, A1 = σx, By =σz + (−1)yσx2 . (7)We thus computed for this state and settings the three different rates of randomnessmentioned above. The result is presented in Figure 1.The randomness rate obtained in case 2 (middle curve) can be certified with the help ofthe following Bell expression:α〈A0B0〉+ 〈A0B1〉+ 〈A1B0〉 − β〈A1B1〉, (8)where the values of α and β depend on V (see [2] for a description of this dependence). Theinset in Figure 1 shows that the advantage provided by using this optimized Bell expressionis however quite limited.The largest amount of randomness is obtained in case 3 when considering the outcomesobserved when all settings are used with the same probability (i. e. p(x, y) = 1/4). Asmentioned in [1], the improvement, of the order of a factor of 2, is certified with the usualCHSH inequality.2.2 Dephasing noiseSecond, we consider measurement of the stateρ(p) = p|φ+〉〈φ+|+ (1− p)(|00〉〈00|+ |11〉〈11|). (9)The optimal violation of the CHSH inequality by this state, S = 2√1 + p2, is provided byusing the following settings [19]:A0 = σz, A1 = σx,By = cosχσz + (−1)y sinχσx, (10)with χ = arctan(p).J.-D. Bancal and V. Scarani 50 0.2 0.4 0.6 0.8 100.20.40.60.811.21.4pRandomness [bits/run]0 0.5 1123pratioFigure 2 Rate of private randomness generation certified by the measure 1, 2 and 3 for a singletstate mixed with dephasing noise. The inset presents the ratio of the curves to the lowest one.The three randomness rates obtained with these state and settings are presented inFigure 2. Similarly to the previous case, strictly more randomness can be certified in case3 than in case 2, and in case 2 than in case 1. The inequality that certifies the largestamount of randomness is again CHSH in case 3, and a different inequality in case 2. One cancheck that this inequality, however, beyond being a correlation inequality presents no specialsymmetry. In particular, it is not of the form (8). Nevertheless, we note that when therandomness is extracted from a single set of settings, using an optimized inequality providesa larger advantage for this dephasing noise than it did in the case of mixture with whitenoise (as shown in the inset of Figure 2).3 ConclusionWe have presented an application of the techniques presented in [1, 2] to the case wherethe measured state is a singlet mixed with either white noise or dephasing noise. Whilea significant advantage in terms of randomness rate can be obtained in both cases whenrandomness is extracted uniformly from all settings, the advantage for extraction from afixed choice of settings is much more significant in the case of dephasing noise.In a practical experiment, characteristics of both white and dephasing noise are expectedto appear [20], as well as various other kind of noises and imperfections [16]. The presentanalysis is not meant to exhaust all the parameter space of a realistic experiment; but itshould be clear that the techniques used here can be extended to describe experiments withall their features.We have focused here on the asymptotic rate of randomness generation. It would beinteresting to extend our analysis to take into account finite statistics, maybe in a way similarto [14] or [21]. This would allow one to quantify how many random bits can be extractedfrom a Bell experiment which involves only a finite number of rounds.TQC’146 More Randomness From Noisy SourcesReferences1 J.-D. Bancal, L. Sheridan, V. Scarani, New J. Phys. 16, 033011 (2014).2 O. Nieto-Silleras, S. Pironio, J. Silman, New J. Phys. 16, 013035 (2014).3 N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, S. Wehner, Rev. Mod. Phys. 86, 419(2014).4 R. Colbeck, Ph.D. thesis, University of Cambridge, arXiv:0911.3814 (2006).5 S. Pironio, A. Acín, S. Massar, A. Boyer de la Giroday, D.N. Matsukevich, P. Maunz, S.Olmschenk, D. Hayes, L. Luo, T.A. Manning, C. Monroe, Nature 464, 1021 (2010).6 U.V. Vazirani, T. Vidick, arXiv:1111.60547 C.A. Miller, Y. Shi, arXiv:1402.04898 M. Coudron, H. Yuen, arXiv:1310.67559 R. Colbeck, R. Renner, Nature Physics 8, 450–454 (2012)10 J. Bouda, M. Pawlowski, M. Pivoluska, M. Plesch, arXiv:1402.097411 K.-M. Chung, Y. Shi, X. Wu, arXiv:1402.479712 R. Gallego, L. Masanes, G. de la Torre, C. Dhara, L. Aolita, A. Acin, Nature Communica-tions 4, 2654 (2013)13 T.P. Le, L. Sheridan, V. Scarani, Phys. Rev. A 87, 062121 (2013).14 S. Pironio and S. Massar, Phys. Rev. A 87, 012336 (2013).15 Y. Z. Law, T.P. Le, J-D. Bancal and V. Scarani, arXiv:1401.4243.16 V. Caprara-Vivoli, P. Sekatski, J.-D. Bancal, C.C.W. Lim, B.G. Christensen, A. Martin,R.T. Thew, H. Zbinden, N. Gisin, N. Sangouard, arXiv:1405.193917 M. Navascues, S. Pironio, A. Acín, Phys. Rev. Lett. 98, 010401 (2007).18 T. Moroder, J-D. Bancal, Y-C. Liang, M. Hofmann, O. Gühne, Phys. Rev. Lett. 111,030501 (2013).19 R. Horodecki, M. Horodecki, P. Horodecki, Phys. Lett. A 200, 340 (1995)20 J. Kofler, S. Ramelow, M. Giustina, A. Zeilinger, arXiv:1307.6475.21 Y. Zhang, S. Glancy, E. Knill, Phys. Rev. A 84, 062118 (2011).

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