Abstract
The intrinsic unpredictability of measurements in quantum mechanics can be used to produce genuine randomness. Here, we demonstrate a random number generator where the randomness is certified by quantum contextuality in connection with the KochenSpecker theorem. In particular, we generate random numbers from measurements on a single trapped ion with three internal levels and certify the generated randomness by showing a bound on the minimum entropy through observation of violation of the KlyachkoCanBiniciogluShumovsky (KCBS) inequality. Concerning the test of the KCBS inequality, we close the detection efficiency loophole for the first time and make it relatively immune to the compatibility loophole. In our experiment, we generate 1 × 10^{5} random numbers that are guaranteed to have 5.2 × 10^{4} bits of minimum entropy with a 99% confidence level.
Introduction
Random number generation is important for many applications^{1,2}. For cryptographic applications, random numbers should have good unpredictability in order to be secure under attack by the adversaries^{3}. Genuine random numbers can never be generated by a classical device because any classical device bears in principle a deterministic description. Quantum mechanics, on the other hand, has intrinsic randomness and thus can be explored to construct a genuine random number generator. There have been many demonstrations of random number generators based on quantum principles^{4,5,6,7,8,9,10,11,12,13,14}.
Selfcertified random number generation is an advance made recently, where the randomness is guaranteed by violation of certain fundamental inequalities^{14,15,16}. In particular, it was proposed in Refs. 14,15 that through violation of the ClauserHornShimonyHolt (CHSH) inequality, one can certify the generated random numbers in a deviceindependent fashion that is secure against the adversaries who have only classical side information^{17}. The first proofofprinciple experiment for this scheme has been recently demonstrated^{14}.
We consider here a scenario where the provider of the device is assumed to be honest. However, we still need to physically certify that the random numbers are generated due to the intrinsic uncertainty of quantum mechanics instead of some uncontrolled classical noise process in the device. In this case, we can use quantum contextuality manifested through the violation of certain KochenSpecker (KS) inequality to certify the generated random numbers^{18,19}. Quantum contextuality is a basic property of quantum mechanics, where the measurement outcomes depend on the specific context of the measurements^{20,21}. Quantum contextuality would be revealed by violations of some KS inequalities and such violations can be observed even in a single indivisible system without any entanglement^{22,23,24,25,26,27}. Because there is no need of entanglement, a certification scheme of random numbers based on the KS theorem can significantly simplify the experimental requirement and generate certified random numbers with a much higher speed^{18}. A proofofprinciple experimental implementation of this idea has been reported with a photonic system quite recently^{18}.
A particular type of the KS inequality, the KlyachkoCanBiniciogluShumovsky (KCBS) inequality^{22}, is convenient for certification of random numbers. Violation of the KCBS inequality has been observed before in a singlephotonic system^{23}. For experimental test of the KCBS inequality, there are two possible loopholes: the detection efficiency loophole if the detectors only register a subset of data due to their inefficiency and the compatibility loophole, which occurs if additional assumptions are required to guarantee that the observables with simultaneous assignment of values in the KCBS inequality are compatible with each other and remain identical when their measurement contexts change. The test of the KCBS inequality with the photonic system is immune to the compatibility loophole^{23}, however, it requires the fairsampling assumption due to the low photon detection efficiency and thus subject to the detection efficiency loophole.
In this paper, we report a random number generator certified by quantum contextuality with a single trapped ion, which allows us to close the detection efficiency loophole for the first time for the KCBS inequality. For the compatibility, we follow basically the same configurations as in Ref. 23, where errors in compatibible measurement settings only reduce the amount of the violations. Even with experimental noise and imperfections, we get significant violations of the KCBS inequality, which lead to lower bounds the minimum entropy of the generated random string. Compared to the experimental certification based on the CHSH inequality^{14}, the generation rate of random numbers is increased by about four orders of magnitudes in our experiment, which is important for practical applications.
The paper is organized as follows. First, we introduce the KCBS inequality and show the experimental violation of this inequality. Then, we introduce the relation between the violation of the KCBS inequality and the minimum entropy of the generated random string for the case of an honest provider and compare the theoretical prediction with our experimental observation. The generated random bits are tested under uniform or biased choice of measurement settings. We conclude the paper by summarizing the results and discussing further improvements of our random number generation scheme.
Results
The KCBS inequality
The KochenSpecker theorem states that the results of quantum mechanics cannot be fully explained by noncontextual classical theories which assume that the measurement outcomes of a physical system are predetermined and independent of their own and other simultaneous compatible measurements^{20,21}. The KCBS inequality illustrates the conflict between quantum mechanics and noncontextual classical theory in the simplest possible system with the Hilbert space dimension d = 3^{22}.
The KCBS inequality is connected with the following simple algebraic equation.
where the value of a_{i} is either 1 or −1. If the values of the observables are predetermined, the average of the left hand of the above equation should be no less than −3, leading to the following inequality:
In quantum mechanics, however, the outcomes of A_{i} do not have predetermined values, which allows violation of the KCBS inequality (2) for a specific state ψ_{0}〉 in systems with d ≥ 3. In the case of d = 3, we denote the bases by 1〉, 2〉 and 3〉 and the observable A_{i}, represented by A_{i} = 1 – 2 v_{i}〉 〈v_{i}, is the projector on the axis v_{i}〉. The maximal violation of the KCBS inequality (2) is achieved for the state along the symmetric axis of the pentagram shown in Fig. 1(a). Here v_{1}〉 = 1〉, v_{2}〉 = 2〉, v_{3}〉 = R_{1} (γ, 0) v_{1}〉, v_{4}〉 = R_{2} (γ, 0) v_{2}〉, v_{5}〉 = R_{1} (γ, 0) v_{3}〉 and , where γ = 51.83° and R_{1,2} denote the rotation operations between 1〉 to 3〉 and between 2〉 to 3〉, respectively. Maximal violation the KCBS inequality is achieved under the state (2), with the corresponding value .
Figure 1(b) shows the scheme for preparation of the initial state ψ_{0}〉 starting from the basis state 3〉 and Fig. 1(c)–(g) describe the implementation of the measurement configurations along the five axes. To ensure context independence, we emphasize that the measurement configuration of A_{i} remains the same when it is measured with either A_{i}_{–1} or A_{i}_{+1} (let A_{0} ≡ A_{5}, A_{6} ≡ A_{1}). For example, the scheme for the measurement A_{2} is exactly the same in the first [Fig. 1(c)] and the second stage [Fig. 1(d)]. To move to the second configuration, we perform a rotation between the states 1〉 and 3〉, which does not influence the state 2〉 that corresponds to the observable A_{2}. Only the observable related to the state 1〉 is changed from A_{1} to A_{3}.
The configuration for the measurement of A_{1} in Fig. 1(c) is not the same as that in Fig. 1(g), which is therefore denoted by . If A_{1} and are not identical, it is possible to violate the inequality (3) even in classical theory. To solve this problem, similarly to Ref. 23, we use a new inequality that includes the observable with the form
Note that the inequality (3) becomes the original KCBS inequality (2) when . Therefore, the difference between two measurements decrease the violation that can be obtained in the experiments^{23}. Another possible way out is to introduce an empirical parameter to upper bounds the violation of compatibility, which would be similar in spirit to a recent work where a parameter in introduced to bound violation of the locality loophole for test of the Bell inequality^{28}. Any imperfection in the initial state preparation or final measurements only leads to a reduction of violation of the KCBS inequality, so a significant violation of this inequality guarantees that the randomness comes from the quantum origin instead of a classical noise process.
Experimental violation of the KCBS inequality
The violation of the KCBS inequality have been observed with single photons^{18,23}, however, those experiments are subject to the detection efficiency loophole. Here, we present the experimental violation of the KCBS inequality in a single trapped ion. Because of the high detection efficiency for the trapped ion, we close the detection efficiency loophole for the first time for this inequality.
We perform the test of the KCBS inequalities (2) with a single trapped ^{171}Yb^{+} ion in a fourrod radiofrequency trap^{26,29}. The qubit states are represented by the two internal levels in the S_{1/2} groundstate manifold, with F = 1, m_{F} = 0〉 ≡ ↑〉 and F = 0, m_{F} = 0〉 ≡ ↓〉. The transition frequency between ↑〉 to ↓〉 is ω_{HF} = (2π)12642.821 MHz, determined by the hyperfine interaction.
The procedure of the experiment consists of Doppler cooling, initialization, coherent operation and detection (see the Method Section). The initial state preparation and the measurement configurations are shown in Fig. 1(b)(g) and they are realized by two microwaves with the frequencies ω_{1} and ω_{2}, which produce Rabi oscillations R_{1} (θ_{1}, ϕ_{1}) and R_{2} (θ_{2}, ϕ_{2}) between 1〉 to 2〉 and between 1〉 to 3〉, respectively. Here, θ_{1,2} and ϕ_{1,2} are controlled by the duration and phase of the microwaves. R_{1} (θ_{1}, ϕ_{1}) and R_{2} (θ_{2}, ϕ_{2}) are have the following explicit forms
For experimental convenience, we transform the observable A_{i} to V_{i} = (1 − A_{i})/2, which is assigned to value v_{i} = 0 when photons are detected or v_{i} = 1 when no photons are detected. With V_{i}, the KCBS inequality (3) is rewritten as
We obtain 〈V_{i}〉 by mapping the axis v_{i} to the state 3〉 and then measuring the probability P_{3〉} (v_{i} = 1) = 〈V_{i}〉 (Fig. 2(b)). For simplicity, let P_{3〉} = P. The correlation terms 〈V_{i}V_{i}_{+1}〉 are obtained by sequential measurements depicted in Fig. 2(c). First, we transfer V_{i} on the state 3〉 and apply the standard fluorescence detection scheme. If we detect photons, the state should not be 3〉 and we assign v_{i} = 0 to the observable V_{i}, where the outcome of the correlation term V_{i}V_{j} vanishes and no further measurements are needed. If we detect no photons, we assign v_{i} = 1 to the V_{i}. Then, we apply the swapping microwave πpulse that converts V_{j} to 3〉 before another round of fluorescence detection. If we observe photons, v_{j} = 0 and if no photons, v_{j} = 1. We asign the value 1 to the correlation term V_{i}V_{j} only when we detect no photons for both rounds of measurements. We obtain the average of the correlation term 〈V_{i}V_{j}〉 = P (v_{i} = v_{i}_{+1} = 1) by repeating the same experimental sequence many times^{26}.
The expectation value is obtained by the scheme shown in Fig. 2(d). If ideally, the correlation shoud be same to 〈V_{1}〉 since V_{1} is projection operator . The state 1〉 at the beginning of Fig. 1(g) corresponds to the observable V_{1}, which is exactly the same configuration as in Fig. 1(c). Therefore, if photons are detected (v_{1} = 0) or not detected (v_{1} = 1) at the place where V_{1} would be measured, photons should be observed () or not be observed () for the shown in Fig. 2 (d). After repeating the sequence of Fig. 2(d), we acquire the probability that no photons are measured (), which gives by definition.
We randomly choose one of the five configurations (c)–(g) of Fig. 1 based on computer generated random numbers and perform the sequential measurements. We change the order of sequential measurements ( V_{i}V_{i}_{+1} or V_{i}_{+1}V_{i}) with equal probability. We occasionally check the overlap of V_{1} and . We repeat the sequences 1 × 10^{5} times and observe 〈χ_{KCBS}〉 = 3.852(0.030), which violates the extended KCBS inequality (3,4) by 31 σ. The detailed results of the measurements are summarized in Table 1. We emphasize that our result of the violation cannot be explained by any noncontextual classical theory which does not exploit the compatibility loophole (the detection loophole is closed in our experiment). In other words, any classical part of the system such as technical noise, imperfections and/or unexpected changes of control parameters can not produce the violation. Therefore, as long as we observe the violation of the inequality, we can ensure that the outcomes of our measurements originate from quantum mechanics.
The relation between violation of the KCBS inequality and the minentropy
We establish the relation between violation of the KCBS inequality (2, 4) and randomness of the generated string from the experiment, similar to the photonic demonstration^{18}. We focus on the scenario with an honest provider of the device^{17} rather than the extreme adversary scenario where the device has been produced by a malicious manufacturer. Even though we trust the device provider, we still need to ensure that the randomness of the generated sequence is caused by quantum uncertainty instead of technical noise^{17}. For this purpose, we assume: (1) the system can be described by quantum theory; (2) the input at lth trial is chosen from a random process that is independent and uncorrelated from the system and its value is revealed to the system only at step l; (3) the outcomes of the corresponding pairs of measurements at step l are compatible (the measurement of one observable does not influence on the marginal distribution of the results of the other observable); (4) the adversary does not have any capability of controlling the inside of the system. The first and the second assumptions here are identical to those made in the certification scheme of Bell's inequality^{14}. The third is the contextuality assumption that replaces the role of locality assumption for the Bell inequality. The fourth is an assumption about the honest provider^{17}.
We consider five sets of measurement configurations S = {A_{1}A_{2}, A_{2}A_{3}, A_{3}A_{4}, A_{4}A_{5}, A_{5}A_{1}}, where A_{i} is the observable with the output a_{i} = ±1 and compatible with A_{i}_{–1} and A_{i}_{+1}. We can rewrite the KCBS inequality (2) as
where P (a_{i} = a_{i}_{+1}A_{i}A_{i}_{+1}) or P (a_{i} ≠ a_{i}_{+1}A_{i}A_{i}_{+1}) is the probability that the output results are the same or different for a chosen measurement setting A_{i}A_{i}_{+1}. Note that we change the sign of the inequality to make the derviation similar to that in Refs. 14,17,30. In our experiment, since we use the observable V_{i} (result v_{i} = 0,1) intead of A_{i} and only distinguish the event of v_{i} = v_{i}_{+1} = 1 from others, the Eq. (5) is modified as
where P (v_{i} = 1V_{i}) is the probability that the output result v_{i} is 1 at a measurement setting V_{i}. The result of terms inside {··· } is ideally zero and nonzero positive value can be occurred by experimental errors or imperfections, which only reduces the amount of violation from the optimal. Therefore, we can conclude that the experimental violations of the inequality (6) arise from solely quantum mechanical origin not any classical mean.
In our relization, we estimate the violation of the inequality (6) by repeating the sequences n times and additional runs n_{cc} of the comparibility check, the measurement setting . The estimation of Eq. (5), obtained from the experimental data, is written as
where N (v_{i} = 1V_{i}) or N (v_{i} = v_{i}_{+1} = 1V_{i}V_{i}_{+1}) is the number of times that the outcome v_{i} or v_{i} and v_{i}_{+1} is one under a measurement setting V_{i} or V_{i} and V_{i}_{+1}, respectively. P (V_{i}) or P (V_{i}V_{i}_{+1}) is the probability with which a measurement configuration V_{i} or V_{i} and V_{i}_{+1} is chosen. Note that positive result of terms inside {···} and [···] originates from the experimental flaws, which only reduces the amount of violation.
The randomness of a single generated bit v_{i} from a measurement setting V_{i} can be characterized by the minentropy , where P (v_{i}V_{i}) is the conditional probability of obtaining v_{i} when the input setting V_{i} and the maximum is taken over all possible values of the output string. The theorem 1 of Ref. 17 shows that the min entropy of the generated string after n trials is bounded by
where is a series of KCBS violation thresholds with the classical bound and the maximum violation and , with r the smallest probability of input choices min_{i}P (V_{i}). The parameter parameter denotes the closeness between the resulting distribution that characterize k successive uses of the device and another extended distribution that is well defined mathematically. The function f is found by semidefinite programming at various expectations L. Fig. 3 presents how the minentropies are affected by the confidence levels, and 1 − δ. When we set a high confidence level, , the bound on the minentropy reduces as expected. Note that the certified minentropy is only determined by measured value and the choice of , independent of experimental details.
Random number results
We perform ten thousand trials to generate random bits as described in the previous section: [Experimental violation of the KCBS inequality]. At each trial, we choose one of the five measurement configurations shown in Fig. 1 (c)−(g) by computergenerated random numbers, perform the sequence composed of Doppler cooling, state initialization and rotations for the chosen configuration and finally record the existence of fluorescence (see Method section). As explained, we obtain a random bit, i.e., 1 (or 0) with fluorescence (or no fluorescence) for each trial. The sequence takes about 10 ms, mainly limited by the waveform loading time to the pulse generator. Note that the random generator based on the CHSH inequality produced a random bit per several min.
Figure 4 shows the minentropies of generated strings discussed in the previous section: [The relation between violation of the KCBS inequality and the minentropy]. We produce a string of length 1 × 10^{5} with uniform choices of the measurement settings, P (V_{i}) = 1/5. As shown in Table 1, we observe the expectation , implying the minentropy with 99% confidence. Note that the other confidence level δ does not have any noticable influence on the bound of minentropy. Here we used the thresholds of KCBS violations .
Fig. 4 shows clearly the advantage of our certification scheme, i.e., we can guarantee the minentropy of the generated random string by only monitoring the violation independent of experimental details. Fig. 4(a) shows the accumulated behavior of the minentropy as the number of experimental trials n increases. The solid lines show the theoretical linear increment of the minentropies and the slopes are determined by only the thresholds . Due to drifts of experimental parameters, the violations are fluctuating from one threshold to another, which accordingly introduces the changes to the minentropy, accordingly. Fig. 4(b) shows details of the transient behavior of the generated random string. We monitor the violation for each batch of n = 1 × 10^{4} trials and estimate the minentropy in the batch. Fig. 4(b) reveals that the minentropies are correlated to the violations and completely determined by the thresholds at given confidence level 99%. Here, we we do not need massive random tests to ensure the amount actual random number in the generated string. The amount of minentropy of our random numbers is guaranteed by the the measured violations , regardless of unexpected changes of experimental parameters.
We also generate random bits with a biased choice of measurement settings, where P (V_{1}) = 1 − 4q, P (V_{2}) = P (V_{3}) = P (V_{4}) = P (V_{5}) = q and q = αn^{−1/2} with α = 6 and n = 10^{5}. We observe basically the same behavior of the minentropy for the generated stream except for a slightly smaller bound due to the nonuniform setting. We get the minentropy bound from 1 × 10^{5} rounds with violation of . For the biased choice of measurement settings, the output entropy (1.35 × 10^{4}) exceeds the input entropy (1.14 × 10^{4}) and we obtain 2.1 × 10^{3} net random bits. For the case of uniform measurement settings, we always need more initial randomness and thus cannot obtain net randomness. This is similar to the random number generation scheme with the CHSH inequality, where to generate net randomness, one always needs to consider nonuniform measurement settings.
Finally, we carry out a series of random tests (see Methods)^{31} to examine the quality of our random numbers obtained by collecting the outcomes of the first measurement in each trial. As expected, our generated random numbers passed all the tests. Fig. 5 shows the summary of the test results. Actually the real randomness of our generated strings is already certified by the KCBS inequality, which is a much stronger statement than claiming that the produced numbers pass all the random tests, since no random tests on finite strings should be considered complete.
Discussion
In summary, we have demonstrated violations of the KCBS inequality using a single trapped ion, with the detection efficiency loophole closed for the first time. We use quantum contextuality to certify randomness of the measurement outcomes. The randomness of our device is ensured by observing violations of the inequality independent of experimental details. With our device, we already obtained a net output entropy. The device can generate random numbers with a higher speed, which is important for practical applications.
Methods
Experiment procedure
The experimental procedure consists of Doppler cooling, initialization, coherent operations and detection. After 1 ms Doppler cooling, the internal state of the ion is initialized to 3〉 by 3 μs standard optical pumping with efficiency 99.1%^{26}. The states are coherently manipulated by the microwaves ω_{1} and ω_{2} that are resonant to the transitions between 1〉 and 3〉 and between 2〉 and 3〉, respectively. The quantum operations of the microwaves ω_{1} and ω_{2} are described by the rotation matrix R_{1} (θ_{1}, ϕ_{1}) and R_{2} (θ_{2}, ϕ_{2}), respectively. Here θ_{1}, θ_{2} and ϕ_{1}, ϕ_{2} are controlled by the duration and the phase of the applied microwaves. The 2 π times for both Rabi oscillations are adjusted to 29.5 μs, that is Ω_{1,2} = (2π) 33.9 kHz in frequency. The maximum probability of offresonant excitation Ω^{2}/(ω_{2} − ω_{1})^{2} is about 1.6 × 10^{−5}, small enough to ensure independence of each Rabi oscillation. The standard fluorescentdetection method enables us to differentiate between one state versus the other two states of a qutrit. We observe on average 10 photons at 369.5 nm for the 1〉 or the 2〉 state and detect no photon for the 3〉 state. The state detection error rates for wrongly registering the state 3〉 and missing the state 3〉 are 0.9% and 1.9%, respectively, with the discrimination threshold n_{ph} = 1. As shown in Fig. 2(b), we transfer the information of obervable A_{i} (A_{j}) by πpulse and apply the measurement sequence. Then we assign the value a_{i} = 1 (a_{j} = 1) on the obsevable A_{i} (A_{j}) when photons detected or a_{i} = −1 (a_{j} = −1) when no photons are detected. After repeating the same experimental procedures, we obtain the 〈A_{i}〉 (〈A_{j}〉). Here we emphasize that our setup is not subject to detection loophole and provide a value of the measurement at every trial.
Random test
We apply the random tests that are appropriate for the size of our random numbers, which are ‘Frequency’, ‘Block Frequency’, Cumulative Sums (Cusums)’, ‘Runs’, ‘LongestRunofOnes in a Block (LROB)’, ‘Rank’, ‘Discrete Fourier Transform Test (FTT)’, ‘Approximate Entropy (AE), ‘Serial.’ The pvalues of all the test, which are the probabilities that an ideal random number generator would produce less random sequence than the tested one. Therefore, a pvalue of 0 simply means that the tested sequence appears to be completely nonrandom, whereas a pvalue of 1 implies that the sequence in test appears to be perfectly random. The pvalues lie in the open interval (0, 1) and if pvalue is larger than a significance level θ, we accept the sequence as random for the test. Typically θ is chosen to be in the range [0.0001, 0.01] and we set θ = 0.01. Note that we use VonNeumann extractor for the output strings to make uniform distributions, which reduces the size of random numbers to one quarter. We also note that the random tests are different from guaranteeing the amount of minentropy in the generated string. In other words, even the data could not pass the random tests but still have the quoted minentropy.
Change history
02 February 2018
Scientific Reports 3: Article number: 1627; published online: 09 April 2013; updated: 02 February 2018. The original version of this Article contained an error in the spelling of the author Yangchao Shen, which was incorrectly given as Shen Yangchao. This error has now been corrected in the PDF and HTML versions of the Article.
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Acknowledgements
We thank Periklis Papakonstantiou, Dominik Scheder and Xiongfeng Ma for helpful discussion. This work was supported in part by the National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301, 2011CBA00302, the National Natural Science Foundation of China Grant 61073174, 61033001, 61061130540. KK acknowledges the support from the Thousand Young Talents program.
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M.U., X.Z., J.Z. and Y.W. developed the experimental setup for the measurements. M.U., X.Z., J.Z., Y.W. and S.Y. carried out the measurements. M.U. analyzed the data and D.L.D., L.M.D. provided the theoretical support. K.K. supervised the experiment. All authors participated in writing the manuscript.
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Um, M., Zhang, X., Zhang, J. et al. Experimental Certification of Random Numbers via Quantum Contextuality. Sci Rep 3, 1627 (2013). https://doi.org/10.1038/srep01627
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