Randomness is a fundamental feature in nature and a valuable resource for applications ranging from cryptography and gambling to numerical simulation of physical and biological systems. Random numbers, however, are difficult to characterize mathematically [1] , and their generation must rely on an unpredictable physical process [2, 3, 4, 5, 6]. Inaccuracies in the theoretical modelling of such processes or failures of the devices, possibly due to adversarial attacks, limit the reliability of random number generators in ways that are difficult to control and detect. Here, inspired by earlier work on nonlocality based [7, 8, 9] and device independent [10, 11, 12, 13, 14] quantum information processing, we show that the nonlocal correlations of entangled quantum particles can be used to certify the presence of genuine randomness. It is thereby possible to design of a new type of cryptographically secure random number generator which does not require any assumption on the internal working of the devices. This strong form of randomness generation is impossible classically and possible in quantum systems only if certified by a Bell inequality violation [15]. We carry out a proof-of-concept demonstration of this proposal in a system of two entangled atoms separated by approximately 1 meter. The observed Bell inequality violation, featuring near-perfect detection efficiency, guarantees that 42 new random numbers are generated with 99% confidence. Our results lay the groundwork for future device-independent quantum information experiments and for addressing fundamental issues raised by the intrinsic randomness of quantum theory. * These authors contributed equally to this work. 1 arXiv:0911.3427v3 [quant-ph] 19 Oct 2010 The characterization of true randomness is elusive. There exist statistical tests used to verify the absence of certain patterns in a stream of numbers [16, 17], but no finite set of tests can ever be considered complete, as there may be patterns not covered by such tests. For example, certain pseudo-random number generators are deterministic in nature, yet produce results that satisfy all the randomness tests [18]. At a more fundamental level, there is no such thing as true randomness in the classical world: any classical system admits in principle a deterministic description and thus appears random to us as a consequence of a lack of knowledge about its fundamental description. Quantum theory is, on the other hand, fundamentally random; yet, in any real experiment the intrinsic randomness of quantum systems is necessarily mixed-up with an apparent randomness that results from noise or lack of control of the experiment. It is therefore unclear how to certify or quantify unequivocally the observed random behaviour even of a quantum process. These considerations are of direct relevance to applications of randomness, and in particular cryptographic applications. Imperfections in random number generators [2, 3, 4, 5, 6] (RNG) can introduce patterns undetected by statistical tests but known to an adversary. Furthermore, if the device is not trusted but viewed as a black box prepared by an adversary, no existing RNGs can establish the presence of private randomness. Indeed, one can never exclude that the numbers were generated in advance by the adversary and copied into a memory located inside the device. Here, we establish a fundamental link between the violation of Bell inequalities and the unpredictable character of the outcomes of quantum measurements and show, as originally proposed by Colbeck [14], that the nonlocal correlations of quantum states can be used to generate certified private randomness. The violation of a Bell inequality [15] guarantees that the observed outputs are not predetermined and that they arise from entangled quantum systems that possess intrinsic randomness. For simplicity, we consider the Clauser-Horn-Shimony-Holt (CHSH) form of Bell inequality [19], but our approach is general and applies to any Bell inequality. We thus consider two separate systems that can each be measured in two different ways, with a measurement on each system resulting in one of two values (Figure 1 ). The binary variables x and y describe the type of measurement performed on each system, resulting in respective binary measurement outcomes a and b. We quantify the Bell inequality violation through the CHSH correlation function [19]  where P (a = b|xy) is the probability that a = b given settings (x, y) and P (a = b|xy) is defined analogously. Systems that admit a local, hence deterministic [20], description satisfy I ≤ 2. Certain measurements performed on entangled states, however, can violate this inequality. In order to estimate the Bell violation, the experiment is repeated n times in succession. The measurement choices (x, y) for each trial are generated by an identical and independent probability distribution P (xy). We denote the final output string after the n runs r = (a 1 , b 1 ; . . . ; a n , b n ) and the input string s = (x 1 , y 1 ; . . . ; x n , y n ). An estimatorÎ of the CHSH expression(1) determined from the observed data is given bŷ where N (a = b, xy) is the number of times that the measurements x, y were performed and that the outcomes a and b were found equal after n realizations, and where N (a = b, xy) is defined analogously. The randomness of the output string r can be quantified by the min-entropy [21] H ∞ (R|S) = − log 2 [max r P (r|s)], where P (r|s) is the conditional probability of obtaining the outcomes r when the measurements s are made. We show (see Appendix A) that the min-entropy of the outputs r is bounded by with probability greater than 1 − δ, where = O( − log δ/(q 2 n)) is a statistical parameter and q = min x,y P (xy) is the probability of the least probable input pair. The function f (I) is obtained using semidefinite programming [22, 23] and presented in Figure 2 . To derive the bound (3), we make the following assumptions: 1) the two observed systems satisfy the laws of quantum theory; 2) they are separated and non-interacting during each measurement step i; 3) the inputs x i , y i are generated by random processes that are independent and uncorrelated from the systems and their value is revealed to the systems only at step i (see Figure 1 ). Other than these assumptions, no constraints are imposed on the states, measurements, or the dimension of the Hilbert space. We do not even assume that the system behaves identically and independently for each trial; for instance, the devices may have an internal memory (possibly quantum), so that the i th measurement can depend on the previous i−1 results and measurement settings. Any value of the min-entropy smaller than (3) is incompatible with quantum theory. The observed CHSH quantityÎ thus provides a bound (see Figure 3 ) on the randomness produced by the quantum devices, independent of any apparent randomness that could arise from noise or limited control over the experiment. This result can be exploited to construct a novel RNG where the violation of a Bell inequality guarantees that the output is random and private from any adversary, even in a device-independent scenario [12, 13] where the internal workings of the two quantum devices are unknown or not trusted (See Appendix B). Some amount of randomness at the inputs is necessary to perform the statistical tests used to estimate the Bell inequality violation. Hence what we describe here is a randomness expansion scheme [14] where a small private random seed is expanded into a longer private random string. The randomness used to choose the inputs needs not be divulged and can be used subsequently for another task. The final random string, the concatenation of the input and output random strings, is thus manifestly longer than the initial one. However, when n becomes sufficiently large, a catalysis effect is possible wherein a seed string of length O( √ n log √ n) produces a much longer random output string of entropy O(n), as illustrated in Figure 3 (see Appendix B). This is possible because I can be adequately estimated without consuming much randomness by using the same input pair most of the time [e.g., (x, y) = (0, 0)] while only seldom sampling from the other possibilities, in which case q 1. This follows from the fact that the CHSH expression depends only the conditional probabilities P (ab|xy), which can be estimated even if x, y are not uniformly distributed. Although the final output string may not be uniformly random (it may not even pass one of the usual statistical tests [16, 17] of randomness), we are guaranteed that its entropy is bounded by (3). With the help of a small private random seed, the output string can then be classically processed using a randomness extractor [24] to convert it into a string of size nf (Î − ) that is nearly uniform and uncorrelated to the information of an adversary. The bound (3) establishes security of our randomness expansion protocol against an adversary that measures his side-information before the randomness extraction step, e.g., against an adversary that has only a short-lived or bounded quantum memory. This is because it applies when conditioned to any measurement performed by the adversary. However, our protocol is not yet proven to be universally-composable against a full quantum adversary, that is, secure against an adversary that stores his side-information in a 3 Inputs Rotations N (0, 0; x, y) N (0, 1; x, y) N (1, 0; x, y) N (1, 1; x, y) Table 1 : Observed number of events N (a, b; x, y) for which the measurement on one atom gave outcome a and the measurement on the other atom gave outcome b, given the binary choices of the measurement bases (x, y) corresponding to π/2 qubit rotations with phase angles (ϕ x , ϕ y ) on the equator of the Bloch sphere. The last column gives the fraction of events where a = b given each input. If the experiment is interpreted as consisting of identical and independent realisations (an assumption not made elsewhere in this paper), the data then indicate a CHSH observable ofÎ = x,y (−1) xy [P (a = b|xy) − P (a = b|xy)] = 2.414 ± 0.058, significantly beyond the localdeterministic threshold of I = 2. quantum memory which can be measured at a later stage. A universally-composable proof would also cover the situation in which the adversary tries to estimate the random numbers after getting partial information about them. Proving universally-composable security of our protocol would also probably lead to much more efficient randomness expansion schemes. Note that the fact that the bound (3) holds for devices that have an internal memory is a significant advance over the device-independent protocols [9, 12, 13] proposed so far. It is the crucial feature that makes our protocol practical. The experimental realization of this scheme requires the observation of a Bell inequality with the detection loophole closed (near-perfect detection of every event), so that the outputs r cannot be deterministically reproduced. The two individual systems should also be sufficiently separated so that they do not interact, but it is not necessary for the two subsystems to be space-like separated (see Appendix C). We realize this situation with two 171 Yb + atomic ion quantum bits (qubits) [25] confined in two independent vacuum chambers separated by about 1 meter. The qubit levels within each atom are entangled through a probabilistic process whereby each atom is entangled with emitted photons and the interference and coincident detection of the two photons heralds successful preparation of a near-maximal entangled state of the two remote atomic qubits through entanglement swapping [26], as described in Fig. 1 and Appendix D. The binary values a and b correspond to subsequent measurement results of each qubit obtained through standard atomic fluorescence techniques (detection error < 3%) [25], and every event is recorded. The respective binary measurement bases x and y are chosen randomly and set by coherent qubit rotation operations prior to measurement. Direct interaction between the atoms is negligible and classical microwave and optical fields used to perform rotations and measurements on one atom have no influence on the other atom (we perform statistical tests to verify that the measurement data is compatible with this hypothesis, see Appendix D.4). To estimate the value of the CHSH inequality we accumulate n = 3, 016 successive entanglement events over the period of about one month, summarized in Appendix D.1 and Table  I . The observed CHSH violation ofÎ = 2.414 represents a substantial improvement over previous results [27, 26]. The probability that a local theory, possibly including an internal memory of past events [28], could produce such a violation is P (Î ≥ 2.414) ≤ 0.00077 (see Appedendix D.3). In 4 the experiment, we chose a uniform random distribution of the initial measurement bases inputs [P (x, y) = 1/4] to minimize the number of runs required to obtain a meaningful bound on the output entropy (see Figure 3) . The observed CHSH violation implies that at least H(R|S) > 42 new random bits are generated in the experiment with a 99% confidence level. This is the first time that one can certify that new randomness is produced in an experiment without a detailed model of the devices. We rely only on a high-level description (atoms confined to independent vacuum chambers separated by one meter) to ensure the absence of interaction between the two subsystems when the measurements are performed. Since no active measures are taken in our experiment to control this interaction, these new random bits cannot be considered private in the strongest adversarial device-independent scenario. The level of security provided by our experiment will nevertheless be sufficient for many applications as it guarantees that almost all failure modes of the devices will be detected. The current experiment does not reach the catalysis regime mentioned above, owing to the low success probability of heralded entanglement generation (2 × 10 −8 ) [26]. However, it should be possible to exceed the catalysis threshold by improving the photon-collection efficiency through the use of nearby optical collection elements or optical cavities [29]. Stepping back to the more conceptual level, note that Eq. (3) relates the random character of quantum theory to the violation of Bell inequalities. This bound can be modified for a situation where we assume only the no-signalling principle instead of the entire quantum formalism (see Figure 2 and 3 and Appedenix A.3). Such a bound lays the basis for addressing in a statistically significant way one of the most fundamental questions raised by quantum theory: whether our world is compatible with determinism (but then necessarily signalling), or inherently random (if signalling is deemed impossible). Acknowledgements We thank Roger Colbeck for sharing his PhD thesis with us. This work was supported by the Swiss NCCR Quantum Photonics, the European ERC-AG QORE, the European QAP, COMPAS projects and ERC Starting grant PERCENT, the Spanish MEC FIS2007-60182 and Consolider-Ingenio QOIT projects, Generalitat de Catalunya and Caixa Manresa, the Interuniversity Attraction Poles Photonics@be Programme (Belgian Science Policy), the U.S. Army Research Office with funds from IARPA, the National Science Foundation (NSF) Physics at the Information Frontier Program, and the NSF Physics Frontier Center at JQI. Experimental realization of private random number generator using two 171 Yb + qubits trapped in independent vacuum chambers. Each atom emits a single photon (to the left) that is entangled with its host atomic qubit and coupled into optical fibers; the interference of the photons on the beamsplitter (BS) and subsequent coincidence detection on photomultiplier tubes (PMT) herald the entanglement of the atomic qubits [26]. After the qubits are entangled, binary random inputs (x, y) are fed to microwave (µ-wave) oscillators that coherently rotate each qubit in one of two ways prior to measurement [26]. Each qubit is finally measured through fluorescence that is collected on PMTs [25], resulting in the binary outputs (a, b). Abstractly, we can view this scheme as composed of two black boxes that receive inputs x, y and produce outputs a, b. In our theoretical analysis, no hypothesis are made about the internal working of the devices, but the classical and quantum information flowing in and out of the boxes is restricted (dotted lines). In particular, the two boxes are free to communicate before inputs are introduced (to establish shared entanglement), but are no longer allowed to interact during the measurement process. Moreover the values of the inputs are revealed to the boxes only at the beginning of the measurement process. In the experiment, no active measures are taken to control the flow of information in and out of the systems. However, once the atoms are entangled, direct interaction between them is negligible. Moreover the value of the chosen measurement bases (x, y), obtained by combining the outputs of several random number generators, is unlikely to be correlated to the state of the atoms before the measurement microwave pulses are applied, satisfying the conditions for the bound (3) on the entropy of the outputs. Figure 2 : Plot of the function f (I). The function f (I) can be interpreted as a bound on the min-entropy per use of the system for a given CHSH expectation I, in the asymptotic limit of large n where finite statistics effects (the parameter in (3)) can be neglected. The function f (I) (curve a) is derived through semidefinite programming using the techniques of [22,23] (semidefinite programming is a numerical method that is guaranteed to converge to the exact result). Curve b corresponds to the analytical lower-bound f (I) ≥ − log 2 1 − log 2 1 + 2 − I 2 4 . Curve c corresponds to the minimal value f (I) = − log 2 (3/2 − I/4) of the min-entropy implied by the no-signalling principle alone. The function f (I) starts at zero at the local threshold value I = 2. Systems that violate the CHSH inequality (I > 2), on the other hand, satisfy f (I) > 0, i.e., have a positive min-entropy. Figure 3: Bound nf (I) on the minimum entropy produced vs. the number of trials n for an observed CHSH violation ofÎ = 2.414, and a confidence level 1 − δ = 99%. The amount of randomness given by the bound (3) depends on the probability with which the inputs of each trial (x i , y i ) are chosen through the parameter q = min x,y [P (x, y)], where P (x, y) is the probability distribution of the inputs. We have plotted the bounds on the entropy implied by quantum theory for a uniform choice of inputs [P (x, y) = 1/4] (curve a) and for a biased choice of inputs given by P (00) = 1 − 3q, P (01) = P (10) = P (11) = q, where q = αn −1/2 with α = 11 (curve b). For a given number n of uses of the devices, the uniform choice of inputs leads to more randomness in the outputs. On the other hand, biased inputs require less randomness to be generated, and the net amount of randomness produced given by the difference between the output and input entropy becomes positive for sufficiently large n. Curve c represents the bound on the entropy implied by the no-signalling principle alone for a uniform choice of inputs. Note that in all cases, a minimal number of uses of the devices (a few thousand) is required to guarantee that some randomness has been produced at the confidence level 1 − δ = 99% The inset shows the net amount of entropy produced (output entropy minus input entropy) for the biased choice of inputs with the observed CHSH violation.
Recently, a quantum randomness expander has been constructed, based on the CHSH inequality REF .
Random Numbers Certified by Bell's Theorem
Nature 464, 1021 (2010)
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