The nonlocal behavior of quantum mechanics can be used to generate guaranteed fresh randomness from an untrusted device that consists of two nonsignalling components; since the generation process requires some initial fresh randomness to act as a catalyst, one also speaks of randomness expansion. Colbeck and Kent proposed the first method for generating randomness from untrusted devices, however, without providing a rigorous analysis. This was addressed subsequently by Pironio et al. [Nature 464 (2010)], who aimed at deriving a lower bound on the min-entropy of the data extracted from an untrusted device, based only on the observed non-local behavior of the device. Although that article succeeded in developing important tools towards the acquired goal, it failed in putting the tools together in a rigorous and correct way, and the given formal claim on the guaranteed amount of min-entropy needs to be revisited. In this paper we show how to combine the tools provided by Pironio et al., as to obtain a meaningful and correct lower bound on the min-entropy of the data produced by an untrusted device, based on the observed non-local behavior of the device. Our main result confirms the essence of the improperly formulated claims of Pironio et al., and puts them on solid ground. We also address the question of composability and show that different untrusted devices can be composed in an alternating manner under the assumption that they are not entangled. This enables for superpolynomial randomness expansion based on two untrusted yet unentangled devices.
The non-local correlations exhibited when measuring entangled particles can be used to certify the presence of genuine randomness in Bell experiments. While non-locality is necessary for randomness certification, it is unclear when and why non-locality certifies maximal randomness. We provide here a simple argument to certify the presence of maximal local and global randomness based on symmetries of a Bell inequality and the existence of a unique quantum probability distribution that maximally violates it. Using our findings, we prove the existence of N-party Bell test attaining maximal global…
The detection of nonlocal correlations in a Bell experiment implies almost by definition some intrinsic randomness in the measurement outcomes. For given correlations, or for a given Bell violation, the amount of randomness predicted by quantum physics, quantified by the guessing probability, can generally be bounded numerically. However, currently only a few exact analytic solutions are known for violations of the bipartite Clauser-Horne-Shimony-Holt Bell inequality. Here, we study the randomness in a Bell experiment where three parties test the tripartite Mermin-Bell inequality. We give tigh…
The rates at which a user can generate device-independent quantum random numbers from a Bell-type experiment depend on the measurements that he performs. By numerically optimising over these measurements, we present lower bounds on the randomness generation rates for a family of two-qubit states composed from a mixture of partially entangled states and the completely mixed state. We also report on the randomness generation rates from a tomographic measurement. Interestingly in this case, the randomness generation rates are not monotonic functions of entanglement.
Genuine randomness can be certified from Bell tests without any detailed assumptions on the working of the devices with which the test is implemented. An important class of experiments for implementing such tests is optical setups based on polarisation measurements of entangled photons distributed from a spontaneous parametric down conversion source. Here we compute the maximal amount of randomness which can be certified in such setups under realistic conditions. We provide relevant yet unexpected numerical values for the physical parameters and achieve four times more randomness than previous…
Device-independent randomness generation and quantum key distribution protocols rely on a fundamental relation between the non-locality of quantum theory and its random character. This relation is usually expressed in terms of a trade-off between the probability of guessing correctly the outcomes of measurements performed on quantum systems and the amount of violation of a given Bell inequality. However, a more accurate assessment of the randomness produced in Bell experiments can be obtained if the value of several Bell expressions is simultaneously taken into account, or if the full set of p…
With the advent of quantum information, the violation of a Bell inequality is used as evidence of the absence of an eavesdropper in cryptographic scenarios such as key distribution and randomness expansion. One of the key assumptions of Bell's Theorem is the existence of experimental "free will", meaning that measurement settings can be chosen at random and independently by each party. The relaxation of this assumption potentially shifts the balance of power towards an eavesdropper. We consider a no-signalling model with reduced "free will" and bound the adversary's capabilities in the task of…
We prove that as conjectured by Ac\'ın et al. [Phys. Rev. A 93, 040102(R) (2016)], two bits of randomness can be certified in a device-independent way from one bit of entanglement using the maximal quantum violation of Gisin's elegant Bell inequality. This suggests a surprising connection between maximal entanglement, complete sets of mutually unbiased bases, and elements of symmetric informationally complete positive operator-valued measures, on one side, and the optimal way of certifying maximal randomness, on the other.
Randomness in Bell test data can be device-independently certified by Bell's theorem without placing assumptions about the experimental devices. The device-independent randomness has very demanding requirement about the experimental devices and relatively lower output randomness. With the same Bell test data we can extract substantially more randomness without using Bell's theorem. To achieve this goal, we introduce a remote state preparation dimension witness and a semi-device-independent randomness certification model which is based on it. This is one important step towards practical use of …
The Clauser-Horne-Shimony-Holt inequality test is widely used as a mean of invalidating the local deterministic theories and a tool of device independent quantum cryptographic tasks. There exists a randomness (freewill) loophole in the test, which is widely believed impossible to be closed perfectly. That is, certain random inputs are required for the test. Following a randomness quantification method used in literature, we investigate the randomness required in the test under various assumptions. By comparing the results, one can conclude that the key to make the test result reliable is to ru…
Noise robustness of three outcome Bell certified quantum randomness
ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology（光子科学研究所，巴塞罗那科学技术学院）；Center for Theoretical Physics, Polish Academy of Sciences（波兰科学院理论物理中心）；Department of Algorithms and System Modeling, Faculty of Electronics, Telecommunications and Informatics, Gdańsk University of Technology（格但斯克理工大学电子、电信与信息学院算法与系统建模系）；ICREA-Institucio Catalana de Recerca i Estudis Avançats（加泰罗尼亚研究与高级研究学院）
We show that paradoxical consequences of violations of Bell's inequality are induced by the use of an unsuitable probabilistic description for the EPR-Bohm-Bell experiment. The conventional description (due to Bell) is based on a combination of statistical data collected for different settings of polarization beam splitters (PBSs). In fact, such data consists of some conditional probabilities which only partially define a probability space. Ignoring this conditioning leads to apparent contradictions in the classical probabilistic model (due to Kolmogorov). We show how to make a completely cons…
The majority of recent works investigating the link between non-locality and randomness, e.g. in the context of device-independent cryptography, do so with respect to some specific Bell inequality, usually the CHSH inequality. However, the joint probabilities characterizing the measurement outcomes of a Bell test are richer than just the degree of violation of a single Bell inequality. In this work we show how to take this extra information into account in a systematic manner in order to optimally evaluate the randomness that can be certified from non-local correlations. We further show that t…
Non-local correlations that obey the no-signalling principle contain intrinsic randomness. In particular, for a specific Bell experiment, one can derive relations between the amount of randomness produced, as quantified by the min-entropy of the output data, and its associated violation of a Bell inequality. In practice, due to finite sampling, certifying randomness requires the development of statistical tools to lower-bound the min-entropy of the data as a function of the estimated Bell violation. The quality of such bounds relies on the choice of certificate, i.e., the Bell inequality whose…
We question the commonly accepted statement that random numbers certified by Bell's theorem carry some special sort of randomness, so to say, quantum randomness or intrinsic randomness. We show that such numbers can be easily generated by classical random generators.
We demonstrate to what extent many copies of maximally entangled two-qubit states enable for generating a greater amount of certified randomness than that can be certified from a single copy. Although it appears that greater the dimension of the system implies a higher amount of randomness, the non-triviality lies in the device-independent simultaneous certification of generated randomness from many copies of entangled states. This is because, most of the two-outcome Bell inequalities (viz., Clauser-Horne-Shimony-Holt, Elegant, or Chain Bell inequality) are optimized for a single copy of two-q…
We present a violation of the CHSH inequality without the fair sampling assumption with a continuously pumped photon pair source combined with two high efficiency superconducting detectors. Due to the continuous nature of the source, the choice of the duration of each measurement round effectively controls the average number of photon pairs participating in the Bell test. We observe a maximum violation of S= 2.01602(32) with average number of pairs per round of ~0.32, compatible with our system overall detection efficiencies. Systems that violate a Bell inequality are guaranteed to generate pr…
Correlations that cannot be reproduced with local variables certify the generation of private randomness. Usually, the violation of a Bell inequality is used to quantify the amount of randomness produced. Here, we show how private randomness generated during a Bell test can be directly quantified from the observed correlations, without the need to process these data into an inequality. The frequency with which the different measurement settings are used during the Bell test can also be taken into account. This improved analysis turns out to be very relevant for Bell tests performed with a fini…
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.
Nowadays contextuality is the hotest topic of quantum foundations and, especially, foundations of quantum information theory. This notion is characterized by the huge diversity of approaches and interpretations. One of the strongest trends in contextual research is to identify contextuality with violation of the Bell inequalities. We call this sort of contextuality {\it Bell test contextuality} (BTC). In this note, we criticize the BTC-approach. It can be compared with an attempt to identify the complex and theoretically nontrivial notion of randomness with a test for randomness (or a batch of…
Franson-type nonlocal correlation is a testing tool for Bell inequality violation using noninterfering interferometers, where coincidence measurements involve an interference fringe of g^((1)) correlation between noninterfering photon pairs. Like the Bell inequality, Franson correlation is also limited to a microscopic regime of entangled photon pairs. Here, randomness-based macroscopic Franson-type nonlocal correlation is presented using polarization-basis coherent superposition of laser light, where probabilistic randomness between bipartite orthonormal bases plays an important role for both…
We develop a framework for certifying randomness from Bell-test trials based on directly estimating the probability of the measurement outcomes with adaptive test supermartingales. The number of trials need not be predetermined, and one can stop performing trials early, as soon as the desired amount of randomness is extractable. It can be used with arbitrary, partially known and time-dependent probabilities for the random settings choices. Furthermore, it is suitable for application to experimental configurations with low Bell violation per trial, such as current optical loophole-free Bell tes…
We demonstrate extraction of randomness from spontaneous-emission events less than 36 ns in the past, giving output bits with excess predictability below $10^{-5}$ and strong metrological randomness assurances. This randomness generation strategy satisfies the stringent requirements for unpredictable basis choices in current "loophole-free Bell tests" of local realism [Hensen et al., Nature (London) 526, 682 (2015); Giustina et al., Phys. Rev. Lett. 115, 250401 (2015); Shalm et al., Phys. Rev. Lett. 115, 250402 (2015)].
We demonstrate that amplification of arbitrarily weak randomness is possible using quantum resources. We present a randomness amplification protocol that involves Bell experiments. We find a Bell inequality which can amplify arbitrarily weak randomness and give a detailed analysis of the protocol involving it. Our analysis includes finding a sufficient violation of Bell inequality as a function of the initial quality of randomness. It has a very important property that for any quality the required violation is strictly lower than possible to obtain using quantum resources. Among other things, …
Certifying optimal device-independent quantum randomness in quantum networks
Hangzhou Dianzi University（杭州电子科技大学）；Zhejiang Provincial Key Laboratory of Sensitive Data Security and Confidentiality Governance（浙江省敏感数据安全与保密治理重点实验室）；The University of Hong Kong（香港大学）
Bell test is one of the most important tools in quantum information science. On the one hand, it enables fundamental test for the physics laws of nature, and on the other hand, it can be also applied in varieties of device independent tasks such as quantum key distribution and random number generation. In practice, loopholes existing in experimental demonstrations of Bell tests may affect the validity of the conclusions. In this work, we focus on the randomness (freewill) loophole and investigate the randomness requirement in a well-known Bell test, the Clauser-Horne test, under various condit…
Experimental violation of Bell-inequalities proves actualization of many futures (~ many-worlds); I show that this is not mere interpretation. To show this self-contained pedagogically, I resolve the EPR paradox by starting with a visually intuitive non-quantum many-worlds model that already has apparent non-locality. A modification leads to a classical-to-quantum transition model. Model-external randomness (a ghost outside the universe throwing a pebble on state-space) stays unchanged, but the modeled observers witness strong Bell-violation. I derive the quantum probability P from classical-t…
We introduce probability estimation, a broadly applicable framework to certify randomness in a finite sequence of measurement results without assuming that these results are independent and identically distributed. Probability estimation can take advantage of verifiable physical constraints, and the certification is with respect to classical side information. Examples include randomness from single-photon measurements and device-independent randomness from Bell tests. Advantages of probability estimation include adaptability to changing experimental conditions, unproblematic early stopping whe…
Unpredictability, or randomness, of the outcomes of measurements made on an entangled state can be certified provided that the statistics violate a Bell inequality. In the standard Bell scenario where each party performs a single measurement on its share of the system, only a finite amount of randomness, of at most $4 log_2 d$ bits, can be certified from a pair of entangled particles of dimension $d$. Our work shows that this fundamental limitation can be overcome using sequences of (nonprojective) measurements on the same system. More precisely, we prove that one can certify any amount of ran…
Colbeck (Thesis, 2006) proposed using Bell inequality violations to generate certified random numbers. While full quantum-security proofs have been given, it remains a major open problem to identify the broadest class of Bell inequalities and lowest performance requirements to achieve such security. In this paper, working within the broad class of spot-checking protocols, we prove exactly which Bell inequality violations can be used to achieve full security. Our result greatly improves the known noise tolerance for secure randomness expansion: for the commonly used CHSH game, full security was…
Recently, a physically realistic protocol amplifying the randomness of Santha-Vazirani sources producing cryptographically secure random bits was proposed; however for reasons of practical relevance, the crucial question remained open whether this can be accomplished under the minimal conditions necessary for the task. Namely, is it possible to achieve randomness amplification using only two no-signaling components and in a situation where the violation of a Bell inequality only guarantees that some outcomes of the device for specific inputs exhibit randomness? Here, we solve this question and…
It is shown that the Einstein, Podolsky and Rosen (EPR) correlations for arbitrary spin-s and the Greenberger, Horne and Zeilinger (GHZ) correlations for three particles can be described by nonlocal joint and conditional quantum probabilities. The nonlocality of these probabilities makes the Bell's inequalities void. A description that exhibits the relation between the randomness and the nonlocality of entangled correlations is introduced. Entangled EPR and GHZ correlations are studied using the Gibbs-Shannon entropy. The nonlocal character of the EPR correlations is tested using the informati…
In this paper I demonstrate that the quantum correlations of polarization (or spin) observables used in Bell's argument against local realism have to be interpreted as {\it conditional} quantum correlations. By taking into account additional sources of randomness in Bell's type experiments, i.e., supplementary to source randomness, I calculate (in the standard quantum formalism) the complete quantum correlations. The main message of the quantum theory of measurement (due to von Neumann) is that complete correlations can be essentially smaller than the conditional ones. Additional sources of ra…
The intrinsic random nature of quantum physics offers novel tools for the generation of random numbers, a central challenge for a plethora of fields. Bell non-local correlations obtained by measurements on entangled states allow for the generation of bit strings whose randomness is guaranteed in a device-independent manner, i.e. without assumptions on the measurement and state-generation devices. Here, we generate this strong form of certified randomness on a new platform: the so-called instrumental scenario, which is central to the field of causal inference. First, we theoretically show that …
We investigate how much randomness can be extracted from a generic partially entangled pure state of two qubits in a device-independent setting, where a Bell test is used to certify the correct functioning of the apparatus. For any such state, we first show that two bits of randomness are always attainable both if projective measurements are used to generate the randomness globally or if a nonprojective measurement is used to generate the randomness locally. We then prove that the maximum amount of randomness that can be generated using nonprojective measurements globally is restricted to betw…
We provide an interesting two-party parity oblivious communication game whose success probability is solely determined by the Bell expression. The parity-oblivious condition in an operational quantum theory implies the preparation non-contextuality in an ontological model of it. We find that the aforementioned Bell expression has two upper bounds in an ontological model; the usual local bound and a non-trivial preparation non-contextual bound arising from the non-trivial parity-oblivious condition, which is smaller that the local bound. We first demonstrate the communication game when both Ali…
Bell inequalities tailored to optimal global randomness certification
ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology（ICFO-光子科学研究所，巴塞罗那科技研究所）；Center for Quantum-Enabled Computing, Center for Theoretical Physics, Polish Academy of Sciences（波兰科学院量子使能计算中心，理论物理中心）；Department of Algorithms and System Modeling, Faculty of Electronics, Telecommunications and Informatics, Gdańsk University of Technology（格但斯克理工大学电子、电信与信息学院算法与系统建模系）；ICREA-Institucio Catalana de Recerca i Estudis Avançats（ICREA-加泰罗尼亚高级研究学院）
In this paper we investigate properties of several randomness generation protocols in the device independent framework. Using Bell-type inequalities it is possible to certify that the numbers generated by an untrusted device are indeed random. We present a selection of certificates which guarantee two bits of randomness for each run of the experiment in the noiseless case and require the parties to share a maximally entangled state. To compare them we study their efficiency in the presence of white noise. We find that for different amounts of noise different operators are optimal for certifyin…
In the mentioned paper we presented results of the estimation of Kolmogorov complexity of sequences of random numbers generated in a famous Bell's experiment, aimed to study the security of QKD. We focused on series of time differences between successive detections of coincidences, and found that randomness cannot be taken for granted. It was then criticized that the theorems that demonstrate the randomness of series produced in Bell's experiments involve series of measurement outcomes, not of measurement times. Here we reply to this objection and present data of series of outcomes, showing th…
We show that by taking into account randomness of realization of experimental contexts it is possible to construct common Kolmogorov space for data collected for these contexts, although they can be incompatible. We call such a construction "Kolmogorovization" of contextuality. This construction of common probability space is applied to Bell's inequality. It is well known that its violation is a consequence of collecting statistical data in a few incompatible experiments. In experiments performed in quantum optics contexts are determined by selections of pairs of angles $(θ_i, θ^\prime_j)$ fix…
Bell inequality violations can be used to certify private randomness for use in cryptographic applications. In photonic Bell experiments, a large amount of the data that is generated comes from no-detection events and presumably contains little randomness. This raises the question as to whether randomness can be extracted only from the smaller post-selected subset corresponding to proper detection events, instead of from the entire set of data. This could in principle be feasible without opening an analogue of the detection loophole as long as the min-entropy of the post-selected data is evalu…
Quantum systems are the ultimate touchstone for the production of random sequences of numbers. Spatially spread entangled systems allow the generation of identical random sequences in remote locations. The impossibility of observing a quantum system, without disturbing it, ensures that the messages encoded using these sequences cannot be eavesdropped. This is the basis of Quantum Key Distribution. It is then of crucial importance knowing whether the sequences generated in the practice by spatially spread entangled states are truly random, or not. Yet, that knowledge is not immediate. One of th…
From dice to modern complex circuits, there have been many attempts to build increasingly better devices to generate random numbers. Today, randomness is fundamental to security and cryptographic systems, as well as safeguarding privacy. A key challenge with random number generators is that it is hard to ensure that their outputs are unpredictable. For a random number generator based on a physical process, such as a noisy classical system or an elementary quantum measurement, a detailed model describing the underlying physics is required to assert unpredictability. Such a model must make a num…
Loophole-free experiments have demonstrated that at least one of three features is false when the violation of Bell's inequalities is observed: Locality, Realism or (what is lesser known) Ergodicity. An experiment is proposed to find out, or at least to get an indication about, which one is false. It is based on recording the time evolution of the rate of series of outcomes that are found not-random in a pulsed Bell's setup. The results of such experiment would be important not only to the foundations of Quantum Mechanics. For, even if the foundational issue remained not fully decided, they wo…
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, and their generation must rely on an unpredictable physical process. 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 bas…
Free will (or randomness) has been studied to achieve loophole-free Bell's inequality test and to provide device-independent quantum key distribution security proofs. The required randomness such that a local hidden variable model (LHVM) can violate the Clauser-Horne-Shimony-Holt (CHSH) inequality has been studied, but a tight bound has not been proved for a practical case that i) the device settings of the two parties in the Bell test are independent; and ii) the device settings of each party can be correlated or biased across different runs. Using some information theoretic techniques, we pr…
It is known from Bell's theorem that quantum predictions for some entangled states cannot be mimicked using local hidden variable (LHV) models. From a computer science perspective, LHV models may be interpreted as classical computers operating on a potentially infinite number of correlated bits originating from a common source. As such, Bell inequality violations achieved through entangled states are able to characterise the quantum advantage of certain tasks, so long as the task itself imposes no restriction on the availability of correlated bits. However, if the number of shared bits is limi…
