Randomness is an indispensable resource in modern science and information technology. Fortunately, an experimentally simple procedure exists to generate randomness with well-characterized devices: measuring a quantum system in a basis complementary to its preparation. Towards realizing this goal one may consider using atoms or superconducting qubits, promising candidates for quantum information processing. However, their unavoidable interaction with the electromagnetic field affects their dynamics. At large time scales, this can result in decoherence. Smaller time scales in principle avoid this problem, but may not be well analysed under the usual rotating wave and single-mode approximation (RWA and SMA) which break the relativistic nature of quantum field theory. Here, we use a fully relativistic analysis to quantify the information that an adversary with access to the field could get on the result of an atomic measurement. Surprisingly, we find that the adversary's guessing probability is not minimized for atoms initially prepared in the ground state (an intuition derived from the RWA and SMA model).
The concept of randomness plays an important role in many disciplines. On one hand, the question of whether random processes exist is fundamental for our understanding of nature. On the other hand, randomness is a resource for cryptography, algorithms and simulations. Standard methods for generating randomness rely on assumptions on the devices that are difficult to meet in practice. However, quantum technologies allow for new methods for generating certified randomness. These methods are known as device-independent because do not rely on any modeling of the devices. Here we review the efforts…
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 …
Random number has many applications, it plays an important role in quantum information processing. It's not difficult to generate true random numbers, the main difficulty is how to certify the random numbers generated by untrusted devices. In [Nature(London) 464, 1021 (2010)], the authors provided us a way to generate certified random number by Bell's theorem. In their scheme, we can use the nonlocal behavior of entangled states to generate certified randomness. But there are entangled states, which admit a local hidden variable model, could not be used in their scheme. We show in our paper th…
Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences（奥地利科学院量子光学与量子信息研究所）；Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna（维也纳大学物理学院维也纳量子科学与技术中心）；Marietta Blau Institute for Particle Physics, Austrian Academy of Sciences（奥地利科学院玛丽亚·布劳粒子物理研究所）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）
Randomness comes in two qualitatively different forms. Apparent randomness can result both from ignorance or lack of control of degrees of freedom in the system. In contrast, intrinsic randomness should not be ascribable to any such cause. While classical systems only possess the first kind of randomness, quantum systems are believed to exhibit some intrinsic randomness. In general, any observed random process includes both forms of randomness. In this work, we provide quantum processes in which all the observed randomness is fully intrinsic. These results are derived under minimal assumptions…
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.
Unconditional Certified Randomness without Structure
University of Washington（华盛顿大学）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；NTT Research（NTT研究所）；Columbia University（哥伦比亚大学）；UC Berkeley（加州大学伯克利分校）
We present an efficient method to extract the amount of true randomness that can be obtained by a Quantum Random Number Generator (QRNG). By repeating the measurements of a quantum system and by swapping between two mutually unbiased bases, a lower bound of the achievable true randomness can be evaluated. The bound is obtained thanks to the uncertainty principle of complementary measurements applied to min- and max- entropies. We tested our method with two different QRNGs, using a train of qubits or ququart, demonstrating the scalability toward practical applications.
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 …
It is known that if two players achieve a superclassical score at a nonlocal game $G$, then their outputs are certifiably random - that is, regardless of the strategy used by the players, a third party will not be able to perfectly predict their outputs (even if he were given their inputs). We prove that for any complete-support game $G$, there is an explicit nonzero function $F_G$ such that if Alice and Bob achieve a superclassical score of $s$ at $G$, then Bob has a probability of at most $1 - F_G ( s )$ of correctly guessing Alice's output after the game is played. Our result implies that c…
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）
The generation of certifiable randomness is one of the most promising applications of quantum technologies. Furthermore, the intrinsic non-locality of quantum correlations allow us to certify randomness in a device-independent way, i.e. one need not make assumptions about the devices used. Due to the work of Curchod et. al., a single entangled two-qubit pure state can be used to produce arbitrary amounts of certified randomness. However, the obtaining of this randomness is experimentally challenging as it requires a large number of measurements, both projective and general. Motivated by these …
The ultimate random number generators are those certified to be unpredictable -- including to an adversary. The use of simple quantum processes promises to provide numbers that no physical observer could predict but, in practice, unwanted noise and imperfect devices can compromise fundamental randomness and protocol security. Certified randomness protocols have been developed which remove the need for trust in devices by taking advantage of nonlocality. Here, we use a photonic platform to implement our protocol, which operates in the quantum steering scenario where one can certify randomness i…
We report on an optical setup generating more than one bit of randomness from one entangled bit (i.e. a maximally entangled state of two-qubits). The amount of randomness is certified through the observation of Bell non-local correlations. To attain this result we implemented a high-purity entanglement source and a non-projective three-outcome measurement. Our implementation achieves a gain of 27$\%$ of randomness as compared with the standard methods using projective measurements. Additionally we estimate the amount of randomness certified in a one-sided device independent scenario, through t…
Metasurface quantum graphs for polarization-independent certified randomness
State Key Laboratory of Optical Quantum Materials, University of Hong Kong（光学量子材料国家重点实验室，香港大学）；Department of Electrical & Electronic Engineering, The University of Hong Kong（电子电气工程学系，香港大学）；Materials Innovation Institute for Life Sciences and Energy (MILES), HKU-SIRI, Shenzhen（生命科学与能源材料创新研究所（MILES），港大-西丽研究院，深圳）；Quantum Science Center of Guangdong-Hong Kong-Macao Great Bay Area（粤港澳大湾区量子科学中心）
Quantum nonlocality offers a secure way to produce random numbers: their unpredictability is intrinsic and can be certified just by observing the statistic of the measurement outcomes, without assumptions on how they are produced. To do this, entangled pairs are generated and measured to violate a Bell inequality with the outcome statistics. However, after a projective quantum measurement, entanglement is entirely destroyed and cannot be used again. This fact poses an upper bound to the number of random numbers that can be produced from each quantum state when projective measurements are emplo…
We study a generalization of the Mermin-Peres magic square game to arbitrary rectangular dimensions. After exhibiting some general properties, these rectangular games are fully characterized in terms of their optimal win probabilities for quantum strategies. We find that for $m \times n$ rectangular games of dimensions $m,n \geq 3$ there are quantum strategies that win with certainty, while for dimensions $1 \times n$ quantum strategies do not outperform classical strategies. The final case of dimensions $2 \times n$ is richer, and we give upper and lower bounds that both outperform the classi…
Randomness is a central feature of quantum mechanics and an invaluable resource for both classical and quantum technologies. Commonly, in Device-Independent and Semi-Device-Independent scenarios, randomness is certified using projective measurements and the amount of certified randomness is bounded by the dimension of the measured quantum system. In this work, we propose a new Source-Device-Independent protocol, based on Positive Operator Valued Measurement (POVM), which can arbitrarily increase the number of certified bits for any fixed dimension. A tight lower-bound on the quantum conditiona…
We propose an application for near-term quantum devices: namely, generating cryptographically certified random bits, to use (for example) in proof-of-stake cryptocurrencies. Our protocol repurposes the existing "quantum supremacy" experiments, based on random circuit sampling, that Google and USTC have successfully carried out starting in 2019. We show that, whenever the outputs of these experiments pass the now-standard Linear Cross-Entropy Benchmark (LXEB), under plausible hardness assumptions they necessarily contain $Ω(n)$ min-entropy, where $n$ is the number of qubits. To achieve a net ga…
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…
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…
Quantum mechanics predicts the existence of intrinsically random processes. Contrary to classical randomness, this lack of predictability can not be attributed to ignorance or lack of control. Here we find the optimal method to quantify the amount of local or global randomness that can be extracted in two scenarios: (i) the quantum steering scenario, where two parties measure a bipartite system in an unknown state but one of them does not trust his measurement apparatus, and (ii) the prepare-and-measure scenario, where additionally the quantum state is known. We use our methods to compute the …
Applications of randomness such as private key generation and public randomness beacons require small blocks of certified random bits on demand. Device-independent quantum random number generators can produce such random bits, but existing quantum-proof protocols and loophole-free implementations suffer from high latency, requiring many hours to produce any random bits. We demonstrate device-independent quantum randomness generation from a loophole-free Bell test with a more efficient quantum-proof protocol, obtaining multiple blocks of $512$ bits with an average experiment time of less than $…
Certified Randomness Generation against Quantum Adversaries via Energy Constraints
Weizmann Institute of Science（魏茨曼科学研究所）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；LMU Munich（慕尼黑大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）
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…
Random numbers are an important resource for applications such as numerical simulation and secure communication. However, it is difficult to certify whether a physical random number generator is truly unpredictable. Here, we exploit the phenomenon of quantum nonlocality in a loophole-free photonic Bell test experiment for the generation of randomness that cannot be predicted within any physical theory that allows one to make independent measurement choices and prohibits superluminal signaling. To certify and quantify the randomness, we describe a new protocol that performs well in an experimen…
A recent sequence of works, initially motivated by the study of the nonlocal properties of entanglement, demonstrate that a source of information-theoretically certified randomness can be constructed based only on two simple assumptions: the prior existence of a short random seed and the ability to ensure that two black-box devices do not communicate (i.e. are non-signaling). We call protocols achieving such certified amplification of a short random seed randomness amplifiers. We introduce a simple framework in which we initiate the systematic study of the possibilities and limitations of rand…
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…
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…
A remarkable aspect of quantum theory is that certain measurement outcomes are entirely unpredictable to all possible observers. Such quantum events can be harnessed to generate numbers whose randomness is asserted based upon the underlying physical processes. We formally introduce, design and experimentally demonstrate an ultrafast optical quantum random number generator that uses a totally untrusted photonic source. While considering completely general quantum attacks, we certify and generate in real-time random numbers at a rate of $8.05\,$Gb/s with a rigorous security parameter of $10^{-10…
Two parties sharing entangled quantum systems can generate correlations that cannot be produced using only shared classical resources. These nonlocal correlations are a fundamental feature of quantum theory but also have practical applications. For instance, they can be used for device-independent (DI) random number generation, whose security is certified independently of the operations performed inside the devices. The amount of certifiable randomness that can be generated from some given non-local correlations is a key quantity of interest. Here we derive tight analytic bounds on the maximum…
By performing local projective measurements on a two-qubit entangled state one can certify in a device-independent way up to one bit of randomness. We show here that general measurements, defined by positive-operator-valued measures, can certify up to two bits of randomness, which is the optimal amount of randomness that can be certified from an entangled bit. General measurements thus provide an advantage over projective ones for device-independent randomness certification.
Braiding of Majorana fermions gives accurate topological quantum operations that are intrinsically robust to noise and imperfection, providing a natural method to realize fault-tolerant quantum information processing. Unfortunately, it is known that braiding of Majorana fermions is not sufficient for implementation of universal quantum computation. Here we show that topological manipulation of Majorana fermions provides the full set of operations required to generate random numbers by way of quantum mechanics and to certify its genuine randomness through violation of a multipartite Bell inequa…
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.
We study how quantum randomness generation based on unbiased measurements on a hydrogen-like atom can get compromised by virtue of the unavoidable coupling of the atom with the electromagnetic field. Concretely, we show that an adversary with access to the quantum EM field, but not the atom, can perform an attack on the randomness of a set of unbiased quantum measurements. We analyze the light-atom interaction in 3+1 dimensions with no single-mode or rotating-wave approximations. In our study, we also take into account the non-pointlike nature of the atom and the exchanges of angular momentum …
When two players achieve a superclassical score at a nonlocal game, their outputs must contain intrinsic randomness. This fact has many useful implications for quantum cryptography. Recently it has been observed (C. Miller, Y. Shi, Quant. Inf. & Comp. 17, pp. 0595-0610, 2017) that such scores also imply the existence of local randomness -- that is, randomness known to one player but not to the other. This has potential implications for cryptographic tasks between two cooperating but mistrustful players. In the current paper we bring this notion toward practical realization, by offering a near-…
The generation of certifiable randomness is the most fundamental information-theoretic task that meaningfully separates quantum devices from their classical counterparts. We propose a protocol for exponential certified randomness expansion using a single quantum device. The protocol calls for the device to implement a simple quantum circuit of constant depth on a 2D lattice of qubits. The output of the circuit can be verified classically in linear time, and is guaranteed to contain a polynomial number of certified random bits assuming that the device used to generate the output operated using …
Nonlocal correlations are useful for device independent (DI) randomness certification [Nature (London) {\bf 464}, 1021 (2010)]. The advantage of this DI protocol over the conventional quantum protocol is that randomness can be certified even when experimental apparatuses are not trusted. Quantum entanglement is the necessary physical source for the nonlocal correlation required for such DI task. However, nonlocality and entanglement are distinct concepts. There exist entangled states which produce no nonlocal correlation and hence are not useful for the DI randomness certification task. Here w…
