Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review Research

Efficient randomness certification by quantum probability estimation

Yanbao Zhang, Honghao Fu, Emanuel H. Knill

Published Jan 7, 2020• 47 citations
Publisher

Abstract

For practical applications of quantum randomness generation, it is important to certify and further produce a fixed block of fresh random bits with as few trials as possible.Consequently, protocols with high finite-data efficiency are preferred.To yield such protocols with respect to quantum side information, we develop quantum probability estimation.Our approach is applicable to device-independent as well as device-dependent scenarios, and it generalizes techniques from previous works [Miller and Shi, SIAM J. Comput. 46, 1304 (2017); Arnon-Friedman et al., Nat. Commun. 9, 459 (2018)].Quantum probability estimation can adapt to changing experimental conditions, allows stopping the experiment as soon as the prespecified randomness goal is achieved, and can tolerate imperfect knowledge of the input distribution.Moreover, the randomness rate achieved at constant error is asymptotically optimal.For the device-independent scenario, our approach certifies the amount of randomness available in experimental results without first searching for relations between randomness and violations of fixed Bell inequalities.We implement quantum probability estimation for device-independent randomness generation in the CHSH Bell-test configuration, and we show significant improvements in finite-data efficiency, particularly at small Bell violations which are typical in current photonic loophole-free Bell tests.

DOI
10.1103/physrevresearch.2.013016

Related papers

Certifying quantum randomness by probability estimation

Yanbao Zhang, Emanuel H. Knill, Peter Bierhorst

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 when goals are achieved, optimal randomness rates, applicability to Bell tests with small violations, and unsurpassed finite-data efficiency. We greatly reduce latencies for producing random bits and formulate an associated rate-tradeoff problem of independent interest. We also show that the latency is determined by an information-theoretic measure of nonlocality rather than the Bell violation.

arXiv:1811.11928•Physical Review A•Oct 31, 2018•30 citations

Quantum Probability Estimation for Randomness with Quantum Side Information

Emanuel H. Knill, Yanbao Zhang, Honghao Fu

We develop a quantum version of the probability estimation framework [arXiv:1709.06159] for randomness generation with quantum side information. We show that most of the properties of probability estimation hold for quantum probability estimation (QPE). This includes asymptotic optimality at constant error and randomness expansion with logarithmic input entropy. QPE is implemented by constructing model-dependent quantum estimation factors (QEFs), which yield statistical confidence upper bounds on data-conditional normalized Rényi powers. This leads to conditional min-entropy estimates for randomness generation. The bounds are valid for relevant models of sequences of experimental trials without requiring independent and identical or stationary behavior. QEFs may be adapted to changing conditions during the sequence and trials can be stopped any time, such as when the results so far are satisfactory. QEFs can be constructed from entropy estimators to improve the bounds for conditional min-entropy of classical-quantum states from the entropy accumulation framework [Dupuis, Fawzi and Renner, arXiv:1607.01796]. QEFs are applicable to a larger class of models, including models permitting experimental devices with super-quantum but non-signaling behaviors and semi-device dependent models. The improved bounds are relevant for finite data or error bounds of the form $e^{-κs}$, where $s$ is the number of random bits produced. We give a general construction of entropy estimators based on maximum probability estimators, which exist for many configurations. For the class of $(k,2,2)$ Bell-test configurations we provide schemas for directly optimizing QEFs to overcome the limitations of entropy-estimator-based constructions. We obtain and apply QEFs for examples involving the $(2,2,2)$ Bell-test configuration to demonstrate substantial improvements in finite-data efficiency.

arXiv:1806.04553•Jun 12, 2018•14 citations

Device-independent randomness generation from several Bell estimators

Olmo Nieto-Silleras, Cédric Bamps, Jonathan Silman +1 more

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 probabilities characterizing the behavior of the device is considered. We introduce protocols for device-independent randomness generation secure against classical side information, that rely on the estimation of an arbitrary number of Bell expressions or even directly on the experimental frequencies of measurement outcomes. Asymptotically, this results in an optimal generation of randomness from experimental data (as measured by the min-entropy), without having to assume beforehand that the devices violate a specific Bell inequality.

arXiv:1611.00352•New Journal of Physics•Jan 23, 2018•33 citations

Experimental Low-Latency Device-Independent Quantum Randomness

Yanbao Zhang, Lynden K. Shalm, Joshua C. Bienfang +10 more

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 random bits with an average experiment time of less than 5 min per block and with a certified error bounded by 2^{-64}≈5.42×10^{-20}.

arXiv:1812.07786•Physical Review Letters•Jan 7, 2020•63 citations

Incorporating Zero-Probability Constraints to Device-Independent Randomness Expansion

Chun‐Yu Chen, Kai-Siang Chen, Kai-Min Chung +3 more

One of the distinguishing features of quantum theory is that its measurement outcomes are usually unpredictable or, equivalently, random. Moreover, this randomness is certifiable with minimal assumptions in the so-called device-independent (DI) paradigm, where a device's behavior does not need to be presupposed but can be verified through the statistics it produces. In this work, we explore various forms of randomness that are certifiable in this setting, where two users can perform two binary-outcome measurements on their shared entangled state. In this case, even though the Clauser-Horne-Shimony-Holt (CHSH) Bell-inequality violation is a pre-requisite for the generation of DI certifiable randomness, the CHSH value alone does not generally give a tight bound on the certifiable randomness. Here, we determine the certifiable randomness when zero-probability constraints are incorporated into the task of DI randomness expansion for the standard local and global randomness and the so-called "blind" randomness. Asymptotically, we observe consistent improvements in the amount of DI certifiable randomness (of all kinds) as we increase the number zero constraints for a wide range of given CHSH Bell violations. However, if we further optimize over the allowed CHSH values, then benefits of these additional constraints over the standard CHSH-based protocol are only found in the case of global and blind randomness. In contrast, in the regimes of finite data, these zero constraints only give a slight improvement in the local randomness rate when compared with all existing protocols.

arXiv:2401.08452•Jan 16, 2024•0 citations

Certified Randomness From Steering Using Sequential Measurements

Brian Coyle, Elham Kashefi, Matty J. Hoban

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., we do not need to 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 difficulties in the device-independent setting, we instead consider the scenario of one-sided device independence where certain devices are trusted, and others are not; a scenario motivated by asymmetric experimental set-ups such as ion-photon networks. We show how certain aspects of previous works can be adapted to this scenario and provide theoretical bounds on the amount of randomness that can be certified. Furthermore, we give a protocol for unbounded randomness certification in this scenario, and provide numerical results demonstrating the protocol in the ideal case. Finally, we numerically test the possibility of implementing this scheme on near-term quantum technologies, by considering the performance of the protocol on several physical platforms.

arXiv:2008.00705•Cryptography•Dec 6, 2019•4 citations
For practical applications of quantum randomness generation, it is important to certify and further produce a fixed block of fresh random bits with as few trials as possible.Consequently, protocols with high finite-data efficiency are preferred.To yield such protocols with respect to quantum side information, we develop quantum probability estimation.Our approach is applicable to device-independent as well as device-dependent scenarios, and it generalizes techniques from previous works [Miller and Shi, SIAM J. Comput. 46, 1304 (2017); Arnon-Friedman et al., Nat. Commun. 9, 459 (2018)].Quantum probability estimation can adapt to changing experimental conditions, allows stopping the experiment as soon as the prespecified randomness goal is achieved, and can tolerate imperfect knowledge of the input distribution.Moreover, the randomness rate achieved at constant error is asymptotically optimal.For the device-independent scenario, our approach certifies the amount of randomness available in experimental results without first searching for relations between randomness and violations of fixed Bell inequalities.We implement quantum probability estimation for device-independent randomness generation in the CHSH Bell-test configuration, and we show significant improvements in finite-data efficiency, particularly at small Bell violations which are typical in current photonic loophole-free Bell tests.
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 when goals are achieved, optimal randomness rates, applicability to Bell tests with small violations, and unsurpassed finite-data efficiency. We greatly reduce latencies for producing random bits and formulate an associated rate-tradeoff problem of independent interest. We also show that the latency is determined by an information-theoretic measure of nonlocality rather than the Bell violation.
We develop a quantum version of the probability estimation framework [arXiv:1709.06159] for randomness generation with quantum side information. We show that most of the properties of probability estimation hold for quantum probability estimation (QPE). This includes asymptotic optimality at constant error and randomness expansion with logarithmic input entropy. QPE is implemented by constructing model-dependent quantum estimation factors (QEFs), which yield statistical confidence upper bounds on data-conditional normalized Rényi powers. This leads to conditional min-entropy estimates for randomness generation. The bounds are valid for relevant models of sequences of experimental trials without requiring independent and identical or stationary behavior. QEFs may be adapted to changing conditions during the sequence and trials can be stopped any time, such as when the results so far are satisfactory. QEFs can be constructed from entropy estimators to improve the bounds for conditional min-entropy of classical-quantum states from the entropy accumulation framework [Dupuis, Fawzi and Renner, arXiv:1607.01796]. QEFs are applicable to a larger class of models, including models permitting experimental devices with super-quantum but non-signaling behaviors and semi-device dependent models. The improved bounds are relevant for finite data or error bounds of the form e−κse^{-κs}e−κs, where sss is the number of random bits produced. We give a general construction of entropy estimators based on maximum probability estimators, which exist for many configurations. For the class of (k,2,2)(k,2,2)(k,2,2) Bell-test configurations we provide schemas for directly optimizing QEFs to overcome the limitations of entropy-estimator-based constructions. We obtain and apply QEFs for examples involving the (2,2,2)(2,2,2)(2,2,2) Bell-test configuration to demonstrate substantial improvements in finite-data efficiency.
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 probabilities characterizing the behavior of the device is considered. We introduce protocols for device-independent randomness generation secure against classical side information, that rely on the estimation of an arbitrary number of Bell expressions or even directly on the experimental frequencies of measurement outcomes. Asymptotically, this results in an optimal generation of randomness from experimental data (as measured by the min-entropy), without having to assume beforehand that the devices violate a specific Bell inequality.
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 random bits with an average experiment time of less than 5 min per block and with a certified error bounded by 2^{-64}≈5.42×10^{-20}.
One of the distinguishing features of quantum theory is that its measurement outcomes are usually unpredictable or, equivalently, random. Moreover, this randomness is certifiable with minimal assumptions in the so-called device-independent (DI) paradigm, where a device's behavior does not need to be presupposed but can be verified through the statistics it produces. In this work, we explore various forms of randomness that are certifiable in this setting, where two users can perform two binary-outcome measurements on their shared entangled state. In this case, even though the Clauser-Horne-Shimony-Holt (CHSH) Bell-inequality violation is a pre-requisite for the generation of DI certifiable randomness, the CHSH value alone does not generally give a tight bound on the certifiable randomness. Here, we determine the certifiable randomness when zero-probability constraints are incorporated into the task of DI randomness expansion for the standard local and global randomness and the so-called "blind" randomness. Asymptotically, we observe consistent improvements in the amount of DI certifiable randomness (of all kinds) as we increase the number zero constraints for a wide range of given CHSH Bell violations. However, if we further optimize over the allowed CHSH values, then benefits of these additional constraints over the standard CHSH-based protocol are only found in the case of global and blind randomness. In contrast, in the regimes of finite data, these zero constraints only give a slight improvement in the local randomness rate when compared with all existing protocols.
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., we do not need to 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 difficulties in the device-independent setting, we instead consider the scenario of one-sided device independence where certain devices are trusted, and others are not; a scenario motivated by asymmetric experimental set-ups such as ion-photon networks. We show how certain aspects of previous works can be adapted to this scenario and provide theoretical bounds on the amount of randomness that can be certified. Furthermore, we give a protocol for unbounded randomness certification in this scenario, and provide numerical results demonstrating the protocol in the ideal case. Finally, we numerically test the possibility of implementing this scheme on near-term quantum technologies, by considering the performance of the protocol on several physical platforms.