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A Comprehensive Review of Quantum Random Number Generators: Concepts, Classification and the Origin of Randomness

Vaisakh Mannalath, Sandeep Mishra, Anirban Pathak

arXiv:2203.00261v3quant-ph

TL;DR

Randomness is important for cryptography and other applications, while physical and deterministic sources have limitations. This review surveys QRNG classes and their randomness foundations using a hierarchy of quantum features. It concludes that QRNG choices involve trade-offs between generation rate and certification, with commercial products available.

  • Problem

    Random numbers are needed across applications, but deterministic generators can be predictable and physical sources can be difficult to quantify or certify.

    Source: §1 · pp. 1–2, §3 · p. 8

  • Method

    The review surveys QRNG classes, compares adversarial models, and analyzes randomness origins through hierarchical quantum axioms and nonclassical theories.

    Source: §1 · p. 3, §7 · p. 27

  • Results

    Homodyne or heterodyne generators have significantly higher bit rates, while device-independent methods offer the highest certifiability with lower generation rates.

    Source: §8 · p. 28

  • Takeaways & Limitations

    Off-the-shelf QRNGs are commercially available and not costly, and the choice among methods depends on user needs and multiple comparison factors.

    Source: §8 · p. 28

    Self-testing protocols require initial genuine random numbers and non-local states, so they expand accessible randomness rather than generating it from nothing.

    Source: §5 · p. 20

Abstract

from arXiv · show

Random numbers are central to cryptography and various other tasks. The intrinsic probabilistic nature of quantum mechanics has allowed us to construct a large number of quantum random number generators (QRNGs) that are distinct from the traditional true number generators. This article provides a review of the existing QRNGs with a focus on their various possible features (e.g., device independence, semi-device independence) that are not achievable in the classical world. It also discusses the origin, applicability, and other facets of randomness. Specifically, the origin of randomness is explored from the perspective of a set of hierarchical axioms for quantum mechanics, implying that succeeding axioms can be regarded as a superstructure constructed on top of a structure built by the preceding axioms. The axioms considered are: (Q1) incompatibility and uncertainty; (Q2) contextuality; (Q3) entanglement; (Q4) nonlocality and (Q5) indistinguishability of identical particles. Relevant toy generalized probability theories (GPTs) are introduced, and it is shown that the origin of random numbers in different types of QRNGs known today are associated with different layers of nonclassical theories and all of them do not require all the features of quantum mechanics. Further, classification of the available QRNGs has been done and the technological challenges associated with each class are critically analyzed. Commercially available QRNGs are also compared.

1. Introduction

Random numbers serve applications from simulation to cryptography, but deterministic generators can be predictable and physical entropy sources can be difficult to certify. This review surveys QRNG designs, their randomness foundations, and their practical trade-offs.

1. Introduction

  • Random numbers support applications from weather prediction and Monte Carlo simulation to cryptography and statistical sampling.Examples in daily use include one-time passwords, bank PINs, and CAPTCHAs.Source: §1 · p. 1
  • PRNGs expand a short input deterministically and can be predicted if the algorithm and initial input are known, while physical TRNG sources are difficult to quantify and certify.Quantum sources offer intrinsic randomness whose quality can be quantified using quantum mechanics.Source: §1 · p. 2
  • Early QRNGs used uncertainty in radioactive decay and electronic shot noise, but faced handling, noise-discrimination, and speed drawbacks.Later approaches used single-photon detection, macroscopic photodetection, and atomic spin noise.Source: §1 · p. 2
  • Self-testing QRNGs use nonlocal correlations to let users test output privacy and randomness without trusting vendors, but their low rates motivate semi-self-testing designs [58].Semi-self-testing approaches seek to balance generation rates and certification.Source: §1 · p. 3
  • The review classifies randomness origins through hierarchical axioms—uncertainty, contextuality, entanglement, nonlocality, and identical-particle indistinguishability—and compares QRNGs under classical and quantum adversaries.It presents the foundational debates as unresolved rather than aiming to settle them.Source: §1 · p. 3
  • Figure 1 presents an explicitly non-exhaustive classification with overlapping generator classes, while the review also surveys commercial QRNG products.The cited examples include products from ID Quantique, Toshiba, PicoQuant, and QuantumCTek.Source: §1 · p. 3, Figure 1 · p. 4

2. Quantifying randomness via entropy

Entropy provides a way to quantify randomness, but practical QRNG settings can violate assumptions behind Shannon entropy. The review describes generalized and conditional entropy measures and explains their relevance to extraction and source assessment.

2. Quantifying randomness via entropy

  • Shannon entropy measures average information from a random variable and increases as its distribution approaches uniformity.It is zero for a single possible outcome and is maximized when all outcomes are equiprobable.Source: §2 · p. 5
  • Shannon entropy applies under an independent-and-identically-distributed assumption in the asymptotic limit, which practical device imperfections may violate.The review therefore introduces generalized entropy measures.Source: §2 · p. 5
  • Rényi entropy is parameterized by order α and reduces to Shannon entropy in the limit α → 1; min-entropy H∞(X) is obtained in the limit α → ∞.Source: §2 · pp. 5–6
  • Conditional min-entropy bounds the probability of guessing subsystem A when subsystem B is fully known; uncorrelated distributions recover ordinary min-entropy [89–91].The text defines it using the joint state ρ_AB and the reduced state of B.Source: §2 · p. 6
  • Quantum theory can bound available min-entropy despite classical noise or an eavesdropper, but practical use requires extracting it into uniform bits.Entropy estimates depend on physical-source analysis, while secure high-rate generation must satisfy practical challenges and estimation assumptions.Source: §2 · pp. 6–7

3. Pseudo and true random number generators

PRNGs expand deterministic seeds into test-like random strings, while TRNGs draw on physical events. The section distinguishes quantum-modelled QRNGs from other TRNGs and notes practical limits in rate and entropy assessment.

3. Pseudo and true random number generators

  • PRNGs deterministically expand a small seed into a longer sequence, but knowledge of the seed and algorithm makes their output predictable.They are useful when speed and reproducibility suffice, but seed compromise threatens security.Source: §3 · pp. 7–8
  • CSPRNGs add forward and backward security requirements, while the cited Blum-Blum-Shub example relies on the computational difficulty of bi-prime factorization.The section also notes linear congruential generators may have correlations missed by standard tests.Source: §3 · pp. 7–8
  • TRNGs use physical processes as entropy sources, but their rates are generally limited and the amount of generated randomness can be difficult to quantify.For circuit noise, difficulty separating shot noise from thermal noise weakens entropy bounds and complicates failure detection.Source: §3 · p. 8
  • The paper reserves QRNG for devices whose working principles can be clearly modelled by quantum mechanics, allowing randomness to be checked against quantum laws.It distinguishes these from classical TRNGs whose apparent unpredictability may reflect incomplete knowledge of the physical system.Source: §3 · pp. 8–9

4. Trusted device QRNGs

Trusted-device QRNGs assume that the source and devices are well characterized; their practical suitability depends on implementation matching the model. They cannot certify that output is genuinely random or beyond an adversary’s control.

4. Trusted device QRNGs

  • Trusted-device QRNGs assume the randomness source and extraction devices are well characterized.The paper divides them into non-optical and optical subclasses.Source: §4 · p. 9
  • Their theoretical simplicity and ease of implementation suit practical use only when the implementation adequately fits the model.The paper notes that controlling quantum systems makes this demanding experimentally.Source: §4 · p. 9
  • A trusted-device QRNG cannot certify whether its output is genuinely random or under an adversary’s control.Its use therefore relies on trusting that the device works as intended.Source: §4 · p. 9

4.1. Non-optical trusted device QRNGs

Non-optical trusted-device QRNGs include radioactive-decay, electronic-noise, and atomic-system approaches. These methods draw on quantum uncertainty, but their implementations face source-specific practical constraints.

4.1. Non-optical trusted device QRNGs

  • Non-optical trusted-device QRNGs include radioactive-decay and atomic systems, as well as electronic-noise sources in circuits.The section surveys their implementations and merits and demerits.Source: §4.1 · p. 9

4.1.1. Radioactive Decay

  • Radioactive-decay QRNGs exploit only Q1, incompatibility and uncertainty, with Geiger-Müller tubes detecting emissions from characterized radioactive sources.The detection pulses form a Poisson distribution whose rate depends on source half-life, position, and detector gas state.Source: §4.1.1 · p. 9
  • Fast-clock methods encode the cycles between detections, while slow-clock methods use counts in a fixed interval; parity or modulo processing can reduce bias.Semiconductor detectors can simplify designs, and an RC circuit can convert an exponential or Poissonian variable to a uniform distribution.Source: §4.1.1 · p. 10, Figure 3 · p. 10
  • Radioactive sources require special care, can reduce detector efficiency over time, and impose dead-time recovery constraints on bit generation and post-processing.GM and semiconductor tubes need time to recover detection ability after an event.Source: §4.1.1 · p. 10

4.1.2. Electronic Noise

  • Electronic-noise QRNGs use circuit elements such as resistors or diodes; shot noise is quantum in origin, but separating it from thermal noise is difficult.Diode voltage spikes can also have memory effects tied to prior charge flow.Source: §4.1.2 · p. 11

4.1.3. Atomic systems

  • Atomic-system proposals include trapped-ion generators and alkali-vapor spin-noise QRNGs, which exploit Q1; trapped-ion setups are complex and generate bits slowly.Rubidium vapor spin noise can be probed optically and separated from background noise under proper conditions; solid-state systems have reported higher rates.Source: §4.1.3 · p. 11

4.2. Optical Trusted Device QRNG

Optical trusted-device QRNGs manipulate and measure light, with approaches grouped by single-photon and macroscopic detectors. Within single-photon schemes, randomness can come from superposition, photon arrival time, spatial mode, or photon-number measurements.

4.2.1.1. Qubit state.

  • Superposition-based qubit QRNGs yield at most one random bit per detected photon, with generation limited to tens of Mbps and detector effects influencing randomness and rate.The schemes use beam splitters or polarizing beam splitters to prepare superposed paths or polarizations; detector dead time can correlate generated bits.Source: §4.2.1.1 · pp. 12–13

4.2.1.2. Temporal mode:.

  • Temporal QRNGs derive randomness from photon arrival times, alleviate detector dead-time effects, and reach around 109 Mbps with 5.5 bits of entropy per detection.Arrival-time precision constrains generation, and exponential arrival distributions require conversion to a more uniform bit sequence.Source: §4.2.1.2 · pp. 13–14

4.2.1.3. Spatial mode:.

  • Spatial-mode QRNGs can produce up to 16 bits per detection, with entropy 0.999856 at 8 Mbps; detector correlations make commercial use impractical.The output depends on the light’s spatial intensity distribution and detector efficiency.Source: §4.2.1.3 · p. 14

4.2.1.4. Multiple photon-number states:.

  • Photon-number measurements yield Poisson-distributed randomness requiring uniformizing post-processing; one implementation reported 1.99 random bits per detection at 144 Mbps.These schemes are sensitive to photon-number statistics and detector efficiencies; photon-count variations have also been used in a mobile phone.Source: §4.2.1.3 · p. 14, §4.2.1.4 · p. 15
  • Solid-state single-photon emitters are a prospect for integrated QRNGs, but challenges from the complex mesoscopic environment remain.Reported emitter materials include diamond nitrogen vacancies, gallium nitride, and hexagonal boron nitride.Source: §4.2.1.4 · p. 15

4.2.2. Macroscopic Photo-detection

Macroscopic photodetection QRNGs measure quantities such as field intensity or amplitude rather than single photons, potentially avoiding single-photon detector dead time. The section describes vacuum-noise, amplified-spontaneous-emission, and Raman-scattering approaches, alongside implementation constraints.

4.2.2.1. Vacuum noise:.

  • Vacuum-noise QRNGs extract randomness from zero-point field fluctuations and can generate more than one bit per measurement; one example reported 3.25 bits.Balanced homodyne detection measures field quadratures, while detector speed limits generation in the shot-noise region; implementations reached 3 Gbps.Source: §4.2.2.1 · pp. 15–16, Figure 5 · p. 16

4.2.2.2. Amplified spontaneous emission:.

  • Amplified-spontaneous-emission QRNGs measure quadratures of phase-randomized weak coherent states, with phase uncertainty potentially exceeding vacuum noise and providing robustness against detector noise.The randomness is attributed to quantum-mechanical phase noise.Source: §4.2.2.2 · p. 16

4.2.2.3. Raman Scattering:.

  • Raman-scattering QRNGs use randomized output-field amplitude or phase; amplitude-based generation avoids interference and uses intensity comparison and correction.The initiating quantum fluctuations are amplified through stimulated Raman scattering to a macroscopic field.Source: §4.2.2.3 · pp. 16–17, Figure 6 · p. 17
  • Integrated trusted QRNGs using substrates such as InP, lithium niobate, and silicon are sensitive to temperature and wavelength and can incur large optical losses.The review compares trusted QRNGs by assumptions, detector use, and bit-generation speed.Source: §4.2.2.3 · p. 17, Table 1 · p. 18

5. Self testing QRNGs

Self-testing QRNGs certify outputs from input-output statistics rather than trusting device details, chiefly through nonlocality tests. Expansion and amplification protocols can certify or increase randomness, but depend on experimental conditions and, in some cases, initial weak randomness.

5. Self testing QRNGs

  • Self-testing QRNGs certify randomness from input-output statistics by witnessing Bell-inequality violation, ruling out local deterministic models under the nonlocality axiom Q4.These tests have high credibility but are experimentally difficult and slow; the devices require randomized strategies and genuine random numbers and nonlocal states initially.Source: §5 · pp. 19–20, Figure 7 · p. 20

5.1. Self testing Random number expansion

  • Around 10 kbps generation rates have been reached in protocols reported secure against quantum and classical adversaries.Earlier, an ion-trap proof of concept generated 42 bits in one month while consuming more randomness than it produced; Bell tests also face locality and detection loopholes.Source: §5.1 · p. 21
  • Contextuality-based generators use incompatible, non-commuting measurements and Q2 rather than Bell nonlocality, avoiding entanglement and space-like separation.They work with untrusted but not adversarial devices; Bell-test settings must also be protected from correlations with hidden variables.Source: §5.1 · p. 21

5.2. Self testing Randomness amplification

  • Randomness amplification uses weak inputs to choose quantum measurement settings and can produce perfectly random bits from arbitrary weak randomness.Earlier work proved amplification above a critical threshold; later work showed arbitrary weak randomness suffices, and practical protocols used limited numbers of independent devices.Source: §5.2 · pp. 21–22

6. Semi-self testing QRNGs

Semi-self-testing QRNGs trust some device components while leaving others uncharacterized, balancing practical performance against credibility. The review covers source-independent, measurement-device-independent, and bounded-dimension approaches, each relying on different assumptions.

6. Semi-self testing QRNGs

  • Semi-self-testing QRNGs trade the high performance and low cost of trusted devices against the high credibility but low performance of fully self-testing devices.They characterize selected parts of the system and ignore others, particularly when addressing device imperfections and noisy channels rather than adversarial settings.Source: §6 · p. 22

6.1. Source device independent QRNGs

  • Source-independent QRNGs use an untrusted source and trusted measurement device, exploiting uncertainty (Q1) to detect noise and requiring an initial random seed.Reported implementations include 8.05 Gbps with composable security parameter 10^-10 and 1.7 Gbit/sec using field quadratures.Source: §6.1 · pp. 22–23, Figure 8 · p. 23

6.2. Measurement-device-independent QRNGs

  • Measurement-device-independent QRNGs trust the source and use randomly chosen states to test an untrusted measurement device, avoiding attacks involving detector inefficiencies.Source imperfections adversely affect the protocol; continuous-variable systems can offer high bit rates but face challenges in entanglement witnessing and measurement tomography.Source: §6.2 · p. 23

6.3. Self-testing QRNGs with bounded dimensions

  • Bounded-dimension QRNGs certify randomness using a dimension witness under a system-dimension bound, with reported generation rates around 23 bps.One example assumes preparation and measurement devices share no classical randomness; other approaches use QRACs, paradoxes, contextuality, state-overlap bounds, or energy bounds.Source: §6.3 · p. 24
  • The reviewed self-testing QRNGs are compared by their assumptions, detector use, and bit-generation speeds.The comparison is presented as a brief history in Table 2.Source: §6.3 · p. 24

7. Origin of randomness in nonclassical theories

The review maps QRNG randomness to a hierarchy of nonclassical axioms and asks which layers are required by different generator classes. It concludes that trusted generators can use only Q1, whereas self-testing approaches may require higher layers.

7. Origin of randomness in nonclassical theories

  • A Q1-only local theory can support trusted generators, including the QUANTIS device whose operation relies on measurement collapse of a superposition.The review also names radioactive-source and shot-noise generators as examples explained by uncertainty relations alone.Source: §7 · pp. 25, 27
  • The framework orders five axioms—Q1 incompatibility and uncertainty, Q2 contextuality, Q3 entanglement, Q4 nonlocality, and Q5 indistinguishability—above classical features C1–C4.Each successive axiom is treated as a superstructure built on the preceding ones; the framework uses toy GPTs to identify which axioms particular QRNGs need.Source: §7 · pp. 26–27
  • Trusted QRNGs can be supported by a local nonclassical theory containing Q1, while self-testing devices based on Bell violations require Q1–Q4.The review identifies uncertainty-based sources such as radioactivity, shot noise, spin noise, vacuum noise, and Raman scattering as Q1-only examples.Source: §7 · p. 27
  • Contextuality-based self-testing can use Q2 in a local theory, whereas Bell-inequality certification of self-testing devices requires the nonlocal Q1–Q4 theory T′4.The review distinguishes this from full quantum mechanics: T′1 does not require Q5.Source: §7 · p. 27

8. Conclusions

The review surveys QRNG methods, their technological challenges, and the origins of their randomness, while comparing practical tradeoffs across generator classes. It reports that commercial products are available in the trusted class, while self-testing and semi-self-testing devices remain in development.

8. Conclusions

  • The review covers quantum-system random-generation methods and traces randomness to nonclassical entropy sources, offering insights toward a near-perfect random-number generator.It also notes that QRNG technologies are more mature than many other quantum technologies and that alternatives exist despite implementation challenges.Source: §8 · pp. 27–29
  • Commercial QRNGs are available off the shelf and belong to the trusted class, while self-testing and semi-self-testing devices remain developmental.Table 3 lists commercial products by bit-generation speed, connectivity, and tests passed.Source: §8 · p. 28, Table 3 · p. 28
  • No QRNG method is best without regard to user needs, because comparisons involve bit rate, cost, ease of use, availability, form factor, and certifiability.The review frames its comparison as a way for users to make informed choices.Source: §8 · p. 28
  • Vacuum-noise and ASE generators offer higher bit rates, commercial products favor form factor and availability, and device-independent methods offer the highest certifiability with lower generation rates.Microchip- or USB-based generators provide portability and accompanying software for nontechnical users.Source: §8 · pp. 28–29

Data availability

No additional data is needed for this work.

Data availability

Authors’ contributions

All authors contributed equally to the writing; AP conceptualized the article and foundational problem, while VM and SM critically analyzed QRNG schemes.

Authors’ contributions

  • AP conceptualized the article and foundational problem, while VM and SM critically analyzed existing QRNG schemes; all authors contributed equally to writing.Source: Authors’ contributions · p. 29
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