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Randomness in Quantum Mechanics: Philosophy, Physics and Technology

Manabendra Nath Bera, Antonio Acín, Marek Kuś, Morgan Mitchell, Maciej Lewenstein

arXiv:1611.02176v2quant-phphysics.hist-ph

TL;DR

This progress report reviews recent developments in quantum randomness, with particular attention to its certification and connection to quantum nonlocality. It surveys philosophical, physical, and technological work, including Bell-based approaches to certifying randomness. The report’s focus is selective rather than a comprehensive review of randomness research.

  • Problem

    The report addresses how randomness should be understood and certified, especially whether quantum randomness is intrinsic rather than due to limited knowledge.

    Source: A. Epistemic and ontic character of… · p. 4, C. Randomness in quantum physics · p. 8

  • Method

    The report synthesizes philosophical, physical, and technological developments, including protocols that use nonlocal correlations to certify randomness.

    Source: §I · p. 3

  • Results

    The report argues that nonlocal quantum correlations can certify randomness under no-signaling, while contextuality can certify randomness but not in a device-independent way.

    Source: §I · p. 3, §2 · p. 11

  • Takeaways & Limitations

    The report connects quantum nonlocality and randomness certification to emerging device-independent quantum information processing and technologies.

    Source: §I · p. 3

    The report does not aim to cover all existing literature on randomness in physics.

    Source: §I · p. 3

Abstract

from arXiv · show

This progress report covers recent developments in the area of quantum randomness, which is an extraordinarily interdisciplinary area that belongs not only to physics, but also to philosophy, mathematics, computer science, and technology. For this reason the article contains three parts that will be essentially devoted to different aspects of quantum randomness, and even directed, although not restricted, to various audiences: a philosophical part, a physical part, and a technological part. For these reasons the article is written on an elementary level, combining very elementary and non-technical descriptions with a concise review of more advanced results. In this way readers of various provenances will be able to gain while reading the article.

I. INTRODUCTION

The report distinguishes apparent randomness from randomness that persists despite full knowledge, then reviews philosophical, physical, and technological developments in quantum randomness. Its focus is recent work linking quantum nonlocality with randomness certification and device-independent quantum information processing and technologies.

I. INTRODUCTION

  • The report distinguishes apparent randomness, caused by incomplete knowledge, from intrinsic randomness, which persists even with full knowledge of a system’s state.The report notes that both can coexist in quantum mechanics when the state is only partially known.Source: §I · pp. 1–2
  • In the philosophical part, the report connects debates about determinism and intrinsic quantum randomness with contextuality, nonlocality, and whether randomness can be certified.It also discusses non-unique solutions of Newton’s equations and their similarities to quantum measurement theory.Source: §I · p. 2
  • The physical part reviews randomness certification, amplification, and expansion, alongside nonlocality, information-theoretic characterization, and device-independent protocols.It also surveys links to quantum secret-key distribution and the special status of quantum mechanics among no-signaling theories.Source: §I · pp. 2–3
  • The technological part reviews certified randomness generation, conventional quantum generators, Bell-violation detection, and commercial quantum protocols, including early device-independent implementations.The report also discusses proof-of-principle experiments using nonlocality.Source: §I · pp. 3–4
  • The report’s novelty is its focus on quantum randomness, its relation to nonlocality and correlations, and emerging device-independent quantum information processing and technologies.It does not aim to cover all literature on randomness in physics.Source: §I · p. 3
  • The report argues that nonlocal quantum correlations can certify randomness securely in a device-independent way, while contextuality can certify randomness but not device-independently.It presents both as routes to certification of intrinsic quantum randomness.Source: §I · p. 3
  • The review adopts the traditional interpretation of quantum mechanics and excludes Bohmian and many-world interpretations from its scope.The report notes that randomness and nonlocality have different meanings in those approaches.Source: §I · p. 4

A. Epistemic and ontic character of probability

The section distinguishes epistemic randomness, attributed to limited knowledge in an otherwise deterministic world, from ontic randomness, treated as independent of knowledge. It connects this distinction to debates about determinism, free will, and the possibility of intrinsic randomness in quantum mechanics.

A. Epistemic and ontic character of probability

  • The distinction matters for cryptography because security may be compromised if apparent unpredictability reflects only an adversary’s limited knowledge.If processes are intrinsically random, the passage says unconditional secure coding schemes are conceivable.Source: A. Epistemic and ontic character of… · p. 4
  • Epistemic and ontic accounts differ over whether randomness comes from incomplete knowledge or is an intrinsic property independent of knowledge.The epistemic account treats the world as basically deterministic; the ontic account treats it as nondeterministic and not reducible to deterministic hidden variables.Source: A. Epistemic and ontic character of… · pp. 4–5
  • The section argues that neither strict determinism nor randomness alone supplies the control or influence needed for free will.It calls intrinsic randomness a possible necessary condition, but not a sufficient one.Source: A. Epistemic and ontic character of… · p. 5

B. Randomness in classical physics

The section examines how determinism and randomness relate in classical physics, where unpredictability can arise without intrinsic randomness. It then turns to quantum mechanics, asking whether its probabilistic outcomes are intrinsic or could reflect hidden variables.

B. Randomness in classical physics

  • Laplacean-style determinism is insufficient to distinguish ignorance-based randomness from randomness intrinsic to events.Unpredictability can arise either from insufficient information or from events that are random in themselves.Source: B. Randomness in classical physics · p. 5
  • Classical unpredictability does not by itself establish intrinsic randomness: limited information and deterministic chaos can make predictions impractical.Classical statistical physics uses hidden variables, while sensitivity to initial conditions and many interacting particles limit practical prediction.Source: B. Randomness in classical physics · p. 6
  • Classical mechanics is deterministic only if initial positions and velocities determine a unique future; non-unique solutions can violate that condition.The passage identifies a Lipschitz condition on forces as sufficient for uniqueness, and says breaking it opens the possibility of non-unique trajectories.Source: B. Randomness in classical physics · p. 6

C. Randomness in quantum physics

  • Quantum mechanics predicts probabilistic outcomes, but the section asks whether hidden variables could make this randomness epistemic rather than intrinsic.It notes that Born’s rule and quantum measurement theory both allow randomness in measurement outcomes.Source: C. Randomness in quantum physics · p. 8

1. Contextuality and randomness

The section contrasts ordinary probabilistic outcomes, which may be explained by hidden-variable ignorance, with contextuality-based arguments for intrinsic randomness. It reviews the Kochen–Specker constraint and contextuality inequalities, while noting limits to their randomness certification.

Contextuality and randomness

  • A spin measurement’s probabilistic outcomes can be indistinguishable from classical probabilities and explained by deterministic hidden-variable models.The cited note imposes no constraint on those hidden variables, which may be nonlocal.Source: §1 · p. 8
  • For Hilbert spaces of dimension at least three, the Kochen–Specker theorem rules out hidden-variable assignments that preserve measurement-independent values and functional relations.The theorem complements Bell’s theorem, which concerns composite systems.Source: §1 · p. 9
  • Single qubits admit non-contextual hidden-variable models for all measurements, a subtle exception the report sets aside when discussing intrinsic randomness.The cited result concerns two-dimensional Hilbert space.Source: §1 · p. 9
  • Contextuality inequalities for non-contextual theories can be violated by quantum states, and state-independent violations have been demonstrated experimentally.A graph-theoretic framework can quantify and characterize contextuality.Source: §1 · p. 9
  • Violating Cabello-like inequalities rules out non-contextual deterministic hidden-variable explanations and implies intrinsic randomness, but this certification is not device independent.Its dependence on the measured non-commuting observables limits its security.Source: §1 · p. 9

2. Nonlocality and randomness

The section explains how Bell correlations connect nonlocality with local measurement randomness, conditional on assumptions such as no-signaling and free measurement choice. It places this account within the report’s broader review of quantum randomness and the quantum-measurement framework.

2. Nonlocality and randomness

  • Entanglement can make local outcomes random even when the global state is pure and completely known, although the report notes this randomness can be described epistemically by deterministic nonlocal models.Local observers cannot access information residing in nonlocal correlations.Source: §2 · pp. 9–10
  • The report frames quantum nonlocality through the EPR debate over locality and realism, and Bell’s theorem’s incompatibility result for theories satisfying both.Bell inequalities bound correlations in spatially separated measurements under local realism.Source: §2 · p. 10
  • Bell inequalities are violated by entangled quantum particles, so assuming no-signaling, their outcomes cannot be predetermined by a local deterministic theory.The report notes experimental violations, including recent loophole-free tests.Source: §2 · p. 10
  • Bell-based randomness claims rely on free measurement choices; relaxing that condition can permit deterministic no-signaling models that violate Bell inequalities.The report also notes the circularity that testing indeterminism requires random measurement adjustments.Source: §2 · p. 11

III. QUANTUM RANDOMNESS AND PHYSICS

  • The physics part reviews measurements, nonlocality, and information-theoretic randomness measures before discussing generation, certification, expansion, and amplification.These are the topics the report identifies for its physics discussion.Source: §III · p. 11

A. Quantum measurements

  • In standard quantum mechanics, measurement outcomes are generally probabilistic, with deterministic outcomes possible when the system is in an eigenstate of the measured observable.The postulates associate observables with measurable quantities, restrict outcomes to eigenvalues, and assign probabilities to measurement results.Source: A. Quantum measurements · p. 11, §1 · p. 11

1. Postulates of QM

  • The postulates describe state specification, measurement and collapse, time evolution, and symmetrization for systems containing identical particles.The report lists these as foundational rules for simple mechanical systems.Source: §1 · pp. 11–12

2. Measurement theory

The section presents quantum measurement as a source of probabilistic outcomes and reviews generalized measurements and uncertainty relations. It also notes that outcome probabilities alone do not generally determine the post-measurement state, and that practical random-number generation requires extraction.

Measurement theory

  • POVM elements are positive semidefinite operators summing to identity, and need not be orthogonal, allowing more elements than the Hilbert-space dimension.The report presents POVMs as a generalization of orthogonal projectors.Source: §2 · p. 12
  • A POVM does not generally determine the post-measurement state, because different measurement operators can yield the same outcome probabilities.The state update depends on the particular measurement operators.Source: §2 · p. 12
  • Quantum measurements on states that are not observable eigenstates can generate random numbers in principle, but practical outputs require randomness extraction.The report says continuous-variable techniques are several orders of magnitude faster than earlier approaches and use known predictability bounds for extraction.Source: §2 · pp. 12–13
  • Uncertainty relations bound measurement uncertainty for non-commuting observables, but randomness can also reflect classical uncertainty in the state.With quantum memory, side information can compensate for measurement randomness; the report also relates uncertainty to contextuality.Source: §2 · p. 13
  • The report associates quantum-mechanical intrinsic randomness with the Born rule and distinguishes it from randomness arising from nonlocal correlations.It identifies measurement processes as the source of inherent randomness in the standard account.Source: §2 · pp. 12–13

1. Two-party nonlocality

A two-party Bell test estimates joint outcome probabilities for choices made by spatially separated observers. Bell inequalities distinguish correlations explainable by local hidden-variable models from quantum correlations that violate those bounds.

1. Two-party nonlocality

  • A Bell test has Alice and Bob choose measurements on a bipartite system, repeat them, and estimate joint probabilities from their settings and outcomes.The observers are spatially separated and perform space-like measurements.Source: §1 · p. 13
  • Local hidden-variable models attribute correlations to a shared cause while each observer’s outcome depends only on their own setting and that cause.The model assumes measurement settings are independent of the hidden variable, and deterministic and general local hidden-variable models are equivalent.Source: §1 · pp. 13–14
  • Quantum correlations can violate Bell inequalities, ruling out a local hidden-variable explanation for those correlations.Bell inequalities are linear bounds that local hidden-variable models must satisfy.Source: §1 · p. 14
  • In the CHSH example, local deterministic strategies can reach the bound of 2, whereas a quantum setting can violate the inequality.The example uses two measurement choices and two outcomes for each observer.Source: §1 · p. 14

2. Multi-party nonlocality and device independent approach

Multi-party nonlocality is described through conditional probabilities for separated parties’ black boxes, with no-signaling constraints and Bell inequalities characterizing correlations. The device-independent approach treats the devices as uncharacterized and uses nonlocal correlations as a resource for randomness applications.

2. Multi-party nonlocality and device independent approach

  • In a device-independent scenario, n separated parties use black boxes with m measurement choices and d outcomes, described by conditional probabilities.The devices’ internal workings are not specified; the approach arose in part after hacking attacks on quantum cryptographic devices.Source: §2 · p. 14
  • No-signaling requires that one party’s measurement choice cannot instantaneously signal to the others.This condition constrains the conditional probabilities describing the parties’ outcomes.Source: §2 · p. 15
  • Local correlations are mixtures of independent parties’ deterministic responses to their settings and a shared hidden variable, and must obey Bell inequalities.Quantum probabilities are given by a trace formula using a multipartite state and local measurement operators; with no local-dimension constraint, measurements can be chosen projective.Source: §2 · p. 15
  • The classical correlations form a polytope whose vertices are deterministic local strategies, while quantum correlations can extend beyond it.Tight Bell inequalities correspond to facets of the classical polytope; characterizing it becomes exponentially complex with the number of parties, even for (n, 2, 2), where it has 2^2n vertices.Source: §2 · p. 15
  • Bell nonlocal correlations are identified as a resource for device-independent random-number generation, certification, expansion, and amplification.The report also describes them as a genuine resource theory when allowed operations are restricted to device-independent local operations.Source: §2 · p. 15

C. Randomness: information theoretic approach

Randomness is a resource in applications including cryptography and simulation, motivating reliable sources and certification tests. This section introduces weak-source models and extraction approaches, including seeded extraction and amplification for particular sources.

C. Randomness: information theoretic approach

  • Random numbers support cryptography, simulations, and gambling, prompting work on reliable sources and certification tests.True RNGs use hard-to-predict physical processes, pseudo-RNGs apply deterministic functions to shorter random seeds, and quantum RNGs use quantum features.Source: C. Randomness: information theoretic… · pp. 15–16
  • Ideal randomness is uniform over the sample space, whereas weak randomness describes a nonuniform distribution that may be biased.Physical processes or measurements are usually biased, according to the report.Source: C. Randomness: information theoretic… · p. 16
  • A weak source is modeled as a set of possible distributions when its probability distribution is unknown, and the least-random distributions determine the source’s randomness.The report focuses on Santha-Vazirani and min-entropy sources for later discussion.Source: C. Randomness: information theoretic… · p. 16
  • Santha-Vazirani sources constrain each bit’s conditional probability given earlier bits, while block sources guarantee randomness over blocks rather than individual bits.Min-entropy sources characterize randomness through the probability of the most likely string.Source: C. Randomness: information theoretic… · p. 16
  • Randomness extraction aims to convert a weak-source string into a possibly shorter string close to perfectly random, but deterministic extraction works only for extractable sources.For many natural sources defined by entropy, nondeterministic extractors are necessary.Source: C. Randomness: information theoretic… · pp. 16–17
  • Two mutually independent ε-free strings from Santha-Vazirani sources can be efficiently processed to extract one almost-perfect bit, with ε′ tending to zero.The stated procedure uses their bitwise products summed modulo 2; this amplification gives ε′ < ε.Source: C. Randomness: information theoretic… · p. 17
  • Min-entropy sources require a uniformly distributed seed for seeded extraction because deterministic extraction is impossible for them.Min-entropy bounds the number of perfectly random bits that can be extracted.Source: C. Randomness: information theoretic… · p. 17

D. Nonlocality, random number generation and certification

The section explains how Bell-inequality violations certify random measurement outcomes, independently of device workings, and relate the amount of violation to the amount of randomness. It also notes that maximal randomness need not require maximal CHSH violation.

D. Nonlocality, random number generation and certification

  • The Bell-test procedure estimates joint outcome probabilities across repeated settings; a violation certifies outcomes as random, private, and device independent.The certification does not depend on the measurement devices’ working principles.Source: D. Nonlocality, random number… · p. 17
  • Observers need free measurement choices while separated, since known settings could let an external agent engineer a Bell violation with deterministic devices.The section presents free choices as a natural assumption for resolving this tension.Source: D. Nonlocality, random number… · p. 17
  • Maximum random-bit production can occur with a non-maximal CHSH violation.The account assumes quantum-mechanical devices when relating nonlocal correlations to generated randomness.Source: D. Nonlocality, random number… · p. 17
  • Min-entropy measures the least-random outcome probability, and Bell-violation bounds can quantify the outcomes’ minimum randomness.For a given violation, the cited approach minimizes the outcome min-entropy over quantum realizations.Source: D. Nonlocality, random number… · p. 18
  • A Bell violation makes the observed correlations incompatible with a deterministic model, guaranteeing some randomness at the local level.The amount is quantified by a convex function that vanishes when there is no violation.Source: D. Nonlocality, random number… · p. 18
  • Bell violation certifies randomness where ordinary random-number generation is difficult to certify.The section characterizes this as randomness without a deterministic classical analogue.Source: D. Nonlocality, random number… · p. 18

E. Nonlocality and randomness expansion

Randomness expansion uses an initial random seed to choose Bell-test settings, estimates the resulting violation, and distills the outcomes into a longer random string. The reviewed protocols include exponential expansion in a simple configuration and unbounded expansion in nested variants.

E. Nonlocality and randomness expansion

  • A seed selects Bell-test settings, which may be biased toward settings producing more randomness to reduce seed consumption and increase output.The general scenario allows multiple devices, measurements, and outcomes, while the description focuses on two devices.Source: E. Nonlocality and randomness expansion · p. 18
  • After N repetitions, insufficient confidence in a Bell violation leads to aborting or collecting more data; the observed violation bounds output min-entropy.The account denotes the generated bits by NG and their entropy lower bound by R ≤ Ng.Source: E. Nonlocality and randomness expansion · p. 18
  • A randomness extractor maps Ng bits with entropy at least R to R bits, using Ne extra seed bits, for expansion ratio R/(Ns + Ne).The extractor is classical post-processing.Source: E. Nonlocality and randomness expansion · p. 18
  • Security proofs address unpredictability to eavesdroppers sharing correlated quantum information, and also consider adversaries constrained only by no-signaling.The reviewed work is described as an important advance in protocol design.Source: E. Nonlocality and randomness expansion · pp. 18–19
  • Any Bell violation can suffice for a randomness-expansion protocol, with exponential expansion in a simple configuration and unbounded expansion in nested variants.These results are attributed to Miller and Shi (2016), Vazirani and Vidick (2012), and Chung et al. (2014); Coudron and Yuen (2013).Source: E. Nonlocality and randomness expansion · p. 19

F. Nonlocality and randomness amplification

Randomness amplification starts from imperfect randomness and uses Bell-violating correlations to produce a higher-quality source. The reviewed protocols reach arbitrarily good randomness under stated asymptotic conditions.

F. Nonlocality and randomness amplification

  • Randomness amplification upgrades an imperfect source, unlike expansion, which starts with perfect random bits and aims to produce a longer list.Both tasks exploit Bell-violating correlations.Source: F. Nonlocality and randomness… · p. 19
  • A four-device amplification scheme tests Bell inequalities and, upon violation, feeds the outputs and remaining SV-source bits into an extractor; otherwise it aborts.The devices are shielded from one another, and local settings are chosen using the SV source.Source: Figure 4 · p. 19
  • A chained-Bell protocol maps SV sources with ϵ < 0.058 to arbitrarily near-perfect randomness asymptotically, requiring infinitely many measurements.The later full-amplification protocol reaches arbitrarily good randomness from any ϵ < 1/2 source, asymptotically requiring infinitely many devices.Source: F. Nonlocality and randomness… · p. 19
  • Amplification protocols use imperfect-source bits for repeated Bell tests and construct final bits after a sufficiently large, statistically confident violation.Unlike expansion, extraction cannot use a perfect-randomness seed; protocols require security proofs against observers correlated with the devices.Source: F. Nonlocality and randomness… · p. 19

IV. QUANTUM RANDOMNESS AND TECHNOLOGY

The technology section contrasts device-independent certification with conventional quantum random-number generation, emphasizing both certification challenges and physical methods for bounding output entropy. Laser phase diffusion offers a fast macroscopic source, while device-independent demonstrations remain slow and consume substantial setting randomness.

IV. QUANTUM RANDOMNESS AND TECHNOLOGY

  • Passing statistical randomness tests does not establish unpredictability: a device can replay a previously random sequence and still pass them.The section therefore treats quantifying adversarial predictability or control as a central challenge.Source: §IV · p. 20
  • A randomness experiment generated 42 bits in about one month (1.6 × 10−5 bits/s), while a second entangled-photon experiment achieved 0.4 bits/s; device-independent techniques remain slow for nearly all applications.Source: §IV · p. 20
  • Device-independent Bell tests consume many random setting bits, and the need for traditional quantum randomness in such protocols remains likely in practice.The section notes disagreement over whether additional physical randomness is needed in principle.Source: §IV · p. 20
  • Entropy bounds require separating intrinsic randomness from thermal and technical noise, which can contaminate physical measurements.Optical systems permit shot-noise calibration through scaling; variance versus mean photocurrent quantifies distinct noise contributions.Source: §IV · pp. 20–21
  • Laser phase diffusion generators reached 68 Gbits/second at the time of writing, using macroscopic optical signals that improve speed and signal-to-noise ratios.The phase randomization arises from spontaneous-emission phase kicks accumulating in a random walk.Source: §IV · p. 21
  • A delayed interferometer converts the randomized laser phase into detector-power variation, while phase uniformity makes the phase distribution insensitive to additional shifts.The phase becomes uniform on [0, 2π), and a two-path setup measures interference between earlier and later output.Source: §IV · p. 21
  • Binary digitization thresholds the interferometer signal, and measured noise distributions support an experimentally guaranteed min-entropy bound.For σnoise = 10 mV and ΔV = 0.5 V, the passage gives single-bit H∞(d|Vnoise) > 0.86 and an extraction route to fully random strings.Source: §IV · pp. 21–22
  • The guarantee rests on a trusted phase-randomness model, a simple interference model, and measured noise effects evaluated with worst-case assumptions.The figure describes the phase-to-power variation and comparator digitization used in the LPD-QRNG.Source: §IV · p. 22, Figure 5 · p. 22

V. QUANTUM RANDOMNESS AND FUTURE

The review surveys philosophical, theoretical-physics, and applied perspectives on quantum randomness. It highlights unresolved questions about certifying randomness and the trade-offs between quantum randomness-generation approaches.

V. QUANTUM RANDOMNESS AND FUTURE

  • The review covers philosophical, physical, and technological implications and new directions in quantum-randomness research.Source: §V · p. 22
  • Randomness certification cannot be proven without assumptions about systems, and identifying the weakest sufficient assumptions remains open.Source: §V · p. 23
  • The relation between non-locality and randomness remains incompletely understood, although small non-locality or entanglement can sometimes certify maximal randomness in Bell experiments.In no-signaling theories, maximal randomness certification is impossible, whereas it is possible in quantum physics.Source: §V · p. 23
  • Device-dependent protocols use precise device knowledge and statistical tests, are comparatively feasible and fast, while device-independent protocols use Bell inequality violations but are harder to implement and slower experimentally.Source: §V · p. 23
  • The authors expect advances across these approaches to yield a wider variety of quantum-powered randomness-generation solutions.Source: §V · p. 23
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