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Bell nonlocality

Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, Stephanie Wehner

arXiv:1303.2849v3quant-ph

TL;DR

Bell nonlocality asks how quantum correlations differ from those reproducible by local classical resources. This review synthesizes concepts and tools for characterizing quantum bounds, nonlocality, dimensions, loopholes, and recent developments, including device-independent computation and closure of the optical detection loophole.

  • Problem

    Bell nonlocality concerns whether quantum predictions can be accounted for by local theories, with non-communicating classical observers unable to reproduce certain entangled correlations.

    Source: B. The limitations of… · p. 4, C. Scope of this review · p. 5

  • Method

    The review develops a broad synthesis of concepts and analytical tools for studying quantum nonlocality, including semidefinite relaxations, statistical divergence, dimension questions, and loophole analyses.

    Source: §1 · pp. 14, 26, 49, F. Quantifying nonlocality · p. 18

  • Results

    The review covers major developments showing that quantum bounds can be easier to compute than classical local bounds, while recent work disproves the tripartite Peres conjecture, enables arbitrary device-independent computation, and closes the optical detection loophole.

    Source: §1 · p. 13, §IX · p. 57

  • Takeaways & Limitations

    Bell nonlocality has become a full sub-field of quantum information science supported by operational tests, computational tools, and experimentally relevant analyses.

    Source: B. The limitations of… · p. 4, F. Quantifying nonlocality · p. 18, §IX · p. 57

    The operational distinction relies on non-communicating observers; if all information about both settings is available to one observer, classical reproduction is possible.

    Source: B. The limitations of… · p. 4

Abstract

from arXiv · show

Bell's 1964 theorem, which states that the predictions of quantum theory cannot be accounted for by any local theory, represents one of the most profound developments in the foundations of physics. In the last two decades, Bell's theorem has been a central theme of research from a variety of perspectives, mainly motivated by quantum information science, where the nonlocality of quantum theory underpins many of the advantages afforded by a quantum processing of information. The focus of this review is to a large extent oriented by these later developments. We review the main concepts and tools which have been developed to describe and study the nonlocality of quantum theory, and which have raised this topic to the status of a full sub-field of quantum information science.

I. INTRODUCTION

This review presents Bell nonlocality as the incompatibility between quantum predictions and local theories, then develops the concepts and tools used to study it from a quantum-information perspective. It introduces the operational Bell scenario, explains the locality condition and its violation, and surveys the review’s technical scope and applications.

I. INTRODUCTION

  • Bell’s theorem shows that quantum predictions for certain entangled-particle experiments cannot be reproduced by any theory satisfying Bell’s locality condition.The review distinguishes Bell’s locality from the commutativity notion used in quantum mechanics and quantum field theory.Source: §I · p. 2, A. Non-locality in a nutshell · p. 3

A. Non-locality in a nutshell

  • A Bell experiment separates two systems, lets Alice and Bob choose measurements x and y, and records outcomes a and b through p(ab|xy).The measurement labels and outcomes are conventional, while repeated trials estimate the joint probability distribution.Source: A. Non-locality in a nutshell · p. 2, Figure 1 · p. 2
  • Locality requires shared past variables λ to account for correlations while making each outcome depend only on its local setting and λ, with measurement choices independent of λ.This factorization does not assume determinism or classical behavior; it expresses the absence of influence between space-like separated regions.Source: A. Non-locality in a nutshell · p. 3
  • The CHSH inequality is obeyed by local models but can be violated by suitable measurements on entangled quantum particles, yielding Bell’s nonlocality result.The review derives the local bound S ≤ 2 and describes quantum predictions that exceed it, with such violations experimentally confirmed.Source: A. Non-locality in a nutshell · p. 3

B. The limitations of non-communicating PhD students

  • Bell violations separate correlations achievable by non-communicating classical observers from those achievable with quantum entanglement and support device-independent certification of entanglement.The distinction relies only on measurement data, without specifying the physical systems or measurements, and disappears if one observer receives both settings.Source: B. The limitations of… · p. 4

C. Scope of this review

  • The review focuses on technical and operational questions about characterizing nonlocal correlations, quantum resources, information processing, experiments, multipartite systems, and related variants of Bell’s theorem.It does not directly address the extensive traditional foundational literature on locality, interpretations of quantum theory, or relativity, and it emphasizes developments from roughly the preceding fifteen years.Source: C. Scope of this review · p. 5, D. Outline · p. 5

II. MATHEMATICAL CHARACTERIZATION OF NONLOCAL CORRELATIONS

This section formalizes Bell behaviors and organizes them into no-signaling, local, and quantum sets, with strict inclusions L ⊂ Q ⊂ NS. It then connects these sets to convexity, hidden-variable representations, and Bell inequalities.

A. General definitions

  • A Bell scenario is represented by behaviors p = {p(ab|xy)}, joint output probabilities for measurement settings x and y, subject to positivity and normalization.The review mainly considers two observers, each with m measurements and Δ possible outcomes.Source: A. General definitions · pp. 5–6

1. No-signaling correlations

  • No-signaling constraints make each party’s marginal independent of the other party’s setting, preventing instantaneous signaling between separated observers.The resulting set NS is an affine space that can be parametrized using a reduced collection of probabilities or correlators.Source: §1 · p. 6

2. Local correlations

  • Local behaviors admit hidden-variable decompositions with local response functions, but some no-signaling behaviors are not local, so L ⊂ NS.Nonlocality is possible only when both the number of outcomes and the number of settings are at least two.Source: §2 · p. 6

3. Quantum correlations

  • The tensor-product and commuting-operator definitions of quantum behaviors satisfy Q ⊆ Q′, while equality remains an open question in general but holds in several finite or binary-input cases.For many applications, the review uses Q for both definitions unless results depend on the distinction.Source: §3 · p. 7
  • Quantum behaviors arise from measurements on a shared quantum state and form a convex set strictly between local and no-signaling behaviors: L ⊂ Q ⊂ NS.Quantum correlations can violate Bell inequalities, while some no-signaling correlations lie outside Q.Source: §3 · p. 7, Figure 2 · p. 7

B. Bell inequalities

  • Bell inequalities are separating hyperplanes: any behavior outside L, Q, or NS violates an inequality valid for the corresponding set.For local behaviors, deterministic points generate the local polytope, and linear programming can test locality or find an inequality witnessing nonlocality.Source: B. Bell inequalities · pp. 7–8

2. Facet Bell inequalities

This section develops facet Bell inequalities, illustrates their computational complexity and examples, and recasts Bell inequalities as nonlocal games. These games provide an equivalent operational language for comparing classical and quantum strategies.

2. Facet Bell inequalities

  • Facet Bell inequalities are the codimension-one faces required for a minimal half-space description of the local polytope, while other Bell inequalities are nonnegative combinations of them.They can be enumerated from deterministic vertices, but computation becomes prohibitively difficult as parties, inputs, or outputs increase.Source: §2 · p. 9
  • Deciding locality is NP-complete for binary-output scenarios with m inputs, making a simple full characterization of the local polytope unlikely.The number of deterministic points grows exponentially with m, limiting both facet enumeration and direct linear-programming approaches.Source: §2 · p. 9

3. Examples

  • In the simplest nontrivial scenario, Δ = 2 and m = 2, CHSH is the only nontrivial facet inequality, and it remains the only facet when Bob has arbitrary inputs and outputs.For larger scenarios, examples include I3322, chained, CGLMP, and families generated through polyhedral or symmetry-based methods.Source: §3 · pp. 9–10

4. Nonlocal games

  • Bell inequalities can be viewed as nonlocal games in which a referee samples questions, players return answers without communication, and the winning probability is a Bell expression.The game value is the maximum winning probability over the chosen resource class.Source: §4 · pp. 10–11
  • XOR games are equivalent to correlation Bell inequalities with binary outcomes, and their optimal quantum strategies are especially well understood.The correspondence follows by expressing XOR winning probabilities through correlators ⟨A_xB_y⟩.Source: §4 · p. 11
  • For CHSH viewed as a game, classical strategies win with probability at most 3/4, whereas quantum resources permit a higher winning probability.The classical bound follows from S ≤ 2; the supplied passage states the quantum bound but truncates its numerical expression.Source: §4 · p. 11
  • Projection games require at most one winning answer for Bob per Alice answer, while k-to-k′ games generalize this by bounding winning-answer counts in both directions.Projection games are the special case k = k′ = 1, and unique games form a related but narrower terminology in some literatures.Source: §4 · p. 11

C. Bell inequality violations

The review presents methods for computing quantum Bell bounds and characterizing the quantum correlation set, from specialized SDPs to general hierarchies. These tools provide lower or upper bounds, with exactness in several important cases but substantial computational limitations in general.

1. Quantum bounds

  • For the CHSH inequality, the operator identity S2 = 4 + [A1, A2][B1, B2] gives ||S|| ≤ 2, establishing its quantum bound.The derivation uses arbitrary ±1-eigenvalued observables.Source: §1 · p. 13
  • Computing quantum bounds is generally difficult: the tripartite problem is NP-hard, whereas correlation-space quantum optimization can be easier than determining the corresponding local bound unless P=NP.The review also describes alternating SDP optimization for fixed states or measurements as a practical heuristic.Source: §1 · pp. 13–14
  • For correlation inequalities, Tsirelson’s vector representation converts quantum membership and optimization into semidefinite programming, yielding efficient algorithms for XOR-game bounds.This approach directly obtains the CHSH and chained-inequality quantum bounds.Source: §1 · p. 13
  • Quantum bounds can be computed by optimizing Bell operators over quantum measurements, but direct finite-dimensional parametrizations generally provide only lower bounds and lack global-convergence guarantees.They remain useful for finding explicit violations, even when optimality is not certified.Source: §1 · pp. 12–13
  • The general SDP hierarchy constructs moment-matrix relaxations Q1 ⊇ Q2 ⊇ . . . ⊇ Q that increasingly approximate quantum correlations from the outside and produce decreasing upper bounds on Sq.Each relaxation is defined by positivity, linear moment constraints, and probabilities embedded in the moment matrix.Source: §1 · p. 14
  • The hierarchy converges asymptotically to Q′, while finite-step optimality or quantum membership can sometimes be certified and low-order levels often approximate quantum bounds well.For correlation inequalities, the first level always gives the optimal solution; in the CHSH scenario, Q1 reaches Tsirelson’s bound.Source: §1 · pp. 14–15, Figure 3 · p. 15

2. No-signaling bounds

No-signaling bounds are efficiently computable because the no-signaling set is a polytope defined by positivity constraints, while its geometry exposes behaviors beyond quantum theory. The review then extends these ideas to multipartite correlations and inequality-free nonlocality proofs.

2. No-signaling bounds

  • No-signaling membership reduces to checking positivity inequalities, and linear programming efficiently computes the no-signaling bound of any Bell expression.The complexity scales polynomially with the number of inputs and outputs.Source: §2 · p. 15
  • The no-signaling set is a polytope whose local deterministic behaviors are vertices, while every other vertex is nonlocal.Its finite facet and vertex descriptions permit standard polytope algorithms.Source: §2 · p. 15

P. The local polytope

  • In the CHSH scenario, the PR box reaches the algebraic maximum s · p = 4 and pCHSH win = 1, exceeding the quantum maximum and proving Q ≠ NS.The no-signaling polytope has 24 vertices, including 8 nonlocal vertices equivalent to PR boxes under relabeling.Source: P. The local polytope · p. 15

D. Multipartite correlations

  • For multipartite systems, Bell, locality, and no-signaling definitions extend through analogous marginal constraints and convex decompositions, but nonlocal correlations become richer and less understood.Different notions of multipartite nonlocality refine the straightforward bipartite extension.Source: D. Multipartite correlations · p. 16
  • The Mermin inequality is violated by GHZ correlations up to the algebraic bound 4, and the corresponding GHZ strategy wins its nonlocal game with probability pwin = 1.The quantum bound of the relevant full-correlation inequalities is achieved by the n-partite GHZ state.Source: D. Multipartite correlations · p. 16, E. Nonlocality without “inequalities” · p. 17

E. Nonlocality without “inequalities”

  • Hardy’s paradox derives nonlocality from three zero-probability constraints plus a positive event, with pHardy > 0 for every entangled two-qubit state except the maximally entangled state.Its maximal quantum value is pHardy = (5√5 − 11)/2 ≈ 9%.Source: E. Nonlocality without “inequalities” · p. 17

F. Quantifying nonlocality

The review develops several ways to quantify Bell nonlocality, while emphasizing that raw Bell-inequality violations require normalization. It also examines how locality and quantum advantage behave in multiple-round and parallel-repetition scenarios.

F. Quantifying nonlocality

  • Normalized Bell-inequality violations can relate nonlocality to operational measures, whereas unnormalized values can be changed arbitrarily by rescaling or offsets.A meaningful violation measure requires both an appropriate normalization and, for ratio-based definitions, absolute values.Source: F. Quantifying nonlocality · pp. 17–18
  • The amount of classical communication needed to reproduce correlations provides an operational nonlocality measure, expressed as C(p) = max_s* s*·p after normalization.Source: F. Quantifying nonlocality · p. 18
  • Kullback-Leibler divergence quantifies statistical strength as asymptotic support for quantum behavior over local behavior per independent trial.It can be viewed as a distance between a behavior p and the set of local behaviors.Source: F. Quantifying nonlocality · p. 18
  • Among bipartite qubit Bell tests, CHSH is the strongest common test, while partially entangled states optimize higher-dimensional tests and Mermin-GHZ is stronger than the considered bipartite tests.Source: F. Quantifying nonlocality · p. 18

G. Multiple-rounds and parallel repetition

  • Multiple-round experiments model repeated inputs and outputs, with independent, memory, and simultaneous strategies differing in whether devices retain histories or act collectively.The general memory model characterizes usual experimental Bell tests, while simultaneous strategies correspond to parallel repetition.Source: G. Multiple-rounds and parallel… · pp. 18–19
  • Memory and collective effects cannot reproduce nonlocal correlations on average, but they can increase apparent Bell violations from statistical fluctuations in some models.For independent and memory local models, conditioning on prior events preserves the Bell inequalities and makes these fluctuations controllable.Source: G. Multiple-rounds and parallel… · p. 19
  • In parallel repetition, collective strategies can outperform repeated single-round strategies for simultaneous games; for CHSH, the local–quantum gap shrinks over many parallel rounds.Source: G. Multiple-rounds and parallel… · p. 19
  • Parallel repetition holds for classical and no-signaling strategies, while quantum perfect repetition is known for XOR-games and repetition also holds for quantum unique games.For some games, including odd cycle games, strong parallel repetition does not always hold classically.Source: G. Multiple-rounds and parallel… · pp. 19–20

III. NONLOCALITY AND QUANTUM THEORY

Quantum nonlocality requires entanglement, but entanglement and nonlocality coincide cleanly only for pure states. For mixed states, locality can persist despite entanglement, and revealing nonlocality may require more settings, filtering, multiple copies, or network scenarios.

A. Nonlocality vs entanglement

  • Nonlocal correlations require entanglement because separable states generate correlations of local form.Source: A. Nonlocality vs entanglement · p. 20
  • Every entangled pure state violates a Bell inequality with suitable local measurements, whereas product states are the only pure states that do not.This includes violation of CHSH for bipartite entangled pure states.Source: A. Nonlocality vs entanglement · p. 20
  • Mixed-state entanglement does not generally imply nonlocality: Werner states can admit local models for projective measurements or POVMs despite being entangled.The relation between mixed-state entanglement and nonlocality remains not fully understood.Source: A. Nonlocality vs entanglement · p. 20
  • Revealing mixed-state nonlocality may require filtering, joint measurements on several copies, or network arrangements, and a state can be nonlocal in one scenario but not another.Joint preprocessing of many copies is closely related to entanglement distillation; the figure distinguishes single-copy, hidden-nonlocality, many-copy, and network scenarios.Source: A. Nonlocality vs entanglement · p. 20, Figure 5 · p. 22

1. Single-copy nonlocality

  • For two-qubit states, CHSH violation is characterized by a correlation-matrix criterion, but some states that fail CHSH violate more complex inequalities such as I3322.Source: §1 · p. 21
  • The two-qubit Werner state violates a Bell inequality with 465 settings per party for p ≳0.7056, beyond the CHSH threshold near 0.707.Source: §1 · p. 21
  • Werner states illustrate a gap between entanglement and single-copy nonlocality: projective-measurement local models hold for p ≤1/2, while entanglement begins for p > 1/3.For the two-qubit family, nonlocality remains unresolved in an intermediate parameter interval even after stronger local-model and Bell-violation bounds.Source: §1 · p. 21
  • Symmetric extensions yield local models when one party uses at most k measurement settings, and an infinite symmetric extension implies separability.Source: §1 · pp. 22–23

2. Hidden nonlocality

The review shows that nonlocality can emerge from entangled states through sequential filtering, joint measurements, networks, and links to distillation and teleportation. These scenarios also reveal that nonlocality need not increase monotonically with entanglement.

2. Hidden nonlocality

  • Local filtering can reveal hidden nonlocality in states that admit local models, including Werner states and states with local models for non-sequential POVMs.Filtering succeeds by post-selecting outcomes before a final Bell test, with the final measurement basis chosen only after successful filtering.Source: §2 · pp. 23–24

3. Multi-copy nonlocality

  • Joint measurements on multiple copies can activate nonlocality, including superactivation of local isotropic states and nonlocality of every entangled isotropic state.For isotropic states, sufficiently many jointly measured copies become nonlocal whenever a single copy is entangled; whether this holds for every entangled state with a local model remains open.Source: §3 · p. 24, Figure 6 · p. 24

4. More general scenarios

  • Networks can activate nonlocality from states admitting local models, because observers may jointly measure several subsystems.For example, Werner states distributed in a star network violate a Bell inequality for p ≳ 0.64, despite admitting local models for projective measurements.Source: §4 · pp. 24–25

5. Entanglement distillation and nonlocality

  • For distillable entangled states, local filtering can be understood as entanglement distillation followed by a standard Bell test, whereas the relation for bound entangled states remains unsettled.PPT states obey important restrictions for several Bell inequalities, but multipartite nonlocality does not imply entanglement distillability.Source: §5 · p. 25

6. Nonlocality and teleportation

  • Although teleportation usefulness and nonlocality can differ in single-copy settings, every state useful for teleportation is a nonlocal resource when multiple copies are jointly measured.The maximal CHSH violation of a two-qubit state also lower-bounds its average teleportation fidelity.Source: §6 · p. 25

7. More nonlocality with less entanglement

  • Partially entangled states can outperform maximally entangled states for several measures of nonlocality, including Bell-inequality violation, detection-loophole resistance, and statistical strength.Some Bell inequalities have violations arbitrarily larger for partially entangled states than for maximally entangled states.Source: §7 · p. 25

B. Nonlocality vs Hilbert space dimension

The review connects nonlocality with Hilbert-space dimension and with resources required to simulate quantum correlations. It develops dimension witnesses, studies unbounded Bell violations, and compares classical communication, nonsignaling resources, and local content.

1. Minimal Hilbert space dimension and dimension witnesses

  • Dimension witnesses certify that some correlations require systems larger than a specified dimension, while the minimal dimension needed to reproduce general quantum correlations is often unknown.Qutrits are required for some ternary-outcome Bell inequalities, and evidence suggests that the I3322 maximum may require infinite dimension.Source: §1 · p. 26
  • Prepare-and-measure dimension witnesses cannot generally separate quantum from classical behavior, because sufficiently high-dimensional classical systems can simulate quantum behavior.This limitation does not apply in the same way to witnesses based on nonlocal correlations.Source: §1 · p. 26

2. Grothendieck’s constant and Bell inequalities with unbounded violation

  • Grothendieck-type methods unify Bell-violation measures, while multipartite and bipartite constructions establish families with unbounded violations.For XOR games, arbitrary-dimensional violations are bounded by Grothendieck’s constant, whereas Mermin inequalities yield violations growing exponentially with the number of parties.Source: §2 · pp. 26–27

C. Simulation of quantum correlations

  • The required nonlocal resource can quantify a state’s nonlocality, with classical communication and nonsignaling resources providing alternative simulation frameworks.Simulation protocols seek to reproduce quantum measurement correlations using shared randomness plus communication or resources such as PR boxes.Source: C. Simulation of quantum correlations · p. 27, §1 · pp. 27–28

1. Simulating the singlet state

  • One classical communication bit exactly simulates local projective measurements on a singlet, while a single PR box simulates the same correlations using a weaker nonsignaling resource.The one-bit model requires infinite shared randomness, which is necessary for finite-communication simulations of this type.Source: §1 · p. 28

2. Other quantum states

  • Simulation costs vary with the state: two bits reproduce binary-output correlations for arbitrary bipartite states, while weakly entangled qubits cannot be simulated with one PR box.For partially entangled qubits, two bits suffice including marginals, but whether one bit suffices remains unknown; one PR box is impossible for θ ≤ π/7.8.Source: §2 · p. 28

3. Elitzur-Popescu-Rohrlich decomposition

  • The EPR2 decomposition separates a quantum distribution into local and nonsignaling parts, with the maximal local weight serving as a nonlocality measure.A distribution attaining the nonsignaling maximum has zero local content, while maximally entangled bipartite states are fully nonlocal under this measure.Source: §3 · pp. 28–29

IV. APPLICATIONS OF QUANTUM NONLOCALITY

The review presents quantum nonlocality as a resource for communication, information processing, cryptography, and device-independent randomness generation. These applications rely on Bell violations together with the no-signaling principle, while their security and performance depend on explicit assumptions about devices and adversaries.

A. Communication complexity

  • Nonlocal correlations reduce communication for certain distributed-computing functions without enabling faster-than-light communication.No-signaling prevents direct communication through the correlations, while shared nonlocal systems can lower communication complexity for selected functions.Source: A. Communication complexity · p. 29

B. Information theory

  • Certain no-signaling and quantum correlations enable zero-error transmission through noisy channels even when the channel alone has zero zero-error capacity.Entanglement does not increase asymptotic vanishing-error capacity, but selected correlations enhance zero-error communication.Source: B. Information theory · p. 29

1. Initial developments

  • Ekert’s CHSH-based protocol uses Bell violations to certify shared randomness and bound an eavesdropper’s information, enabling secret-key extraction after error correction and privacy amplification.Realistic imperfections make the violation nonmaximal and key bits imperfectly correlated, so finite leakage must be handled by post-processing.Source: §1 · p. 30
  • Device-independent cryptography removes detailed assumptions about device implementation by certifying security directly from Bell-inequality violations.The devices may be tested without specifying their measurements, and can in principle even be prepared by an eavesdropper.Source: §1 · pp. 30–31

2. Randomness vs non-locality

  • Bell violation constrains an eavesdropper’s guessing probability: at maximal quantum CHSH violation, pguess(A|EX = x) ≤ 0.79, while maximal randomness requires maximal no-signaling violation, S = 4.Source: §2 · p. 32
  • Randomness is not determined solely by the magnitude of nonlocality: maximal local or global randomness can occur arbitrarily close to the local region or with arbitrarily little entanglement.The cited results connect quantitative randomness more closely to extremality of correlations than to nonlocality alone.Source: §2 · p. 32

3. Device-independent randomness generation

  • Device-independent randomness-generation protocols measure devices repeatedly, estimate a Bell parameter, and extract a private uniform string; protocols can achieve quadratic or superpolynomial randomness expansion.The protocols are noise-robust for arbitrary Bell inequalities and have proof-of-principle experimental demonstrations, while some superpolynomial schemes require high CHSH violation and are not noise-tolerant.Source: §3 · p. 33
  • These protocols assume quantum or no-signaling behavior, an independent random seed, non-interacting devices, and no direct leakage, while otherwise leaving device dimensions and measurements unspecified.Relaxations include allowing limited cross-talk between devices, enabling possible use of high-data-rate systems such as Josephson phase qubits on one chip.Source: §3 · p. 33

4. Device-independent quantum key distribution

Device-independent key distribution and broader nonlocality research extend Bell-based certification beyond cryptographic protocols. The review describes security limits, self-testing, general no-signaling properties, and emerging resource-theoretic approaches to comparing and transforming nonlocal correlations.

4. Device-independent quantum key distribution

  • DIQKD protocols measure entangled systems, estimate Bell violation and error rates, then use error correction and privacy amplification to distill a secure key.The security analysis is device-independent apart from assumptions such as quantum or no-signaling behavior, an independent seed, device separation, and no direct leakage.Source: §4 · pp. 34–35
  • Complete DIQKD security has been proven for a constant-noise protocol generating a linear amount of key, but the best achievable noise-rate and key-rate trade-off remains open.Earlier results include polynomially inefficient, noise-intolerant security and analyses restricted to memoryless devices or limited adversary models.Source: §4 · pp. 34–35
  • Repeated use of untrusted devices can expose previously generated keys through device memory, making protocol reuse vulnerable in dishonest-provider scenarios.Countermeasures include refined protocols and encryption schemes designed to prevent devices from leaking information about earlier keys.Source: §4 · p. 35
  • Transmission loss remains a fundamental obstacle to loophole-free DIQKD over distances relevant to quantum key distribution.Heralded amplifiers, quantum relays, repeaters, cavity interactions, and improved post-processing have been proposed, but experimental realization remains challenging.Source: §4 · p. 35

D. Other device-independent protocols

  • Maximal CHSH violation self-tests a two-qubit singlet and anticommuting measurements, allowing device-independent characterization of unknown states and measurement devices.Robust self-testing extends this conclusion to near-maximal violations and can remain valid when devices possess quantum memory.Source: D. Other device-independent protocols · pp. 35–36

A. Properties of no-signaling correlations

  • No-signaling theories share information-theoretic features with quantum theory, including no-cloning, monogamy, measurement complementarity, uncertainty, and intrinsically random outcomes.For PR-box correlations, assuming perfect cloning or compatible measurements would permit signaling, contradicting the no-signaling premise.Source: A. Properties of no-signaling… · pp. 36–37

B. Nonlocality measures, interconversion and distillation

  • PR boxes form a useful bipartite unit because any bipartite no-signaling box can be simulated to arbitrary precision from them, but no analogous multipartite unit is known.Some multipartite no-signaling boxes, including quantum nonlocal correlations, cannot be approximated from arbitrarily many PR boxes.Source: B. Nonlocality measures,… · p. 37
  • Nonlocality can be distilled for some boxes, including arbitrarily weakly nonlocal boxes yielding PR-box correlations, while isotropic correlations resist known complete distillation proofs.These results motivate measures such as distillable nonlocality and broader resource theories based on local operations and wirings.Source: B. Nonlocality measures,… · p. 37

C. Consequences of superstrong non-locality

The review examines how super-quantum correlations differ from quantum ones through information-processing tasks, physical principles, generalized probabilistic theories, and multipartite nonlocality.

1. Information-theoretic consequences

  • Unlimited PR boxes collapse communication complexity: one classical bit lets Alice and Bob compute any binary function.The inner-product construction underlies the result, and even weak nonlocality can produce this collapse after distillation.Source: §1 · p. 38
  • Super-quantum correlations can outperform classical and quantum strategies in nonlocal computation, while quantum correlations offer no advantage for uniformly distributed inputs.With partial input knowledge, however, quantum correlations can outperform classical ones.Source: §1 · p. 38
  • PR-box correlations let Bob recover either of Alice’s two bits perfectly after receiving only one classical bit, violating information causality.Information causality is satisfied classically and quantumly, but not by correlations exceeding Tsirelson’s bound.Source: §1 · pp. 38–39

2. Physical consequences

  • Macroscopic locality admits a classical limit and equals Q1, yet remains strictly larger than the quantum set and can coexist with information-causality violations.Uncertainty relations can also be recast as limits on information retrieval, linking nonlocality with uncertainty across physical theories.Source: §2 · p. 39
  • Bipartite local quantum systems with no signaling yield only quantum correlations, whereas multipartite locally quantum systems can produce correlations stronger than any quantum correlations.These results motivate studying nonlocality within generalized probabilistic theories, which extend classical and quantum theories into broader operational frameworks.Source: §2 · p. 40, D. Nonlocality in generalized… · p. 40

D. Nonlocality in generalized probabilistic theories

  • Generalized probabilistic theories connect stronger nonlocality to local state-space geometry and permit states that encode more information than their Hilbert-space dimension allows.Some such theories are locally almost indistinguishable from quantum mechanics while generating maximally nonlocal correlations.Source: D. Nonlocality in generalized… · p. 40

VI. MULTIPARTITE NONLOCALITY

  • Multipartite nonlocality is richer and less completely understood than bipartite nonlocality, motivating definitions of locality and genuine multipartite nonlocality.Genuine multipartite nonlocality requires all parties to share a common nonlocal resource, rather than merely containing a nonlocal subset.Source: §VI · p. 40, A. Defining multipartite nonlocality · pp. 40–41

1. Genuine multipartite nonlocality `a la Svetlichny

  • Svetlichny’s framework models correlations as mixtures where at most two parties share nonlocal resources, distinguishing 2-way from genuinely 3-way nonlocal correlations.Its operational interpretation allows arbitrary communication between two parties while excluding communication with the third.Source: §1 · p. 41

2. Beyond Svetlichny’s model

The review develops alternatives to Svetlichny’s multipartite definition that exclude signaling bipartite components, then surveys Bell inequalities for detecting genuine multipartite nonlocality.

2. Beyond Svetlichny’s model

  • Svetlichny’s decomposition permits signaling bipartite distributions, creating physically and operationally problematic models of multipartite nonlocality.The issue arises because the bipartite terms are unconstrained apart from normalization.Source: §2 · p. 41
  • No-signaling and time-ordered decompositions remove these shortcomings by constraining the bipartite components used in the multipartite model.Time ordering ensures that whichever party receives a measurement first cannot be signaled to by a later party.Source: §2 · p. 42
  • The three genuine-multipartite definitions are nonequivalent: Svetlichny violation guarantees genuine tripartite nonlocality, but weaker definitions detect additional tripartite correlations.The review therefore treats the choice of definition as consequential for what counts as genuine multipartite nonlocality.Source: §2 · p. 42

B. Detecting genuine multipartite nonlocality

  • Svetlichny’s Bell inequality detects genuine tripartite nonlocality, and its violation also implies genuine tripartite entanglement.The inequality holds for all distributions admitting Svetlichny’s bipartition decomposition.Source: §1 · pp. 42–43

1. Svetlichny’s inequality

  • Svetlichny’s inequality is built from CHSH expressions, with one party’s setting selecting which CHSH game the other two parties effectively play.This construction explains why the bound applies across all bipartitions.Source: §1 · p. 42

2. Generalizations to more parties, measurements and dimensions

  • The inequality has been generalized to arbitrary numbers of parties, measurements, and local dimensions, alongside inequalities for more refined multipartite notions.These extensions preserve the strategy of constructing multipartite tests from lower-party Bell expressions.Source: §2 · p. 43

C. Monogamy

The review relates multipartite nonlocality to monogamy, entanglement, and structured multipartite states, showing both strong constraints and state-specific forms of nonlocal behavior.

C. Monogamy

  • CHSH nonlocality is monogamous: if Alice and Bob violate CHSH, Alice and Charlie cannot, while no-signaling correlations obey |⟨B_AB⟩| + |⟨B_AC⟩| ≤ 4.This monogamy underlies device-independent security arguments and extends beyond CHSH to essentially all bipartite Bell inequalities.Source: C. Monogamy · p. 43

1. Multipartite nonlocality vs multipartite entanglement

  • All pure entangled multipartite states are nonlocal, while genuine multipartite nonlocality requires genuine multipartite entanglement and can certify it device-independently.Whether every pure genuinely multipartite entangled state is genuinely multipartite nonlocal depends on the chosen nonlocality definition and remains unresolved in general.Source: §1 · pp. 43–44

2. Greenberger-Horne-Zeilinger states

  • GHZ correlations can be maximally nonlocal yet fail to be genuinely multipartite nonlocal, although other measurements on GHZ states reach the maximal quantum Svetlichny violation S3 = 4√2 > 4.The violation is understood by Charlie remotely preparing Alice and Bob’s state for the relevant CHSH game.Source: §2 · p. 44

3. Graph states

  • Graph-state stabilizers generate Bell inequalities maximally violated by every graph state, with violations growing exponentially for tree-graph families.For four-qubit cluster states, the associated inequality reaches the algebraic maximum and is not violated by the four-qubit GHZ state.Source: §3 · pp. 44–45
  • Cluster-state relations produce a perfect contradiction between quantum and deterministic local predictions, yielding a strong entanglement witness.The construction generalizes to cluster states with arbitrary numbers of qubits and local dimensions, but the displayed inequality has a noted three-party entanglement loophole unless symmetrized.Source: §3 · p. 45
  • All connected-graph states exhibit fully genuine multipartite nonlocality, although the connection between graph-state nonlocality and computational power is not clearly established.The review distinguishes these stronger genuine-nonlocality results from the broader Bell-nonlocality proofs for graph states.Source: §3 · p. 46

4. Nonlocality of other multipartite quantum states

This section examines the diverse nonlocal behavior of multipartite entangled states and its experimental demonstrations across several physical systems. It also addresses how asymmetric detection efficiencies affect loophole-free Bell tests and the range of efficiencies tolerated under different conditions.

4. Nonlocality of other multipartite quantum states

  • W states reveal robust multipartite nonlocality: measurements by n − 2 parties can leave a Bell pair for CHSH testing, and nonlocality persists after losing k ≪ n particles.For W states, tr_k(|W_n⟩⟨W_n|) ≈ |W_{n−k}⟩⟨W_{n−k}|, giving high persistency of nonlocality; this loss resistance also distinguishes W from GHZ correlations.Source: §4 · p. 46
  • Fully bound entangled multipartite states can violate Bell inequalities, showing that nonlocality does not imply distillable entanglement and refuting the multipartite Peres conjecture.Earlier Mermin-inequality violations involved states becoming distillable when parties grouped, whereas a fully bound entangled example violates a Bell inequality.Source: §4 · p. 46

1. Photons

  • Bell violations have been experimentally demonstrated with photonic GHZ, W, four-photon GHZ, cluster, and genuinely multipartite GHZ states.Photonic experiments also progressed from the first CHSH violation to space-separated and locality-loophole-closed tests, although early photon experiments remained vulnerable to detection or other loopholes.Source: §1 · pp. 47–48

2. Atoms

  • Atomic Bell experiments improved detection efficiency and separation, reaching 20m with an event-ready entanglement-swapping scheme that is robust to photon losses and coincidence-time loopholes.The 20m separation still did not close the locality loophole, for which approximately 300m was required under the stated measurement procedure.Source: §2 · p. 48, Figure 7 · p. 48

3. Hybrid schemes and other systems

  • Nonlocality has also been observed in atom-photon and photon–collective-atomic-excitation systems, as well as superconducting qubits through CHSH violations and GHZ paradoxes.These results extend experimental demonstrations beyond standalone photonic and atomic platforms.Source: §3 · p. 48

1. Detection Loophole

  • Detection inefficiency can let local models fake Bell violations, so no-click outcomes must not be discarded without fair-sampling assumptions or explicitly incorporated into the Bell inequality.For CHSH, a local strategy can reproduce maximal S = 4 correlations when efficiencies are below 2/3, while a maximally entangled two-qubit state requires efficiency above approximately 82.8%; partially entangled states can lower the threshold toward 2/3.Source: §1 · pp. 49–50
  • For atom–photon Bell tests with ηA < 1 and ηB = 1, Alice’s detector efficiency can reach 50% for CHSH and approximately 43% with a three-setting inequality.These results apply when Alice holds the photon and Bob the atom.Source: §1 · p. 51
  • With d measurement settings and d-dimensional systems, detection efficiency as low as 1/d can be tolerated, and this threshold is optimal (V´ertesi et al., 2010).Source: §1 · p. 51
  • If one measurement per party has unit efficiency, CHSH violation remains possible as the other two efficiencies approach zero.This asymmetric arrangement suits Bell tests combining high-efficiency homodyne measurements with low-efficiency photo-detection.Source: §1 · p. 51

2. Locality loophole

The review examines how locality, finite statistics, detection efficiency, and measurement design constrain Bell tests and their interpretation. It surveys experimental strategies and theoretical tools for addressing these limitations, including random settings, statistical methods, high-efficiency detectors, and continuous-variable schemes.

2. Locality loophole

  • Locality loophole tests require spacelike separation and measurement choices independent of the distant setting and hidden variables, but complete closure is impossible in principle.Aspect et al. used rapidly switched analyzers, while later experiments used independent quantum random generators; precise event times and hidden pre-determination remain unresolved concerns.Source: §2 · pp. 51–52

3. Finite statistics

  • Finite-statistics analyses based on standard deviations can lack operational meaning and assume Gaussian sampling and memoryless devices.Probability bounds for local models, martingales, and asymptotically optimal methods address finite data while allowing arbitrary memory effects.Source: §3 · p. 52

C. Towards a loophole-free Bell test

  • Photonic Bell tests face limited detector efficiency: superconducting detectors exceeded 95%, while detection-loophole-free demonstrations reached 75% efficiency per party without closing locality.Because these detectors operate on microsecond timescales, a loophole-free Bell violation would require separation of roughly 300m.Source: C. Towards a loophole-free Bell test · pp. 52–53
  • PrecERTifying a photon before local measurement has been proposed as another route to high detection efficiency, although its experimental implementation is challenging.Source: C. Towards a loophole-free Bell test · p. 53

2. Continuous variable systems.

  • Homodyne measurements can approach unit efficiency, but continuous outcomes must be binned into discrete values for standard Bell inequalities.Existing proposals remain experimentally difficult because some require unrealizable states, others yield very small violations, and post-selection can reopen the detection loophole.Source: §2 · p. 53
  • Hybrid homodyne and photodetection schemes can produce relatively high CHSH violations using experimentally producible states.Other proposals use parity or nonlinear local measurements, but such measurements remain beyond current technology.Source: §2 · p. 53

3. Atom-atom and atom-photon entanglement

The review surveys experimental routes to Bell tests, alignment-free protocols, stronger correlation constraints, alternative nonlocal models, steering, semi-quantum games, and recent developments in Bell nonlocality.

3. Atom-atom and atom-photon entanglement

  • Heralded photon-to-spin mapping can avoid the detection loophole, but closing the locality loophole still requires roughly 300 meters for trapped-atom implementations.Atom-atom schemes reached separations of 20 meters, while efficient photon collection makes direct atom-photon approaches more delicate.Source: §3 · pp. 53–54

D. Bell tests without alignment

  • With three unbiased measurements per party, the probability of violating a Bell inequality becomes one, and experiments show robustness to losses and finite statistics.For comparison, uniformly random measurements violate CHSH with probability ∼28%, rising to ∼42% for unbiased measurements.Source: D. Bell tests without alignment · p. 54

A. Bi-locality and more general correlation scenarios

  • Bilocality requires independent source variables, making it strictly stronger than locality and motivating nonlinear inequalities for entanglement-swapping tests.The bilocal correlation set is generally nonconvex, so linear Bell inequalities do not characterize it completely.Source: A. Bi-locality and more general… · pp. 54–55

B. No-go theorems for nonlocal models

1. Models `a la Leggett

  • Leggett-type models impose more definite individual properties than quantum theory, but their marginal constraints yield inequalities violated by quantum predictions and are not device-independent.More general no-signaling models with more accurate individual predictions are likewise incompatible with quantum correlations.Source: §1 · p. 55

2. Superluminal signaling models

  • Finite-speed signaling models either differ from quantum mechanics beyond a distance bound or become theoretically incompatible with quantum correlations under average no-signaling assumptions.Preferred-frame models also predict disappearing nonlocal correlations in before-before configurations, a prediction falsified experimentally.Source: §2 · p. 56

C. Steering

  • Steering is intermediate between entanglement and Bell nonlocality: entanglement is necessary but insufficient for steering, and steering is necessary but insufficient for nonlocality.Steering tests require trust in Bob’s measurements but treat Alice’s device as a black box; loophole-free steering is less demanding than loophole-free Bell testing.Source: C. Steering · p. 56

D. Semi-quantum games

  • Semi-quantum games use quantum rather than classical inputs and can reveal nonlocal behavior for every entangled state through an appropriate game.They are closely related to entanglement witnesses and support applications to detecting entanglement and steering.Source: D. Semi-quantum games · pp. 56–57

Appendix: Guide to Bell inequalities

The appendix organizes Bell inequalities by parties, measurements, and outcomes, while recording where complete tight classifications are known and where the landscape remains incomplete.

Appendix: Guide to Bell inequalities

  • Bell inequalities are classified by the number of parties n, measurements m, and outcomes ∆, with references supplied rather than explicit inequality formulas.The guide covers bipartite, multipartite, arbitrary-output, and genuine-multipartite-nonlocality scenarios.Source: Appendix: Guide to Bell inequalities · p. 57

a. Binary outputs: (2, m, 2)

  • For binary-output bipartite scenarios, CHSH is the only tight inequality at m=2, while I3322 is an additional tight inequality at m=3.For m=4,5 complete lists are unknown, and for m≥6 substantially less is known.Source: a. Binary outputs: (2, m, 2) · p. 57

b. Arbitrary number of outputs: (2, m, ∆)

a. Binary outputs: (n, m, 2)

  • All tight correlation inequalities are known for m=2 in multipartite binary-output scenarios, while complete non-correlation classifications are available for n=3,m=2.For arbitrary n and m, tight inequalities have also been provided, but the appendix does not claim a complete classification.Source: a. Binary outputs: (n, m, 2) · p. 58

b. Arbitrary number of outputs: (n, m, ∆)

  • Multipartite arbitrary-output scenarios include variance-based, Mermin-type, and functional inequalities, with tightness established only for selected parameter ranges.Mermin generalizations are known to be tight for ∆≤8 in the (3,2,∆) scenario.Source: b. Arbitrary number of outputs: (n, m,… · p. 58

c. Bell inequalities detecting genuine multipartite nonlocality

  • Svetlichny’s inequality initiated tests of genuine multipartite nonlocality, later generalized across party and outcome configurations alongside alternative definitions of the concept.The simplest (3,2,2) scenario has received additional inequalities beyond Svetlichny’s original construction.Source: c. Bell inequalities detecting genuine… · p. 58
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