Operational No-Signalling Constraints

• Operational No-Signalling Constraints are rigorous restrictions on correlations and transformations that prevent faster-than-light communication while preserving causal consistency.
• They are formulated via conditional probabilities and sheaf-theoretic frameworks, linking quantum behavior to spacetime causal structures and free-choice models.
• They play a key role in quantum information processing, cryptographic protocols, and establishing the boundary between classical and nonlocal quantum correlations.

Operational No-Signalling Constraints are formal, physically motivated restrictions on the possible correlations and transformations in operational theories, ensuring that no combination of local actions and measurements can be used to transmit information outside relativistic causality–in particular, precluding any operational superluminal signalling. Originally justified by analogy with special relativity’s prohibition of faster-than-light communication, the operational no-signalling (ONS) constraints generalize this principle to arbitrary spacetime configurations, multipartite systems, nonlocal boxes, generalized contextuality scenarios, and quantum communication protocols. ONS thus provides the foundational boundary between physically admissible correlations and theories that would permit logical paradoxes, causal loops, or the breakdown of reference-frame invariance (Eckstein et al., 29 Dec 2025).

1. Formal Definition and General Framework

Let $M$ be a spacetime with causal relation $\prec$ (e.g., Minkowski), and consider $n$ agents, each with an input $(X_i,p_i)$ (input value $x_i$ at spacetime point $p_i$) and an output $(A_i,q_i)$ (output value $a_i$ at $q_i$), with $p_i \prec q_i$. The empirical statistics—detector clicks, observed outcomes, etc.—are encoded by conditional probabilities

$P\bigl((a,q)\,|\,(x,p)\bigr)$

for all tuples of inputs $x=(x_1,\dots,x_n)$ and outputs $a=(a_1,\dots,a_n)$ at their respective spacetime points.

Operational separation of outputs $q^{G} = \{q_j\,|\,j\in G\}$ from inputs $p^{F} = \{p_i\,|\,i\in F\}$ requires existence of a gathering point $Q$ such that $q_j\preceq Q$ for all $j\in G$ but $p_i\nprec Q$ for all $i\in F$.

The operational no-signalling constraint states: $$q^G\;\text{op.\ separated from }p^F \implies \sum_{\substack{a_i\,(i\notin G)}} P\bigl((a,q)|(x,p)\bigr) = \sum_{\substack{a_i\,(i\notin G)}} P\bigl((a,q)|(x',p)\bigr)$$ for all $x, x'$ that agree outside $F$—i.e., marginal distributions of outputs in $G$ cannot depend on the freely chosen inputs in $F$ (Eckstein et al., 29 Dec 2025).

These constraints subsume both the traditional Bell-type (spacelike) no-signalling requirements and temporal no-signalling (prohibiting influence outside the forward light-cone of the input), in arbitrary spacetimes and for arbitrary multipartite settings.

2. Sheaf-Theoretic and Algebraic Structure

ONS constraints are elegantly formulated in the sheaf-theoretic language. Consider a measurement scenario with set $X$ of observables, $O$ possible outcomes per measurement, contexts $\mathcal{C}\subset \mathcal{P}(X)$ (maximal jointly measurable sets), and atomic events $(C,s)$ with $s\in O^C$. An empirical model is a family $\{e_C\}_{C\in\mathcal{C}}$ of probability distributions. No-signalling is then encoded as compatibility, or gluing, of these local distributions: $$\sum_{s|_U=u} e_C(s) = \sum_{t|_U=u} e_D(t) \quad\forall\, u \in O^U,\; C, D \in \mathcal{C}$$ for overlaps $U=C\cap D$ (Abramsky et al., 2014).

Operator-theoretic frameworks generalize ONS further: for contextuality scenarios modeled by hypergraphs or operator systems, the set of no-signalling distributions is characterized as the set of states on a suitable maximal tensor product of operator systems or algebras. Marginal consistency (the core of ONS) becomes a linear relation among these operator-valued assignments (Anoussis et al., 18 May 2024).

3. Equivalence with Free Choice (Lambda-Independence)

A profound result is the equivalence between ONS and "free choice" of measurements, formalized as $\Lambda$-independence. An empirical model is no-signalling if and only if it admits a decomposition

$$e_C(o_C) = \sum_{\lambda\in O^X} \mu(\lambda)\;\delta_{\lambda|_C,\,o_C},$$

where $\mu$ is a signed probability measure ($\sum_{\lambda}\mu(\lambda)=1$, possibly $\mu(\lambda)<0$). Here, the choice of context is stochastically independent from the hidden variable $\lambda$. Any positive measure would force locality; allowing signed (possibly negative) measures captures all no-signalling models, including nonlocal and even super-quantum boxes (Abramsky et al., 2014, Abramsky et al., 2014).

This equivalence establishes that ONS is not just a relativistic causality constraint but a global statistical consistency condition underpinning all operational (non-signalling) empirical models, with free choice and no-signalling two faces of the same principle.

4. Operational and Foundational Implications

a) Information Processing and Communication

Operational no-signalling constraints are central to quantum information processing:

• They bound the set of correlations attainable in Bell experiments (defining polytopes that underlie all Bell inequalities and their facets) (Cope et al., 2018).
• They guarantee that nonlocal, yet no-signalling, devices (PR boxes, nonlocal boxes, etc.) cannot be used for superluminal communication or for "broadcasting" nonlocal resources via local operations (Joshi et al., 2011).
• In quantum channel theory, ONS determines the structure of quantum no-signalling correlations (bipartite CPTP maps with linear constraints), which in turn control assisted zero-error capacities and classical simulation costs via semidefinite programming (Duan et al., 2014).

b) Physical Realizability and Causal Structure

ONS is necessary and sufficient for compatibility with the demands of relativistic causality (no operational causal loops or backward-in-time signalling) in Minkowski spacetime, under the assumption of Poincaré invariance (Eckstein et al., 29 Dec 2025, Horodecki et al., 2016). Violation of ONS entails either logical paradox (causal loop) or operational breakdown of relativity.

Notably, ONS is strictly stronger than necessary in some spacetime configurations, especially for multipartite experiments. There exist configurations where only a subset of the traditional no-signalling constraints is required to preclude causal loops (Horodecki et al., 2016), and recent causal modelling frameworks rigorously delineate sufficiency and necessity of ONS for various interventions (Vilasini et al., 2023).

c) No-Signalling and Epistemic Agency

ONS is, fundamentally, an operational or epistemic constraint: it describes what agents can or cannot achieve in communication protocols, rather than imposing an ontic prohibition on all causal influence (as critiqued in the context of deterministic hidden-variable theories) (Walleczek et al., 2014, Walleczek et al., 2015). Operational non-signalling ensures that only agent-controlled interventions matter for compatibility with special relativity; instantaneous hidden influences that cannot be controlled or measured do not violate ONS (Walleczek et al., 2014, Walleczek et al., 2015, Sen et al., 2020).

5. Extensions and Strengthenings

Recent research enriches ONS in several directions:

• Contextual no-disturbance and no-disturbance to dilated systems: Extending ONS to encompass all measurement contexts and requirement of compatibility upon dilation (e.g., inclusion of ancilla systems), one obtains necessary and sufficient conditions for quantum realizability of joint probabilities: ONS plus time-orientation (arrow-of-time) constraints select precisely the valid quantum states (Frembs et al., 2022, Frembs et al., 2019).
• Generalized Probabilistic Theories: In GPTs, ONS is recovered as secrecy constraints between commuting agent algebras and transformation monoids. Commutation of local sub-algebras or dynamical maps is the primitive from which ONS and non-signalling follow (Kraemer et al., 2017).
• Dynamical ONS in Spacetime: The dynamical version of ONS demands that local measurements/operations cannot affect detection probabilities outside the forward light-cone. This imposes stringent constraints even on quantum dynamics (precluding naive Schrödinger evolution for single particles) and sets the bar for any consistent post-quantum theory (Eckstein et al., 2019).
• Jamming and Partial No-Signalling: ONS admits "jamming" phenomena where one party can affect joint (but not marginal) correlations of other spacelike parties without enabling signalling, provided the spacetime arrangement admits no operational separation for the relevant marginals (Eckstein et al., 29 Dec 2025, Vilasini et al., 2023). This generalizes the traditional notion of monogamy and undercuts the sufficiency of standard no-signalling constraints in complex networks.

6. Negative Probabilities and Simulation

ONS is precisely the set of empirical models that admit representations as signed hidden-variable models. Every ONS correlation can be simulated by sampling over a signed probability measure (possibly negative weights) on deterministic strategies, followed by empirical post-processing; this operationalizes the long-standing concept of negative probabilities and aligns all ONS correlations with classical simulation protocols involving signed events (Abramsky et al., 2014).

The existence of such representations demonstrates, in particular, why ONS is strictly broader than quantum correlations—the latter require positivity and further physicality constraints (e.g., complete positivity, complete non-negativity) that ONS alone does not impose.

7. Significance, Limitations, and Research Directions

Operational No-Signalling Constraints unify the study of nonlocality, contextuality, and the landscape of admissible correlations in spacetime, providing a precise and generalizable boundary between physically allowable and paradox-inducing theories (Eckstein et al., 29 Dec 2025, Abramsky et al., 2014, Anoussis et al., 18 May 2024). They:

• Rigorously encode the physical principle prohibiting usable superluminal communication, independent of particular quantum or classical realizations.
• Underlie all device-independent protocols, multipartite network constraints, and cryptographic security proofs in spacetime-dependent scenarios (Gisin et al., 2019).
• Highlight the necessity of careful distinction between operational and ontic notions of causality, particularly in the interpretation of Bell-type experiments and generalizations of quantum theory (Walleczek et al., 2014, Sen et al., 2020).

Limitations and ongoing research include precise characterization of ONS in nontrivial spacetimes (e.g., black hole horizons, where gathering points may not be defined), the sufficiency of ONS in highly causally connected networks, the connection to higher-order affects and interventions, and the boundary with quantum realizability within the broader ONS polytope (Eckstein et al., 29 Dec 2025, Vilasini et al., 2023, Frembs et al., 2022).

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References

• "No-Signalling Is Equivalent To Free Choice of Measurements" (Abramsky et al., 2014)
• "The operational no-signalling constraints and their implications" (Eckstein et al., 29 Dec 2025)
• "An Operational Interpretation of Negative Probabilities and No-Signalling Models" (Abramsky et al., 2014)
• "Operator systems, contextuality and non-locality" (Anoussis et al., 18 May 2024)
• "The Non-Signalling theorem in generalizations of Bell's theorem" (Walleczek et al., 2014)
• "Non-broadcasting of non-signalling boxes via operations which transform local boxes into local ones" (Joshi et al., 2011)
• "Operational locality in global theories" (Kraemer et al., 2017)
• "From no-signalling to quantum states" (Frembs et al., 2022)
• "No-signalling, Contextuality, and the Arrow of Time" (Frembs et al., 2019)
• "Impossible measurements revisited" (Borsten et al., 2019)
• "Superdeterministic hidden-variables models I: nonequilibrium and signalling" (Sen et al., 2020)
• "The standard no-signalling constraints in Bell scenarios are neither sufficient nor necessary for preventing superluminal signalling with general interventions" (Vilasini et al., 2023)
• "Relativistic Causality vs. No-Signaling as the limiting paradigm for correlations in physical theories" (Horodecki et al., 2016)
• "Bell Inequalities From No-Signalling Distributions" (Cope et al., 2018)
• "No-Signalling Assisted Zero-Error Capacity of Quantum Channels and an Information Theoretic Interpretation of the Lovasz Number" (Duan et al., 2014)
• "Operational causality in spacetime" (Eckstein et al., 2019)