One-Sided No-Signaling Theories

• One-sided no-signaling theories are a framework that models composite systems where a trusted quantum subsystem coexists with untrusted no-signaling components.
• They distinguish between axiomatic (foundational) and effective (operational) non-signaling, detailing how nonlocal influences occur without enabling controllable, superluminal signals.
• The framework underpins device-independent cryptography and the study of post-quantum correlations through rigorous formulations like HNSQ and one-sided CHSH inequalities.

One-sided no-signaling theories constitute a rigorous framework for modeling composite physical systems in which only a subset of subsystems is governed by quantum mechanics, while others are constrained solely by the no-signaling principle. These frameworks have become central in quantum foundations, device-independent cryptography, and the paper of post-quantum correlations, elucidating the separation between information transfer and operational signal transmission across spacelike separations. One-sided no-signaling theories offer a precise operational middle ground, admitting nonlocal influences or assemblage structures without enabling superluminal communication or message transmission, and their structure and properties shed light on quantum/non-quantum boundary phenomena.

1. Interpretations of the Non-Signaling Theorem

The non-signaling theorem is foundational to reconciling quantum correlations with relativity. Two principal interpretations are distinguished:

Axiomatic (Foundational) Non-Signaling:

This formulation enforces that outcome probabilities of one party are completely independent of remote measurement choices, even conditional on hidden variables $\lambda$, i.e., $p(a|x, y, \lambda) = p(a|x, \lambda)$ and $p(b|x, y, \lambda) = p(b|y, \lambda)$. Consequently, for any input distribution $p(x)$, the marginal $p(b|x)$ remains $x$-independent, fully suppressing any channel capacity: $I(X; B) = 0$, $C = 0$. Any hidden, possibly superluminal, influence is rigorously precluded from carrying operational signals (“no-go” axiom) (Walleczek et al., 2015).

Effective (Operational) Non-Signaling:

This interpretation allows for nonlocal influences in the ontic (hidden variable) model but imposes two communication-theoretic constraints: 1. Non-Transfer-Control (NTC): Local agents cannot modulate remote marginal probabilities by their measurement choices; the mapping $\lambda \to B$ is uncontrollable, so $C_\mathrm{eff} = \max_{p(x)} I(X; B) = 0$ even if $C_{\mathrm{th}}>0$. 2. Non-Signification-Control (NSC): Even if perfectly correlated patterns appear, no code or key is available for agents to assign semantic meaning; thus, there is no effective mutual information $I(M; \hat{M})$ between Alice’s message $M$ and Bob’s decoded output.

Shannon signals—message-encoded, agent-controlled transmissions—are forbidden to traverse spacelike gaps, while non-Shannon signals—raw, uncontrolled detection events—may occur. This dichotomy is essential for distinguishing mere information transfer from actionable communication.

2. Structure and Realization of One-Sided No-Signaling Assemblages

A one-sided no-signaling assemblage arises in scenarios where one subsystem (trusted) admits a fully quantum description and the remainder are untrusted and only guaranteed to obey no-signaling constraints. Formally, given untrusted devices $A_i$ and a trusted quantum system $B$ (e.g., $\mathbb{C}^d$), an assemblage is a family $$\{\sigma_{a_1\cdots a_n\mid x_1\cdots x_n}\} \subset B(\mathbb{C}^d)$$, positive and normalized, satisfying

$$\sum_{a_j,\,j\notin I} \sigma_{a_1\dots a_n|x_1\dots x_n} = \sigma_{a_I|x_I},$$

for all subsets $I$, with $\sigma_B = \sum_{\vec a} \sigma_{\vec a|\vec x}$ independent of $\vec x$ (Banacki et al., 2021). Assemblages are quantum-realizable if

$\sigma_{\vec a|\vec x} = \mathrm{Tr}_{A_1\cdots A_n}[(M^{(1)}_{a_1|x_1}\!\otimes \cdots \otimes M^{(n)}_{a_n|x_n} \otimes \mathds{1}_B) \rho_{A_1\cdots A_n B}],$

where the quantum set is strictly contained within the convex set of all no-signaling assemblages, reflecting the existence of “post-quantum” steering phenomena (Ramanathan et al., 2020).

One-sided frameworks are constructively realized in experiments such as quantum steering: untrusted parties (steering participants) affect the quantum state of a central trusted system via measurement choices, but only the trusted system’s statistics are quantumly characterized.

3. Relaxations, Bell-Type Inequalities, and Indeterminism

One-sided no-signaling models can be rigorously analyzed by relaxing constraints only for selected subsystems. In a bipartite scenario where, e.g., Bob’s side alone admits relaxation of determinism ($I_2$), signaling ($S_{1\to 2}$), and measurement independence ($M_2$), the simultaneous constraints yield the one-sided relaxed CHSH inequality: $$\langle xy\rangle + \langle x y'\rangle + \langle x' y\rangle - \langle x' y'\rangle \leq B(I_2, S_{1\to 2}, M_2)$$ with

$$B(I_2, S_{1\to 2}, M_2)= \begin{cases} 4-(1-I_2)(2-M_2), & S_{1\to 2}<1-2I_2,\, M_2<2, \ 4-(1-S_{1\to 2})(2-M_2), & S_{1\to 2}\geq 1-2I_2,\, M_2<2, \ 4, & \text{otherwise}. \end{cases}$$

Critically, one-sided indeterminism alone (randomness on Bob’s side with $S_{1\to 2}=0$ and $M_2=0$) cannot by itself violate the CHSH bound of $2$, whereas either one-sided signaling ($S_{1\to 2}>0$) or one-sided measurement dependence ($M_2>0$) suffices to simulate any amount of nonlocality (Paul et al., 2013).

This result confirms, in the one-sided regime, the conjectured minimality of resources: random outputs without signaling do not engender Bell violations, but either controllable signaling or input dependence does.

4. Hybrid No-Signaling-Quantum Correlations and the HNSQ Set

Hybrid no-signaling-quantum (HNSQ) correlations formalize one-sided no-signaling theories in multipartite settings, interpolating between standard quantum correlations ($Q$) and the full set of no-signaling boxes ($NS$). In HNSQ$(n_{\mathrm{ns}}+n_q, m, k)$, $n_{\mathrm{ns}}$ untrusted subsystems are only required to be no-signaling, while $n_q$ subsystems are quantum-trusted. Correlations $P(\mathbf{a}, \mathbf{b} | \mathbf{x}, \mathbf{y})$ in HNSQ arise via: $$P(\mathbf{a}, \mathbf{b} | \mathbf{x}, \mathbf{y}) = \mathrm{Tr}\left[ (M^{(1)}_{b_1|y_1} \otimes \cdots \otimes M^{(n_q)}_{b_{n_q}|y_{n_q}}) \sigma_{\mathbf{a}|\mathbf{x}} \right],$$ where $$\sigma_{\mathbf{a}|\mathbf{x}} \in nsA(n_{\mathrm{ns}}, m, k, \prod d_i)$$ is a no-signaling assemblage (Banacki et al., 2021).

Strict inclusions exist: $$\mathrm{LOC} \subsetneq Q \subsetneq \mathrm{HNSQ} \subsetneq NS$$. This hierarchy distinguishes the enhanced structure of HNSQ: while $NS$ includes boxes unattainable even by generalized quantum procedures (e.g., PR boxes), HNSQ admits “super-quantum” boxes under the restriction that only some parties are constrained by quantum mechanics, and the remainder by no-signaling.

Optimization over HNSQ is achieved by an outer semidefinite programming (SDP) hierarchy that combines operator norm bounds with PPT-symmetric extension tests, extending the Doherty–Parrilo–Spedalieri approach.

5. Extremal Points, Self-Testing, and Device-Independent Applications

In the fully no-signaling model, all extremal nonlocal vertices (e.g., PR boxes) are post-quantum—they cannot be realized by quantum mechanics, nor even by arbitrary sequential quantum strategies (Ramanathan et al., 2020). However, with a single trusted qubit, quantum mechanics can realize extremal points of the no-signaling assemblage polytope. For example, using the tripartite GHZ state and suitable projective measurements, one constructs an inflexible, exposed extremal assemblage, saturating a unique linear steering functional.

In HNSQ, certain extremal tripartite points—such as boxes maximally violating self-testing bipartite Bell inequalities for the quantum subset and having fixed deterministic behavior on the no-signaling side—admit honest quantum realizations (Banacki et al., 2021). Self-testing properties in this regime imply that, for specific facet Bell inequalities (e.g., Sliwa 6, 20, 45), the maximal HNSQ and quantum violations coincide, providing device-independent certification even against super-quantum adversaries.

In one-sided device-independent cryptography, particularly key distribution, the realization of self-testable extremal HNSQ (or quantum-assemblage) points ensures that adversaries restricted to HNSQ (i.e., even beyond quantum but still no-signaling on untrusted devices) cannot gain information beyond trivial guessing rates. The Devetak–Winter bound,

$r \geq H_{\min}(A|E) - H(A|B),$

remains valid under HNSQ adversaries, enabling composably secure protocols without full device trust on all subsystems (Banacki et al., 2021, Ramanathan et al., 2020).

6. Operational Criteria and Model-Building Guidelines

To ensure that no superluminal communication can occur even in the presence of ontic (hidden-variable, one-sided nonlocal) influences:

• Agent controllability must be clearly assigned to variables; only those accessible to an experimenter allow operational information transfer.
• Transfer capacity $C_\mathrm{eff}$ (the controlled mutual information $I(X; B)$, maximized over $p(x)$) must be enforced to vanish: $C_\mathrm{eff} = 0$.
• Signification capacity (i.e., the existence of a pair of encoding and decoding maps leading to $I(M; \hat{M})>0$ across the nonlocal link) must also be strictly forbidden.

Any model must ensure that hidden-variable evolution $\lambda \to B$ is either inherently stochastic or decoupled from preparational control, and that any observed pattern is “un-signifiably” rich—foreclosing the possibility of a finite code or shared key enabling message extraction. Thus, models remain consistent with quantum predictions, allow for ontic nonlocal transfers, yet operationally guarantee no superluminal signaling (Walleczek et al., 2015).

7. Implications and Theoretical Significance

One-sided no-signaling theories delineate a regime where physical reality may admit nonlocal influences, yet operational restrictions imposed by agent-level control and semantic assignment preclude the transmission of actionable messages faster than light. These theories illuminate the structure of post-quantum correlations and serve as the basis both for rigorous cryptographic security analyses and for elucidating the boundaries between classical, quantum, and supra-quantum/non-signaling theories. By separately quantifying transfer and signification control and distinguishing Shannon from non-Shannon signals, one-sided no-signaling frameworks unify foundational, operational, and practical perspectives on nonlocality and signal prohibition in quantum theory.

Theory/Set	Trusted Parties	Untrusted Parties	Realizable Extremal Points
Q (Quantum)	All	—	Some (Tsirelson bound)
NS (No-Signaling)	—	All	All (post-quantum, e.g. PR)
HNSQ (Hybrid)	$\ge 1$ (quantum subsystem)	$\ge 1$ (no-signaling)	Some (self-tested in quantum/HNSQ)

A plausible implication is that any increase in the set of trusted (quantum) subsystems, while keeping the remainder merely no-signaling, expands the space of attainable correlations beyond the quantum boundary yet restricts it relative to the unconstrained no-signaling set, with direct ramifications for security, certification, and foundational analysis.