---
title: One-Sided No-Signaling Theories
url: https://www.emergentmind.com/topics/one-sided-no-signaling-theories
type: topic
---

# One-Sided No-Signaling Theories

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 study 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) [1501.07177].

**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 $\vec x$ [2107.06878]. 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 [2004.14782].

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 [1304.7409].

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 [2107.06878].

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 [2004.14782]. 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 [2107.06878]. 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 [2107.06878, 2004.14782].

## 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 [1501.07177].

## 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.

Source: https://www.emergentmind.com/topics/one-sided-no-signaling-theories