2x2 Non-Signaling Correlations in CHSH

• Two-input–two-output non-signaling correlations are bipartite probability distributions with binary inputs and outputs that obey positivity, normalization, and stringent no-signaling constraints.
• They exhibit a convex polytope structure with 24 extremal boxes, including 16 local deterministic and 8 PR-type vertices, which underpin the CHSH bounds across local, quantum, and no-signaling models.
• Additional structural insights from resource wirings and quantum-logical reconstructions further delineate the transformative limitations and operational distinctions between classical, quantum, and supraquantum correlations.

Two-input–two-output non-signaling correlations are bipartite conditional probability distributions $P(a,b\mid x,y)$ with binary inputs $x,y\in\{0,1\}$ and binary outputs $a,b\in\{0,1\}$, or equivalently $a,b\in\{\pm1\}$ under the standard sign map. They are constrained by positivity, normalization, and the requirement that each party’s marginal be independent of the distant input. In the CHSH scenario, they provide the canonical comparison between local hidden-variable, quantum, and general no-signaling models, with the standard inclusion $\mathcal{L}\subset\mathcal{Q}\subset\mathcal{NS}$ and the corresponding bounds $S\le2$, $S\le2\sqrt{2}$, and $S\le4$ for the CHSH functional (Uzan, 2016).

1. Formal scenario and parameterizations

The basic object is the conditional distribution

$P(a,b\mid x,y),\qquad x,y,a,b\in\{0,1\},$

subject to

$$P(a,b\mid x,y)\ge 0,\qquad \sum_{a,b}P(a,b\mid x,y)=1$$

for every input pair. The no-signaling conditions are

$x,y\in\{0,1\}$0

and

$x,y\in\{0,1\}$1

These equalities ensure that Alice’s marginal does not depend on Bob’s input and Bob’s marginal does not depend on Alice’s input (Uzan, 2016).

A standard reparameterization uses $x,y\in\{0,1\}$2-valued outputs. With $x,y\in\{0,1\}$3 and $x,y\in\{0,1\}$4, the correlators are

$x,y\in\{0,1\}$5

Together with the local biases

$x,y\in\{0,1\}$6

they reconstruct the distribution as

$x,y\in\{0,1\}$7

In the binary-input, binary-output case, normalization and no-signaling reduce the full space to an 8-dimensional affine set (Sikora et al., 2015).

2. Geometry, extremal boxes, and the CHSH landscape

In the $x,y\in\{0,1\}$8 scenario, the no-signaling set is a convex polytope. Its extremal structure consists of $x,y\in\{0,1\}$9 vertices: $a,b\in\{0,1\}$0 local deterministic boxes and $a,b\in\{0,1\}$1 nonlocal PR-type vertices. The local deterministic vertices are of the form

$a,b\in\{0,1\}$2

while the PR family is

$a,b\in\{0,1\}$3

with $a,b\in\{0,1\}$4 (Sikora et al., 2015).

The canonical PR box is defined by

$a,b\in\{0,1\}$5

It has uniform marginals and satisfies

$a,b\in\{0,1\}$6

so that the CHSH parameter

$a,b\in\{0,1\}$7

reaches the algebraic maximum $a,b\in\{0,1\}$8 (Shukla et al., 30 Sep 2025).

The geometric relations among the standard sets are

$a,b\in\{0,1\}$9

Here $a,b\in\{\pm1\}$0 is the local polytope, $a,b\in\{\pm1\}$1 is the no-signaling polytope, and $a,b\in\{\pm1\}$2 is convex but not a polytope. In CHSH terms, local models satisfy $a,b\in\{\pm1\}$3, quantum models satisfy Tsirelson’s bound $a,b\in\{\pm1\}$4, and no-signaling boxes permit $a,b\in\{\pm1\}$5 (Shukla et al., 30 Sep 2025).

3. Marginals, nontrivial no-signaling constraints, and quantum-logical reconstruction

A recurring structural point is that product correlators alone do not characterize no-signaling. In the four-dimensional space with coordinates $a,b\in\{\pm1\}$6, the no-signaling region is simply the unit cube, with facet inequalities

$a,b\in\{\pm1\}$7

Accordingly, nontrivial no-signaling inequalities require marginals as well as products. One canonical example is

$a,b\in\{\pm1\}$8

and the same work derives $a,b\in\{\pm1\}$9 CHSH-like inequalities and $\mathcal{L}\subset\mathcal{Q}\subset\mathcal{NS}$0 six-term inequalities involving correlators and single-party expectations (Seevinck, 2011).

The logical reconstruction of box-world events leads to a different, explicitly nonclassical perspective. For the two-box world, the experimentally meaningful propositions generate a logic with $\mathcal{L}\subset\mathcal{Q}\subset\mathcal{NS}$1 elements. This logic is an atomistic orthomodular poset, but not a lattice; states on it are in one-to-one correspondence with all non-signaling boxes; and it is set-representable in the sense of quantum logics (Tylec et al., 2013). The same reconstruction emphasizes several contrasts with Hilbert-space quantum theory: the logic is finite, measurements must be destructive unless an additional state-update rule is postulated, and Heisenberg-type uncertainty relations fail because the logic is concrete (Tylec et al., 2013).

These results separate two levels of description. At the probabilistic level, every non-signaling box is allowed. At the level of event structure, however, the theory already departs from both classical probability simplices and Hilbert-lattice quantum logic, which is why the $\mathcal{L}\subset\mathcal{Q}\subset\mathcal{NS}$2 box world is not a straightforward generalization of the two-qubit system (Tylec et al., 2013).

4. Quantum realizability, incompatibility, and structural restrictions

One influential line of work argues that physical no-signaling is stronger than the purely statistical condition

$\mathcal{L}\subset\mathcal{Q}\subset\mathcal{NS}$3

On that view, local alternatives such as $\mathcal{L}\subset\mathcal{Q}\subset\mathcal{NS}$4 versus $\mathcal{L}\subset\mathcal{Q}\subset\mathcal{NS}$5 and $\mathcal{L}\subset\mathcal{Q}\subset\mathcal{NS}$6 versus $\mathcal{L}\subset\mathcal{Q}\subset\mathcal{NS}$7 are incompatible, whereas any $\mathcal{L}\subset\mathcal{Q}\subset\mathcal{NS}$8 is jointly measurable with any $\mathcal{L}\subset\mathcal{Q}\subset\mathcal{NS}$9. The experimentally meaningful propositions therefore form a partial Boolean algebra whose quantum-logical representation is non-commutative, with Hermitian dichotomic observables obeying

$S\le2$0

With the CHSH operator

$S\le2$1

the identity

$S\le2$2

implies $S\le2$3, hence $S\le2$4 for every state $S\le2$5. Under this corrected interpretation of physical no-signaling, PR-box correlations are excluded as physically realizable (Uzan, 2016).

Within standard quantum theory, measurement incompatibility does not have a uniform operational effect across all measurement classes. In the two-input, two-output Bell scenario, any pair of incompatible dichotomic projective measurements at both parties is sufficient to demonstrate Bell nonlocality. By contrast, there exists a class of incompatible qubit POVMs,

$S\le2$6

with the corresponding Bob observables given in rotated $S\le2$7-$S\le2$8 directions, such that for $S\le2$9 the measurements are incompatible but never produce CHSH violation for any shared state; by Fine’s theorem the resulting behaviors are local (Ghosh et al., 2023).

A complementary restriction is obtained by enlarging a no-signaling theory with one global Bell measurement. When local subsystems admit three dichotomic measurements $S\le2\sqrt{2}$0, adding a Bell-basis measurement and imposing the associated parity equalities forces the local state space to be exactly the qubit Bloch ball. Combined with the theorem that local quantum measurements plus no-signaling imply quantum correlations, this rules out supraquantum correlations and recovers precisely the two-qubit quantum set; in any CHSH slice, the Tsirelson bound follows (Czekaj et al., 2018).

5. Resource ordering, wirings, and optimization frameworks

Non-signaling boxes form not only a geometric set but also a resource theory under wirings. A wiring is any local protocol in which Alice and Bob use finitely many boxes, possibly in adaptive order, choose later inputs from local transcripts, and finally output a new effective box. For this resource theory, maximal correlation

$S\le2\sqrt{2}$1

is monotone under wirings, and so are the hypercontractivity ribbon and maximal-correlation ribbon (Beigi et al., 2014).

This immediately constrains transformations among isotropic boxes

$S\le2\sqrt{2}$2

for which

$S\le2\sqrt{2}$3

Without shared randomness, $S\le2\sqrt{2}$4 cannot be generated from finitely many copies of $S\le2\sqrt{2}$5 whenever $S\le2\sqrt{2}$6. With shared randomness, impossibility is proved for $S\le2\sqrt{2}$7. The same monotonicity yields a continuum of wiring-closed sets

$S\le2\sqrt{2}$8

inside the no-signaling landscape (Beigi et al., 2014).

Optimization over classical, quantum, and no-signaling correlations also admits a unified conic formulation. In the two-party $S\le2\sqrt{2}$9 case,

$S\le4$0

This turns the value of a nonlocal game into a linear conic program; the no-signaling value is an LP, while outer approximations to the quantum value are SDPs. For CHSH, the Feige–Lovász doubly nonnegative relaxation coincides with the quantum value, and the resulting bounds satisfy

$S\le4$1

for any $S\le4$2 game (Sikora et al., 2015).

6. Operational simulations, information-theoretic coordinates, and ongoing tensions

Recent work has introduced an explicitly operational notion of beyond-quantum simulation. A four-qubit oracle with subsystems $S\le4$3 for Alice and $S\le4$4 for Bob uses a Bell pair on $S\le4$5, a Toffoli gate controlled by the unprimed input qubits, and restricted access to the oracle so that the primed qubits are initialized as $S\le4$6 and the unprimed qubits are confined to computational-basis states. Under this restricted access, the oracle is analytically non-signaling at the level of observed $S\le4$7, reproduces exact PR correlations in the computational basis for an entanglement-free classical-oracle construction, and in experiment achieves $S\le4$8, with $S\le4$9 in the diagonal basis and exact $P(a,b\mid x,y),\qquad x,y,a,b\in\{0,1\},$0 in selected bases at the formal level (Shukla et al., 30 Sep 2025).

A different organizing principle maps each $P(a,b\mid x,y),\qquad x,y,a,b\in\{0,1\},$1 behavior to the pair $P(a,b\mid x,y),\qquad x,y,a,b\in\{0,1\},$2, where $P(a,b\mid x,y),\qquad x,y,a,b\in\{0,1\},$3 is the maximal CHSH value over relabelings and $P(a,b\mid x,y),\qquad x,y,a,b\in\{0,1\},$4 is the mutual information for uniform inputs. In this two-dimensional representation, local, quantum, and post-quantum regions become visibly distinct, the isotropic family obeys $P(a,b\mid x,y),\qquad x,y,a,b\in\{0,1\},$5, and the Tsirelson point $P(a,b\mid x,y),\qquad x,y,a,b\in\{0,1\},$6 appears as a singular point on the no-signaling lower boundary through a change of concavity, without invoking quantum mechanics in the derivation (Perito et al., 2020).

The causal structure assumed for the box is also decisive. If one input bit is constrained by Deutsch’s closed-time-like-curve consistency condition, the PR relation $P(a,b\mid x,y),\qquad x,y,a,b\in\{0,1\},$7 together with $P(a,b\mid x,y),\qquad x,y,a,b\in\{0,1\},$8 gives the deterministic box

$P(a,b\mid x,y),\qquad x,y,a,b\in\{0,1\},$9

and Alice’s marginal then depends on Bob’s input, so non-signaling is violated. The same mechanism produces signaling in tripartite Svetlichny and Mermin boxes under analogous input-output identifications (Chakrabarty et al., 2011).

Taken together, these results partition the subject into several non-equivalent notions: abstract no-signaling polytopes, operator-realizable quantum correlations, resource transformations under wirings, and restricted-access simulations that reproduce PR-type observed statistics. This suggests that the central disputes in the literature concern not the formal definition of the $$P(a,b\mid x,y)\ge 0,\qquad \sum_{a,b}P(a,b\mid x,y)=1$$0 no-signaling set, but which additional structural, causal, or operational assumptions count as part of a physically realizable correlation model (Uzan, 2016).