Post-Quantum Correlations

• Post-quantum correlations are no-signaling behaviors that surpass quantum mechanical limits, exemplified by the canonical PR box achieving the algebraic CHSH maximum.
• They emerge in diverse settings such as Bell nonlocality, contextuality, steering assemblages, and continuous-variable systems, challenging conventional quantum boundaries.
• Research investigates principles like information causality, macroscopic locality, and exclusivity to delineate quantum from post-quantum regimes, influencing network simulations and device emulation.

Post-quantum correlations are operational behaviors that obey no-signaling or its contextual analogue, yet cannot be realized within standard quantum mechanics. In bipartite Bell scenarios they occupy the gap between the quantum set and the full no-signaling polytope, with Popescu–Rohrlich boxes furnishing the canonical example by attaining the algebraic CHSH maximum while remaining no-signaling (Ringbauer et al., 2014). The notion now spans several technically distinct settings: Bell nonlocality, contextuality, almost-quantum models, steering assemblages, network nonlocality, continuous-variable multimode correlations, and operational reconstructions based on quasiprobabilities or post-selected dynamics (Fahmi, 2024, Sainz et al., 2024, Pozas-Kerstjens et al., 2023, Cherian et al., 2018, Oliveira, 6 May 2026).

1. Operational definition and geometric placement

In the standard bipartite $2\text{-}2\text{-}2$ Bell scenario, a behavior is specified by probabilities $p(a,b|x,y)$. Classical correlations admit a local hidden-variable decomposition, quantum correlations admit a Hilbert-space realization with local POVMs and Born’s rule, and no-signaling correlations satisfy marginal constraints independent of the distant input (Ringbauer et al., 2014, Ahanj, 2017). This yields the strict nesting

$$\mathrm{LHV} \subsetneq \mathrm{Quantum} \subsetneq \mathrm{No\text{-}signaling},$$

with post-quantum correlations defined as elements of $$\mathrm{No\text{-}signaling}\setminus\mathrm{Quantum}$$ (Ahanj, 2017).

The canonical quantitative witness is the CHSH expression

$$S = E_{00}+E_{01}+E_{10}-E_{11},\qquad E_{xy}=\sum_{a,b}ab\,p(a,b|x,y),$$

for which local models satisfy $|S|\le 2$, quantum theory satisfies Tsirelson’s bound $|S|\le 2\sqrt2$, and no-signaling allows $|S|=4$ (Oliveira, 6 May 2026, Ringbauer et al., 2014). In the XOR formulation used in information-causality analyses, the same hierarchy appears as $S\le 3$ classically, $S\le 2+\sqrt2$ quantumly, and $p(a,b|x,y)$0 algebraically (Ringbauer et al., 2014).

The same operational pattern recurs outside Bell nonlocality. In contextuality scenarios, one replaces spacelike separation by compatibility or no-disturbance constraints on sequential tests, and post-quantum correlations are those compatible behaviors that exceed the quantum optimum while still respecting exclusivity-type constraints (Amselem et al., 2011). In generalized EPR settings and steering, the relevant objects are assemblages rather than ordinary conditional probabilities; a post-quantum assemblage is no-signaling but has no quantum realization, even when every Bell correlation extractable from it remains quantum in the original scenario (Sainz et al., 2024, Zjawin et al., 2024). In continuous-variable multimode settings, the no-signaling set again strictly exceeds the quantum set, and post-quantum correlations are no-signaling behaviors excluded by the Robertson–Schrödinger uncertainty relation (Cherian et al., 2018).

A common misconception is that “post-quantum” means signaling or relativistically inconsistent. The defining point is the opposite: the correlations remain no-signaling, but are stronger or structurally different than those admitted by quantum realizations (Ringbauer et al., 2014, Shukla et al., 30 Sep 2025).

2. Canonical examples and families

The PR box remains the benchmark example. In one standard parametrization it satisfies

$p(a,b|x,y)$1

with uniformly random marginals, thereby reaching the algebraic CHSH maximum while preserving no-signaling (Ringbauer et al., 2014). Many analyses of principles such as information causality, local orthogonality, and macroscopic locality use PR boxes or their anisotropic deformations as reference extremal points (Ringbauer et al., 2014, Ahanj, 2017).

A more structured post-quantum family is the almost-quantum set $p(a,b|x,y)$2, which satisfies all standard kinematic axioms of quantum correlations except full operator commutativity between different parties. In the formulation based on a permutation postulate, local projectors need commute only on the given state,

$p(a,b|x,y)$3

rather than as operator identities on the whole Hilbert space (Fahmi, 2024). This makes $p(a,b|x,y)$4 strictly larger than the quantum set while remaining strictly inside no-signaling (Fahmi, 2024).

Network scenarios furnish qualitatively different examples. In the minimal triangle scenario with no inputs and binary outputs, symmetric distributions can be parameterized by

$p(a,b|x,y)$5

The point $p(a,b|x,y)$6, $p(a,b|x,y)$7, $p(a,b|x,y)$8 is identified as the triangle analogue of a PR box: it is compatible with no-signaling and independence of the three sources, but excluded by both classical and quantum triangle realizations (Pozas-Kerstjens et al., 2023).

Post-quantum behavior can also arise in less abstract dynamical models. A standard one-dimensional coined discrete-time quantum walk with ordinary Hadamard coin and nearest-neighbor shift yields admissible post-quantum coin-position Bell correlations when the initial coin “state” is replaced by a Hermitian trace-one but nonpositive operator

$p(a,b|x,y)$9

while admissibility is enforced directly at the level of observable statistics (Oliveira, 6 May 2026). For $$\mathrm{LHV} \subsetneq \mathrm{Quantum} \subsetneq \mathrm{No\text{-}signaling},$$0, the construction gives an admissible CHSH value $$\mathrm{LHV} \subsetneq \mathrm{Quantum} \subsetneq \mathrm{No\text{-}signaling},$$1 at $$\mathrm{LHV} \subsetneq \mathrm{Quantum} \subsetneq \mathrm{No\text{-}signaling},$$2, beyond Tsirelson’s bound, with smallest joint probability $$\mathrm{LHV} \subsetneq \mathrm{Quantum} \subsetneq \mathrm{No\text{-}signaling},$$3 (Oliveira, 6 May 2026).

A different class appears in photonic simulations of beyond-quantum nonlocality via non-signaling quantum oracles. There the effective bipartite statistics reproduce the PR-box pattern

$$\mathrm{LHV} \subsetneq \mathrm{Quantum} \subsetneq \mathrm{No\text{-}signaling},$$4

using a four-qubit photonic circuit with restricted input access and intrinsic randomness suppressing operational signaling (Shukla et al., 30 Sep 2025). This suggests that post-quantum behaviors may arise as effective correlations of larger non-signaling devices, even when the underlying implementation is entirely quantum-mechanical.

3. Principles proposed to delimit the quantum set

A major line of research asks which physical principles exclude post-quantum correlations without simply postulating the quantum formalism. Information causality is among the most studied candidates. In the standard random-access-code task, if Alice sends $$\mathrm{LHV} \subsetneq \mathrm{Quantum} \subsetneq \mathrm{No\text{-}signaling},$$5 classical bits, the accessible information at Bob’s side must satisfy

$$\mathrm{LHV} \subsetneq \mathrm{Quantum} \subsetneq \mathrm{No\text{-}signaling},$$6

or equivalently, for the lower bound used experimentally,

$$\mathrm{LHV} \subsetneq \mathrm{Quantum} \subsetneq \mathrm{No\text{-}signaling},$$7

Classical and quantum correlations satisfy the principle, whereas sufficiently strong post-quantum no-signaling boxes violate it (Ringbauer et al., 2014).

Information causality is powerful but incomplete. In isotropic CHSH slices it recovers Tsirelson’s bound asymptotically, yet anisotropic regions of the no-signaling polytope can contain post-quantum correlations that still satisfy the usual IC tests (Ringbauer et al., 2014). This is one reason the search broadened to other principles such as macroscopic locality and local orthogonality (Ahanj, 2017).

Graph-theoretic exclusivity provides another route. For an exclusivity graph $$\mathrm{LHV} \subsetneq \mathrm{Quantum} \subsetneq \mathrm{No\text{-}signaling},$$8, the classical, quantum, and single-copy exclusivity-principle bounds of a linear functional are respectively the independence number $$\mathrm{LHV} \subsetneq \mathrm{Quantum} \subsetneq \mathrm{No\text{-}signaling},$$9, the Lovász number $$\mathrm{No\text{-}signaling}\setminus\mathrm{Quantum}$$0, and the fractional packing number $$\mathrm{No\text{-}signaling}\setminus\mathrm{Quantum}$$1 (Amselem et al., 2011). A 2024 graph-theoretic result establishes that, given the set of correlations for the complementary experiment $$\mathrm{No\text{-}signaling}\setminus\mathrm{Quantum}$$2, the exclusivity principle restricts the maximum set of correlations for the original experiment to the anti-blocking set. Within that framework, if all quantum behaviors are accessible in Nature, the exclusivity principle implies that no post-quantum behaviors can be realized (Nogueira et al., 2024).

The almost-quantum program motivated a different class of principles. One result argues that isotropy and homogeneity of flat space, imposed as invariance of Born-rule probabilities under local rotations and translations, are sufficient and necessary to collapse the almost-quantum model back to standard quantum mechanics in both bipartite and multipartite systems (Fahmi, 2024). In that analysis, the space-symmetry requirement promotes commutation-on-state to full operator commutation, eliminating the almost-quantum excess over $$\mathrm{No\text{-}signaling}\setminus\mathrm{Quantum}$$3 (Fahmi, 2024).

A plausible implication is that the boundary between quantum and post-quantum correlations may depend not only on information-theoretic constraints but also on geometric or kinematic symmetry principles. The literature does not present these principles as equivalent; rather, they probe different structural facets of the quantum set (Ringbauer et al., 2014, Fahmi, 2024, Nogueira et al., 2024).

4. Beyond Bell nonlocality: contextuality, steering, and assemblages

Post-quantum correlations are not confined to Bell nonlocality. In contextuality, one asks whether compatible tests admit a noncontextual hidden-variable decomposition. For a specific 10-vertex exclusivity graph, the noncontextual bound is $$\mathrm{No\text{-}signaling}\setminus\mathrm{Quantum}$$4, the quantum maximum is $$\mathrm{No\text{-}signaling}\setminus\mathrm{Quantum}$$5, and the maximum over all general theories satisfying compatibility and exclusivity is also $$\mathrm{No\text{-}signaling}\setminus\mathrm{Quantum}$$6 (Amselem et al., 2011). This equality,

$$\mathrm{No\text{-}signaling}\setminus\mathrm{Quantum}$$7

means that in that scenario no post-quantum theory obeying the same compatibility and exclusivity constraints can outperform quantum contextuality (Amselem et al., 2011). The experiment reported $$\mathrm{No\text{-}signaling}\setminus\mathrm{Quantum}$$8, implying noncontextual content $$\mathrm{No\text{-}signaling}\setminus\mathrm{Quantum}$$9 (Amselem et al., 2011).

Steering and generalized EPR scenarios introduce a finer-grained layer. In tripartite steering, there exist post-quantum assemblages $$S = E_{00}+E_{01}+E_{10}-E_{11},\qquad E_{xy}=\sum_{a,b}ab\,p(a,b|x,y),$$0 such that every tripartite Bell correlation

$$S = E_{00}+E_{01}+E_{10}-E_{11},\qquad E_{xy}=\sum_{a,b}ab\,p(a,b|x,y),$$1

remains quantum for all measurements on the trusted subsystem (Sainz et al., 2024). This shows that post-quantum steering does not necessarily imply directly observable post-quantum Bell nonlocality in the same geometry (Sainz et al., 2024).

Generalized bipartite EPR scenarios sharpen this further. In Bob-with-input, measurement-device-independent, and channel-EPR settings, there exist non-signaling bipartite assemblages that are post-quantum as resources, yet produce only quantum bipartite Bell correlations in the original scenario (Zjawin et al., 2024). The paper on activation of post-quantumness in these settings constructs larger networks and tailored Bell inequalities whose quantum bound is violated whenever the original assemblage is post-quantum (Zjawin et al., 2024).

These results collectively undermine the idea that post-quantumness is always detectable at the level of ordinary two-party conditional probabilities. In several frameworks it first appears at the level of assemblages or channels, and only becomes Bell-visible after network embedding or an activation protocol (Sainz et al., 2024, Zjawin et al., 2024).

5. Networks, activation, and hidden post-quantumness

Network configurations reveal post-quantum structure inaccessible in standard Bell tests. In the triangle scenario, non-signaling plus source independence define a network analogue of the no-signaling set, and inflation techniques provide outer approximations to the classical and quantum subsets (Pozas-Kerstjens et al., 2023). The symmetric point $$S = E_{00}+E_{01}+E_{10}-E_{11},\qquad E_{xy}=\sum_{a,b}ab\,p(a,b|x,y),$$2 lies beyond both known classical and quantum thresholds, establishing post-quantum nonlocality in the minimal triangle scenario (Pozas-Kerstjens et al., 2023).

Activation results show that hidden post-quantum resources can become Bell-detectable in larger networks. For tripartite post-quantum steering, a four-party network with Alice, Bob, Charlie, and Dani allows one to self-test a reference assemblage on one link, perform an entangling measurement at Charlie, and convert any steering inequality

$$S = E_{00}+E_{01}+E_{10}-E_{11},\qquad E_{xy}=\sum_{a,b}ab\,p(a,b|x,y),$$3

for quantum assemblages into a Bell-type inequality on network probabilities (Sainz et al., 2024). The central theorem states that a tripartite assemblage is post-quantum if and only if it can generate post-quantum correlations in that network (Sainz et al., 2024).

An analogous activation phenomenon holds in bipartite generalized EPR scenarios. There, one supplements the post-quantum assemblage with a self-tested auxiliary quantum assemblage and uses an entangling measurement to map an EPR functional into a Bell functional. For quantum assemblages the resulting Bell value is nonnegative, whereas the target post-quantum assemblage yields a negative value, thus witnessing post-quantumness at the level of observable correlations (Zjawin et al., 2024).

A plausible implication is that “post-quantum steering but Bell-quantum” is not a terminal category but a scenario-dependent one. Once the causal structure is enlarged, the latent post-quantumness can be activated into outright Bell post-quantumness (Sainz et al., 2024, Zjawin et al., 2024).

6. Continuous variables, operational emulation, and accessibility

Continuous-variable multimode scenarios show that post-quantum correlations are not limited to finite alphabets. A family of $$S = E_{00}+E_{01}+E_{10}-E_{11},\qquad E_{xy}=\sum_{a,b}ab\,p(a,b|x,y),$$4-mode Gaussian-mixture behaviors was constructed so that nonlocality is detected by Cavalcanti–Foster–Reid–Drummond inequalities, while post-quantumness is detected by violation of the Robertson–Schrödinger uncertainty relation

$$S = E_{00}+E_{01}+E_{10}-E_{11},\qquad E_{xy}=\sum_{a,b}ab\,p(a,b|x,y),$$5

For the three-mode case, the CFRD inequality reduces to

$$S = E_{00}+E_{01}+E_{10}-E_{11},\qquad E_{xy}=\sum_{a,b}ab\,p(a,b|x,y),$$6

while the product-choice uncertainty condition is

$$S = E_{00}+E_{01}+E_{10}-E_{11},\qquad E_{xy}=\sum_{a,b}ab\,p(a,b|x,y),$$7

The overlap of CFRD-violating and RS-violating regions yields explicit continuous-variable post-quantum nonlocal correlations (Cherian et al., 2018).

Operational emulation is another recurring theme. In the quantum-walk construction, the extended nonpositive coin operator admits a two-component quasiprobability decomposition

$$S = E_{00}+E_{01}+E_{10}-E_{11},\qquad E_{xy}=\sum_{a,b}ab\,p(a,b|x,y),$$8

with $$S = E_{00}+E_{01}+E_{10}-E_{11},\qquad E_{xy}=\sum_{a,b}ab\,p(a,b|x,y),$$9 for $|S|\le 2$0 (Oliveira, 6 May 2026). Any joint probability is then reconstructed as

$|S|\le 2$1

where $|S|\le 2$2 come from actual standard quantum-walk experiments starting from physical coin states (Oliveira, 6 May 2026). The cost is increased sampling overhead, with quasiprobability $|S|\le 2$3-norm $|S|\le 2$4 controlling the variance (Oliveira, 6 May 2026).

A distinct experimental route uses local loss and post-selection to simulate effective no-signaling correlations beyond Tsirelson’s bound in photonic Bell tests. In that setup, polarization-dependent loss and normalization on successful detections produce effective conditional distributions

$|S|\le 2$5

which can exceed the quantum CHSH limit and even approach the PR value as the loss imbalance increases (Ringbauer et al., 2014). The experiment measured $|S|\le 2$6 at $|S|\le 2$7, already above $|S|\le 2$8, and $|S|\le 2$9 at high loss (Ringbauer et al., 2014).

Accessibility can differ sharply from existence. In the quantum-walk case, Schmidt-aligned position observables reveal admissible post-quantum CHSH values, but coarse-grained diagonal-in-position measurements at $|S|\le 2\sqrt2$0 reduce the best admissible value to $|S|\le 2\sqrt2$1, eliminating even ordinary Bell violation (Oliveira, 6 May 2026). This suggests that post-quantum behavior may exist in principle while remaining operationally inaccessible under realistic measurement constraints (Oliveira, 6 May 2026).

7. Open structure and recurring controversies

Several controversies in the field concern whether any single principle can characterize the quantum set. Information causality excludes many post-quantum points but not all, particularly in anisotropic regions (Ringbauer et al., 2014). Macroscopic locality and local orthogonality likewise leave residual post-quantum gaps in Hardy and Cabello nonlocality arguments (Ahanj, 2017). In multipartite settings, Hardy-type examples show that correlations can be post-quantum while satisfying all bipartite information principles and even all GYNI inequalities, reinforcing the claim that any complete principle must be intrinsically multipartite (Das et al., 2012).

Another recurring issue is whether post-quantum models are necessarily exotic at the level of dynamics. Several constructions answer negatively. Standard quantum walks with unmodified unitary nearest-neighbor dynamics, classical or quantum non-signaling oracles implemented as ordinary circuits, and photonic loss-postselection schemes all generate effective post-quantum statistics without modifying the underlying microscopic quantum hardware (Oliveira, 6 May 2026, Shukla et al., 30 Sep 2025, Ringbauer et al., 2014). What changes is the admissible preparation rule, the causal embedding, or the operational reconstruction.

Finally, the relationship between quantum, classical, and post-quantum correlations is subtler than a simple strength ordering. Genuine multipartite quantum correlations can exist without genuine multipartite classical correlations in the sense of vanishing $|S|\le 2\sqrt2$2-body covariances, even for macroscopic odd-$|S|\le 2\sqrt2$3 systems (0705.1969). This suggests that the taxonomy of correlation classes is structurally richer than the Bell hierarchy alone. A plausible implication is that progress on post-quantum correlations will continue to depend on combining geometric, operational, and resource-theoretic viewpoints rather than on any single scalar measure or single Bell inequality (0705.1969, Nogueira et al., 2024).