Unexpected consequences of Post-Quantum theories in the graph-theoretical approach to correlations

1. Problem Statement

The paper addresses the problem of quantum correlations: identifying the fundamental physical principles that distinguish quantum mechanics (QT) from more general "post-quantum" theories. While the Non-Signaling (NS) principle allows for correlations stronger than quantum mechanics (e.g., Popescu-Rohrlich boxes), it fails to single out the quantum set.

The authors focus on the Exclusivity Principle (EP), which posits that for any set of pairwise exclusive measurement events, the sum of their probabilities must be $\leq 1$. The specific challenge addressed is whether the EP, when applied to complementary experiments (a concept introduced by Yan), is sufficient to rule out all post-quantum behaviors for any experimental scenario, not just specific self-complementary graphs. Previous work (Cabello, 2019) proved this for general graphs but relied on complex constructions involving infinite copies of experiments. This paper aims to provide a simpler, more direct proof using graph-theoretical tools.

2. Methodology

The authors utilize the Cabello-Severini-Winter (CSW) framework, which maps contextuality scenarios to graph theory:

• Exclusivity Graph ($G$): Vertices represent measurement events; edges connect exclusive events.
• Behavior Sets:
  • $NC(G)$: Classical (non-contextual) behaviors (convex hull of deterministic assignments).
  • $Q(G)$: Quantum behaviors (realizable via projectors on a Hilbert space).
  • $E_1(G)$: Behaviors satisfying the single-copy Exclusivity Principle.
• Yan's Construction: The authors analyze a composite system formed by an experiment $G$ and its complementary experiment $\bar{G}$ (where the exclusivity graph is the complement $\bar{G}$).
  • They consider composite events $f_{ij} = e_i \wedge e'_j$.
  • The EP applied to the set of diagonal events $\{f_{ii}\}$ yields the inequality: $\sum_i p(e_i)p(e'_i) \leq 1$, assuming independence between $G$ and $\bar{G}$.
• Anti-Blocking Theory: The core mathematical tool is the concept of anti-blocking sets. For a set of vectors $A$, the anti-blocking set $abl(A)$ is defined as $\{b \geq 0 \mid \sum a_i b_i \leq 1, \forall a \in A\}$.

3. Key Contributions and Results

A. Proposition 1: The Anti-Blocking Connection

The authors establish a fundamental technical result: If the set of correlations for a complementary experiment $\bar{G}$ is $X(\bar{G})$, then the maximum set of correlations allowed by the EP for the original experiment $G$ is exactly the anti-blocking set of $X(\bar{G})$.
$$Y(G) = abl(X(\bar{G}))$$
This generalizes previous findings, showing that the EP acts as a geometric constraint defined by the anti-blocking operation relative to the complementary experiment's behavior set.

B. Corollary 1: Duality of Classical and EP Sets

Using the self-duality property of anti-blocking sets for convex corners ($abl(abl(D)) = D$), the authors prove:

• If the complementary experiment allows all $E_1$ behaviors, the EP restricts the original experiment to Classical ($NC$) behaviors.
• Conversely, if the complementary experiment is restricted to Classical ($NC$) behaviors, the EP allows all $E_1$ behaviors for the original experiment.

C. Remark 2: Recovery of ATC's Result

The authors show that if the complementary experiment is restricted to Quantum behaviors ($Q(\bar{G})$), the EP singles out the Quantum set ($Q(G)$) for the original experiment. This recovers a result by Amaral, Terra, and Cabello (ATC, 2014) but frames it within a theory-independent anti-blocking formalism.

D. Main Result: The "Unexpected Consequence"

The central contribution is a proof by contradiction regarding post-quantum theories:

1. Assumption: Assume the EP holds universally and that Nature is capable of realizing all quantum behaviors (the "No-Restriction Hypothesis").
2. Hypothesis: Suppose there exists a post-quantum behavior $w'$ (where $w' \in E_1(\bar{G}) \setminus Q(\bar{G})$) accessible in a complementary experiment $\bar{G}$.
3. Consequence: By Proposition 1, the EP constraint for the original experiment $G$ becomes $Y(G) = abl(U(\bar{G}))$, where $U(\bar{G})$ contains $w'$.
4. Derivation: Because $w'$ is post-quantum, there exists a genuinely quantum behavior $p \in Q(G)$ such that $\sum p_i w'_i > 1$. Consequently, this valid quantum behavior $p$ is excluded from the set $Y(G)$.
5. Conclusion: If post-quantum correlations exist in any complementary experiment, the EP forces the exclusion of some genuinely quantum correlations in the original experiment.

Since experimental evidence strongly suggests that all quantum behaviors are realizable in Nature, the authors conclude that post-quantum correlations must be forbidden to preserve the consistency of quantum mechanics under the EP.

4. Significance

• Simplification of Proof: The paper resolves the "quantum correlations problem" for arbitrary graphs using Yan's construction and anti-blocking theory, avoiding the complex, infinite-copy constructions required by Cabello (2019).
• Generalization: It generalizes the result that the EP forbids post-quantum sets from specific self-complementary graphs to any exclusivity graph.
• Fundamental Limitation: It provides a strong argument that the Exclusivity Principle is a fundamental physical principle that naturally limits Nature to quantum correlations, provided that all quantum behaviors are physically realizable.
• Self-Duality Characterization: The work highlights that Quantum Theory is uniquely characterized (within the CSW framework) by the self-duality property $Q(\bar{G}) = abl(Q(G))$. Any theory satisfying this property cannot strictly contain or be strictly contained by Quantum Theory without violating the EP or the realizability of quantum states.

In summary, the paper demonstrates that the Exclusivity Principle, when applied through the lens of complementary experiments and anti-blocking sets, acts as a "filter" that eliminates post-quantum theories to preserve the integrity of the quantum set.