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This theory paper shows that "almost-quantum" correlations, a set that lies beyond quantum mechanics in the NPA hierarchy, can beat quantum and commuting-operator models in a zero-error communication task, while quantum models stay pinned to the shared-randomness value of 1/N. Its results are mathematical and foundational, with no hardware or experimental implications, but they sharpen which "post-quantum" principles separate real quantum theory from its nearest relatives. For quantum information readers, the main takeaways are the closed-form semidefinite programming solutions and the bound on one-way zero-error cost for equality testing.
An Unbounded Separation of Almost-Quantum and Commuting-Operator Correlations in Zero-Error Communication
arXiv:2610.41448v1 Announce Type: new Abstract: Almost-quantum correlations form the $\mathtt{1+AB}$ level of the Navascu\'es--Pironio--Ac\'in hierarchy. They strictly contain the quantum set, are characterized by a single semidefinite program, arise in Sorkin's quantum measure theory, and satisfy several principles proposed to single out quantum theory. We show that they are nevertheless unboundedly stronger than quantum correlations in a zero-error communication task. Alice and Bob receive labels $s,t\in[N]$, $N\ge3$; after one bit from Alice, Bob must decide whether $s=t$, never erring but allowed to abstain. For every model of quantum correlations, from finite-dimensional tensor products to commuting operators, the optimal worst-case probability of a conclusive answer is exactly $1/N$, the value attained with shared randomness, so the bound does not depend on the resolution of Tsirelson's problem. An explicit family of almost-quantum correlations attains $1/19$ for every $N$. For certificates of equality from binary-outcome boxes, we solve the underlying semidefinite programs in closed form. The almost-quantum optimum equals $N^2/(17N^2-80N+64)$ for $N\ge8$ and tends to $1/17$. The first level of the hierarchy, which coincides with the set singled out by macroscopic locality, attains $N/[8(N-2)]$, which tends to $1/8$; its optimal boxes lie outside the almost-quantum set for every $N\ge5$. For longer messages, entanglement saves at most one bit over shared randomness, whereas almost-quantum correlations reduce the one-way zero-error cost of deciding equality of $n$-bit strings from $n$ bits to at most $13$. The quantum bound also yields a linear witness of post-quantumness on a face of the nonsignalling polytope.
Abstract:Almost-quantum correlations form the $\mathtt{1+AB}$ level of the Navascués--Pironio--Acín hierarchy. They strictly contain the quantum set, are characterized by a single semidefinite program, arise in Sorkin's quantum measure theory, and satisfy several principles proposed to single out quantum theory. We show that they are nevertheless unboundedly stronger than quantum correlations in a zero-error communication task. Alice and Bob receive labels $s,t\in[N]$, $N\ge3$; after one bit from Alice, Bob must decide whether $s=t$, never erring but allowed to abstain. For every model of quantum correlations, from finite-dimensional tensor products to commuting operators, the optimal worst-case probability of a conclusive answer is exactly $1/N$, the value attained with shared randomness, so the bound does not depend on the resolution of Tsirelson's problem. An explicit family of almost-quantum correlations attains $1/19$ for every $N$. For certificates of equality from binary-outcome boxes, we solve the underlying semidefinite programs in closed form. The almost-quantum optimum equals $N^2/(17N^2-80N+64)$ for $N\ge8$ and tends to $1/17$. The first level of the hierarchy, which coincides with the set singled out by macroscopic locality, attains $N/[8(N-2)]$, which tends to $1/8$; its optimal boxes lie outside the almost-quantum set for every $N\ge5$. For longer messages, entanglement saves at most one bit over shared randomness, whereas almost-quantum correlations reduce the one-way zero-error cost of deciding equality of $n$-bit strings from $n$ bits to at most $13$. The quantum bound also yields a linear witness of post-quantumness on a face of the nonsignalling polytope.
arXiv:2610.41448 [quant-ph]
(or arXiv:2610.41448v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2610.41448 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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