Finite classical-communication simulation problem for quantum communication and Bell nonlocality

For each integer $d\ge 2$ , determine whether there exists a classical protocol using a finite amount of classical communication and unlimited shared randomness that exactly reproduces both (i) the statistics of arbitrary $d$ -dimensional quantum communication and (ii) all correlations obtainable from measurements on two entangled $d$ -dimensional quantum systems. When such protocols exist, determine the required communication cost; when they do not, prove that no finite classical communication suffices.

References

Primary source

arXiv

Additional references

Quantum communication and Bell nonlocality require infinite classical communication to simulate — arXiv

Posted 1 month ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

A September preprint claims finite classical simulations exist through qutrits but are impossible from four dimensions onward, yet nobody has independently verified the result.

Known results

Toner and Bacon (2003) gave a one-bit protocol for the singlet correlation, a restricted two-qubit case.
Regev and Toner (2009) gave a two-bit simulation for two-outcome measurements on shared bipartite $d$ -dimensional states.
Earlier work reported lower bounds of $5$ messages for qutrit communication and $10$ for ququart communication, with finite upper bounds unknown.

September 3, 2026 claimed classification

Carlos de Gois, Thyago S. R. Santos, and Carlos Vieira claim that $2$ classical bits exactly simulate qubit communication and all entangled-qubit correlations, that qutrits admit an explicit $357$ -bit protocol, and that no finite protocol exists for either task when $d\ge 4$ . The arXiv preprint is unrefereed and has no independent verification.

Community submission (unverified)

A submitted proof argues for the same qutrit/ququart transition, reducing impossibility to a fidelity kernel on complex projective space and a zero-slope property at zero fidelity; it presents the $357$ -bit figure as an upper bound, not an optimum. Its analytic ingredients are explicitly left source-dependent.

Current status (as of September 2026): the claimed $2$ -bit qubit result and $357$ -bit qutrit protocol are reported, while impossibility for $d\ge 4$ and the full classification remain unverified.

Sources

ProofFinite Classical-Communication Simulation of Quantum Communication and Bell Nonlocality DR. ARIE-ARIADNE DEWATSON A proof-oriented reconstruction and audit of the dimensional classification claimed in arXiv:2609.04182v1See full solutionHide full solution

We study exact simulation of finite-dimensional quantum communication and bipartite quantum correlations by classical protocols with unlimited shared randomness and a finite classical communication alphabet. The classification claimed in arXiv:2609.04182v1 exhibits a sharp transition between qutrits and ququarts. Qubit communication and qubit Bell correlations admit exact finite simulations; an explicit qutrit construction uses fewer than 2^357 messages, hence at most 357 bits in a fixed-length encoding; whereas no finite exact classical simulation is claimed for any dimension d >= 4. The impossibility argument reduces the problem to the fidelity kernel on complex projective space, uses fixed-fidelity averaging and a projective-harmonic spectral decomposition, and derives a zero-slope property at zero fidelity. The latter contradicts the quantum kernel K(t)=t. We also explain the reduction to Bell correlations.

This manuscript is a source-faithful reconstruction of the supplied audit and the cited preprint. In particular, the 357-bit figure is an explicit upper bound, not an optimality claim. The higher-order level-set step and the cited auxiliary analytic/geometric results should be regarded as source-dependent ingredients unless independently verified from their original references.

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