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Dutil’s multiparty quantum typicality conjecture and related smoothing conjectures

Let ρA1⋯Am\rho^{A_1\cdots A_m} be an mm-party quantum state and let ε>0\varepsilon>0. Writing ρK\rho^K for the marginal on every nonempty subset K⊆{1,…,m}K\subseteq\{1,\ldots,m\}, the conjecture asserts that there exists a quantum state σA1⋯Am\sigma^{A_1\cdots A_m} such that P(ρ,σ)≤εP(\rho,\sigma)\leq\varepsilon and, simultaneously for every nonempty K⊆{1,…,m}K\subseteq\{1,\ldots,m\}, Hmin⁡(σK)≥Hmin⁡ε(ρK)H_{\min}(\sigma^K)\geq H_{\min}^{\varepsilon}(\rho^K), where PP is purified distance and Hmin⁡εH_{\min}^{\varepsilon} is the ε\varepsilon-smooth min-entropy.

References

Primary source

arXiv

Additional references

functional-analysisquantum-information
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Progress summary

Refreshed
Claimed solved

A September 2026 preprint claims to disprove the multiparty conjectures with explicit quantum counterexamples, but no independent confirmation is supplied.

The conjecture asks whether one nearby state can simultaneously realize the smooth min-entropy improvements of every nonempty marginal of an mm-party quantum state. Earlier work established important two-party and restricted cases, while leaving the general quantum problem open.

Known results

  • Two-party quantum case: proved with purified-distance bound P(ρ,σ)≤∣K∣2εP(\rho,\sigma)\leq |\mathcal K|\sqrt{2\varepsilon}.
  • Commuting marginals and non-overlapping marginal families: proved.
  • Classical case: proved with trace-distance bound (2m−1)ε(2^{m}-1)\varepsilon.
  • A separate 2012 paper claims the two-party typicality case and an arbitrary-kk-norm extension.

September 2026 claimed counterexample

Benoit Collins’s preprint The Failure of Simultaneous Quantum Typicality claims explicit counterexamples at the smallest failing party numbers, based on the four-qubit Higuchi–Sudbery state, with maximal trace-distance separation under the conjectured marginal-purity conditions. If correct, it simultaneously settles the related smoothing conjectures negatively; the supplied evidence does not independently verify the claim.

Current status (as of September 2026): The general multiparty conjecture is claimed disproved by Collins’s preprint, while the two-party and restricted cases are established and the new counterexamples remain unverified.

Sources

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