The general quantum simultaneous smoothing conjecture: a single state realizing all smooth one-shot marginal divergences of a multipartite quantum state

Can a single subnormalised quantum state simultaneously realize several smooth one-shot divergence requirements on different, generally non-commuting marginals of a multipartite state — the quantum analogue of intersecting classical typical sets? Formulated by Drescher and Fawzi in 2013 as one-shot multiparty typicality (simultaneous min-entropy smoothing) and restated in Sen's fully smooth one-shot multipartite covering and decoupling paper (arXiv:2410.17893, 2025), the conjecture is a recognized bottleneck for union/intersection arguments in one-shot network quantum information theory. Telescoping and mean-zero decomposition techniques bypass it for specific covering and decoupling tasks but do not settle it. The min-entropy formulation of the conjecture (Conjecture III.1 of Drescher and Fawzi) was refuted in 2026: arXiv:2609.25726 exhibits multipartite families, built from the four-qubit Higuchi–Sudbery state, for which no single subnormalised state within a dimension-independent distance realizes the marginal min-entropy bounds. That refutation does not settle the divergence form stated in arXiv:2410.17893 (Section 2), which remains open: the known positive results there cover only commuting marginals, two parties, and non-overlapping marginals, and no impossibility result for the divergence form is known.

Background

In one-shot quantum information theory one analyses information-processing tasks executed a single time, and the operative quantities are smoothed entropic functionals. A subnormalised quantum state is a positive semidefinite operator with trace at most 1, and the epsilon-smoothing of an entropy or divergence is obtained by optimizing it over all subnormalised states within trace distance epsilon of the original state; for example, the smooth Renyi-2 divergence $D_2^\epsilon(\alpha\|\beta)$ minimizes the order-2 Renyi divergence $D_2(\alpha'\|\beta)$ over all subnormalised $\alpha'$ with $\|\alpha'-\alpha\|_1 \le \epsilon\,\mathrm{Tr}\alpha$. Smoothing models discarding atypical events of total probability at most epsilon, the exact one-shot analogue of classical typicality. For orders $p>1$ the various smooth Renyi divergences are equivalent, up to dimension-independent corrections polynomial in $\log\epsilon^{-1}$, to the smooth max-divergence $D_\infty^\epsilon$ (the same smoothing applied to the max-relative entropy $D_\infty(\alpha\|\beta)$, the least $\lambda$ with $\alpha \le 2^\lambda\beta$), so the choice of order is a convention (as summarized in arXiv:2410.17893).

Arguments in network information theory — for example simultaneous decoding by several receivers, or confidentiality against several eavesdroppers — require a single distribution to be typical in several ways at once. Classically this is handled by unions and intersections of typical sets, where the intersection of two probability subdistributions is simply $(p\cap q)(x)=\min\{p(x),q(x)\}$, which is close to both $p$ and $q$ and inherits every individual smoothing. For quantum states there is no analogue of this intersection, fundamentally because density matrices on overlapping subsystems need not commute: this is the simultaneous smoothing bottleneck, called 'a famous open problem in network quantum information theory' in arXiv:2410.17893 and originating with Drescher and Fawzi (arXiv:1312.7642), who asked whether the largest eigenvalues of all marginals of a multipartite state can be smoothed simultaneously while staying close to the original state, and proved it only when the marginals commute, when there are just two parties, or when the marginals to optimize are non-overlapping, remarking that new techniques seem needed in general.

The bottleneck is specific to covering-type tasks — soft covering (alias the convex-split lemma, the workhorse for achievability in quantum network information theory) and decoupling. The packing-type analogue was removed by Sen's one-shot quantum joint typicality lemma (arXiv:1806.07278), and the covering-side gap was explicitly pointed out by Ding, Gharibyan, Hayden and Walter (arXiv:1809.10149). Cheng, Gao and Berta (arXiv:2304.12056) and Colomer and Winter (arXiv:2304.12114, published in Communications in Mathematical Physics, 2024) introduced a telescoping cum mean-zero decomposition that bypasses simultaneous smoothing for particular multipartite convex-split and decoupling problems, and Sen (arXiv:2410.17893) simplified and extended it to the first fully smooth one-shot multipartite covering and decoupling theorems, yielding new one-shot inner bounds such as the generalised quantum Slepian–Wolf problem and the polyhedral rate region for the quantum multiple access channel with limited entanglement assistance. In the source paper's running example, the CMG covering problem (named after Chong, Motani and Garg, the discoverers of a well-known inner bound for the classical interference channel; it is the covering ingredient for inner bounds on private classical communication over a quantum wiretap interference channel), a direct fully smooth proof would need a single subnormalised state $\bar\sigma^{X'XY'YE}$ close to the given control state $\sigma^{X'XY'YE}$ that realizes, through its own unsmoothed marginals, eight simultaneous requirements of the form $D_2(\bar\sigma^{S}\|\bar\sigma^{S_1}\otimes\bar\sigma^{S_2}) \le D_2^\epsilon(\sigma^{S}\|\sigma^{S_1}\otimes\sigma^{S_2}) + g(\epsilon^{-1})$ for the relevant marginal triples $S=S_1S_2$ (such as $X'E$, $Y'E$, $X'Y'E$, and their primed variants). The telescoping route instead decomposes one trace distance into a sum of terms and smooths each term separately, thereby sidestepping — but not settling — the conjecture that such a single state exists in general.

Problem Statement

Resolve the general quantum simultaneous smoothing conjecture, in the divergence form stated in arXiv:2410.17893 (Section 2), closely related to the one-shot multiparty typicality conjecture of Drescher and Fawzi (arXiv:1312.7642, Conjecture III.1):

given an arbitrary multipartite quantum state $\sigma$ on finite-dimensional systems (in the source formulation $\sigma^{X'XY'YE}$), an error parameter $\epsilon>0$, and an arbitrary finite family of smooth one-shot divergence requirements on chosen marginals of $\sigma$ — the eight requirements $D_2(\bar\sigma^{S}\|\bar\sigma^{S_1}\otimes\bar\sigma^{S_2}) \le D_2^{\epsilon}(\sigma^{S}\|\sigma^{S_1}\otimes\sigma^{S_2}) + g(\epsilon^{-1})$ of the fully smooth CMG covering lemma being the canonical but non-mandatory instance — does there exist a single subnormalised state $\bar\sigma$ with $\|\bar\sigma-\sigma\|_1 \le f(\epsilon)$, where $f(\epsilon)\to 0$ as $\epsilon\to 0$, $f$ and $g$ are independent of the Hilbert-space dimensions, and $g$ is a 'not too large' function of $\epsilon^{-1}$ — explicitly, one bounded above by a polynomial in $\log\epsilon^{-1}$? Following Conjecture III.1 of arXiv:1312.7642, dependence of the overheads on the number of parties and on the number of requirements in the family is permitted; what must not occur is dependence on the Hilbert-space dimensions.

The original formulation of Drescher and Fawzi (Conjecture III.1 of arXiv:1312.7642) asks the same simultaneous-smoothing question in min-entropy form: for any number of parties $m$ there is a function $g_m$ with $g_m(\epsilon)\to 0$ as $\epsilon\to 0$, which may depend on $m$ but must be independent of the local dimensions, such that for every $m$-party state $\rho$ there is a single subnormalised state within distance $g_m(\epsilon)$ of $\rho$ (in purified distance, equivalently in trace distance) whose min-entropy on every nonempty marginal $S$ is at least the $\epsilon$-smooth min-entropy of $\rho$ on $S$. The two formulations are instances of the same simultaneous smoothing question but are not known to be literally equivalent: the standard conversions between smooth Renyi-$p$ divergences for $p>1$ relate divergence orders, and do not by themselves convert divergence requirements against products of $\bar\sigma$'s own marginals into min-entropy requirements on marginals. The min-entropy formulation (Conjecture III.1) is now known to be false (arXiv:2609.25726), so the open content is the divergence form. A resolution must therefore settle the divergence-form existence question for general non-commuting, overlapping marginals — an arbitrary finite family of smooth one-shot divergence requirements on chosen marginals of $\sigma$, as formulated above, rather than only one fixed instance.

Either of the following constitutes a resolution:

(i) a proof that such a simultaneous smoothing state always exists for arbitrary non-commuting, overlapping marginals — with no commutativity, bounded-party-number, or non-overlap assumption — with overheads $f$ and $g$ independent of the Hilbert-space dimensions and with $g$ bounded above by a polynomial in $\log\epsilon^{-1}$; or

(ii) a refutation: an explicit family of states and requirement families for which every candidate $\bar\sigma$ satisfying all the divergence requirements with an admissible $g$, bounded above by a polynomial in $\log\epsilon^{-1}$ and independent of the Hilbert-space dimensions, either has $\|\bar\sigma-\sigma\|_1$ bounded away from zero by a dimension-independent constant or requires a $g$ outside that admissible class — establishing a genuine separation from the classical case, where the iterated intersection $(p\cap q)(x)=\min\{p(x),q(x)\}$ of the individual optimizers satisfies all requirements simultaneously at a closeness overhead growing only with the number of requirements, which is permitted.

Partial positive results restricted to commuting marginals, two parties, or non-overlapping marginals, and task-specific bypasses such as telescoping cum mean-zero decomposition, do not close this problem.

References

1. P. Sen, Fully smooth one shot multipartite covering and decoupling of quantum states via telescoping, arXiv:2410.17893 (v2, 2025), DOI 10.48550/arXiv.2410.17893, https://arxiv.org/abs/2410.17893
2. L. Drescher and O. Fawzi, On simultaneous min-entropy smoothing, IEEE International Symposium on Information Theory (ISIT) 2013, pp. 161-165, DOI 10.1109/ISIT.2013.6620208, arXiv:1312.7642, DOI 10.48550/arXiv.1312.7642, https://arxiv.org/abs/1312.7642
3. H.-C. Cheng, L. Gao, and M. Berta, Quantum broadcast channel simulation via multipartite convex splitting, arXiv:2304.12056 (2023); published in Communications in Mathematical Physics 406 (2025), DOI 10.1007/s00220-024-05191-4, https://arxiv.org/abs/2304.12056
4. P. Colomer and A. Winter, Decoupling by local random unitaries without simultaneous smoothing, and applications to multi-user quantum information tasks, Communications in Mathematical Physics (2024), DOI 10.1007/s00220-024-05156-7, arXiv:2304.12114, https://arxiv.org/abs/2304.12114
5. P. Sen, Fully smooth one shot multipartite soft covering of quantum states without pairwise independence, arXiv:2410.17908, DOI 10.48550/arXiv.2410.17908, https://arxiv.org/abs/2410.17908
6. P. Sen, Matrix Chernoff concentration bounds for multipartite soft covering and expander walks, arXiv:2504.04067, DOI 10.48550/arXiv.2504.04067, https://arxiv.org/abs/2504.04067
7. P. Sen, Unions, intersections and a one-shot quantum joint typicality lemma, Sadhana 46:57 (2021), DOI 10.1007/s12046-020-01555-3, arXiv:1806.07278, DOI 10.48550/arXiv.1806.07278, https://arxiv.org/abs/1806.07278
8. D. Ding, H. Gharibyan, P. Hayden, and M. Walter, A quantum multiparty packing lemma and the relay channel, IEEE Transactions on Information Theory 66:3500-3519 (2020), DOI 10.1109/TIT.2019.2960500, arXiv:1809.10149, DOI 10.48550/arXiv.1809.10149, https://arxiv.org/abs/1809.10149
9. H.-C. Cheng, L. Gao, C. Hirche, H.-W. Huang, and P.-C. Liu, Sharp estimates of quantum covering problems via a novel trace inequality, arXiv:2507.07961 (2025), DOI 10.48550/arXiv.2507.07961, https://arxiv.org/abs/2507.07961
10. G. Gour, Convex split lemma without inequalities, arXiv:2502.06526 (2025), DOI 10.48550/arXiv.2502.06526, https://arxiv.org/abs/2502.06526
11. G. Gour, Optimal trace inequalities for single-shot quantum information, arXiv:2604.14617 (2026), DOI 10.48550/arXiv.2604.14617, https://arxiv.org/abs/2604.14617
12. B. Regula, L. Lami, and N. Datta, Tight relations and equivalences between smooth relative entropies, arXiv:2501.12447 (2025), DOI 10.48550/arXiv.2501.12447, https://arxiv.org/abs/2501.12447
13. M. Hayashi, H.-C. Cheng, and L. Gao, Resolvability of classical-quantum channels, IEEE Transactions on Information Theory 71:6061-6074 (2025), DOI 10.1109/TIT.2025.3570569, arXiv:2410.16704, https://arxiv.org/abs/2410.16704
14. B. Collins, The failure of simultaneous quantum typicality, arXiv:2609.25726 (2026), DOI 10.48550/arXiv.2609.25726, https://arxiv.org/abs/2609.25726