The Failure of Simultaneous Quantum Typicality
Abstract
We establish exact thresholds for the failure of simultaneous quantum typicality: three parties for mixed states and four for pure states. For the four-qubit Higuchi--Sudbery state, every sequence satisfying the expected purity bounds on three overlapping marginals, with sufficiently small fixed positive entropy slack, is asymptotically perfectly distinguishable from the tensor powers of the input: its trace distance from them tends to the maximum value $2$. This maximal separation persists, with a common positive slack, throughout an open neighborhood in the full four-qubit state space. Tracing out one party yields a mixed tripartite example. The proof connects simultaneous quantum smoothing with polynomial concentration on CP1. Symmetry reduces the problem to a frame inequality on Symn(ℂ2), where a linear entropy deficit competes with a quadratic cost of polynomial concentration. Bernstein--Markov estimates and a subharmonic barrier control this cost, yielding an exponential obstruction to simultaneous typicality. These results disprove Dutil's multiparty quantum typicality conjecture, formulated in 2011, and the one-shot simultaneous min-entropy-smoothing conjectures of Drescher--Fawzi and Colomer--Winter. Together with the positive results for smaller systems, they determine the smallest numbers of parties at which simultaneous typicality can fail.
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BibTeX
@misc{collins2026failure,
title = {{The Failure of Simultaneous Quantum Typicality}},
author = {Benoit Collins},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25726},
eprint = {2609.25726},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25726},
}
RIS
TY - UNPB
TI - The Failure of Simultaneous Quantum Typicality
AU - Benoit Collins
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25726
UR - https://arxiv.org/abs/2609.25726
AB - We establish exact thresholds for the failure of simultaneous quantum typicality: three parties for mixed states and four for pure states. For the four-qubit Higuchi--Sudbery state, every sequence satisfying the expected purity bounds on three overlapping marginals, with sufficiently small fixed positive entropy slack, is asymptotically perfectly distinguishable from the tensor powers of the input: its trace distance from them tends to the maximum value $2$. This maximal separation persists, with a common positive slack, throughout an open neighborhood in the full four-qubit state space. Tracing out one party yields a mixed tripartite example. The proof connects simultaneous quantum smoothing with polynomial concentration on CP^1. Symmetry reduces the problem to a frame inequality on Sym^n(ℂ^2), where a linear entropy deficit competes with a quadratic cost of polynomial concentration. Bernstein--Markov estimates and a subharmonic barrier control this cost, yielding an exponential obstruction to simultaneous typicality. These results disprove Dutil's multiparty quantum typicality conjecture, formulated in 2011, and the one-shot simultaneous min-entropy-smoothing conjectures of Drescher--Fawzi and Colomer--Winter. Together with the positive results for smaller systems, they determine the smallest numbers of parties at which simultaneous typicality can fail.
ER -