Equal-weight decomposition of quantum channels with bounded Kraus rank
For integers $m, n \ge 2$ consider quantum channels $\Phi : M_m(\mathbb{C}) \to M_n(\mathbb{C})$, that is, completely positive trace-preserving linear maps, and their Kraus rank, the least number of Kraus operators (equivalently, the rank of the Choi matrix). Ruskai proved that the channels of Kraus rank at most $m$ form the closure of the set of extreme points of the convex set of channels $M_m(\mathbb{C}) \to M_n(\mathbb{C})$. A channel whose Choi matrix has rank $mn$ is not a convex combination of fewer than $n$ channels of Kraus rank at most $m$. The decomposition below exists for $n = 2$ (Ruskai, extending Ruskai, Szarek and Werner's result for $m = n = 2$). Kumar and Wolf proved it for $m = 2$ and every $n$, for every classical-quantum and every quantum-classical channel, and for a set of channels with nonempty interior in every dimension. They also proved that for $m = n = 3$ every channel is a convex combination, with weights that need not be equal, of three channels of Kraus rank at most $3$, and that in general every channel is the average of finitely many channels of Kraus rank at most $m$, with more than $n$ terms allowed.
Problem. For every pair of integers $m, n \ge 2$ and every quantum channel $\Phi : M_m(\mathbb{C}) \to M_n(\mathbb{C})$, are there quantum channels $\Phi_1, \dots, \Phi_n : M_m(\mathbb{C}) \to M_n(\mathbb{C})$, not necessarily distinct, each of Kraus rank at most $m$, such that $$\Phi = \frac{1}{n} \sum_{j=1}^{n} \Phi_j \, ?$$
Source: Ruskai (2007), Conjecture 2, the Audenaert–Ruskai conjecture, based on numerical evidence of Audenaert; Ruskai's printed rank bound is the output dimension, and the form with the input dimension is the strong Ruskai–Audenaert conjecture of Kumar and Wolf (2026), Conjecture 1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MP — Mathematical Physics
- Secondary
- FA, OA
- Keywords
- AIM Open Applied Problems, convex decompositions, Kraus rank, quantum channels
References
@misc{ruskai2007,
author = {Ruskai, Mary Beth},
title = {Some open problems in quantum information theory},
year = {2007},
eprint = {0708.1902},
archivePrefix = {arXiv},
}
@misc{kumar2026,
author = {Kumar, Niranjan and Wolf, Michael M.},
title = {The {R}uskai--{A}udenaert conjecture \& equipartitions of positive operators},
year = {2026},
eprint = {2607.23066},
archivePrefix = {arXiv},
}
@article{ruskai2002,
author = {Ruskai, Mary Beth and Szarek, Stanislaw and Werner, Elisabeth},
title = {An analysis of completely-positive trace-preserving maps on {$M_2$}},
journal = {Linear Algebra Appl.},
volume = {347},
number = {1-3},
year = {2002},
pages = {159--187},
doi = {10.1016/S0024-3795(01)00547-X},
eprint = {quant-ph/0101003},
archivePrefix = {arXiv},
}
@misc{aim-ruskai-audenaert-decomposition,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Equal-weight decomposition of quantum channels with bounded Kraus rank}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/309-ruskai-audenaert-decomposition.md},
note = {Entry 309 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00470, "Equal-weight decomposition of quantum channels with bounded Kraus rank", clopen.solutions.