Are the standard quantum entropy inequalities complete?
For $N \ge 1$, an $N$-party state is a density matrix $\rho$ on $\mathcal{H}_1 \otimes \cdots \otimes \mathcal{H}_N$ with finite-dimensional factors. Its entropy vector is $(S(\rho_I))_{\varnothing \ne I \subseteq [N]}$, where $\rho_I$ is the reduced state of the parties in $I$ and $S(\sigma) = -\operatorname{tr}(\sigma \log_2 \sigma)$. Let $\Gamma_N^Q$ be the closure of the conic hull of the entropy vectors of all $N$-party states, over all local dimensions. Writing $h(\varnothing) = 0$, let $\Sigma_N$ be the cone of vectors $h$ with $h(I) \ge 0$ and, for all pairwise disjoint $I, J, K \subseteq [N]$, $$\begin{aligned} h(I) + h(J) &\ge h(IJ), \\ h(I) + h(IJ) &\ge h(J), \\ h(IJ) + h(JK) &\ge h(J) + h(IJK), \\ h(IJ) + h(JK) &\ge h(I) + h(K), \end{aligned}$$ where juxtaposition denotes union. These are subadditivity, the Araki–Lieb inequality, strong subadditivity (Lieb and Ruskai) and weak monotonicity, so $\Gamma_N^Q \subseteq \Sigma_N$. Pippenger proved that for $N \le 3$ the closure of the set of entropy vectors is exactly the cone cut out by strong subadditivity and weak monotonicity. Linden and Winter found a four-party inequality, valid for all states satisfying certain linear constraints on their entropies, that does not follow from these inequalities, and Cadney, Linden and Winter found infinitely many such constrained inequalities.
Problem. Is $\Gamma_N^Q = \Sigma_N$ for every $N \ge 4$? If not, exhibit a homogeneous linear inequality satisfied by the entropy vectors of all $N$-party states, for some $N$, that is not implied by the inequalities defining $\Sigma_N$.
Source: Pippenger (2003), who proved the equality for at most three parties and raised it for four or more.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- IT — Information Theory
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, entropy cones, entropy inequalities, von Neumann entropy
References
@article{pippenger2003,
author = {Pippenger, Nicholas},
title = {The inequalities of quantum information theory},
journal = {IEEE Trans. Inform. Theory},
volume = {49},
number = {4},
year = {2003},
pages = {773--789},
doi = {10.1109/TIT.2003.809569},
}
@article{lieb1973,
author = {Lieb, Elliott H. and Ruskai, Mary Beth},
title = {Proof of the strong subadditivity of quantum-mechanical entropy},
journal = {J. Math. Phys.},
volume = {14},
number = {12},
year = {1973},
pages = {1938--1941},
doi = {10.1063/1.1666274},
}
@article{linden2005,
author = {Linden, Noah and Winter, Andreas},
title = {A new inequality for the von {N}eumann entropy},
journal = {Comm. Math. Phys.},
volume = {259},
number = {1},
year = {2005},
pages = {129--138},
doi = {10.1007/s00220-005-1361-2},
eprint = {quant-ph/0406162},
archivePrefix = {arXiv},
}
@article{cadney2012,
author = {Cadney, Josh and Linden, Noah and Winter, Andreas},
title = {Infinitely many constrained inequalities for the von {N}eumann entropy},
journal = {IEEE Trans. Inform. Theory},
volume = {58},
number = {6},
year = {2012},
pages = {3657--3663},
doi = {10.1109/TIT.2012.2185036},
eprint = {1107.0624},
archivePrefix = {arXiv},
}
@misc{aim-quantum-entropy-cone,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Are the standard quantum entropy inequalities complete?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/035-quantum-entropy-cone.md},
note = {Entry 035 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-00375, "Are the standard quantum entropy inequalities complete?", clopen.solutions.