For nn-party quantum states, every known linear inequality obeyed by the von Neumann entropies of the marginals — strong subadditivity and weak monotonicity — follows from the Lieb–Ruskai theorem. The classical analogue of this situation was overturned by Zhang and Yeung's 1998 non-Shannon inequality, but a quantum counterpart has never been found: it is open whether any linear inequality valid for the entropy vectors of all quantum states is independent of strong subadditivity and weak monotonicity, i.e. whether the closed quantum entropy region Γ‾n\overline{\Gamma}_n is strictly contained in the polyhedral von Neumann cone Ξn\Xi_n. The problem is posed in Christian Majenz's 2018 PhD thesis, which proved a new constrained inequality of this type and asked whether its constraints can be removed; it descends from Linden and Winter's 2005 constrained inequality and their conjectured unconstrained version. A 2025 status paper confirms the question is open.

Background

The von Neumann entropy H(ρ)=−tr(ρlog⁡ρ)H(\rho)=-\mathrm{tr}(\rho\log\rho) of a density matrix is the quantum generalization of Shannon entropy, and for an nn-party quantum state ρ\rho one collects the entropies of all 2n−12^n-1 nonempty marginals into an entropy vector h(ρ)∈R2n−1h(\rho)\in\mathbb{R}^{2^n-1}. The set Γn\Gamma_n of all such vectors has, by a theorem of Pippenger, a topological closure Γ‾n\overline{\Gamma}_n that is a convex cone, called the quantum entropy cone; the closure Σ‾n\overline{\Sigma}_n of the set of entropy vectors of classical (diagonal) states is the classical entropy region, contained in Γ‾n\overline{\Gamma}_n because every classical distribution is a quantum state.

Two families of linear inequalities are known to hold for all quantum states: strong subadditivity, H(AB)+H(BC)≥H(ABC)+H(B)H(AB)+H(BC)\ge H(ABC)+H(B) for all tripartite states (Lieb–Ruskai 1973), and weak monotonicity, H(AB)+H(BC)≥H(A)+H(C)H(AB)+H(BC)\ge H(A)+H(C), obtained from strong subadditivity by purifying the state. The closed polyhedral cone Ξn\Xi_n cut out by all instances of these two families is called the von Neumann cone; its dual cone (the set of linear functionals that are nonnegative on the cone) is the conic hull of the inequality coefficient vectors, and elements of that dual are called von Neumann-type inequalities. Since Γ‾n⊆Ξn\overline{\Gamma}_n\subseteq\Xi_n, every von Neumann-type inequality is valid for all quantum entropy vectors; the question is whether the converse fails — whether the cone of linear inequalities valid for all quantum states is strictly larger than Ξn∘\Xi_n^\circ — which is equivalent to whether Γ‾n⊊Ξn\overline{\Gamma}_n\subsetneq\Xi_n.

The classical precedent makes the question sharp. For Shannon entropy, the Shannon-type inequalities (monotonicity plus subadditivity) were long the only known ones, until Zhang and Yeung exhibited an unconstrained non-Shannon-type inequality for four random variables, proving the classical region strictly smaller than the Shannon cone. Their proof, and all later ones, rest on the copy lemma, i.e. on the ability to duplicate a random variable — a step with no quantum analogue because of the no-cloning theorem (the impossibility of copying an unknown quantum state).

For quantum states only constrained inequalities beyond the von Neumann type are known: inequalities that hold for all states satisfying additional linear entropy constraints. Linden and Winter (2005) proved the first one, I(C:D)≥I(C:AB)I(C:D)\ge I(C:AB), valid for four-party states satisfying the three constraints I(C:A∣B)=I(C:B∣A)=I(A:B∣D)=0I(C:A|B)=I(C:B|A)=I(A:B|D)=0 (strong subadditivity saturated with pivot systems BB, AA and DD), and conjectured an unconstrained robust version with positive constants κ1,κ2,κ3\kappa_1,\kappa_2,\kappa_3:

κ1I(A:C∣B)+κ2I(B:C∣A)+κ3I(A:B∣D)+I(C:D)−I(C:AB)≥0\kappa_1 I(A:C|B)+\kappa_2 I(B:C|A)+\kappa_3 I(A:B|D)+I(C:D)-I(C:AB)\ge 0

for all four-party states, where I(A:B∣C)=H(AC)+H(BC)−H(ABC)−H(C)I(A:B|C)=H(AC)+H(BC)-H(ABC)-H(C) is the conditional mutual information. Cadney, Linden and Winter (2012) then produced infinitely many constrained inequalities. Majenz's 2018 thesis added a further constrained inequality — I(C:AB)≤I(C:D)+I(A:B∣D)I(C:AB)\le I(C:D)+I(A:B|D) for states in which A ⁣− ⁣B ⁣− ⁣CA\!-\!B\!-\!C and B ⁣− ⁣A ⁣− ⁣CB\!-\!A\!-\!C are exact quantum Markov chains (a state ρABC\rho_{ABC} is a quantum Markov chain of this form when I(A:C∣B)=0I(A:C|B)=0) — and proved it independent of all previously known inequalities. Adjacent results delimit the question without answering it: the von Neumann entropy cone of stabilizer states (states fixed by a maximal abelian subgroup of the nn-qubit Pauli group) satisfies the extra Ingleton inequality (a four-party entropy inequality from classical network coding that is not of Shannon type), but only for that restricted class; for Rényi entropies (the one-parameter family Hα(ρ)=11−αlog⁡tr(ρα)H_\alpha(\rho)=\frac{1}{1-\alpha}\log\mathrm{tr}(\rho^\alpha), reducing to the von Neumann entropy as α→1\alpha\to1) with α>1\alpha>1 no nontrivial homogeneous inequalities exist, and for 0<α<10<\alpha<1 non-negativity is the only one; and recent work (2022–2025) has computed extreme rays of the subadditivity cone, built inner bounds on the quantum cone, and released a quantum analogue (qITIP) of the classical ITIP linear-programming prover specifically to aid the search for non-von-Neumann-type inequalities.

Problem Statement

Does there exist an integer n≥4n\ge 4 and a linear inequality on the von Neumann entropies of the 2n−12^n-1 nonempty marginals of an arbitrary nn-party quantum state — valid for all quantum states, with no side constraints on the entropy vector — that is not a conic consequence of strong subadditivity and weak monotonicity? Equivalently: is the closed quantum entropy cone Γ‾n\overline{\Gamma}_n strictly contained in the von Neumann cone Ξn\Xi_n for some nn? (For n≤3n\le 3 the two cones are known to coincide; the question is open from n=4n=4 upward.) A positive answer may be given, in particular, by resolving the Linden–Winter conjecture: exhibiting positive constants κ1,κ2,κ3\kappa_1,\kappa_2,\kappa_3 such that

κ1I(A:C∣B)+κ2I(B:C∣A)+κ3I(A:B∣D)+I(C:D)−I(C:AB)≥0\kappa_1 I(A:C|B)+\kappa_2 I(B:C|A)+\kappa_3 I(A:B|D)+I(C:D)-I(C:AB)\ge 0

holds for every four-party quantum state, or by removing the exact quantum-Markov-chain constraints from Majenz's inequality I(C:AB)≤I(C:D)+I(A:B∣D)I(C:AB)\le I(C:D)+I(A:B|D) while keeping it valid — in either case together with a proof that the resulting inequality is independent of all von Neumann-type inequalities. A negative answer would be a proof that Γ‾n=Ξn\overline{\Gamma}_n=\Xi_n for all nn, i.e. that strong subadditivity and weak monotonicity generate every linear entropy inequality satisfied by quantum states.

References

  1. Christian Majenz, "Entropy in Quantum Information Theory -- Communication and Cryptography", PhD thesis, University of Copenhagen, 2018. DOI 10.48550/arXiv.1810.10436, https://arxiv.org/abs/1810.10436
  2. Noah Linden and Andreas Winter, "A New Inequality for the von Neumann Entropy", Communications in Mathematical Physics 259(1):129–138, 2005. DOI 10.1007/s00220-005-1361-2, https://doi.org/10.1007/s00220-005-1361-2
  3. Josh Cadney, Noah Linden and Andreas Winter, "Infinitely Many Constrained Inequalities for the von Neumann Entropy", IEEE Transactions on Information Theory 58(6):3657–3663, 2012. DOI 10.1109/TIT.2012.2185036, https://arxiv.org/abs/1107.0624
  4. Shao-Lun Huang, Tobias Rippchen and Mario Berta, "Quantum Entropy Prover", 2025. arXiv:2501.16025, https://arxiv.org/abs/2501.16025; published in 2025 IEEE International Symposium on Information Theory (ISIT), DOI 10.1109/ISIT63088.2025.11195418
  5. Matthias Christandl, Bergfinnur Durhuus and Lasse Harboe Wolff, "Tip of the Quantum Entropy Cone", 2023. arXiv:2306.00199, https://arxiv.org/abs/2306.00199; published as Physical Review Letters 131, 240201 (2023), DOI 10.1103/PhysRevLett.131.240201
  6. Temple He, Veronika E. Hubeny and Massimiliano Rota, "Inner bounding the quantum entropy cone with subadditivity and subsystem coarse grainings", 2023. arXiv:2312.04074, https://arxiv.org/abs/2312.04074; published as Physical Review A 109, 052407 (2024), DOI 10.1103/PhysRevA.109.052407
  7. Temple He, Veronika E. Hubeny and Massimiliano Rota, "On the relation between the subadditivity cone and the quantum entropy cone", 2022. arXiv:2211.11858, https://arxiv.org/abs/2211.11858; published as Journal of High Energy Physics 08 (2023) 018, DOI 10.1007/JHEP08(2023)018
  8. Temple He, Veronika E. Hubeny and Massimiliano Rota, "Algorithmic construction of SSA-compatible extreme rays of the subadditivity cone and the N=6{\sf N}=6 solution", 2024. arXiv:2412.15364, https://arxiv.org/abs/2412.15364; published as Journal of High Energy Physics 06 (2025) 055, DOI 10.1007/JHEP06(2025)055
  9. Noah Linden, František Matúš, Mary Beth Ruskai and Andreas Winter, "The Quantum Entropy Cone of Stabiliser States", 2013. arXiv:1302.5453, https://arxiv.org/abs/1302.5453; published in Proceedings of the 8th Conference on Theory of Quantum Computation, Communication and Cryptography (TQC 2013), LIPIcs vol. 22, pp. 270–284, DOI 10.4230/LIPIcs.TQC.2013.270
  10. Noah Linden, Milán Mosonyi and Andreas Winter, "The structure of Renyi entropic inequalities", 2012. arXiv:1212.0248, https://arxiv.org/abs/1212.0248; published as Proceedings of the Royal Society A 469(2158):20120737, 2013, DOI 10.1098/rspa.2012.0737
  11. Ning Bao, Newton Cheng, Sergio Hernández-Cuenca and Vincent P. Su, "The Quantum Entropy Cone of Hypergraphs", 2020. arXiv:2002.05317, https://arxiv.org/abs/2002.05317; published as SciPost Physics 9, 067 (2020), DOI 10.21468/SciPostPhys.9.5.067
  12. Nothando Khumalo, Aman Mehta, William Munizzi and Prineha Narang, "Navigating the Quantum Resource Landscape of Entropy Vector Space Using Machine Learning and Optimization", 2025. arXiv:2511.16724, https://arxiv.org/abs/2511.16724
  13. Zhiwei Song, Lin Chen, Yize Sun and Mengyao Hu, "A complete picture of the four-party linear inequalities in terms of the 0-entropy", 2021. arXiv:2105.07679, https://arxiv.org/abs/2105.07679; published as IEEE Transactions on Information Theory 69(4):2385–2399, 2023, DOI 10.1109/TIT.2022.3227914
  14. Christian Majenz, "Constraints on Multipartite Quantum Entropies", Master's thesis, Albert-Ludwigs-Universität Freiburg, 2014 (posted to arXiv in 2018). arXiv:1810.12845, https://arxiv.org/abs/1810.12845