Which linear information inequalities ('laws of information theory') are obeyed by the von Neumann entropies of subsystems of multipartite quantum states is a foundational open problem of quantum information theory, open since Pippenger's formulation. Gross and Walter (J. Math. Phys. 54, 082201 (2013)) proved, via Wigner-function classical models, that stabilizer states satisfy every balanced classical information inequality in von Neumann entropy, and that the Rényi-2 entropies of Gaussian states satisfy every balanced inequality valid for multivariate normal distributions; they explicitly left open (i) whether any homogeneous linear inequality beyond weak monotonicity and strong subadditivity holds for all quantum states, (ii) whether the von Neumann entropy of Gaussian states inherits the balanced inequalities that its Rényi-2 entropy provably satisfies, and (iii) whether entropy-faithful state-to-probability maps exist beyond stabilizer and Gaussian states. The forward literature (2013-2026) leaves all three cores open: no unconditional quantum entropy inequality beyond strong subadditivity and weak monotonicity is known (only constrained and non-homogeneous near-apex inequalities), the Gaussian von Neumann question has no dedicated follow-up, and the classical-model construction is obstructed but unresolved.

Background

For a density operator ρ\rho, the von Neumann entropy is S(ρ)=−tr(ρlog⁡ρ)S(\rho) = -\mathrm{tr}(\rho\log\rho), the quantum analogue of Shannon entropy. Given an nn-partite state ρ\rho on a tensor-product Hilbert space, each subsystem I⊆{1,…,n}I \subseteq \{1,\dots,n\} has a reduced state ρI\rho_I, and the entropy vector (S(ρI))I(S(\rho_I))_{I} collects the entropies of the whole and all its parts. The set of realizable entropy vectors has a convex-cone closure, called the quantum entropy cone; a linear information inequality is a homogeneous inequality ∑IνIS(ρI)≥0\sum_I \nu_I S(\rho_I) \ge 0 valid for all states, i.e., a description of the cone's facets. Classically, Shannon entropies obey strong subadditivity (SSA) H(XIK)+H(XJK)−H(XK)−H(XIJK)≥0H(X_{IK}) + H(X_{JK}) - H(X_K) - H(X_{IJK}) \ge 0 and monotonicity, and Chan showed that the balanced information inequalities together with monotonicity completely determine the classical entropy cone (an inequality is balanced, also called correlative, if ∑I∋iνI=0\sum_{I \ni i} \nu_I = 0 for every ii, which holds iff it is tight on product states); Zhang and Yeung moreover exhibited a non-Shannon-type balanced inequality, so the classical entropy region is not polyhedral and has infinitely many facets (Matúš). Quantumly, von Neumann entropy violates monotonicity (entanglement), satisfies weak monotonicity S(ρIK)+S(ρJK)−S(ρI)−S(ρJ)≥0S(\rho_{IK}) + S(\rho_{JK}) - S(\rho_I) - S(\rho_J) \ge 0, and famously still satisfies SSA (Lieb and Ruskai). Pippenger's question - whether von Neumann entropy obeys any further 'laws' - remains unanswered: the only known unconditional inequalities are weak monotonicity, SSA, and their conic combinations, while Linden and Winter and Cadney, Linden and Winter constructed infinitely many constrained inequalities, valid only under linear constraints on the entropy vector.

Gross and Walter attacked the question through states with classical phase-space models. A stabilizer state is a joint eigenstate of a maximal Abelian subgroup of the generalized Pauli (Weyl) operators, the computational backbone of quantum error correction; for odd local dimension its discrete Wigner function - a phase-space quasiprobability distribution - is a genuine probability distribution, and the marginal for subsystem II is exactly the Wigner function of ρI\rho_I. For stabilizer states the reduced states have flat spectra, so all Rényi entropies (where the Rényi-2 entropy is S2(ρ)=−log⁡trρ2S_2(\rho) = -\log\mathrm{tr}\rho^2) equal the von Neumann entropy, giving S(ρI)=H(XI)−∣I∣S(\rho_I) = H(X_I) - |I| for the phase-space random variables: every balanced classical inequality, and even Ingleton-type linear rank inequalities (linear four-party entropy inequalities such as Ingleton's, which hold for entropy vectors of Abelian groups and of random variables linearly determined by common ones, but fail for general random variables; Chan and Yeung's group-theoretic correspondence), transfers to stabilizer states. Independently, Linden, Matúš, Ruskai and Winter reached the same conclusions by group-theoretic constructions and showed the four-party stabilizer cone is exactly SSA + weak monotonicity + Ingleton. A multi-mode Gaussian state is the continuous-variable analogue, a state with positive Gaussian Wigner function encoded by a covariance matrix Σ\Sigma satisfying the uncertainty relation Σ+iΩ≥0\Sigma + i\Omega \ge 0, where Ω\Omega is the symplectic form (the fixed nondegenerate antisymmetric matrix encoding the canonical commutation relations); its subsystem entropies are spectral functions of the symplectic eigenvalues of the principal submatrices ΣI\Sigma_I (the symplectic eigenvalues are the moduli of the eigenvalues of iΩIΣIi\Omega_I \Sigma_I). The Wigner function of a Gaussian state is a multivariate normal distribution, and Gross and Walter showed S2(ρI)S_2(\rho_I) equals a differential Rényi-α\alpha entropy of the corresponding phase-space variables minus a per-mode constant (the α→1\alpha \to 1 case is a formula long used in Gaussian quantum information and exploited by Adesso, Girolami and Serafini), so Rényi-2 entropies of Gaussian states satisfy every balanced inequality valid for multivariate normal distributions (a classical region itself only partially mapped, by Shadbakht and Hassibi), and Gaussian states violate Ingleton exactly as multivariate normals do.

Problem Statement

Determine which linear information inequalities - the 'laws of information theory' - survive quantization in von Neumann entropy, at the two levels of generality posed by Gross and Walter (2013), together with the classical-model mechanism that underlies every known positive result. The common objective is a characterization of the linear constraints on von Neumann entropy vectors, with three linked components: (1) General quantum states: (1a) does there exist a homogeneous linear inequality ∑IνIS(ρI)≥0\sum_I \nu_I S(\rho_I) \ge 0 for von Neumann entropies of nn-partite states, for some nn, that holds for all quantum states of arbitrary finite local dimension and is not a conic consequence of weak monotonicity and strong subadditivity; and (1b) in full generality, characterize the closure of the set of von Neumann entropy vectors (the quantum entropy cone), including deciding whether it is polyhedral. (2) Gaussian states: do the von Neumann entropies of nn-mode Gaussian states satisfy every balanced information inequality that is valid for multivariate normal distributions - the exact statement that Gross and Walter proved for Rényi-2 entropies and explicitly left open for the von Neumann entropy? (3) Classical models: does there exist a map assigning to each quantum state ρ\rho (in some class strictly larger than stabilizer states, or for Gaussian states with the von Neumann entropy in place of Rényi-2) a probability distribution, such that for every subsystem II the Shannon entropy of the corresponding marginal equals S(ρI)S(\rho_I) up to a state-independent additive constant cIc_I? The three components serve one objective: component (2) is a concrete restriction of component (1) to Gaussian states and balanced inequalities, where a positive answer is plausible via phase-space classical models; component (3) asks for the mechanism that would explain such an answer, since every known inequality transfer (stabilizer and Gaussian Rényi-2) arises from such a map; accordingly, the map sought in (3) is a uniform classical-model rule defined across the whole state class, not a state-by-state patchwork of distributions assembled after the fact. An answer to any component - a new unconditional inequality with proof, an explicit Gaussian covariance-matrix counterexample to a balanced inequality valid for multivariate normals, a transfer theorem, or a construction/no-go for entropy-faithful classical models - constitutes progress on the stated objective; the completion boundary is a settled answer to (1a) - a new unconditional inequality with proof (for some admissible number of parties, valid for all local dimensions) or a proof that no such inequality exists - or a settled answer to (2), in each case at the stated generality and not a numerical study or a result confined to a fixed small nn; the fuller characterization (1b) and the classical-model construction (3) are further objectives that advance the work without lying on this completion boundary.

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