Additional linear inequalities for the von Neumann entropy with four or more parties

Determine whether the von Neumann entropy for systems with N ≥ 4 subsystems obeys further nontrivial linear inequalities beyond strong subadditivity and weak monotonicity, and characterize any such inequalities.

Background

The authors review standard properties of classical and quantum entropies, noting that strong subadditivity (submodularity) and weak monotonicity are known constraints on the von Neumann entropy.

They highlight that while for N = 3 these are the only linear constraints, it remains unknown whether additional unconstrained linear inequalities exist for N ≥ 4, aside from various constrained inequalities valid only under special conditions.

References

An open question in quantum information theory is whether the von Neumann entropy obeys further constraints, particularly in the form of linear inequalities. For \mathsf{N} = 3, it is known that eq:submod and eq:wmo are the only linear constraints on \mathsf{S}, but the question remains open for \mathsf{N} \geq 4.

eq:submod:

F(X)+F(Y)≥F(X∩Y)+F(X∪Y) .F(X) + F(Y) \geq F(X \cap Y) + F(X \cup Y)\,.

eq:wmo:

SX+SY≥SX∖Y+SY∖X .S{X} + S{Y} \geq S{X \setminus Y} + S{Y \setminus X}\,.

— Combinatorial properties of holographic entropy inequalities  (2601.09987 - Grimaldi et al., 15 Jan 2026) in Subsection 1.1 (Broad overview), paragraph “Classical and quantum entropies”