Tip of the Quantum Entropy Cone.

Relations among von Neumann entropies of different parts of an N-partite quantum system have direct impact on our understanding of diverse situations ranging from spin systems to quantum coding theory and black holes. Best formulated in terms of the set Σ_{N}^{} of possible vectors comprising the entropies of the whole and its parts, the famous strong subaddivity inequality constrains its closure Σ[over ¯]_{N}^{}, which is a convex cone. Further homogeneous constrained inequalities are also known. In this Letter we provide (nonhomogeneous) inequalities that constrain Σ_{N}^{} near the apex (the vector of zero entropies) of Σ[over ¯]_{N}^{}, in particular showing that Σ_{N}^{*} is not a cone for N≥3. Our inequalities apply to vectors with certain entropy constraints saturated and, in particular, they show that while it is always possible to upscale an entropy vector to arbitrary integer multiples it is not always possible to downscale it to arbitrarily small size, thus answering a question posed by Winter.