Tip of the Quantum Entropy Cone

DOI10.1103/PHYSREVLETT.131.240201arXiv2306.00199MaRDI QIDQ6510368

Matthias Christandl, Lasse Harboe Wolff, Bergfinnur Durhuus

Abstract: 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 $Sigm{a}_N^{*}$ of possible vectors comprising the entropies of the whole and its parts, the famous strong subaddivity inequality constrains its closure $overlineSigm{a}_N^{*}$, which is a convex cone. Further homogeneous constrained inequalities are also known. In this work we provide (non-homogeneous) inequalities that constrain $Sigm{a}_N^{*}$ near the apex (the vector of zero entropies) of $overlineSigm{a}_N^{*}$, in particular showing that $Sigm{a}_N^{*}$ is not a cone for $Ngeq3$. Our inequalities apply to vectors with certain entropy constraints saturated and, in particular, they show that while it is always possible to up-scale an entropy vector to arbitrary integer multiples it is not always possible to down-scale it to arbitrarily small size, thus answering a question posed by A. Winter. Relations of our work to topological materials, entanglement theory, and quantum cryptography are discussed.

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