Tip of the Quantum Entropy Cone

• URL: http://arxiv.org/abs/2306.00199v2
• Date: Tue, 2 Jan 2024 15:59:56 GMT
• Title: Tip of the Quantum Entropy Cone
• Authors: Matthias Christandl, Bergfinnur Durhuus, Lasse Harboe Wolff
• Abstract summary: Relations among von Neumann entropies of different parts of an $N$-partite quantum system have direct impact on our understanding of diverse situations. We 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.
• Score: 1.1606619391009658
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $\Sigma^*_N$ of possible vectors comprising the entropies of the whole and its parts, the famous strong subaddivity inequality constrains its closure $\overline\Sigma^*_N$, which is a convex cone. Further homogeneous constrained inequalities are also known. In this work we provide (non-homogeneous) inequalities that constrain $\Sigma_N^*$ near the apex (the vector of zero entropies) of $\overline\Sigma^*_N$, in particular showing that $\Sigma_N^*$ is not a cone for $N\geq 3$. 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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