Tip of the Quantum Entropy Cone

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 $Σ^*_N$ of possible vectors comprising the entropies of the whole and its parts, the famous strong subaddivity inequality constrains its closure $\overlineΣ^*_N$, which is a convex cone. Further homogeneous constrained inequalities are also known. In this work we provide (non-homogeneous) inequalities that constrain $Σ_N^*$ near the apex (the vector of zero entropies) of $\overlineΣ^*_N$, in particular showing that $Σ_N^*$ 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.

Figures (2)

• Figure 1: The conditions of Theorem \ref{['thm:most general constrained inequality']} are here represented with each circle denoting a constituent system $X_i$. The double lines indicate that the mutual information between the two systems is $0$, and it is assumed that the total state is pure.
• Figure 2: The solid figure represents the set of permissible values for $( H(A),H(C),H(ABC) )$ satisfying $III_{XY}=0$ for all $X,Y$, given the the inequalities (\ref{['eq: def ssa']}) and (\ref{['eq: def weak monotonicity']}) and Corollary \ref{['cor: constrained inequality N constarints']}. We have further made the projection $H(A)=H(B)$ to get a $3$-dimensional surface. The dashed lines span a part of $\Sigma_3$ ruled out by Corollary \ref{['cor: constrained inequality N constarints']}, and $O$ denotes the apex of $\Sigma_3$. The ray $\ell$ is the top edge in the figure.

Theorems & Definitions (9)

• Theorem 1
• Lemma 1
• Lemma 2
• Lemma 3
• proof : Proof of Theorem \ref{['thm:most general constrained inequality']}.
• Corollary 1
• proof : Proof of Lemma \ref{['lemma: lower bound V1111']}
• proof : Proof of Lemma \ref{['lemma: lower bound Va111']}
• proof : Proof of Lemma \ref{['lemma:product inequality']}