On the relation between the subadditivity cone and the quantum entropy cone

Abstract

Given a multipartite quantum system, what are the possible ways to impose mutual independence among some subsystems, and the presence of correlations among others, such that there exists a quantum state which satisfies these demands? This question and the related notion of a \textit{pattern of marginal independence} (PMI) were introduced in arXiv:1912.01041, and then argued in arXiv:2204.00075 to be central in the derivation of the holographic entropy cone. Here we continue the general information theoretic analysis of the PMIs allowed by \textit{strong subadditivity} (SSA) initiated in arXiv:1912.01041. We show how the computation of these PMIs simplifies when SSA is replaced by a weaker constraint, dubbed \textit{Klein's condition} (KC), which follows from the necessary condition for the saturation of subadditivity (SA). Formulating KC in the language of partially ordered sets, we show that the set of PMIs compatible with KC forms a lattice, and we investigate several of its structural properties. One of our main results is the identification of a specific lower dimensional face of the SA cone that contains on its boundary all the extreme rays (beyond Bell pairs) that can possibly be realized by quantum states. We verify that for four or more parties, KC is strictly weaker than SSA, but nonetheless the PMIs compatible with SSA can easily be derived from the KC-compatible ones. For the special case of 1-dimensional PMIs, we conjecture that KC and SSA are in fact equivalent. To make the presentation self-contained, we review the key ingredients from lattice theory as needed.

arXiv commentAdded Corollary 1, Theorem 8, Theorem 9 and Conjecture 1. 65 pages, 10 figures, published version

References in corpus (17)

• Gauge Theory Correlators from Non-Critical String Theory
• Holographic Derivation of Entanglement Entropy from AdS/CFT
• A Covariant Holographic Entanglement Entropy Proposal
• Holographic duality from random tensor networks
• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Holographic Mutual Information is Monogamous
• The Holographic Entropy Cone
• The Holographic Entropy Cone for Five Regions
• Holographic entropy relations
• A new inequality for the von Neumann entropy
• The holographic entropy arrangement
• Infinitely many constrained inequalities for the von Neumann entropy
• Stabilizer information inequalities from phase space distributions
• The Quantum Entropy Cone of Hypergraphs
• The holographic entropy cone from marginal independence
• Hypergraph min-cuts from quantum entropies
• All Inequalities for the Relative Entropy

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• Testing holographic entropy inequalities in 2+1 dimensions
• Properties of the contraction map for holographic entanglement entropy inequalities
• A gap between holographic and quantum mechanical extreme rays of the subadditivity cone
• Inner bounding the quantum entropy cone with subadditivity and subsystem coarse grainings
• Algorithmic construction of SSA-compatible extreme rays of the subadditivity cone and the $\mathsf{N}=6$ solution
• Purely GHZ-like entanglement is forbidden in holography
• Exploring the holographic entropy cone via reinforcement learning