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Journal of High Energy Physics

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

Temple He, Veronika E. Hubeny, Massimiliano Rota

Published Aug 4, 2023• 12 citations
Publisher

Abstract

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 pattern of marginal independence (PMI) were introduced in [1], and then argued in [2] to be central in the derivation of the holographic entropy cone. Here we continue the general information theoretic analysis of the PMIs allowed by strong subadditivity (SSA) initiated in [1]. We show how the computation of these PMIs simplifies when SSA is replaced by a weaker constraint, dubbed 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.

DOI
10.1007/jhep08(2023)018

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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 pattern of marginal independence (PMI) were introduced in [1], and then argued in [2] to be central in the derivation of the holographic entropy cone. Here we continue the general information theoretic analysis of the PMIs allowed by strong subadditivity (SSA) initiated in [1]. We show how the computation of these PMIs simplifies when SSA is replaced by a weaker constraint, dubbed 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.
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.
A bstract We continue the study of the quantum marginal independence problem, namely the question of which faces of the subadditivity cone are achievable by quantum states. We introduce a new representation of the patterns of marginal independence (PMIs, corresponding to faces of the subadditivity cone) based on certain correlation hypergraphs, and demonstrate that this representation provides a more efficient description of a PMI, and consequently of the set of PMIs which are compatible with strong subadditivity. We then show that these correlation hypergraphs generalize to arbitrary quantum systems the well known relation between positivity of mutual information and connectivity of entanglement wedges in holography, and further use this representation to derive new results about the combinatorial structure of collections of simultaneously decorrelated subsystems specifying SSA-compatible PMIs. In the context of holography, we apply these techniques to derive a necessary condition for the realizability of entropy vectors by simple tree graph models, which were conjectured in [1] to provide the building blocks of the holographic entropy cone. Since this necessary condition is formulated in terms of chordality of a certain graph, it can be tested efficiently.
We show via explicit construction that for six or more parties, there exist extreme rays of the subadditivity cone that can be realized by quantum states, but not by holographic states. This is a counterexample to a conjecture first formulated in Hernández-Cuenca [The holographic entropy cone from marginal independence, .], and implies the existence of deep holographic constraints that restrict the allowed patterns of independence among various subsystems beyond the universal quantum mechanical restrictions. Published by the American Physical Society 2024
Abstract The holographic entropy cone characterizes the relations between entanglement entropies for a spatial partitioning of the boundary spacetime of a holographic CFT in any state describing a classical bulk geometry. We argue that the holographic entropy cone, for an arbitrary number of parties, can be reconstructed from more fundamental data determined solely by subadditivity of quantum entropy. We formulate certain conjectures about graph models of holographic entanglement, for which we provide strong evidence, and rigorously prove that they all imply that such a reconstruction is possible. Our conjectures (except only for the weakest) further imply that the necessary data is remarkably simple. In essence, all one needs to know to reconstruct the holographic entropy cone, is a certain subset of the extreme rays of this simpler “subadditivity cone”, namely those which can be realized in holography. This recasting of the bewildering entanglement structure of geometric states into primal building blocks paves the way to distilling the essence of holography for the emergence of a classical bulk spacetime.
<!--HTML-->This talk will explain recent puzzling revelations in the ongoing efforts to obtain a useful characterization of entanglement structure of geometric states in a holographic CFT, via the so-called holographic entropy cone (HEC). The relations between subsystem entanglement entropies which delimit this cone are known explicitly for only a rather coarse subdivision of the system (specified by N spatial regions, for up to N = 5). We argue that, subject to a certain graph theoretic conjecture, the task of finding the HEC for arbitrary N can be recast in terms of a much simpler combinatorial one which effectively reduces to the connectivity of entanglement wedges. More specifically, the N-party HEC can be reconstructed by solving the holographic marginal independence problem (HMIP) for a finer subdivision N′ ≥ N, which technically amounts to identifying which extreme rays of this subadditivity cone are realizable holographically. Curiously, despite the fact that subadditivity is a universal property which states that total correlation cannot be negative, the non-trivial facets of the HEC constructed therefrom nevertheless cannot be recast as correlation measures.
Strong subadditivity (SSA) on a common light cone gives the Casini-Huerta entropic proof of the three-dimensional FFF-theorem. We ask whether holographic entropy inequalities beyond SSA similarly constrain renormalization group flows for which every intermediate theory admits a semiclassical holographic description. For small, disjoint deformations of a common light-cone region, we show that the second-order response of a broad class of balanced holographic inequalities depends only on pairwise correlations already controlled by SSA. A six-party example shows that the full finite inequality nevertheless contains genuinely multipartite information, so its disappearance is a limitation of the second-order expansion rather than of the inequality itself. Two natural finite constructions do not recover the missing information. We nevertheless find two ways in which information beyond SSA survives. A five-party inequality bounds the rate at which a conditional correlation grows as one region is enlarged. Separately, a continuum limit of the odd-cyclic inequalities gives a constraint on the angular shape dependence of entanglement entropy, and Lorentz symmetry relates this constraint to radial evolution. Thus holographic entropy inequalities beyond SSA do constrain entanglement along RG flows, although we do not obtain a second universal analogue of the FFF-function.