The Holographic Entropy Cone for Five Regions

TL;DR

This paper fully characterizes the holographic entropy cone for five regions, establishing all known inequalities and confirming no additional inequalities are needed, thus advancing understanding of quantum entanglement in holographic theories.

Contribution

It provides a complete construction of the five-region holographic entropy cone using graph models, confirming the existing inequalities are sufficient and no new inequalities are necessary.

Findings

Complete description of the $ ext{C}_5$ cone

All holographic entropy inequalities for five regions are proven to be complete

Graph models for all extreme rays are constructed

Abstract

Even though little is known about the quantum entropy cone for $N\geq4$ subsystems, holographic techniques allow one to get a handle on the subspace of entropy vectors corresponding to states with gravity duals. For static spacetimes and $N$ boundary subsystems, this space is a convex polyhedral cone known as the holographic entropy cone $\mathcal{C}_N$ for $N$ regions. While an explicit description of $\mathcal{C}_N$ was accomplished for all $N\leq4$ in the initial study, the information given about larger $N$ was only partial already for $\mathcal{C}_5$. This letter provides a complete construction of $\mathcal{C}_5$ by exhibiting graph models for every extreme ray orbit generating the cone defined by all proven holographic entropy inequalities for $N=5$. The question of whether there exist additional inequalities for $5$ parties is thus settled with a negative answer. The conjecture that $\mathcal{C}_5$ coincides with the analogous cone for dynamical spacetimes is supported by demonstrating that the information quantities defining its facets are primitive.

Figures

Equations

$$S_A=\underset{\mathcal{A}}{\min }\frac{\text{area }\mathcal{A}}{4G_{\mathrm{N}}},$$

$$I_2(\text{A:BC})\ge I_2(\text{A:B})+I_2(\text{A:C}),$$

Full text

1 Introduction

The most broadly studied limit of the AdS/CFT correspondence conjectures a holographic duality between certain strongly coupled gauge theories and classical gravity Maldacena1999 ; Aharony1999a . More explicitly, for a pair of gauge-gravity dual theories, the AdS/CFT dictionary poses a specific spacetime geometry as the gravitational counterpart of a given state in the Hilbert space of the quantum theory. On grounds of such a duality, it is of interest to determine which quantum states are dual to classical bulk geometries. A remarkable finding from the study of holographic entanglement is that, regardless of the theory, quantum states with particular patterns of correlations do not admit smooth geometric duals Hayden2013 ; Balasubramanian2014 .

At the heart of this result lies the Ryu-Takayanagi (RT) proposal, which states that for static bulk geometries the entanglement entropy $S_{\textrm{A}}$ of a spatial region $A$ of the boundary conformal field theory is given by Ryu2006 ; Nishioka2009 ; Rangamani2017

[TABLE]

where the minimization is performed over all bulk codimension-$2$ surfaces homologous to $A$ and such that $\partial A=\partial\mathcal{A}$. The Hubeny-Rangamani-Takayanagi (HRT) prescription gives the covariant generalization of RT that applies to arbitrary dynamical spacetimes Hubeny2007 ; Rangamani2017 . That the RT formula should reproduce the results of the von Neumann entropy for arbitrary partitions of a quantum state establishes a necessary condition for the existence of a smooth bulk dual. The discovery that there exist valid holographic entropy inequalities which are not true in quantum theory means that this necessary condition is not met by arbitrary quantum states. In particular, this is the case for the inequality known as monogamy of mutual information (MMI) Hayden2013 ,

[TABLE]

defined here in terms of the mutual information $I_{2}(\textrm{A:B})=S_{\textrm{A}}+S_{\textrm{B}}-S_{\textrm{AB}}$, and where A, B and C stand for three disjoint regions. This inequality has been proven true holographically for arbitrary dynamical spacetimes Hayden2013 ; Wall2014 , yet is easily violated quantum mechanically (e.g. by the GHZ state).

It follows that characterizing entanglement properties via entropy inequalities provides a powerful criterion to determine whether a quantum state can possibly be geometric (i.e. whether it can be holographically dual to a classical geometry). The formalization of this idea was carried out in Bao2015 , which introduced what is known as the holographic entropy cone to parametrize the space of allowed entropies for geometric states. The purpose of this work is to continue the systematic study and enumeration of holographic entropy inequalities. The picture for $4$ and fewer regions was completed in Bao2015 , where partial results were also found for $5$ regions. Building on this previous work, the complete construction of the holographic entropy cone for $5$ regions is produced here. A direct corollary is that no additional entropy inequalities are needed for the completion of the $5$-party cone.

2 Framework and Approach

Let $\Sigma$ be a spacelike slice of the spacetime manifold of a quantum field theory in a state which admits a holographic description in terms of a smooth bulk geometry. Consider $N\in\mathbb{Z}^{+}$ arbitrary nonempty codimension-$1$ disjoint subsets $X_{i}\subset\Sigma$, where $i\in[N]\equiv\{1,\dots,N\}$. Any such $X_{i}$ will be referred to as a monochromatic region of colour $i$. One also defines the set of polychromatic indices $\wp_{N}$ as the power set of $[N]$ with the empty set removed. The latter has cardinality $D=2^{N}-1$ and its elements $I\in\wp_{N}$ are used to label polychromatic regions $X_{I}\equiv\bigcup_{i\in I}X_{i}$. Denoting the entanglement entropy of each region $X_{I}$ by $S_{I}$, one may construct a $D$-tuple $\boldsymbol{\mathbf{S}}\equiv\left\{S_{I}\,\mid\,I\in\wp_{N}\right\}$. Canonically ordering its entries by increasing cardinality of $I$ and then lexicographically, $\boldsymbol{\mathbf{S}}\in\mathbb{R}^{D}$ defines an entropy vector.

Every entropy $S_{I}$ of a collection of regions can be computed holographically to leading order in the central charge of the boundary theory using the HRT prescription Hubeny2007 . For static bulk geometries for which this construct reduces to the RT formula (1.1) Ryu2006 , the space $\mathcal{C}_{N}\subset\mathbb{R}^{D}$ of all physically realizable holographic entropy vectors $\boldsymbol{\mathbf{S}}\in\mathbb{R}^{D}$ is known as the holographic entropy cone $\mathcal{C}_{N}$ for $N$ regions. It was shown in Bao2015 that this space is indeed a convex cone which is closed, rational and polyhedral. The Farkas-Minkowski-Weyl theorem recasts polyhedrality into the existence of two dual representations of such convex cones Grunbaum2003a :

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Facet representation: $\mathcal{C}_{N}$ can be constructed as the intersection of a finite number of half-spaces specified by entropy inequalities of the form $\boldsymbol{\mathbf{S}}\cdot\boldsymbol{\mathbf{Q}}_{j}\geq 0$, where $\boldsymbol{\mathbf{Q}}_{j}\in\mathbb{R}^{D}$. The minimal collection $\{\boldsymbol{\mathbf{Q}}_{j}\in\mathbb{R}^{D}\}$ of such vectors is unique and geometrically defines the support hyperplanes or facets of the cone.

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Extreme ray representation: $\mathcal{C}_{N}$ can be finitely generated as the conical hull of a set of vectors. The minimal collection of such vectors is unique and consists of the extreme rays $\{\boldsymbol{\mathbf{e}}_{k}\in\mathcal{C}_{N}\}$ of the cone, i.e. the vectors in $\mathcal{C}_{N}$ which cannot be conically spanned by other vectors in $\mathcal{C}_{N}$.

Importantly, since $\mathcal{C}_{N}$ is closed, for every extreme ray $\boldsymbol{\mathbf{e}}_{k}\in\mathcal{C}_{N}$ there exists a bulk geometry and a choice of boundary regions such that their corresponding entropy vector $\boldsymbol{\mathbf{S}}\propto\boldsymbol{\mathbf{e}}_{k}$ Bao2015 . Also, by virtue of being rational, the facet vectors and extreme rays of $\mathcal{C}_{N}$ can be written with integer coordinates. In particular, by the non-negativity of entanglement entropy, every $\boldsymbol{\mathbf{e}}_{k}$ has non-negative integer entries, as will be seen.

Constructing the holographic entropy cone $\mathcal{C}_{N}$ for $N$ regions amounts to finding a representation of it. A collection of proven entropy inequalities for $N$ parties does not necessarily provide a complete representation of $\mathcal{C}_{N}$. More specifically, supposing that such collection of inequalities represents a cone $\tilde{\mathcal{C}}_{N}$, that they are true entropy inequalities only guarantees that $\mathcal{C}_{N}\subseteq\tilde{\mathcal{C}}_{N}$. Proving that the facets of the two cones in fact coincide is better done in the dual description in terms of extreme rays. In particular, if for every extreme ray of $\tilde{\mathcal{C}}_{N}$ one is able to find a geometry whose entropy vector lies on it, then convexity immediately implies that $\mathcal{C}_{N}=\tilde{\mathcal{C}}_{N}$.

This strategy was implemented in Bao2015 to construct the holographic entropy cones for $N\leq 4$. For $N=5$, the authors successfully found and proved by contraction five new entropy inequalities, but left as an open question whether this set was complete. A thorough understanding of $\mathcal{C}_{5}$ has thus been lacking. In this work, the complete representation of the holographic entropy cone $\mathcal{C}_{5}$ for $5$ regions is provided by explicit construction of its extreme rays. One of the outcomes is that there are no new holographic inequalities for $5$ parties, so that the facets of $\mathcal{C}_{5}$ are precisely certain upliftings of known inequalities for $N\leq 3$ and the five new ones proven in Bao2015 .

The construction of the extreme rays of $\mathcal{C}_{5}$ is given here in terms of graph models as introduced in Bao2015 . The key theorem behind this combinatorial approach is that $\boldsymbol{\mathbf{S}}\in\mathcal{C}_{N}$ if and only if there exists a graph model that realizes $\boldsymbol{\mathbf{S}}$. In other words, the holographic entropy cone and the analogously defined graph-model entropy cone are identical. A graph model for $N$ parties is an undirected graph $(V,E)$ with $V$ vertices and $E$ edges, where a subset $\partial V\subseteq V$ is coloured by a map $c:\partial V\to[N]$. As the nomenclature suggests, a vertex coloured by $i$ stands as the graph representative of the monochromatic region $X_{i}$ in the boundary theory. The elements of $\partial V$ are thus called boundary vertices, while those in the complement $V\smallsetminus\partial V$ are called bulk vertices. Edges are assigned non-negative edge capacities by a weight map $E\to\mathbb{R}_{\geq 0}$. Then, the entropy $S_{I}$ of a polychromatic subset of boundary vertices $\partial V_{I}\equiv c^{-1}[I]\subset\partial V$ is given by the maximum flow between multisources $V_{I}$ and multitargets $\partial V\smallsetminus\partial V_{I}$ which respects the edge capacities. By the max-flow min-cut theorem, this is equivalent to the prescription that defines $S_{I}$ as the total weight in the minimum cut which disconnects source from sink. Physically, the latter is equivalent to the RT prescription, while the former corresponds to the bit-thread formulation of entanglement Freedman2017 .

3 The Holographic Entropy Cone for Five Regions

The action of the symmetric group $S_{N}$ which relabels the regions $X_{i}$ clearly leaves $\mathcal{C}_{N}$ invariant. This symmetry extends to an $S_{N+1}$ symmetry which implements the exchange of any $X_{i}$ with the purifier $O\equiv\Sigma\smallsetminus\bigcup_{i\in[N]}X_{i}$. Henceforth, statements about symmetries refer to the extended symmetry group $S_{N+1}$. The following subsections detail the description of $\mathcal{C}_{5}$ in its two representations.

3.1 Facets

The starting point of the strategy described above is a set of true inequalities for $N=5$ which is to be proven complete. This set consists of $372$ inequalities, which reduce to just $8$ when quotiented by symmetry. Table 1 shows a representative inequality for each symmetry orbit111For the explicit construction of $N=5$ objects, monochromatic indices are assigned alphabetic values rather than positive integers, and polychromatic indices are written by juxtaposition of letters rather than tuples.. The first three are upliftings of well-known inequalities for $N\leq 3$, whereas the last five are new to $N=5$. Inequality 1 is the trivial uplifting of subadditivity, whose orbit includes instances of the Araki-Lieb inequality too. Inequalities 1 and 1 are two different upliftings of MMI, which can be more compactly written in terms of the tripartite information as $I_{3}(\textrm{A:B:C})\leq 0$ and $I_{3}(\textrm{A:BC:DE})\leq 0$, respectively. Inequality 1 is the $5$-party instance of an infinite family of cyclic entropy inequalities222In the literature, this inequality is often quoted with manifest cyclicity of the subsystem labels. For consistency with the notation used for the other inequalities, symbols and their labels are ordered lexicographically here. Bao2015 . Like inequality 1, the remaining four were proven by contraction for the RT case in Bao2015 . This set of inequalities defines a cone in entropy space which will be shown to be precisely the holographic entropy cone $\mathcal{C}_{5}$ for $5$ regions in the next section. A natural question, however, is how to arrive at these inequalities, in particular the last five, in the first place. As for now, only Hubeny2018b succeeded in algebraically deriving these as candidate inequalities using the formalism of the holographic entropy arrangement Hubeny2018a ; Hubeny2018b .

It is worth remarking that, as defined, $\mathcal{C}_{N}$ is the space of holographic entropy vectors for states with time-reflection symmetry to which the RT prescription applies. In principle, lifting this restriction to the fully covariant HRT case could allow for a larger space of entropy vectors, the HRT holographic entropy cone $\mathcal{C}_{N}^{\textrm{HRT}}\supseteq\mathcal{C}_{N}$. While the original RT-based proof of strong subadditivity Headrick2007 was extended to dynamical setups and proofs of MMI Hayden2013 ; Wall2014 , it has been argued that the same methods may not be generalizable to non-static proofs for the $5$-party inequalities Rota2018 . Alternative bit-thread-based proofs of MMI Hubeny2018 ; Cui2018 may lend themselves to generalizations to larger-$N$ inequalities and covariance, but this is yet to be explored. The validity of inequalities 1$-$1 for dynamical spacetimes thus remains an open question which has only been verified in specific setups333In $2+1$ bulk dimensions, Czech2019 has recently proven that any inequality for which a contraction map exists is not only valid for RT, but also for HRT. In particular, this implies $\mathcal{C}_{N}^{\textrm{HRT}}=\mathcal{C}_{N}$ for $N\leq 5$ in $2+1$ bulk dimensions. Erdmenger2017 ; Bao2018b ; Caginalp2019 . However, a suggestive indication that $\mathcal{C}_{N}^{\textrm{HRT}}$ is no larger than $\mathcal{C}_{N}$ is precisely the algebraic derivation of these from the holographic entropy arrangement, which is defined for arbitrary spacetimes. More importantly, all facets of $\mathcal{C}_{5}$ can be shown to be primitive quantities as defined in Hubeny2018a ; Hubeny2018b , thus corresponding to phase transitions of entangling surfaces for arbitrary geometric states. Explicitly, using the proto-entropic formalism and notation for building blocks established in Hubeny2018b , Table 2 provides a set of configurations which suffice to generate the information quantities associated to inequalities 1$-$1 as primitive, respectively. It is remarkable that, besides canonical building blocks, only the non-adjoining configurations $\EuScript{C}_{4}^{\ast}$ and $\EuScript{C}^{\circledast}_{5}$ are needed to generate all facets of the polyhedron for $N=5$ up to symmetries (see Table 2 for notation). Note also the necessity of considering non-simply connected boundary topologies with enveloping, for otherwise the $I_{n}$ theorem would preclude the construction of these quantities as primitive Hubeny2018a . The configurations in Table 2 strongly support the conjecture in Hubeny2018b that the holographic entropy cone and polyhedron are indeed the same object.

3.2 Extreme Rays

The cone specified above by its $372$ facets admits a dual description in terms of $2267$ extreme rays. The latter can be grouped into $19$ distinct symmetry orbits, such that one may focus on a single representative ray per orbit. Table 3 shows one such choice of representatives444Semicolons separate entropy entries corresponding to polychromatic indices of different cardinality. The ordering is given by: ( A, B, C, D, E ; AB, AC, AD, AE, BC, BD, BE, CD, CE, DE ; ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE ; ABCD, ABCE, ABDE, ACDE, BCDE ; ABCDE )., while Fig. 1 provides every graph model needed to construct the holographic entropy cone $\mathcal{C}_{5}$ for $5$ regions555In producing these graphs, the choice has been to fix the number of boundary vertices to $5+1$, such that the colouring map is bijective for the $5$ regions and their purifier $O$. An explicit construction of an associated wormhole geometry can be accomplished by operations that bring the graph to a suitable form without changing its entropies, as explained in Bao2015 .. The first seven rays continue the pattern of being realizable by star graphs, which prove sufficient to generate all extreme rays for $N\leq 4$. However, the other twelve exhibit much richer structure, both in terms of reduced symmetry and non-planarity.

4 Conclusion

The holographic entropy cone $\mathcal{C}_{N}$ is now known for all $N\leq 5$. Besides the infinite family of cyclic inequalities, an understanding of the general $N$ case remains elusive. Early explorations of $N=6$ reveal that $\mathcal{C}_{6}$ consists of at least $19$ valid (i.e. proven by contraction), linearly independent orbits of holographic entropy inequalities. The following is an example of one such $6$-party inequality666This inequality has been proven using the contraction-map method introduced in Bao2015 . It can also be generated as a primitive quantity using the building blocks from Hubeny2018b , thus demonstrating that it is a facet of $\mathcal{C}_{6}$. The full construction of $\mathcal{C}_{6}$ is work in progress.:

$S_{\textrm{AB}}+S_{\textrm{ABC}}+S_{\textrm{ACD}}+S_{\textrm{ADE}}+S_{\textrm{BCD}}+S_{\textrm{BDE}}+S_{\textrm{CDE}}+S_{\textrm{CDF}}+S_{\textrm{DEF}}+S_{\textrm{ABCE}}\geq S_{\textrm{A}}+S_{\textrm{B}}+S_{\textrm{AC}}+S_{\textrm{BC}}+S_{\textrm{CD}}+2S_{\textrm{DE}}+S_{\textrm{DF}}+S_{\textrm{ABE}}+S_{\textrm{ABCD}}+S_{\textrm{CDEF}}+S_{\textrm{ABCDE}}$.

Any constructive approach to exploring $\mathcal{C}_{N}$ for larger $N$ must overcome the difficulty of dealing with an entropy space of $2^{N}-1$ dimensions. Already the dual description problem, for which no efficient algorithm is known, can only be feasibly solved up to symmetry Bremner . Moreover, most aspects of the problem suffer a combinatorial explosion which is doubly exponential in $N$ and any hope to proceed constructively must be accompanied by a strategy to tame the combinatorics. In particular, it is indispensable to turn the tables regarding the large degree of redundancy in the structure of $\mathcal{C}_{N}$ and use its symmetry to one’s advantage. Nevertheless, it would ultimately be desirable to understand $\mathcal{C}_{N}$ for arbitrary $N$. This will most likely require reducing the problem to an algebraic question rather than a combinatorial one, potentially along the lines of the formalism in Hubeny2018a ; Hubeny2018b .

Acknowledgements.

It is a pleasure to thank Xi Dong, Gary Horowitz, Veronika Hubeny, Mukund Rangamani, Massimiliano Rota, Gabriel Treviño Verastegui and Wayne Weng for useful discussions. This research was supported by fellowship LCF/BQ/AA17/11610002 from “la Caixa” Foundation (ID 100010434), NSF grant PHY-1801805, and the Center for Scientific Computing from the CNSI, MRL: an NSF MRSEC (DMR-1720256) and NSF CNS-1725797.