# Curvature from multipartite entanglement in quantum gravity states

Simone Cepollaro<EMAIL_ADDRESS>Scuola Superiore Meridionale,
Largo S. Marcellino 10, 80138 Napoli, Italy; INFN, Sezione di Napoli, Italy;
Goffredo Chirco<EMAIL_ADDRESS>Gianluca Cuffaro
<EMAIL_ADDRESS>Vittorio D’Esposito<EMAIL_ADDRESS>Dipartimento di Fisica “Ettore Pancini”, Università di Napoli Federico II,
Napoli, Italy; INFN, Sezione di Napoli, Italy.

###### Abstract

We investigate the multipartite entanglement of a uniformly curved quantum
$3D$ space region with boundary, realised in terms of spin networks defined on
a graph with non trivial $SU(2)$ holonomies, in the framework of loop quantum
gravity. The presence of intrinsic curvature in the region is encoded in
closure (topological) defects associated with _tag_ -spins attached to the
vertices of the graph. For such states, we generalise the bulk-to-boundary
mapping as to include the space of tags in an extended boundary space: bulk
information is _shared_ among generically entangled boundary surfaces and
intrinsic curvature degrees of freedom. We model the quantum region on a
tripartite system composed by two (complementary) boundary subregions and the
set of bulk tags. Via replica techniques, we can compute the typical value of
the logarithmic negativity of the reduced boundary, described as an open
quantum system, in a large spin regime.

We find three entanglement regimes, depending on the ratio between the number
of tags (curvature) and the area of the dual surface at the boundary. These
are well described by the _generalised Page curve_ of a tripartite random
state. In particular, we find area scaling behaviour for negativity in case of
small curvature, while for large curvature the negativity vanishes, suggesting
an effective thermalization of the boundary. Remarkably, the PPT character of
the mixed boundary state corresponds to a change in the effective topology of
the network, with the two boundary subregions becoming disconnected.

Keywords

Quantum gravity, random tensor networks, multipartite entanglement,
logarithmic negativity

††preprint: APS/123-QED

##  I Introduction

In the last two decades, the idea that continuum spacetime geometry may be
emerging from entanglement has radically changed all approaches to quantum
gravity [1, 2, 3, 4, 5, 6, 7, 8, 9]. Beside the AdS/CFT framework [10], where
the correspondence between geometry and entanglement is a necessary instance
of the gauge/gravity duality[11, 12, 13, 14, 15], non-perturbative and
background independent approaches to quantum gravity recently offered a
different playground to investigate the roots of such interplay.

In loop quantum gravity [16, 17, 18], and related generalizations [19, 20],
like spin foam theory and group field theory, a beautiful quantization of
space at one time, is realised in terms of _quantum spin networks_ [21, 22,
23]. These are typically $SU(2)$ gauge symmetric tensor networks, defined by
_graphs_ with edges labelled by $SU(2)$ spin _irreps_ and vertices dressed by
intertwining operators [18].

For given graph, the spin-networks Hilbert space essentially capture the
_kinematics_ of general relativity in its first order formulation [24], which
is eventually described in analogy with the conventional Hilbert space of an
$SU(2)$ lattice Yang-Mills theory [25, 26].

In this sense, spin networks provide a phenomenal quantitative tool to
investigate the operational content of the gravitational field, reduced on
$3D$ spacetime slices, at the quantum scale [27, 28, 29, 30, 31, 32, 33, 34,
35, 36, 37]. Indeed, geometric and topological features of $3D$ geometry get
encoded into purely algebraic and combinatorial degrees of freedom, hence
local, non-local and holographic features of the entanglement structure of
quantum spin networks are expected to reflect the signatures of Einstein’s
equations and the invariance for diffeomorphisms at the quantum level.

In this light, spin networks entanglement has become a central resource to
characterize physical vacuum states of the theory [38, 39, 40, 41],
investigating _local_ holographic properties of quantum spacetime geometry
[42, 43, 44] and ultimately to study the emergence of classical spacetime
geometry from its quantum description, with a renovate interplay of techniques
and tools from quantum information theory, information geometry, quantum many-
body theory and quantum computation theory more recently [45, 46, 47, 48, 49,
50, 51, 52, 53, 54, 55, 56].

Most recent work in the field, in particular, has focused on quantum $3D$
geometry states with $2D$ boundaries, corresponding classically to space-time
spatial slices with _corners_ [57], thereby looking at quantum spin-networks
with boundaries as _boundary maps_ with corner states encoding the geometric
and topological information stored in the bulk correlation structure. In this
setting, different measures of multipartite entanglement have been proposed to
investigate encoding and decoding of bulk information in the boundary/bulk
mapping [58, 59, 60] and its holographic behaviour.

In [61, 62, 63, 64, 65, 66], the holographic character of the spin-networks
boundary/bulk mapping has been investigated within a quantum typical regime
starting from a _random_ tensor-network description of $3D$ quantum geometry
states in the large spin limit [67, 9, 68]. Along this line, in [60], the
authors first investigated the measure of _entanglement negativity_ , along
the lines of [69, 70, 71, 72], to extend the study of the hierarchy of
boundary correlations to mixed states, beyond the bipartite pure-state setting
of entanglement entropy.

In this work, we consider a synthetic description of a uniformly curved
quantum $3D$ space region with boundary, realised in terms of open _tagged_
spin-networks states in loop quantum gravity [73, 74]. We show that the
effective topology of this region is reflected in the multipartite
entanglement of its _tagged_ spin-networks state description. In presence of
intrinsic curvature, we generalise the bulk-to-boundary mapping description as
to include the space of bulk defects in a generalised boundary space. In the
resulting _extended boundary_ state, bulk information is _shared_ among
entangled boundary spins and tags: surface and intrinsic curvature of the
quantum space region are entangled.

We model the generalised boundary mapping on a tripartite random state and
compute the logarithm negativity for the bipartite reduced boundary, focusing
on the universal typical behaviour of the large spin regime.

We find that the multipartite entanglement of the system depends not only on
the bulk quantum correlations among the intertwiner states comprising the bulk
of the spin network, but in an essential way, on the topology of the spin
network state.

In particular, we show that the degree of quantum correlations among boundary
subregions depends on the dimension of the bulk tags system, which plays the
role of a hidden environment from the viewpoint of a generic observer
measuring correlations on the boundary. When the bulk curvature environment is
smaller than the boundary system, the entanglement negativity of the boundary
subregions displays the expected area law behaviour. However, when the
environment is larger, the negativity vanishes suggesting an effective
thermalization of the boundary. Remarkably, the PPT character of the mixed
boundary state corresponds to change in the effective topology of the network
in the large spin regime, with the two complementary boundary subregions
becoming disconnected.

The paper is organized as follows. Section II introduces loop quantum gravity
quantum geometry states as superpositions of quantum spin networks. In Section
III, this notion is generalised to include boundaries and non-trivial
topology. In particular, here we define quantum states associated with regions
of bounded $3D$ space with non-vanishing intrinsic curvature. Section IV
introduces the notion of tagged spin networks, which is the starting point of
our characterisation of the bounded quantum geometry as a mixed state. In
Section V, we define the tripartition $\\{A_{1},A_{2},B\\}$ of our system,
with $A_{1}$, $A_{2}$ corresponding to complementary subregions of the
boundary and $B$ referring to the set of tags produced by the integral
(partial tracing) over the non-trivial loop holonomies in the bulk. For such a
tripartite system, we quantify the entanglement of the boundary via a measure
of _typical_ entanglement negativity [75, 76, 77, 78, 79, 80, 81] by
considering the boundary mapping on a bulk state _prepared at random_. The
random measurement combined with the boundary map projection allows us to
limit the partial tracing to the “curvature” degrees of freedom, while
(partially) retaining the information on the graph structure. We compute
typical $k$-th order Rényi log-negativity for even $k$ via replicas by mapping
momenta of the $A_{1},A_{2}$ reduced density matrix to partition functions of
a classical generalised Ising model [69, 72], hence looking for the minimal
free energy configurations of the model. In Section VI, we characterise the
bulk states contributions to the free energy of the classical model. The
analysis is thereby restricted to states with small intertwiner entanglement.
Section VII contains an explicit example of open spin network state - a
cluster of tags - with four vertices. For such a simplified setting, we can
explicitly compute the typical logarithmic negativity of the mixed boundary
state in the large spin regime and characterise the entanglement phase diagram
of the boundary state. This section contains the main results of the paper.
Finally, we close in Section VIII with a summary of the results and some
closing remarks.

##  II Spin Networks states

Loop quantum gravity (LQG) space of 3D geometry is realized as a space of
square-integrable functions,

$H_{\gamma}=L^{2}\quantity[SU(2)^{\times E}/SU(2)^{\times V}]\,,$ (1)

with support on closed oriented graphs $\gamma$ comprised by $V$ vertices
connected by $E$ edges. States in $H_{\gamma}$ depend on one group element
$g_{e}\in SU(2)$ for each edge $e$ of the graph and are assumed to be
invariant under the $SU(2)$-action at each vertex $v$, that is, $\forall h\in
SU(2)$,

$\displaystyle\psi_{\gamma}:SU(2)^{\times E}$ $\displaystyle\to$
$\displaystyle\mathbb{C}$ (2) $\displaystyle\\{g_{e}\\}_{e\in\gamma}$
$\displaystyle\mapsto$
$\displaystyle\psi_{\gamma}(\\{g_{e}\\})=\psi_{\gamma}(\\{h_{t(e)}\,g_{e}\,h_{s(e)^{-1}}\\})\,,\
$

where $t(e)$ and $s(e)$ respectively refer to the target and source vertices
of the edge $e$ [82].

For $SU(2)$ is compact, by the Peter–Weyl theorem, functions of $d$ group
elements $g$ can be decomposed into irreducible representations (irreps) of
the group,

$f(\\{g\\})=\sum_{\\{j\\}}\sum_{\\{m\\}\\{n\\}}f^{\\{j\\}}_{\\{m\\}\\{n\\}}\prod_{i=1}^{d}d_{j_{i}}D^{j_{i}}_{m_{i}n_{i}}(g_{i}),$
(3)

with $\\{j_{i}\\}\in\mathbb{N}/2$ labelling irreps of $SU(2)$; indices $m_{i}$
($n_{i}$) labelling a basis in the vector space
$V^{j_{i}}=\text{span}\\{\ket{j_{i},m_{i}}\\}$, carrying the representation
$j_{i}$, of dimension $d_{j_{i}}=2j_{i}+1$, and
$D^{j_{i}}_{m_{i}n_{i}}(g_{i})=\bra{j_{i},n_{i}}g_{i}\ket{j_{i},m_{i}}$ the
Wigner matrix representing the group element $g_{i}$.

This allows to write $SU(2)$-invariant functions $\psi_{\gamma}$ as
_superpositions_ of gauge symmetric tensor networks with fixed spins
$\\{j\\}_{\gamma}$ and the connectivity of the graph $\gamma$. Two vertices of
the graph, say $v$ and $w$, are connected by edge states labeled by a spin $j$
irreducible representation (irrep) of $SU(2)$ and dressed with a group element
$g_{e}\in SU(2)$, that is

$\ket{e_{vw}(g_{e})}\equiv\sum_{\\{m\\}\\{n\\}}\frac{(-1)^{j-n}}{\sqrt{d_{j}}}\,D^{j}_{mn}(g_{e})\,\ket{j,m}_{v}\otimes\overline{\ket{j,n}}_{w}$
(4)

in the Hilbert space
$H_{vw}=\Big{[}V_{v}^{j}\otimes\overline{V}_{w}^{j}\Big{]}\,$ . At each vertex
$v$ of the graph, an intertwiner operator $\iota_{v}$ enforces $SU(2)$ gauge
invariance via a projection

$\iota_{v}:\,\,\bigotimes_{e\in v}\,V_{j_{e}}\to V_{0}\,.$ (5)

corresponding to a recoupling of the edges spins at the vertex in the
$SU(2)$-symmetric (singlet) representation with $j=0$. This associates to each
vertex the degeneracy space of $V_{0}$ for fixed $\\{j_{e}\\}$,

$H_{v}=\text{Inv}_{SU(2)}\Big{[}\bigotimes_{e\in
v}V^{j_{e}}\Big{]}=\text{span}\\{\ket{\iota_{v}}\\}\,,$ (6)

with $\ket{\iota_{v}}$ an orthonormal basis in $H_{v}$.

A _spin network_ state $\ket{\gamma,\\{j_{e}\\},\\{\iota_{v}\\}}$ is thereby
defined as the assignment of representation labels $\\{j_{e}\\}$ to each edge
of $\gamma$ and the choice of a vector $\ket{\\{\iota_{v}\\}}$ in
$H_{V}=\otimes_{v}^{V}H_{v}$ for the vertices, corresponding to the
contraction (see Fig 1)

$\ket{\gamma,\\{j_{e}\\},\\{i_{v}\\}}=\left(\bigotimes_{e=1}^{E}\bra{e(g_{e})}\right)\,\ket{\\{\iota_{v}\\}}.$
(7)

Such spin-networks define an orthogonal basis in $H_{\gamma}$, for the
following isomorphism holds

$H_{\gamma}\approx\bigoplus_{\\{j\\}}{H_{V}}.$ (8)

Any quantum $3D$ geometry state $\psi_{\gamma}$ can be constructed from the
_contraction_ of a generic state $\ket{\psi}$ in $H_{V}$ with the set of $E$
edge states, via a sum over spins

$\ket{\psi_{\gamma}(\\{g_{e}\\})}=\bigoplus_{\\{j_{e}\\}}\left(\bigotimes_{e=1}^{E}\bra{e(g_{e})}\right)\,\ket{\psi},$
(9)

where the contraction amounts to a sum over bulk indices.

Figure 1: Open spin network with support on the graph $\gamma_{R}$. The
boundary is comprised by the set of uncontracted edges with dangling magnetic
indices pictured as empty circles.

##  III Spin-networks with boundary and Non-trivial Topology

Imagine to cut a generic _bounded_ region $R$ out of quantum 3D space. The
corresponding state will be defined with support on an _open_ graph
$\gamma_{R}\subset\gamma$, which we can take to be comprised by the sets of
vertices $\mathcal{V}_{R}$ and edges $\mathcal{E}_{R}$, with
$\absolutevalue{\mathcal{V}_{R}}=V_{R}$ the number of vertices and
$\absolutevalue{\mathcal{E}_{R}}=E_{R}$ the number of edges respectively. The
_boundary_ of $R$, $\partial R$, is the set $\mathcal{E}_{\partial R}$ of
$E_{\partial R}$ edges which have only one end attached to a vertex laying in
$R$. Such boundary edges are characterised by an open (uncontracted) dangling
index $m$, while bulk edges connecting vertices within $R$ are fully
contracted. For states with support on open graphs, the contraction in (9)
defines a _mapping_ of the state $\ket{\psi}\in H_{V_{R}}$ on the boundary
space of uncontracted indices,

${H}_{\partial R}=\bigoplus_{\\{j_{e}\\}}\bigotimes_{e\in\mathcal{E}_{\partial
R}}V^{j_{e}}.$ (10)

In particular, the gauge invariant property of the LQG wavefunctions defined
on closed graphs reduces to $SU(2)$-covariance of the bounded region
wavefunction,

$\psi_{R}(\\{g_{e}\\}_{e\in\mathcal{E}})=\psi_{R}(\\{g_{e}\,h_{s(e)^{-1}}\\}_{e\in\mathcal{E}_{\partial
R}})\,.\ $ (11)

In the boundary mapping, the information on the bulk holonomies, as well as
the correlation structure of $\psi\in H_{V_{R}}$ gets encoded in the
coefficients of the resulting boundary state $\ket{\psi_{R}}$. We are
interested in the information concerning the _bulk curvature_ and, in
particular, in the way this affects (gets reflected in) the correlation
structure of the boundary state $\ket{\psi_{R}}$.

Bulk curvature is generally associated to the presence of loops of non-trivial
holonomies in the graph. However, the topology of the graph is only partially
relevant to characterise such a degree of freedom. Indeed, thanks to the local
gauge invariance of $\psi_{R}$, the structure of the bulk of $\gamma_{R}$ can
be drastically simplified via a partial gauge-fixing of the bulk holonomies
[45, 73, 74]. In particular, for any open region $R$, the gauge-invariant
Hilbert space on $\gamma_{R}$ is isomorphic to the gauge-invariant reduced
space defined on a new graph $\Gamma_{R}$ consisting of a single vertex
intertwining the external edges of the boundary $\partial R$ together with a
number of (independent) loops $L$ fixed by the combinatorics of the region,
that is $L=E_{R}-V_{R}+1$. If $L=0$ there are no loops and $\gamma_{R}$ has
trivial topology. In particular, this implies that $R$ has vanishing
_intrinsic_ curvature.

More generally, the Hilbert space of the reduced graph $\Gamma_{R}$ will
consist of the intertwiner space of the single bulk vertex times the product
of the boundary representations spaces, that is

$H_{\Gamma_{R}}=L^{2}[SU(2)^{\times(E+2L)}/SU(2)]\\\
=\bigoplus_{\\{j_{\ell,e}\\}}H_{v}^{L}\otimes\bigotimes_{e\in\mathcal{E}_{\partial
R}}V^{j_{e}}\,,$ (12)

where the degeneracy space at the vertex now also involves the irreps on the
loop

$H_{v}^{L}\equiv\text{Inv}_{SU(2)}\left[\bigotimes_{\ell=1}^{L}\left(V^{j_{\ell}}\otimes\overline{V}^{j_{\ell}}\right)\otimes\bigotimes_{e\in\mathcal{E}_{\partial
R}}V^{j_{e}}\right]\,.$ (13)

The space in (12) provides a very synthetic description of the region $R$ in
terms of a single “loopy” vertex with both its boundary degrees of freedom and
bulk nontrivial holonomies. The gauge reduction isomorphism does not produce
any coarse graining on physical degrees of freedom. However, the information
of the combinatorics of the original graph is lost. As a consequence, the
correspondence between the original graph and its flower is many-to-one [74].

Now, imagine to operate a similar reduction-by-gauge-fixing on a _collection_
of subregions, so as to end up with a set of loopy vertices glued together via
edges dressed with a trivial holonomy. In this way, we construct a new spin
network state with support on an open graph $\tilde{\gamma}_{R}$ associated to
an extended $3D$ space region with distributed _intrinsic_ curvature (see Fig.
2). Such an intermediate level of description allows to localize and keep
track of the nontrivial holonomies $\\{g_{\ell}\\}$ (classically closure
defects) responsible for the curvature at the vertices of the graph, without
trivializing the topology of the graph, which is going to play a role in the
entanglement structure of the boundary.

##  IV tags from spin-network kirigami

Figure 2: a) Reduction by gauge fixing of the holonomies along the maximal
spanning tree connecting three vertices of an open spin network; b) Resulting
non-trivial holonomy on a loop attached to a coarse-grained vertex; c) Bounded
quantum $3D$ region with uniformly distributed loops realised as the gluing of
a set of coarse-grained vertices.

Along this line, let us consider for simplicity the case of a uniformly curved
region of quantum $3D$ space, given by an open-graph state
$\ket{\psi_{R}}=\ket{\psi_{R}(\\{g_{\ell}\\})}$ with all $V_{R}$ vertices
carrying one single loop holonomy $g_{\ell}$. Further, let us assume for now
that each vertex can have at most one open edge. This very simplified setting
is what we need to neatly separate boundary from bulk degrees of freedom, as
shown in Fig.(3). The first step in this sense consists in integrating out the
loop holonomies. Following [83], for each spin $k$ carried by the loop, one
can separate the loop from the edges at each vertex by unfolding the
intertwiner into two different intertwiners connected by a virtual link
labelled by an intermediate spin $J$: one intertwiner recoupling the boundary
edges spins $j_{e}$ at $v$ into $J$ and a second 3-valent intertwiner
recoupling the two copies of the spin k into $J$ at the virtual vertex
$\tilde{v}$. Accordingly, the intertwiner space in (13) now decomposes as

$H_{v}^{L}=\bigoplus_{J,\\{k\\}}\Bigg{\\{}\text{Inv}_{SU(2)}\Big{[}\bigotimes_{e\in
v}V^{j_{e}}\otimes V^{J}\Big{]}\\\
\otimes\,\text{Inv}_{SU(2)}\left[V^{J}\otimes\left(V^{k}\otimes\overline{V}^{k}\right)\right]\Bigg{\\}}.$
(14)

A basis in $H_{v}^{L}$ is given by (see for instance [84])

$\ket{J,k,\iota,\\{j_{e}\\}[g_{\ell}]}=D^{k}(g_{\ell})\,C^{k,J|k}_{\tilde{m},M|m}\,\innerproduct{J,M}{\iota}$
(15)

with

$C^{k,J|k}_{\tilde{m},M|m}=\innerproduct{k,m}{(k,\tilde{m})\otimes(J,M)}$ (16)

the Clebsh-Gordan coefficient associated to the 3-valent intertwiner at
$\tilde{v}$.

At each vertex of the graph, the integration over the loop holonomy only
involves the Clebsh-Gordan coefficient and the Wigner matrix of the loop at
the level of the basis in (15). One can then compute the vertex density matrix
$\rho_{k}$ as

$\rho_{k}=\int\mathrm{d}g_{\ell}\ket{J,k,\iota,\\{j_{e}\\}[g_{\ell}]}\bra{J,k,\iota,\\{j_{e}\\}[g_{\ell}]}$
(17)

and show that the sum of $\rho_{k}$ over the loop spin $k$ gives exactly the
identity matrix $\rho=\innerproduct{J,M}{\iota}\innerproduct{\iota}{J,M}$,
with $\ket{\tau}=\innerproduct{J,M}{\iota}$ a _basis_ in the _reduced_ vertex
space [84]

$H_{v}^{J}=V^{J}\otimes M_{J}^{\\{j_{e}\\}}\,,$ (18)

with
$M_{J}^{\\{j_{e}\\}}\equiv\text{Inv}_{SU(2)}\Big{[}V^{J}\otimes\bigotimes_{e\in
v}V^{j_{e}}\Big{]}\,$.

As a result of the integration, curvature is locally encoded into a virtual
spin $J$, or _tag_ , attached to the vertex. We can see the vector
$\ket{J,M}\in V^{J}$ as the result of the recoupling of all the spins $j_{e}$
at the vertex into a non-vanishing overall spin $J$, a topological defect
which breaks local gauge invariance.111Given a tagged basis $\ket{\tau_{v}}$
at each vertex, we can define a _tagged_ _spin network_ basis
$\ket{\gamma,\\{j_{e}\\},\\{\iota_{v}\\},\\{J_{v}\\}}=\left(\bigotimes_{e=1}^{E}\bra{e(g_{e})}\right)\,\ket{\\{\tau_{v}\\}}$
(19) for the whole state on $\tilde{\gamma}_{R}(V_{R},E_{R},E_{\partial R})$.

Starting from the tagged intertwiner description, we now define a
_homogeneously curved_ quantum $3D$ bounded space geometry as the _extended_
boundary map

$\ket{\psi_{\tau}}=\left(\bigotimes_{e=1}^{E}\bra{e(g_{e})}\right)\,\ket{\psi_{b}}$
(20)

where the generic tagged state $\ket{\psi_{b}}$ is defined, for fixed spins
$\\{J,j_{e}\\}_{v}$, as

$\ket{\psi_{b}}=\sum_{\\{M_{v}\\}\\{\iota_{v}\\}}C(\\{J,j_{e}\\})_{\\{M_{v}\\},\\{\iota_{v}\\}}\bigotimes_{v\in\mathcal{V}_{R}}\ket{\tau_{v}}$
(21)

in the tensor product space of $V_{R}$ tagged intertwiner spaces
$H_{V_{R}}\equiv\bigotimes_{v}^{V_{R}}H_{v}^{J_{v}}$. Differently from (9),
the projected state now lives in an _extended_ boundary space,

$H_{\tau}\equiv\bigotimes_{v\in\mathcal{V}_{R}}V^{J_{v}}\otimes\bigotimes_{e\in\mathcal{E}_{\partial
R}}V^{j_{e}}\,$ (22)

which comprises the tensor product of tags and boundary-spins representation
spaces.222In fact, one can think of the tag factor in (22) as a disjoint
_inner_ boundary associated to closure defects at the bulk vertex. This is
compatible with a picture of the tags as the result of a set of holes cut out
of planar spin network punctured by uncontracted bulk edges. In the standard
picture, such holes correspond to loops which recouple the puncturing edges.

The tagged open graph state is written as

$\ket{\psi_{\tau}}=\sum_{\\{M_{v},m_{e}\\}}\mathcal{C}(\\{\iota_{v},J_{v},j_{e}\\})_{\\{M_{v},m_{e}\\}}\\\
\bigotimes_{v\in\mathcal{V}_{R}}\ket{J_{v},M_{v}}\bigotimes_{e\in\mathcal{E}_{\partial
R}}\ket{j_{e},m_{e}},$ (23)

with coefficients
$\mathcal{C}(\\{\iota_{v},J_{v},j_{e}\\})_{\\{M_{v},m_{e}\\}}$ encoding the
information on the quantum correlations among bulk intertwiners, the
connectivity, and the topological defects of the graph. We expect states in
(23) to be generically highly entangled, with information about bulk geometry
encoded into the boundary state coefficients $\mathcal{C}$ in a complicated
way. In the extended boundary system, such information is shared by boundary
spins and bulk tags.

Now, consider an observer having access only to the actual boundary system.
Such an observer is precluded from measuring the information on the bulk,
hence they will typically describe the system via a mixed boundary state.
Concretely, this amounts to a partial tracing over the bulk. In our analysis,
we describe the coarse-grained viewpoint of the boundary observer by
introducing a _random measurement_ on the state $\ket{\psi_{\tau}}$. We then
investigate the boundary-system/bulk-environment coupling via a combination of
random projections and partial tracing over the tags.

##  V Negativity of a tripartite boundary-tags system

Figure 3: a) A single loopy vertex ; b) vertex unfolding: isolate the
nontrivial loop holonomy; c) partial trace via integration over the loop
holonomy; d) tagged vertex.

Let us consider the extended boundary space (22). For convenience, let us
assume curvature, in the form of closure defects, to be homogeneously
distributed throughout the graph $\tilde{\gamma}_{R}$. Accordingly, we set all
the spins of the graph $\\{j_{e}\\}$ and $\\{J_{v}\\}$ to be equal to the
single value $j$ and $J$ respectively, generally with $j\neq J$.

A _random measurement_ on the bulk state $\ket{\psi_{b}}$, prepared on
$\bigotimes_{v}\,H_{v}^{J}$, is realized by a projection onto a set of
independent and individually (Haar) random vertex states $\ket{f_{v}}$ in the
_extended_ vertex space

$K_{v}^{J}=V^{J}\otimes\bigotimes_{j_{e}\in v}V^{j_{e}}\otimes
M_{J}^{\\{j_{e}\\}}\,,$ (24)

which comprises the intertwiner space at the vertex, the spin irrep spaces of
the edges connecting the vertex with the rest of the graph, as well as the tag
spin space. In particular, we can obtain the state $\ket{f_{v}}$ by acting on
a reference state $\ket{0}$ in $K_{v}^{J}$ at $v$ with a Haar random unitary,
that is $\ket{f_{v}}=U_{v}\ket{0}$ [67].

The generic random vertex state is explicitly written as

$\ket{f_{v}}=\sum_{\iota_{v},\\{n_{e}\\}}f^{v}_{\iota_{v},\\{n_{e}\\}}\,\innerproduct{J,M}{\iota_{v}}\otimes\bigotimes_{e\in
v}\ket{j_{e},n_{e}}\,$ (25)

The randomised extended boundary density matrix is then simply defined along
with (20) by a trace over bulk indices, which contract
$\rho_{b}=\outerproduct{\psi_{b}}{\psi_{b}}$ and
$\rho_{E}=\outerproduct{E}{E}$, for $\ket{E}=\bigotimes_{e}^{E}\ket{e}$, with
the further insertion of the _random projector_
$\Pi=\bigotimes_{v}^{V_{R}}\Pi_{v}=\bigotimes_{v}^{V_{R}}\outerproduct{f_{v}}{f_{v}}$:

${\rho}_{\tau}=\text{Tr}\left[\,\rho_{b}\,\cdot\,\rho_{E}\,\Pi\right]\,.$ (26)

The bulk state $\rho_{b}$ and the maximally entangled edges state $\rho_{E}$
enter as product states in the trace, while the information on the tags is
enclosed in the bulk state.

We shall define a tripartition of the extended boundary system into three
subsystems $A_{1}$, $A_{2}$, $B$, with $A_{1}$, $A_{2}$ complementary regions
of the external boundary, and $B$ the set of tags in the bulk. This
corresponds to the following factorization of the extended boundary Hilbert
space

$H_{\tau}=\bigotimes_{e\in A_{1}}V^{j_{e}}\otimes\bigotimes_{e\in
A_{2}}V^{j_{e}}\otimes\bigotimes_{i\in B}V^{J_{i}}\,.$ (27)

Starting from (27), we focus on the reduced state obtained tracing over the
tag spins in $B$,

$\rho_{A_{1}A_{2}}=\text{Tr}_{B}[\rho_{\tau}]\,.$ (28)

We evaluate the entanglement of the mixed state $\rho_{A_{1}A_{2}}$ by
computing the _logarithmic negativity_ of the _partial transpose_ of the state
[28, 85, 69, 72, 60]. In the bipartite subsystem $A_{1}A_{2}$, with Hilbert
space ${H}_{A_{1}}\otimes{H}_{A_{2}}$, given an orthonormal basis
$\ket{i}_{A_{1}}$ in $\mathcal{H}_{A_{1}}$ and similarly $\ket{j}_{A_{2}}$ in
$\mathcal{H}_{A_{2}}$, the partial transpose of $\rho_{A_{1}A_{2}}$ with
respect to the subsystem $A_{2}$ is defined as

$(\rho_{i_{A_{1}}j_{A_{2}},k_{A_{1}}l_{A_{2}}})^{T_{A_{2}}}=\rho_{i_{A_{1}}l_{A_{2}},k_{A_{1}}j_{A_{2}}}\,.$
(29)

The eigenvalues of the partially transposed reduced density matrix
$\rho_{A_{1}A_{2}}^{T_{A_{2}}}$ are real since the partial transposition is an
Hermitian and trace-preserving map. Yet $\rho_{A_{1}A_{2}}^{T_{A_{2}}}$ is not
completely positive, _i.e._ it may have negative eigenvalues. The presence of
negative eigenvalues in the partial transpose _witnesses_ quantum correlations
in $\rho_{A_{1}A_{2}}$. Concretely, the amount of entanglement of the state
can be quantified by counting the number of negative eigenvalues of
$\rho_{A_{1}A_{2}}^{T_{A_{2}}}$ [80], as

$N(\rho_{AB})\equiv\frac{\lVert\rho_{AB}^{T_{B}}\rVert_{1}-1}{2}=\sum_{i:\lambda_{i}<0}\absolutevalue{\lambda_{i}}\,,$
(30)

where $\lVert\cdot\rVert_{1}$ is the trace norm, and the logarithmic
negativity is defined as

${E}_{N}(\rho_{A_{1}A_{2}})\equiv\log\lVert\rho_{A_{1}A_{2}}^{T_{A_{2}}}\rVert_{1}\,.$
(31)

Both $N$ and $E_{N}$ are entanglement monotone under general positive partial
transpose (PPT) preserving operations [75]. In order to compute (31), the
first key ingredient is to first look at the $k$-th Rényi negativity of
$\rho_{A_{1}A_{2}}$,333Notice that the trace operation Tr in (32) now involves
sums over boundary indices (dangling edges and tags). The denominator in (32)
directly comes from the normalization of $\rho_{AA_{2}}$.

$N_{k}(\rho_{A_{1}A_{2}})=\Tr\quantity[\quantity(\rho_{A_{1}A_{2}}^{T_{A_{2}}})^{k}\Big{/}\quantity(\text{Tr}\quantity[\rho_{A_{1}A_{2}}])^{k}]\,,$
(32)

hence, eventually recover the logarithmic negativity by taking the $k\to 1$
limit of the logarithm of the analytic continuation of the momenta for even
$k$ [69]. In particular, since we took a random measurement on the bulk state,
we will need to compute the Rènyi negativity in _expected value_ , with
respect to the uniform Haar measure $\mu$, namely

$\mathbb{E}_{\mu}\left[N_{k}(\rho_{A_{1}A_{2}})\right]\equiv\overline{N_{k}(\rho_{A_{1}A_{2}})}\,.$
(33)

Following [69, 67, 60], we know that, due to concentration of the trace, a
regime of large spins allows us to approximate the Rènyi negativity by the
ratio of expected values of the $k$-th moment and the $k$-th power of the
partition function of ${\rho}_{A_{1}A_{2}}^{\,T_{A_{2}}}$, that is

$\overline{N_{k}(\rho_{A_{1}A_{2}})}\simeq\frac{\overline{\Tr\left[\quantity({\rho}_{A_{1}A_{2}}^{\,T_{A_{2}}})^{k}\right]}}{\overline{\left(\Tr\left[\quantity({\rho}_{A_{1}A_{2}})\right]\right)^{k}}}\equiv\frac{\overline{Z^{(k)}_{1}}}{\overline{Z^{(k)}_{0}}}\,.$
(34)

Moreover, in the large spin regime, we have

$N_{k}(\rho_{A_{1}A_{2}})\simeq\overline{N_{k}(\rho_{A_{1}A_{2}})}\,,$ (35)

that is, the function is well approximated by its _typical value_.

Figure 4: a) Tripartition of the spin network extended boundary for a simple
graph comprised by two 4-valent vertices glued via an edge; b) Partial tracing
of the density matrix over the tags space.

The expected values of the partition functions in (34) are computed via the
standard _replica_ technique. We first linearize the partial transpose matrix
as follows,

$\Tr\left[\left({\rho}_{A_{1}A_{2}}^{\,T_{A_{2}}}\right)^{k}\right]=\Tr\left[{\rho}_{A_{1}A_{2}}^{\,\otimes
k}\,P_{A_{1}}\quantity(X)\otimes P_{A_{2}}\quantity(X^{-1})\right]\\\
\qquad=\text{Tr}\left[{\rho}_{R}^{\,\otimes k}\,P_{A_{1}}\quantity(X)\otimes
P_{A_{2}}\quantity(X^{-1})\otimes P_{B}\quantity(\mathbbm{1})\right]\,.$ (36)

Here, we generally denote $P_{I}(\sigma)$ as a unitary representation of the
permutation $\sigma$, $I=A_{1},A_{2},B$, with $X$, $X^{-1}$ and $\mathbbm{1}$
the cyclic, anti-cyclic and identity permutations. On the spins and tags
dangling indices, the operator $P_{B}\quantity(\mathbbm{1})$ realises the
partial trace over the tags on $\rho_{b}$, while $P_{A_{1}}\quantity(X)\otimes
P_{A_{2}}\quantity(X^{-1})$ act on the $k$ copies of uncontracted spins in
$\rho_{E}^{\otimes k}$ implementing the replica trick and the partial
transposition respectively.

For the linearity of the trace, the expectation value can be carried out
before taking the trace and it will concern only the random tensors. From
(26), we have

$\begin{split}\overline{Z_{1}^{(k)}}&=\Tr[\rho_{b}^{\otimes
k}\otimes\rho_{E}^{\otimes
k}\,\quantity(\bigotimes_{v}\overline{\quantity(\ket{f_{v}}\bra{f_{v}})^{\otimes
k}})\\\ \quad\hfil&\cdot P_{A_{1}}\quantity(X)\otimes
P_{A_{2}}\quantity(X^{-1})\otimes P_{B}\quantity(\mathbbm{1})\Bigg{]}\,,\\\
\overline{Z^{(k)}_{0}}&=\Tr\quantity[\rho_{b}^{\otimes
k}\otimes\rho_{E}^{\otimes
k}\,\quantity(\bigotimes_{v}\overline{\quantity(\ket{f_{v}}\bra{f_{v}})^{\otimes
k}})]\,,\end{split}$ (37)

where the overall trace now runs over both bulk and boundary indices.

By Schur’s lemma, the average of the $k$ copies of the vertex state in (37)
results in a sum over unitary representations of the permutation group $g_{v}$
acting on the $k$ copies of the extended vertex space $K_{v}^{J}$ [71, 69,
60],

$\overline{\quantity(\ket{f_{v}}\bra{f_{v}})^{\otimes
k}}=\frac{\quantity(D_{v}-1)!}{\quantity(D_{v}+k-1)!}\sum_{g_{v}\in
S_{k}}P_{v}(g_{v})\,,$ (38)

with dimension $D_{v}=\text{dim}(K_{v}^{J})$.

By performing the average, individually on each independent random vertex, we
obtain444 $\overline{Z^{(k)}_{0}}$ has the same form with $X$ and $X^{-1}$
replaced by $\mathbbm{1}$.

$\overline{Z^{(k)}_{1}}=\mathcal{C}\Tr[\rho_{b}^{\otimes
k}\otimes\rho_{E}^{\otimes k}\,\quantity(\bigotimes_{v}\sum_{g_{v}\in
S_{k}}P_{v}(g_{v}))\\\ \quad\cdot P_{A_{1}}\quantity(X)\otimes
P_{A_{2}}\quantity(X^{-1})\otimes P_{B}\quantity(\mathbbm{1})\Bigg{]}\,,$ (39)

with
$\mathcal{C}=\prod_{v}\quantity[\frac{\quantity(D_{v}-1)!}{\quantity(D_{v}+k-1)!}]$.

Figure 5: Permutation operator $P_{v}(g_{v})$ action factorizes on the indices
of the extended vertex state. In the example in figure the operator acts as
the identity connecting the set of internal indices of $v$ to the same set in
$\bar{v}$, while $P_{B}\quantity(\mathbbm{1})$ realises the partial trace over
the tags.

On the bulk indices at each vertex, the permutation operator $P_{v}(g_{v})$
factorizes into _four_ independent operators (see fig. 5),

$P_{v}(g_{v})=P_{v,0}(g_{v})\,\otimes\bigotimes_{e^{i}_{vw}\in\mathcal{E}_{R}}P_{v,i}(g_{v})\\\
\otimes\bigotimes_{e^{i}_{v\bar{v}}\in\mathcal{E}_{\partial
R}}P_{v,i}(g_{v})\,\otimes P_{v,\tau}(g_{v})\,,$ (40)

where $P_{v,0}(g_{v})$ acts on $k$ copies of the multiplicity intertwiner
space, $\bigotimes_{e^{i}_{vw}\in\mathcal{E}_{R}}P_{v,i}(g_{v})$ acts on $k$
copies of the internal edges,
$\bigotimes_{e^{i}_{v\bar{v}}\in\mathcal{E}_{\partial R}}P_{v,i}(g_{v})$ acts
on the boundary semi-edges,555 We indicate by $\bar{v}$ a virtual vertex
connected to $v$ by the open boundary edge $e^{i}_{v\bar{v}}$. and finally
$P_{v,\tau}(g_{v})$ acting on $k$ copies of the _recoupled tag spin at the
vertex_.

Accordingly, the trace in (39) factorizes over the Hilbert spaces of a)
internal edges, b) boundary spins, c) tags spins and d) bulk intertwiners. As
a result, we can eventually rewrite the $k$th power of the normalized
partition function in (34) as

$\frac{\overline{Z^{(k)}_{1}}}{\overline{Z^{(k)}_{0}}}=\sum_{\\{g_{v}\\}}e^{-A^{(k)}_{1/0}}\,,$
(41)

where $A^{(k)}_{1/0}\equiv A^{(k)}_{1}/A^{(k)}_{0}$ defines the
(normalized)666with
$A^{(k)}_{0}=\sum_{e^{i}_{vw}\in\mathcal{E}_{R}}\Delta(g_{v},g_{w})\log
d_{j^{i}_{vw}}\\\ +\sum_{e^{i}_{v\bar{v}}\in A_{1}\cup
A_{2}}\Delta(g_{v},\mathbbm{1})\log d_{j^{i}_{v\bar{v}}}+A_{b}+\xi\,.$ (42)
action of a classical _generalized_ Ising-like model:

$A^{(k)}_{1}\big{[}\\{g_{v}\\}\big{]}=A_{b}+\xi+\sum_{e^{i}_{vw}\in\mathcal{E}_{R}}\Delta(g_{v},g_{w})\log
d_{j^{i}_{vw}}\\\ +\sum_{e^{i}_{v\bar{v}}\in A_{1}}\Delta(g_{v},X)\log
d_{j^{i}_{v\bar{v}}}+\sum_{e^{i}_{v\bar{v}}\in A_{2}}\Delta(g_{v},X^{-1})\log
d_{j^{i}_{v}}\,,$ (43)

with $\xi$ a constant term and

$A_{b}=-\log\\{\Tr[\rho_{b}^{\otimes k}\otimes
P_{B}\quantity(\mathbbm{1})\,\cdot\\\
\cdot\quantity(\bigotimes_{v}P_{v,0}(g_{v})^{\,}\otimes
P_{v,\tau}(g_{v}))\Bigg{]}\Bigg{\\}}$ (44)

the bulk state contribution to the action, which accounts for the presence of
curvature measured by the tags term in $\rho_{b}$. In the analogy, the
_generalized spins_ correspond to the permutation operators $P_{v}(g_{v})$
sitting on the vertices of the open spin network graph $\tilde{\gamma}_{R}$
acting on the $k$ replicas of the vertex indices space. The permutations
fields $X$, $X^{-1}$ and $\mathbbm{1}$ play the role of boundary conditions
set on the virtual one-valent vertices ($\bar{v}$) attached to the dangling
spins and tags of the extended boundary.

The two-body interaction terms measure the length of the permutation-loops
across the replicas, weighted by the (log of the) dimension of the
representation carried by the edges of the graph. These lengths are
conveniently described as Cayley distances
$\Delta\quantity(g,h)=k-\chi(g^{-1}\,h)$ on the permutation group $S_{k}$,
where $\chi(g)$ indicates the number of cycles in a permutation $g$ [69,
60].777This is $k$ for the identity and $1$ for a $k$-cycle. Then, we see that
$\Delta\quantity(g,h)$ gives the minimal number of swaps to go from $g$ to
$h$, for $g,h\in S_{k}$.

The calculation of typical value of Rényi negativity is then mapped to the
minimization of the action (43). In general, the three boundary conditions
will propagate inside the spin network graph $\tilde{\gamma}_{R}$. Depending
on the couplings and the connectivity of the graph, different configurations
of the model will be associated to different sets of domains for
$X,X^{-1},\mathbb{1}$ in the bulk. As usual, we expect maximal domains to be
associated to equilibrium configurations. In particular, maximal domains
correspond to minimal domain walls.

By describing interactions via a Cayley distance we see that the Rényi
negativity should depend directly on the _area_ of such minimal domain walls
in the network, given by the number of edges of $\tilde{\gamma}_{R}$
associated to a non-vanishing Cayley distance between permutations in the two
domains. In particular, thanks to the existence of a well defined loop quantum
gravity area operator acting on the edges of the spin network, we can map
domain walls areas to actual quantum geometry surfaces in our $3D$ space
region $R$.

##  VI Generalized Ising Model: modelling Bulk contributions

The tensor product form of the edges and bulk density matrices in (39)
corresponds to a factorization of the typical Rényi negativity into a
combinatorial contribution associated to the distribution of maximally
entangled edge states $e_{vw}$ and a bulk contribution carrying physical
correlations among intertwiners. The latter, in particular, contains the
information on the intrinsic curvature of the $3D$ geometry, in the form of
tags, as apparent from (21). We would expect such information to be highly
mixed in the bulk. However, after the partial tracing over the loop
holonomies, the $SU(2)$-covariance of the tagged-bulk state induces
$SU(2)$-invariance on the associated density matrix, which necessarily reads
as a tensor product for fixed tags spins $\\{J_{v}\\}$, that is

$\rho_{b}=\bigotimes_{v}^{V_{R}}\frac{\mathbbm{1}}{2J_{v}+1}\otimes\rho_{\\{J_{v}\\}}$
(45)

which corresponds to a totally mixed state on the tags space and a generally
non-trivial density matrix $\rho_{\\{J_{v}\\}}\in\text{End}[H_{V_{R}}]$ [84,
86].

From the form of $\rho_{b}$ in (45), we have

$\rho_{b}^{\otimes
k}=\bigotimes_{v=1}^{V_{R}}\quantity(\frac{\mathbbm{1}}{2J_{v}+1})^{\otimes
k}\otimes\rho_{\\{J_{v}\\}}^{\otimes k}\,,$ (46)

then the bulk term $A_{b}$ generally factorizes as follows

$\displaystyle A_{b}$
$\displaystyle=-\log\\{\Tr\left[\rho_{\\{J_{v}\\}}^{\otimes
k}\otimes\bigotimes_{v}\quantity(\frac{\mathbbm{1}}{2J_{v}+1})^{\otimes
k}\right.$ (47)
$\displaystyle\qquad\left.\otimes\,P_{B}\quantity(\mathbbm{1})\left(\bigotimes_{v}P_{v,0}(g_{v})^{\,}\otimes
P_{v,\tau}(g_{v})\right)\right]\Bigg{\\}}$
$\displaystyle\qquad=A_{\iota}+A_{\tau}$

with a bulk intertwiner contribution $A_{\iota}$ defined as

$\begin{split}A_{\iota}\equiv-\log\left\\{\Tr\quantity[\rho_{\\{J_{v}\\}}^{\otimes
k}\quantity(\otimes_{v}P_{v,0}(g_{v}))]\right\\}\,,\end{split}$ (48)

and a contribution of the tags, $A_{\tau}$, given by

$A_{\tau}\equiv-\log\\{\Tr[\bigotimes_{v}\quantity(\frac{\mathbbm{1}}{D_{J_{v}}})^{\otimes
k}\otimes\,P_{B}\quantity(\mathbbm{1})\,\cdot\\\
\cdot\,\big{(}\otimes_{v}P_{v,\tau}(g_{v})\big{)}\Bigg{]}\Bigg{\\}}\,,$ (49)

where we used $D_{J_{v}}=2J_{v}+1$, the dimension of the tag space. The result
of the trace in (49) is

$\quantity(\prod_{v\in\mathcal{V}_{R}}D_{J_{v}}^{-k})\Tr\quantity[\bigotimes_{\langle
v\bar{v}\rangle}\quantity(\sum_{g_{v}\in S_{k}}P_{v}(g_{v}))\otimes
P_{B}(\mathbbm{1})]\\\
=\sum_{\\{g_{v}\\}}\prod_{v}D_{J_{v}}^{-\Delta(g_{v},\mathbbm{1})}\,.$ (50)

Therefore, the tags contribution to the action reads

$A_{\tau}=\sum_{v}\Delta(g_{v},\mathbbm{1})\log D_{J_{v}}\,,$ (51)

which ends up being equivalent to a pairwise interaction term from an inner
edge. The full action can be written as

$\displaystyle
A^{(k)}_{1}=\xi+A_{\iota}+\sum_{e^{i}_{vw}\in\mathcal{E}_{R}}\Delta(g_{v},g_{w})\log
d_{j^{i}_{vw}}$ $\displaystyle+\sum_{e^{i}_{v\bar{v}}\in
A_{1}}\Delta(g_{v},X)\log d_{j^{i}_{v}}+\sum_{e^{i}_{v\bar{v}}\in
A_{2}}\Delta(g_{v},X^{-1})\log d_{j^{i}_{v}}$
$\displaystyle+\sum_{v}\Delta(g_{v},\mathbbm{1})\log D_{J_{v}}$ (52)

The bulk contribution $A_{\iota}$ is nontrivial as long as the bulk
intertwiners are correlated. In fact, it vanishes only if we consider
$\rho_{J_{v}}$ to be a product state in $\text{End}[H_{V_{R}}]$ as shown in
[60]. On the other hand, both edges and tags contributions have to do with the
topology of the graphs support $\tilde{\gamma}_{R}$, connectivity and
topological defects respectively.

As a general result of our analysis, we can then formally divide the action in
three terms as follows

$A^{(k)}_{1}=A_{\text{topology}}^{(k)}+A_{\text{phys}}^{(k)}$ (53)

with $A_{\text{topology}}^{(k)}=A_{\text{edges}}^{(k)}+A_{\text{tags}}^{(k)}$
and $A_{\text{phys}}=A_{\iota}$ referring to the physical quantum correlations
among bulk intertwiners.

In the following analysis, we will consider the case where the intertwiners
contribution is small, thereby looking at the correlations induced solely by
the combinatorial structure of the graph, with a focus on the role of the
tags. We leave the study of the intertwiners contribution in this setting for
future work.

##  VII Logarithmic Negativity and entanglement phases

By neglecting $A_{\iota}$, we are left with action $A_{1}^{(k)}$ with two-body
interactions only, which favours neighbouring “spins” to be parallel. Given
$d_{j}=2j+1$ and $D_{J}=2J+1$ the dimension of edge spins and tag Hilbert
space respectively, we define

$\log d_{j}\equiv\beta\quad,\qquad\log D_{J}\equiv\beta_{t}$ (54)

the interaction strengths intended as inverse temperatures, in analogy with
the Ising model. Thereby, we can rewrite the action
${A}_{\text{topology}}^{(k)}$ as

$\displaystyle{A}_{\text{topology}}^{(k)}=\beta$
$\displaystyle\Biggr{[}\sum_{e^{i}_{vw}\in\mathcal{E}_{R}}\Delta(g_{v},g_{w})+\sum_{e^{i}_{v\bar{v}}\in
A_{1}}\Delta(g_{v},X)$ $\displaystyle+\sum_{e^{i}_{v\bar{v}}\in
A_{2}}\Delta(g_{v},X^{-1})\Biggr{]}+\beta_{t}\sum_{v\in\mathcal{V}_{R}}\Delta(g_{v},\mathbbm{1})$
$\displaystyle=\beta\,H_{e}+\beta_{t}\,H_{\text{tags}}$ (55)

with generalised hamiltonians $H_{e},H_{\text{tags}}$ expressing the energy
cost of the configuration. In general, we expect that the boundary conditions
$X$, $X^{-1}$ and $\mathbb{1}$ percolate inside the network, flipping the
generalized spins on the bulk vertices. This creates some internal domains
filled with one of the three boundary conditions. Nonvanishing contributions
to the action come from edges that connect different domains, which will be
related to a non-zero Cayley distance. Given

$N_{k}(\rho_{A_{1}A_{2}})\simeq\sum_{\\{g_{v}\\}}e^{-A^{(k)}_{\text{topology}/0}}\,,$
(56)

in the large spin (the strong coupling or “low temperature”) regime, the
leading contribution to $N_{k}$ corresponds to dominant $\\{g_{v}\\}$
configurations that minimize the action. Such configurations correspond to
maximal uniform spin domains separated by extremal domain walls which minimize
the energy cost (minimal surface areas) [60].

Concretely, we then proceed minimizing the action in (55) via a heuristic
argument based on the property of the Cayley distance on the permutation group
$S_{k}$. Given two permutations $g,h\in S_{k}$, the Cayley distance
$\Delta(g,h)$ defines a metric on the group, corresponding to the minimum
number of swaps acting on $g$ to give $h$. For some special permutations, the
Cayley distance depends nicely on the order on the group $k$. For example, in
the case of the (anti-) cyclic permutations we deal with, we have

$\displaystyle\Delta(\mathbbm{1},X)=\Delta(\mathbbm{1},X^{-1})=k-1$ (57)
$\displaystyle\Delta(X,X^{-1})=\begin{cases}k-1,\quad k\,\text{odd}\\\
k-2,\quad k\,\text{even}\end{cases}$ (58)

With this metric, we can define a geodesic on the group $S_{k}$ between $g$
and $k$ as the set of permutations $\\{\pi\\}$ such that

$\Delta(g,\pi)+\Delta(\pi,h)=\Delta(g,h)$ (59)

In particular, it is known that the set of permutations that are
simultaneously geodesics between $X$, $X^{-1}$ and $\mathbb{1}$ exist, and it
is in a bijection with the set of non crossing partitions $NC(k)$ of the set
$\\{1,\dots,k\\}$, with cardinality given by the Catalan numbers $C_{k}$ [69].
Defining by $\tau$ the permutations on such geodesics, we have that

$\displaystyle\Delta(\mathbb{1},\tau)=\left\lfloor\frac{k}{2}\right\rfloor,\quad\Delta(X,\tau)=\Delta(X^{-1},\tau)=\left\lceil\frac{k}{2}\right\rceil-1$
(60)

Given the homogeneity of our states, the minimization of the energy costs
amounts to a minimization of the Cayley distances between any pairs of
connected vertices.

Let us consider a single vertex $v$. To such a vertex the random averaging
assigns a permutation element $g_{v}$, which will eventually interact with the
three permutations we choose as boundary conditions, namely $X$, $X^{-1}$ and
$\mathbb{1}$. Recalling the symmetry between $X$ and $X^{-1}$, we expect that
only three possible minimal energy configurations can result from such
interaction:

  * •

$g_{v}=X,X^{-1}$, in which case we minimize the distance between the vertex
$v$ and the neighboring vertices respectively contained in the $X$ or $X^{-1}$
domain;

  * •

$g_{v}=\mathbb{1}$, in which case we minimize the distance between the vertex
$v$ and the neighboring vertices contained in the $\mathbb{1}$ domain;

  * •

$g_{v}=\tau$, in which we minimize simultaneously the distance between the
$X$, $X^{-1}$ and the $\mathbb{1}$ domains.

Notice that the latter case is peculiar to the tripartite model only, since it
can occur only when all three permutations interact in a single vertex [60,
69, 67].

###  VII.1 Example: Cluster of four tags

Figure 6: Example: tagged open spin network graph corresponding to a cluster
of four-valent tagged vertices, with $T=4$ and $E_{\partial R}=4$.

Let us consider the example of an open spin network state with support on a
graph $\tilde{\gamma}_{R}$ with $E_{\partial R}=4$ boundary edges, and
$V_{R}=4$ tagged 4-valent vertices in the bulk (see Fig. 6). In the following
we indicate with $T$ the number of tags, which in the present case is equal to
the number of vertices, namely $T=V_{R}$. We refer to this setting as a
cluster of tags, corresponding to a uniformly curved quantum $3D$ space region
$R$. Via random averaging, we get four permutations operators on the spin
network, one for each tagged vertex. Following the discussion above, we have
three possible configurations with minimal energy. The corresponding actions
(for even $k>0$) reads

  1. 1.

cluster colored with $X(X^{-1})$:

$\mathcal{A}_{k}^{(X)}=\beta\,S\,(k-2)+\beta_{t}\,T\,(k-1)\,$ (61)

Here $S$ is the _minimal_ number of edges cuts comprising the surface
separating two domains with $X$ and $X^{-1}$.

  2. 2.

cluster colored with $\mathbb{1}$:

$\mathcal{A}_{k}^{(\mathbb{1})}=\beta E_{\partial R}\,(k-1)$ (62)

  3. 3.

cluster colored with $\tau$:

$\mathcal{A}_{k}^{(\tau)}=\beta E_{\partial
R}\,\left(\frac{k}{2}-1\right)+\beta_{t}T\,\frac{k}{2}$ (63)

Such configurations define the three equilibrium phases of our simple model,
corresponding to three different entanglement regimes for the boundary state
$\rho_{A_{1}A_{2}}$. The parameter space of such regimes is completely
specified by the following three inequalities:

$\displaystyle\mathcal{A}_{k}^{(\mathbb{1})}<\mathcal{A}_{k}^{(\tau)}\iff\beta
E_{\partial R}<\beta_{t}T$ (64)
$\displaystyle\mathcal{A}_{k}^{(\tau)}<\mathcal{A}_{k}^{(X)}\iff\beta
E_{\partial R}<\beta_{t}T+2\beta S$ (65)
$\displaystyle\mathcal{A}_{k}^{(X)}<\mathcal{A}_{k}^{(\mathbb{1})}\iff\beta
E_{\partial R}>\beta_{t}T+\beta S\left(1-\frac{1}{k-1}\right)$ (66)

where the last condition (66) is relevant to establish a hierarchy between the
three configurations, while it does not play a role in the search for the
minimal energy cost. Notably, this rules out from the parameter space the only
condition which depends explicitly on the order of the permutation group $k$.

Consider the (positive definite) quantity $\beta E_{\partial R}$, namely the
set of edges comprising the external surface of our $3D$ region weighted with
the factor $\beta$. If $\beta E_{\partial R}<\beta_{t}T$, as $2\beta S>0$,
also (65) is satisfied, then
$\mathcal{A}_{k}^{(\mathbb{1})}<\mathcal{A}_{k}^{(X)},\mathcal{A}_{k}^{(\tau)}$.
The configuration with a bulk colored with $g_{v}=\mathbb{1}\,\forall\,v$
dominates. In particular, this corresponds to the formation of a domain wall
around the cluster of tags. The domain wall sharply divides the spin network
boundary from the bulk of the region $R$. We refer to such a regime as the
_hole_ regime (see Fig. 7).

Figure 7: Domain walls configuration in the hole regime.

Notice that, due to the bound on the maximum dimension of the tag spin [74],
in the present example we can have at most

$J^{(max)}=4j\,\implies\,\beta_{t}^{(max)}=\beta+\log 4\,.$ (67)

Then, in a large spins regime, we can generally consider
$\beta_{t}^{(max)}\backsim\beta$ in facts. Thereby, in order for the _hole_
regime condition to be verified we must have

$T>E_{\partial R}\,.$ (68)

More generally, the hole regime will be favoured by tags recoupling a large
number of edge spins, so for coarse-grained vertices with high valence.

When the number of tagged vertices in the bulk and boundary edges is balanced
we exit the hole regime and we need to consider $\beta E_{\partial
R}\in[\beta_{t}T,\,\beta_{t}T+2\beta S]$. From eq. 64, we see that in this
range of values $\mathcal{A}_{k}^{(\tau)}<\mathcal{A}_{k}^{(\mathbb{1})}$. At
the same time, eq. 65 implies that
$\mathcal{A}_{k}^{(\tau)}<\mathcal{A}_{k}^{(X)}$ hence the minimal
configuration is the one in which the geodesic permutation $\tau$ takes all
the bulk, with the action $\mathcal{A}_{k}^{(\tau)}$ dominating. We call this
setting the _island_ regime. The island domain wall stands between the three
boundary conditions domains preventing them to access the bulk of $R$ (see
Fig. 8). Differently from the hole regime, the connectivity of the graph is
not canceled in this case. Tags and boundary spins are connected through the
island.

Figure 8: Domain walls configuration in the island regime.

Finally, for $\beta E_{\partial R}>\beta_{t}T+2\beta S$, namely when the tags
environment is small compared with the boundary system, from eq. 64 and eq. 65
it follows that
$\mathcal{A}_{k}^{(X)}<\mathcal{A}_{k}^{(\mathbb{1})},\mathcal{A}_{k}^{(\tau)}$.
In this regime, the minimal energy configuration is associated to an effective
bipartition of $R$ into two domains, one colored with the $X$ permutation and
the other with the $X^{-1}$ permutation (see Fig. 9).888The configurations in
which on all the vertices in $R$ there is $X$ or $X^{-1}$ belong to this
regime too. In these two extreme cases the domain wall $S$ is expelled from
the bulk, and invades some of the boundary, in particular the region
corresponding to the Hilbert space with minimal dimension among
$\mathcal{H}_{A_{1}}$ and $\mathcal{H}_{A_{2}}$, which in turns is equivalent
to consider the minimal area between the areas of the regions $A_{1}$ and
$A_{2}$. We refer to this setting as the bipartite regime.

###  VII.2 Entanglement Phases

We can now compute the typical logarithmic negativity of the random mixed
state $\rho_{A_{1}A_{2}}$ in the three regimes considered in the given
example.

In the hole regime, from the action in (62), we get

${E}_{N}^{(\text{hole})}(\rho_{A_{1}A_{2}})\simeq-\lim_{k\to
1}\mathcal{A}_{k}^{(\mathbb{1})}\\\ =-\lim_{k\to 1}\beta E_{\partial
R}(k-1)=0\,.$ (69)

A vanishing negativity tells us that the boundary state has Positive Partial
Transpose (PPT), hence, according to the Peres criterion [75], that boundary
state could be separable. This suggests that $\rho_{A_{1}A_{2}}$ is well
described by a maximally mixed state. The boundary system is almost fully
entangled with the large number of degrees of freedom environment with little
quantum correlation between $A_{1}$ and $A_{2}$. This effectively corresponds
to a thermalization of the boundary state, in strong analogy with the results
in [87, 72], where the typicality of the random state, in the large spin
regime, similarly played a crucial role.

Figure 9: Domain walls configuration in the bipartite regime.

Moving to the island regime, the _minimal_ action reads

$\mathcal{A}_{k}^{(\tau)}=\beta E_{\partial
R}\left(\frac{k}{2}-1\right)+\beta_{t}T\frac{k}{2}$ (70)

and the typical value of the logarithmic negativity in the $k\to 1$ limit is
given by

$\displaystyle E_{N}^{(\text{island})}(\rho_{A_{1}A_{2}})\simeq-\lim_{k\to
1}\left[\mathcal{A}_{k}^{(i)}+\log C_{k}\right]=$ $\displaystyle=-\lim_{k\to
1}\left[\beta E_{\partial
R}\left(\frac{k}{2}-1\right)+\beta_{t}T\frac{k}{2}+\log C_{k}\right]=$
$\displaystyle=\frac{1}{2}(\beta E_{\partial
R}-\beta_{t}T)+\log\frac{8}{3\pi}\,,$ (71)

with the Catalan numbers $C_{k}$ and $C_{1}=\frac{8}{3\pi}$ here taking into
account the degeneracy in the choice of the geodesic coloring $\tau$ for the
bulk.

We have then a transition from an unentangled phase to an area law behavior,
as expected by the topological character of the entanglement contributions. In
fact, it is possible to write $E_{N}^{(\text{island})}(\rho_{A_{1}A_{2}})$ as
a generalisation of the Ryu-Takayanagi formula for entanglement entropy [69].
By defining the minimal surfaces that separate the entanglement wedges
relative to each subsystem as $\absolutevalue{\gamma_{A_{1}}}$,
$\absolutevalue{\gamma_{A_{2}}}$ and $\absolutevalue{\gamma_{B}}$, it is
straightforward to see that

$\absolutevalue{\gamma_{A_{1}}}=E_{A_{1}}\,,\quad\absolutevalue{\gamma_{A_{2}}}=E_{A_{2}}\,,\quad\absolutevalue{\gamma_{B}}=T$
(72)

where $E_{A_{1}}$, $E_{A_{2}}$ are the number of edges in $A_{1}$ and $A_{2}$
respectively.

Hence, since $E_{A_{1}}+E_{A_{2}}=E_{\partial R}$ we can write eq. 71 as

$E_{N}^{(\text{island})}(\rho_{A_{1}A_{2}})=\frac{1}{2}\Bigr{[}\beta(\absolutevalue{\gamma_{A_{1}}}+\absolutevalue{\gamma_{A_{2}}})+\beta_{t}\absolutevalue{\gamma_{B}}\Bigr{]}\\\
+\log\frac{8}{3\pi}$ (73)

This is a generalisation of the RT formula found in [60] to the case of non-
homogeneous mixed spin network states (two temperatures). As for the previous
works, it can be expressed as a mutual information between the subsystem
$A_{1}$ and $A_{2}$:

$\displaystyle E_{N}^{(\text{island})}$
$\displaystyle=\frac{1}{2}\Bigr{[}S(\rho_{A_{1}})+S(\rho_{A_{2}})-S(\rho_{A_{1}A_{2}})\Bigr{]}+\log\frac{8}{3\pi}$
$\displaystyle=\frac{1}{2}I(A_{1}:A_{2})+\log\frac{8}{3\pi}$ (74)

In the island regime, then, the logarithmic negativity is _always_ non-zero,
and scales with the total area of the boundary of $R$, namely $\beta
E_{\partial R}$, minus a negative correction that depends linearly on the
number of tags. This implies that the boundary state is necessarily entangled.
In particular, the presence of the environment reflects in a saturation of the
entanglement of the two boundary subsystems. The logarithmic negativity only
depends on the size of the system and of the environment but it is independent
of the specific bipartition of the boundary.

Finally, in the bipartite regime, the action with minimal energy is given by

$\mathcal{A}_{k}^{(X)}=\beta S(k-2)+\beta_{t}T(k-1)\,$ (75)

and the typical logarithmic negativity reads

$\displaystyle E_{N}^{(\text{bipartite})}(\rho_{A_{1}A_{2}})\simeq-\lim_{k\to
1}\mathcal{A}_{k}^{(X)}=$ (76) $\displaystyle=-\lim_{k\to 1}\left[\beta
S(k-2)+\beta_{t}T(k-1)\right]=$ $\displaystyle=\beta S$

In this regime, negativity scales with the area of the internal domain wall
$S$. In particular, we see from (76) that in this regime we could achieve a
PPT state for the boundary only by setting $S$ to 0, where a PPT state would
correspond to a disconnected graph as expected.

Such a behaviour is consistent with a limit toward a bipartite _pure_ boundary
state setting corresponding to vanishing tags (trivial topology) [63].

Altogether, the effective topology of the bounded region is reflected in the
multipartite entanglement of its _tagged_ spin-networks state description.
Beside the toy model example, the proposed analysis can be easily generalised
for graphs with generic number of vertices, with more than one open edge
attached to each boundary vertex and with clusters of tags restricted to a
subregion of the bulk graph. The universal character of the result is indeed
associated to the typical behaviour of the randomised system in the large spin
regime.

##  VIII Discussion

In the framework of loop quantum gravity, quantum geometry states
corresponding to _bounded_ regions of $3D$ space with _uniform curvature_ can
be synthetically described via superpositions of _tagged_ spin-networks with
boundary. Curvature naturally builds up at the spin-network vertices in the
form of topological defects described by spin tags, as the result of the
_partial tracing_ over nontrivial $SU(2)$ holonomies of the Ashtekar
connection along the network.

In presence of a boundary, a quantum geometry wave-function generally behaves
as a map encoding the bulk information into the boundary Hilbert space. In the
proposed analysis, we generalize such a bulk-to-boundary mapping as to include
the space of bulk topological defects, effectively described as extra (inner)
boundary degrees of freedom.999In fact, we can think of bulk tags as the
recoupled dangling bulk spins dual to the surface of holes cut out of planar
spin network graph. In the resulting _extended boundary_ state, bulk
information is _shared_ among generically entangled boundary spins and tags:
the surface and the intrinsic curvature of the quantum space region are
entangled. Remarkably, the degree of quantum correlations among boundary
subregions is affected by the presence of the bulk curvature, which plays the
role of a hidden environment for the mixed outer boundary state.

The effect of the curvature on the boundary entanglement is the focus of our
analysis. We model the generalised boundary mapping on a tripartite
($A_{1},A_{2},B$) system and consider the reduced boundary state
$\rho_{A_{1}A_{2}}$ obtained via a trace over the space of tags (system $B$).
We further characterise the coarse viewpoint of a boundary observer by
considering a _random measurement_ on the bulk state, thereby ultimately
dealing with a _random mixed state_ for the boundary.

The entanglement of the bipartite mixed boundary is quantified via a measure
of logarithmic negativity, along the lines of the recent results in [69, 72,
60]. Concretely, we restrict the analysis to a class of states characterised
by graphs coloured by the same spin $j$ on each edge and dressed with a tag of
corresponding recoupled spin $J$ at each vertex. We call this setting a
_cluster of tags_. The full computation is carried on for a simple cluster of
four tags is given in Section VII.1. By using standard replica techniques for
random tensor network, this computation is ultimately mapped to the evaluation
of the minimal action of a classical generalised Ising-like statistical model
on the spin network graph, in the limit of vanishing temperature (see [60] and
references therein).

As a first general result, we find that the Ising-like model action dual to
the random spin network generally splits into two main contributions:

$A^{(k)}_{1}=A_{\text{topology}}^{(k)}+A_{\text{phys}}^{(k)}\,$ (77)

that is a contribution $A_{\text{topology}}^{(k)}$ which accounts for the
entanglement induced by the _topology_ of the spin network graph and a second
one, $A_{\text{phys}}^{(k)}$, encoding the bulk intertwiner entanglement. The
former, in particular, further splits into two well distinct contributions,

$A_{\text{topology}}^{(k)}=A_{\text{edges}}^{(k)}+A_{\text{tags}}^{(k)}\,,$

respectively associated to the _connectivity_ of the bulk network (trivial
topology), induced by maximally entangled edge states, and to the presence of
the tags. The last term is therefore specific to the case of spin network
states with topological defects.

In a topologically trivial setting, as showed in [60], the first term in (77)
is associated to an area law scaling of the entanglement of the boundary,
while bulk intertwiners entanglement can be interpreted in analogy with [67].
We leave the characterisation of the random bulk contribution for future work,
while focusing on the nontrivial topological contribution to the action.

Within the typical regime, the entanglement phases of the random boundary
state can be described solely in terms of the relative dimensions of the three
subsystems $A_{1}$, $A_{2}$ and $B$,101010Where
$\text{dim}(A_{1})=(d_{j})^{E_{A_{1}}}$,
$\text{dim}(A_{2})=(d_{j})^{E_{A_{2}}}$ and $\text{dim}(B)=(d_{J})^{T}$
respectively. via the parameter

$q=\beta_{t}T/\beta E_{\partial
R}=\frac{\log[\text{dim}(B)]}{\log[\text{dim}(A_{1})\text{dim}(A_{2})]}\,,$
(78)

which expresses the ratio of bulk curvature over boundary surface, with
$E_{\partial R}=E_{A_{1}}+E_{A_{2}}$. Two main entanglement phases are
separated by a critical point at $q=1$.

For $1-2S/E_{\partial R}<q<1$, the curvature environment mediates the
entanglement of the boundary. The logarithmic negativity, modulo degeneracy,
scales with the area of the cluster boundary with a negative correction which
depends linearly on the number of tags. We have

$\displaystyle E_{N}^{(\text{island})}(\rho_{A_{1}A_{2}})$
$\displaystyle\propto\frac{1}{2}\beta E_{\partial R}(1-q)$ (79)
$\displaystyle=\frac{1}{2}\Bigr{[}\beta(\absolutevalue{\gamma_{A_{1}}}+\absolutevalue{\gamma_{A_{2}}})+\beta_{t}\absolutevalue{\gamma_{B}}\Bigr{]}$

with $\absolutevalue{\gamma_{A_{1}}}=E_{A_{1}}$
,$\absolutevalue{\gamma_{A_{2}}}=E_{A_{2}}$ and
$\absolutevalue{\gamma_{B}}=T$, the minimal surfaces separating the
entanglement wedges relative to each subsystem. This is a generalisation of
the Ryu-Takayanagi formula for entanglement entropy for the tripartite
setting, in agreement with [60, 69] and it can be expressed as a mutual
information between the subsystem $A_{1}$ and $A_{2}$.

Figure 10: In red the generalized Page curve of the logarithmic negativity for
the random mixed state $\rho_{A_{1}A_{2}}$. In red, how the Page curve would
appear for random pure states.

Finally, for $q<1-2S/E_{\partial R}$, when the curvature environment is much
smaller than the boundary system, we tend to a bipartite setting and the
typical logarithmic negativity scales with the area of the surface $S$
separating $A_{1}$ and $A_{2}$ through the boundary. In this case, we find

$E_{N}^{(\text{bipartite})}(\rho_{A_{1}A_{2}})\propto\beta S\,.$ (80)

Notice that the value of $S$ cannot be tuned in our analysis, as the surface
is fixed by the spin network connectivity for the given graph. However,
assuming some regularity of the graph, one can geometrically relate this value
to the bipartition of the boundary. For instance, assuming a generic tagged
graph with a large density of vertices, dual to a 3-ball, we can imagine the
minimal surface $S$ to be approximated by the intersection of the sphere and a
generic plane, which divides $A_{1}$ from $A_{2}$ on the boundary of $R$. In
this case, we can approximately set

$S\approx j^{2}\frac{E_{A_{1}}E_{A_{2}}}{E_{\partial R}}$ (81)

Therefore, we see that the island condition is favoured when the dimension of
the two subsystems $A_{1}$ and $A_{2}$ is well balanced, since $S$ is
maximized by high values of both $E_{A_{1}}$ and $E_{A_{2}}=E_{\partial
R}-E_{A_{1}}$. Conversely, from the condition $q<1-2S/E_{\partial R}$, the
bipartite regime is favoured when the two boundary subregions $A_{1}$ and
$A_{2}$ are extremely asymmetric in extension (smaller $S$).

In particular, we can describe the transition from the island to the bipartite
regime by using (81) in the condition
$\mathcal{A}_{k}^{(bipartite)}<\mathcal{A}_{k}^{(island)}$. We find

$\left(\frac{E_{A_{1}}}{E_{\partial
R}}\right)^{2}-\frac{E_{A_{1}}}{E_{\partial R}}+\left(1-q\right)>0$ (82)

Thereby, for $0<q<1$, we can identify two critical points between the island
and the bipartition phases,

$c^{*}_{1,2}(T)=\frac{1}{2}\pm\frac{1}{2}\sqrt{q}\,.$ (83)

The behaviour of the negativity is well described by the generalized Page
curve in fig. 10. As a general feature of a tripartite random state, we find
that the degree of correlation of the two boundary subregions $A_{1},A_{2}$
depends on the degree of quantum correlations the two boundary systems have
with the curvature environment. By varying the relative size of the two
subsystems, the logarithmic negativity shows an initial increase and a final
decrease in analogy with the Page curve. When $T=0$ (trivial topology),
$c^{*}_{1}(0)=c^{*}_{2}(0)=1/2$ and we get back the standard Page curve [88]
(no tripartition). As soon as $T\neq 0$, the intermediate island regime forms
and the negativity has a plateau. In this regime the logarithmic negativity
only depends on the size of the system and of the environment but not on how
the system is partitioned.

On the other hand, for $q>1$, corresponding to a large tags environment with
respect to the boundary system, the logarithmic negativity vanishes. The
boundary system is almost fully entangled with the environment with little
quantum correlation between $A_{1}$ and $A_{2}$. In particular, the area law
behaviour is lost. This suggests that large curvature is associated to a
_maximally_ mixed boundary state $\rho_{A_{1}A_{2}}$. Such a thermalization of
the boundary due to curvature is in strong analogy to the quantum gravity
formulation of the hoop conjecture proposed in [87], where again the
typicality of the random state, in the large spin regime, plays a crucial
role.

If we consider that the large spin regime we have been working in is
necessarily associated with a semiclassical limit for our quantum geometry
states, then we can tentatively interpret the formulas found for the
entanglement negativity in terms of areas of a bounded $3D$ region of a
Riemannian manifold and look for a direct relation with its Ricci curvature.
Indeed, we know that the value of the scalar curvature $\mathcal{R}$ of a
Riemannian $n$-manifold $\mathcal{M}$ at a point $p$ can be quantified by the
ratio of the area of the $(n-1)$-dimensional boundary of a ball of radius
$\varepsilon$ in $\mathcal{M}$ to that of a corresponding ball in the
Euclidean space. For small $\varepsilon$, the ratio is given by [89, 90]

$\frac{\text{Area}(\partial
B_{\varepsilon}(p)\subset\mathcal{M})}{\text{Area}(\partial
B_{\varepsilon}(0)\subset\mathbb{R}^{n})}=1-\frac{\mathcal{R}}{6n}\varepsilon^{2}+O(\varepsilon^{3})\,.$
(84)

Now, in the semiclassical limit, for $n=3$, we can easily associate the area
of the boundary of a flat three ball region with the sum of the areas of the
triangles dual to the open edges spins comprising the boundary of $R$ (no
tags), that is $\text{Area}(\partial B_{\varepsilon}(0))\simeq\beta
E_{\partial R}(\varepsilon)$, where $E_{\partial R}(\varepsilon)$ counts the
number of boundary edges at a certain radius $\varepsilon$, now measured via
some distance on the spin network graph. In case of the curved three ball in
$\mathcal{M}$, however, we expect this value to be modified by the presence of
tags. In particular, if we assume

$\text{Area}(\partial B_{\varepsilon}(p))\simeq\beta E_{\partial
R}(\varepsilon)+\kappa\,\beta_{t}T(\varepsilon)\,$ (85)

namely a linear correction in the number of tags (positive or negative
depending on the sign of $\kappa$), where $T(\varepsilon)$ indicates the
number of tags within a certain radius $\varepsilon$. Thereby, the ratio in
(84) gives us a direct relation

$q(\varepsilon)\simeq-\frac{\mathcal{R}}{18\,\kappa}\varepsilon^{2}$ (86)

between the entanglement phase parameter and the Ricci curvature. For
instance, in this light, we find a nice relation between the area scaling
negativity (or mutual information) in (79) and the geometric relation in (84).
Such a relation is consistent with the interpretation of our results and hints
toward a characterisation of the curvature in purely information theoretic
terms [91].

Let us then conclude with a few further remarks. A first remark is in order
concerning the non-gauge invariant character of the degrees of freedom
considered in our analysis. Spin network’s tags are not different from
boundary dangling spins, whose presence indeed breaks gauge invariance at the
boundary. The entanglement we describe among boundary spins and tags cannot be
measured in terms of gauge invariant observables, hence it is considered as
non physical in loop quantum gravity [92]. Nevertheless, we expect such a
topological layer of entanglement to become essential in modelling the
emergence of spacetime connectivity as soon as we allow both edge and tags
spins to be physical at some higher energy scale, where we expect gauge
invariance to break. This perspective would require an extension of the
kinematical description of quantum geometry in loop quantum gravity, which is
in part provided by the second quantized formalism of group field theory [20].
Differently, one can take the present analysis as a toy model for an ultimate
description of the entanglement structure of _edge modes_ degrees of freedom
at the interface of subsystems [93, 94, 95]. In this sense, we expect the
described universal features of the interface entanglement in the large spin
regime to be general.

Figure 11: Milne spacetime sliced by $3D$ surfaces with hyperbolic embedding.
Close to Milne’s big bang, the quantum realisation of such surfaces have
vanishing negativity, $E_{N}(\rho_{A_{1}A_{2}})\to 0$. As time increases, the
negativity of the $2D$ cosmological horizon grows and tends to bipartite
entanglement entropy, $E_{N}(\rho_{A_{1}A_{2}})\to S(\rho_{A_{1}A_{2}})$ at
large times.

A final remark has to do with the interplay of intrinsic and extrinsic
geometry in loop quantum gravity. As showed in [83, 96], in loop quantum
geometry states, non-trivial Ashtekar-Barbero holonomies can be thought
alternatively as intrinsic geometry or _extrinsic curvature_. In particular,
it is possible to compensate the defects of the intrinsic geometry appearing
while coarse-graining spin networks by Lorentz boosts which change the local
embedding of the spin network description of quantum $3D$ space in the
$(3+1)$-D geometry. This suggests a natural relation between embedding and
entanglement to which we shall dedicate further investigation in future work.
For instance, one could figure the phase transition previously described,
between a maximally entangled $(q\ll 1)$ to a maximally mixed $(q\gg 1)$
boundary state, as being described by a sequence of quantum $3D$ spacelike
slices with a gradient in curvature. If we embed such slices, via boost, in
the hyperbolic space, we would get a transition from almost flat bounded
regions with large corner surface corresponding to a maximally entangled
boundary state to highly curved regions with small corner surface
corresponding to maximally mixed boundary state. A similar setting naively
recalls the geometry of the _Milne spacetime_ cosmology [97] (see Fig. 11).
Intriguingly, in this analogy, the proposed analysis would suggest that, close
to the big bang, quantum $2D$ boundary surfaces would be described by
maximally mixed (or thermal) states, while as the universe expands and
flattens, quantum correlations between subregions of the surface would
increase. An accurate analysis of a similar scenario is left for future work.

###### Acknowledgements.

The authors would like to thank Alioscia Hamma and Daniele Oriti for useful
discussions on the preliminary results of the work. G.C. would like to thank
in particular Michele Arzano for suggesting the relation between a set of
boost embedded quantum 3D space regions and Milne geometry.

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