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Superposed Random Spin Tensor Networks and their Holographic Properties
Simon Langenscheidt

Munich 2022

Superposed Random Spin Tensor Networks and their Holographic Properties

Master’s thesis
Faculty of physics
Ludwig–Maximilians–University
Munich
submitted by
Simon Langenscheidt
from Munich
Munich, August 27, 2024

Acknowledgements I wish to express my gratitude to

  * •

My supervisors Dr. Daniele Oriti and Eugenia Colafranceschi for their
continuous support, feedback and direction during this project, as well as
providing me with a PhD student position.

  * •

Professor Ulrich Schollwöck for his interest in our work and being the second
referee for this thesis’ defense.

  * •

My many university colleagues from the physics and TMP program as well as the
geometry groups for many helpful discussions and moral support.

  * •

My family and particularly my parents for their support throughout the years
it took me to get here.

Supervisor: Daniele Oriti

We study criteria for and properties of boundary-to-boundary holography in a
class of spin network states defined by analogy to projected entangled pair
states (PEPS). In particular, we consider superpositions of states
corresponding to well-defined, discrete geometries on a graph. By applying
random tensor averaging techniques, we map entropy calculations to a random
Ising model on the same graph, with distribution of couplings determined by
the relative sizes of the involved geometries. The superposition of tensor
network states with variable bond dimension used here presents a picture of a
genuine quantum sum over geometric backgrounds. We find that, whenever each
individual geometry produces an isometric mapping of a fixed boundary region
$C$ to its complement, then their superposition does so iff the relative
weight going into each geometry is inversely proportional to its size.
Additionally, we calculate average and variance of the area of the given
boundary region and find that the average is bounded from below and above by
the mean and sum of the individual areas, respectively. Finally, we give an
outlook on possible extensions to our program and highlight conceptual
limitations to implementing these.

###### Table of contents

  1. 0 Introduction
    1. 1 Spin network states as models of quantum geometries
      1. 1 Quantisation of a single tetrahedron
      2. 2 Lattice gauge theories and their loop representation
      3. 3 Spin network states in canonical Loop Quantum Gravity
      4. 4 Spin foam models and Group Field Theories
      5. 5 Spin network states as tensor networks
    2. 2 Holography and (Random) tensor networks
      1. 1 Random tensor networks
    3. 3 Bulk-to-boundary holography for single spin sectors
      1. 1 Bulk-to-boundary isometry
  2. 1 RSTN with multiple spin sectors
    1. 1 Superposition of spins
      1. 1 Calculating average entropy
      2. 2 The random Ising model
      3. 3 Characterising holographic states
      4. 4 Example calculation
    2. 2 Averages of observables
  3. 2 Extensions and Limitations
    1. 1 Alternative distributions
    2. 2 Superposition of graph structures
    3. 3 GFT Dynamics in holographic calculations
      1. 1 The free field theory as weak imposition of constraints
      2. 2 Expressing the trace as a GFT integral
      3. 3 Extension of the spin foam picture
      4. 4 Implementing dynamically distinguishable quanta
  4. 3 Conclusion
  5. 4 Appendix
    1. 1 Average over all tetrahedra
    2. 2 Fixed-spin isometry
    3. 3 Example calculations

##  Chapter 0 Introduction

Within approaches to providing quantised UV completions to general relativity
(GR), there are several distinct directions. Some prefer to stay in the
continuum, such as Asymptotic Safety, some discretise degrees of freedom with
the intent of recovering a continuum regime, as in Loop Quantum Gravity(LQG),
while yet others recover low-energy phenomenology in more indirect ways, like
through consistency conditions and effective actions in String Theory.
However, certain features of gravitational theories are shared among them.
From general thermodynamical and information theoretical arguments, one
expects black holes, quantum or not, to fulfil a Bekenstein-Hawking bound10

$S\leq\frac{A}{4G_{N}}$ (1)

for an appropriately chosen entropy. This states, through microstate counting
interpretations of entropies, that the number of degrees of freedom of a black
hole should not scale with volume, as it would in most systems, but rather
with the area bounding said volume. This implies a form of so-called
(informational) holography: Information on the degrees of freedom of the
system can be or is encoded on the boundary of a region, from where it may be
recovered. This is termed after optical holography, where volumetric
information (phase differences) can be encoded on a photoplate boundary, from
where it may be reconstructed using the right techniques, displaying the
original 3-dimensional image from a 2-dimensional data carrier.
Other forms of more concrete holography have been developed which feature a
more direct relation to gravity: Starting from Maldacena31, String theorists
were able to construct several examples of gravitational systems in Anti-
deSitter spaces whose degrees of freedom, when suitably mapped to the
conformal boundary, give rise to a set of operators generating a conformal
field theory. This AdS/CFT correspondence has also been extended to 3D
euclidean gravity or asymptotically flat, 4D spacetimes24; 23, where the
asymptotic symmetry is given by variants of the BMS algebra, extending
conformal symmetry. What all of these examples have in common is that they
describe a relation between a bulk gravitational theory and a seperate set of
degrees of freedom on an asymptotic boundary, describing the same physics.
However, a different type of holography for finite regions of space/spacetime
appears to exist in a variety of contexts, in particular through entropy
bounds. For example, recent work in classical gravity suggests that corner
charges of GR provide an encoding of bulk information20;
freidelEdgeModesGravity2020a; 22, which applies to any finite region of space
with boundary. In condensed matter physics, the ground states of local
Hamiltonians on lattice systems are often found to be short-range entangled15:
The entropy of any finite region scales at most as the size of its boundary.
States with such properties are highly desirable: They provide a more
manageable subset of the space of states to begin searches for ground states
in, but also have properties like exponentially decaying correlations between
regions mimicking a local lightcone structure through Lieb-Robinson bounds32;
11.
As such, given the many insights from local and global holography in quantum
gravity (QG) contexts, it is natural to characterise classes of quantum
geometries which feature this behaviour on a small scale. A natural starting
point of this investigation are spin network states, which form both a
conceptual and concrete basis of spatial quantum geometry. These states were
originally envisioned by Penrose40; 1 and later recovered in the LQG canonical
quantisation programme44. They also feature in any spin foam approach to
constructing QG path integrals41 as well as their completions in Group Field
Theories (GFT)28; 35; 37 as basis states of discretised geometries with
definite area and volume values.
The goal of this work is to elaborate on criteria for holographic mappings
between patches of the boundary of a finite spatial region to exist. This
region is modeled by a superposition of spin network states. So far, only spin
network states with a fixed graph structure and definite, fixed spin values
have been considered, where areas take definite values. This thesis extends
existing results to a superposition of different spin values, covering a much
larger class of states. In this way, the work presented here provides a first
step into studying generic local holography in multiple QG approaches, with
possible applications to semiclassical states where a large number of
different spin network states are superposed.
The general structure of this work is as follows: After a brief introduction
of spin networks, their structure as glued vertex states and tensor networks,
we review notions of holography and the work done in the ’no-superposition’
case. We then perform a similar analysis on states which are superpositions of
individual tetrahedra with different areas, but fixed given glueing
arrangement. Importantly, we make a typicity111By a typicity statement we mean
propositions of the form ”The typical object from class X has property P”, as
given by a probability distribution. Such a distribution, for simplicity,
might be approximated by a Gaussian, in which case the main parameters are the
mean and variance. Then, a typicity statement is better the lower the variance
is, as the probability of finding an object in class X with the given property
deviating from the mean is very low.  statement about holographic properties
of what we will call Random Spin Tensor Networks (RSTN). We find that it is
convenient to write the purity of boundary states as a probability average in
a distribution of geometries, determined by relative weights of the individual
geometries in the state. We illustrate a few examples and determine a
condition on said distribution to ensure holography. A short calculation of
the area value of a holographic boundary region reveals a simple formula for
these special superpositions. Finally, we discuss some natural extensions of
the superposition of spins setting and highlight limitations in pursueing
these.

###  1 Spin network states as models of quantum geometries

To start, we review the notion of spin networks as abstract combinatorial
objects, as well as their use in labeling states in various fields of physics,
particularly in relation to lattice gauge theory and quantum gravity. In doing
so, we introduce the key concept of a spin network state.
For a given compact Lie group $G$, typically chosen to be $SU(2)$, a standard
definition of a spin network can be given as follows: It consists of

  1. 1.

A graph $\gamma$ with vertex and edge sets $(V,E)$.

  2. 2.

A colouring of its edges by representations $\rho$ of $G$.

  3. 3.

A colouring of its vertices by intertwining maps $\iota$ of the
representations on the adjacent edges. Equivalently, these are invariant
tensors $\text{Inv}_{G_{Diag}}(\bigotimes_{\alpha}V_{\rho_{\alpha}})$ in the
tensor product of representation spaces associated to the edges at the vertex.

Figure 1: Example of a simple spin network labeling a state. Links are labeled
by representations of $SU(2)$, here in terms of their dimension $d_{j}$.
Intertwiners are situated at vertices, but not specified graphically.

It is worth emphasizing the question of boundary edges. To allow a spin
network to have a boundary, one either allows the edge set to contain
semiedges/semilinks starting from any vertex, or introduces a colouring of the
graph by adding a second vertex set of boundary vertices. In the former, the
semiedges do not end on a vertex, meaning we do not work with graphs in the
usual sense. The latter amends this by attaching a 1-valent vertex to every
boundary semilink. Then, a boundary edge/link is identified by its end being a
boundary vertex. Practically, these might be distinguished by a coloring of
the vertices into grey and red. Since these two variants are translateable
into each other, we will at times switch between them when convenient.
Additionally, boundary semiedges should be labeled by relevant vectors from
the representations associated to these edges. For the example of $SU(2)$,
this will typically be an azimuthal quantum number $n\in[-j:j]$ labeling the
standard basis of the spin-$j$ representation.

Figure 2: An example graph with boundary/bulk colouring in grey/red. Removing
the grey vertices results in an open graph with semiedges.

This definition can be generalised and restricted in many different ways46; 4.
In particular, one might restrict the set of graphs to be of a fixed valency
or to be bipartite or oriented. On the other hand, one may replace the compact
Lie group $G$ by other objects with a similar representation theory, such as
quantum groups or even Hopf algebras. Furthermore, one may furnish the graph
with some imbedding into a manifold or additionally with a framing.
Historically, the notion of spin networks was invented by Penrose40; 1 as a
suggestion that continuum structures such as Euclidean space and continuous
probabilities might be limiting cases of discrete substructures. In
particular, a set of 3-valent $SU(2)$-spin networks with a large number of
boundary links were shown to admit limits in which they describe continuum
angles in 3-dimensional Euclidean space. However, they have since found use in
a variety of contexts independent of these ideas. Before presenting these
contexts, we will present an independent motivation for them as
representatives of quantum (twisted) simplicial geometry by starting from the
quantisation of a single tetrahedron. We believe that this is the most direct
way to reach spin network states without appealing to specific problems in
other approaches.

####  1 Quantisation of a single tetrahedron

The goal of this section is to quantise the classical degrees of freedom of a
3-simplex and its embedding into a larger simplicial complex by analogy to the
standard quantisation of a point particle.9 We will in this way produce a
first idea of quantum geometry which naturally admits spin network states as a
basis.

Figure 3: A simple example of the type of classical geometry we wish to
quantise. 5 abstract simplices are glued along their faces. The geometric
properties of the full space are given by the areas of the faces (here labeled
by A) and a connection for transporting local frames around. Such a transport
is exemplified along the white line crossing the face between the central
simplex and a peripheral one: The connection is flat inside the simplices,
meaning that transport is trivial before and after crossing the face. At the
face, the initial frame $e$ is rotated into $e^{\prime}$ using a group element
from the rotation group. Flatness on interiors implies that these types of
connections are specified completely through a group element on each face.
These connections together with the areas specify a metric on the entirety of
the simplices and thus the whole space.
Importantly, the space is not smooth, but still piecewise linear.

In the classical realm, we consider spaces with some discrete geometric
features: It shall be a space made by attaching tetrahedra to each other at
their boundaries, giving the faces area values and providing a connection or
parallel transport of local frames in the space. Such geometries are sometimes
referred to as twisted21. Each tetrahedral cell will be supposed flat in its
interior, so the parallel transport is trivial, while transport across faces
will induce nontrivial rotations of frames.
Additionally, each face of the cells will have an area value, which, together
with the flatness of the interior, will allow one to construct a unique metric
and geometry on the whole cell. Providing areas for the faces induces a metric
on each cell through embeddings into Euclidean space. An embedding of a
classical tetrahedron into Euclidean space may be described easily by its 4
normal vectors $J_{\alpha}\in\mathbb{R}^{3}$ ($\alpha\in[1:4]$), each with
length given by the area of its associated equilateral face. These 4 normals
are not independent, for they satisfy

$\sum_{\alpha}\vec{J}_{\alpha}=0$ (2)

which is conventionally referred to as the geometric closure constraint.
Notably, we do not impose the nondegeneracy condition that the normal vectors
should not be collinear or coplanar. In some states, the tetrahedron might
rather look like a triangle or even line.
This embedding defines a metric on the tetrahedron. There may eventually be
nontrivial glueings with other tetrahedra across faces. If we consider a local
coordinate vector frame $\\{\vec{e_{i}}\\}$ at the center of it, we may
describe the nontrivial embedding into $\mathbb{R}^{3}$ by how this frame is
transported to the faces - a flat geometry will have trivial transport. A
frame such as this may only be rotated into another frame by this operation,
so when transporting a frame from the interior of a tetrahedron to another’s,
we will have an associated
$\overrightarrow{e^{\prime}_{i}}=\overrightarrow{R_{f}e_{i}}$ with rotations
$R_{f}\in SO(3)$.
So, we consider the state of a classical tetrahedral cell in our complex to be
given by the 4 parallel transports across its faces, as well as the normals to
them. This encodes how the simplices of different sizes and orientations,
which are intrinsically all identical, are glued together to form a larger
space. In a way, we are prescribing the geometric data attached to an
immersion

$\Delta_{3}\hookrightarrow\Sigma$ (3)

of a 3-simplex into the 3-dimensional space $\Sigma$ we construct, as well as
an embedding of each simplex into Euclidean space. By providing said data, we
can then actually build the space $\Sigma$ just from the abstract simplices
alone. Quantising the data then leads to a quantum version of the same space.
The quantisation procedure proceeds in analogy to that of a point particle:
The classical phase space of a particle is given by $P=T^{\ast}Q$, the
cotangent bundle of the configuration space $Q$. The algebra of quantities of
interest is given by functions on $P$, equipped with the Poisson bracket
induced by the canonical symplectic structure on $P$. A particular focus lies
on the subalgebra of polynomials in some coordinates on $P$, such as the
positions $q$ and momenta $p$. Quantisation realises/represents (a deformation
of) this Poisson algebra as operators on a separable Hilbert space of states
$\mathbb{H}$. There are in principle inequivalent representations of this,
which turn out to all be unitarily equivalent to one another. One typical
choice is to let $\mathbb{H}=L^{2}(Q)$ and represent polynomial functions
$f(x,p)$ by multiplication and derivative operators, respectively. Another,
called the Segal-Bargmann representation, uses different starting coordinates
on $P$, $(z,q)$ where $z=x+ip$ is a complex coordinate. The Hilbert space in
question is then $L^{2}_{hol}(P)$, where only holomorphic functions of $z$ are
used222Technically, there is an additional Gaussian weight $e^{-|z|^{2}}$ in
the inner product, which is irrelevant to our discussion. We will quantise the
tetrahedron by taking the normal vectors $J$ as coordinates analogous to $p$,
while the role of $q$ will be played by the parallel transports. These group
elements (we will choose them rather from the rotation group’s double cover
Spin(3) = SU(2)) play the role of coordinates on $Q=SU(2)^{4}$. The usefulness
of this follows from the differential geometric fact that $T^{\ast}G\cong
G\times\mathfrak{g}$ for any Lie group, where $\mathfrak{g}$ is the Lie
algebra of the group. See each normal vector as an element of
$su(2)\cong\mathbb{R}^{3}$. From this, we may identify the 4 normals as giving
half of the phase space, corresponding to momenta. As soon as we glue together
multiple tetrahedra, the parallel transports will provide curvature defects
between faces, allowing for local geometry that is not globally flat.
Accepting $P=T^{\ast}(SU(2)^{4})$ as our classical phase space, we can easily
quantise using $\mathbb{H}=L^{2}(SU(2)^{4})$. Upon quantisation, each of the
normal vectors becomes a basis for its corresponding Lie algebra component.
However, we still have the closure constraint to impose, which was neglected
so far - as well as an $SU(2)_{\text{Diagonal}}$-invariance of the parallel
transports,
$(g_{1},g_{2},g_{3},g_{4})\sim(hg_{1}h^{-1},hg_{2}h^{-1},hg_{3}h^{-1},hg_{4}h^{-1})$.
It is easy to interpret this constraint when looking at an exponential element
of $SU(2)^{4}$:

$g(t_{\alpha})=\exp(-i\sum^{4}_{\alpha=1}\vec{t}_{\alpha}\vec{J}_{\alpha})\stackrel{{\scriptstyle\vec{t}_{\alpha}\equiv\vec{t}}}{{=}}\exp(-i\vec{t}\sum^{4}_{\alpha=1}\vec{J}_{\alpha})$
(4)

Figure 4: (a) Normal vectors to a tetrahedron, with length specifying the area
of the faces. (b) Dual edges labeled by parallel transports.

by taking $\vec{t}_{\alpha}\equiv\vec{t}$, the closure constraint becomes the
condition that the diagonal subgroup acts trivially, as required. In other
words, we rather consider the configuration space and, therefore, the Hilbert
space, to be

$Q=\frac{SU(2)^{4}}{SU(2)_{Diag}}\qquad\mathbb{H}=L^{2}(\frac{SU(2)^{4}}{SU(2)_{Diag}})$
(5)

where we choose the normalised Haar measure of $SU(2)\cong S_{3}$ on the
homogeneous space to provide us with the means of integral norms on $Q$.
So, the state of a quantum tetrahedron or 3-simplex can be specified as a
square integrable function on 4 $SU(2)$-variables which is invariant under
diagonal conjugation. By the Peter-Weyl decomposition of $L^{2}(G)$333An
analogue of the Fourier decomposition of square integrable functions on the
circle, which, for compact Lie groups, splits a function into a sum over
representations of the group. for compact groups such as the one we are
considering25, we may write a function of 4 group elements as

$f(g_{1},g_{2},g_{3},g_{4})=\sum_{j_{1},j_{2},j_{3},j_{4}\in\mathbb{Z}^{+}}\sum^{j_{\alpha}}_{m_{\alpha},n_{\alpha}=-j_{\alpha}}f^{j_{1},j_{2},j_{3},j_{4}}_{m_{1},m_{2},m_{3},m_{4};n_{1},n_{2},n_{3},n_{4}}D^{j_{1}}_{m_{1};n_{1}}(g_{1})D^{j_{2}}_{m_{2};n_{2}}(g_{2})D^{j_{3}}_{m_{3};n_{3}}(g_{3})D^{j_{4}}_{m_{4};n_{4}}(g_{4})$
(6)

which we abbreviate, due to its cumbersome length, by the following notation:
Whenever 4-tuples associated to a vertex appear, they are written as a
boldface representative, e.g. $(j_{1},j_{2},j_{3},j_{4})=\mathbf{j}$. Once we
work with $N$ tetrahedra, we will often collect these into yet another vector,
denoted by $\vec{\textbf{j}}=(\mathbf{j}^{1},\dots,\mathbf{j}^{N})$. In fact,
the more common normalisation of coefficients is given as

$f(\mathbf{g})=\sum_{\mathbf{j}}\sum_{\mathbf{m},\mathbf{n}}f^{\mathbf{j}}_{\mathbf{m},\mathbf{n}}d_{\mathbf{j}}D^{\mathbf{j}}_{\mathbf{m},\mathbf{n}}(\mathbf{g})$
(7)

where the dimensions of the representations of $SU(2)$, labeled by half-
integers $j_{\alpha}$, have been factored out of the coefficients. In both of
these expressions, $m,n$ label a basis of the representation space of the
$j$-representation, $V^{j}$. The functions $D^{j}_{m;n}(g)$ form such a basis
in the $L^{2}$-functions on the group and are known as Wigner matrices.
If the closure constraint is satisfied, there exist relations between the
coefficients $f^{j}_{m;n}$. To find and resolve these, it is useful to look at
the Hilbert space decomposition given by Peter-Weyl:

$\mathbb{H}\cong\bigoplus_{\mathbf{j}}V^{\mathbf{j}}\otimes\overline{V^{\mathbf{j}}}$
(8)

One may impose the closure constraint by replacing the second factor in each
sector/direct summand by
$\mathcal{I}_{\mathbf{j}}=\text{Inv}(\overline{V^{\mathbf{j}}})$ (dimension
$\mathcal{D}_{\mathbf{j}}$), the maximal subspace invariant under the diagonal
group action. In terms of coefficients, this amounts to specifying a
dependence

$f^{\mathbf{j}}_{\mathbf{m},\mathbf{n}}=\sum_{\iota}f^{\mathbf{j}}_{\mathbf{m},\iota}C^{\mathbf{j},\iota}_{\mathbf{n}}\implies
f(\mathbf{g})=\sum_{\mathbf{j},\mathbf{m},\iota}f^{\mathbf{j}}_{\mathbf{m},\iota}d_{\mathbf{j}}\sum_{\mathbf{n}}C^{\mathbf{j},\iota}_{\mathbf{n}}D^{\mathbf{j}}_{\mathbf{m},\mathbf{n}}(\mathbf{g})$
(9)

with the $3j$-symbol of Wigner, $C^{\mathbf{j},\iota}_{\mathbf{n}}$. The
functions

$\langle\mathbf{g}\ket{\mathbf{j}\mathbf{m}\iota}=\psi_{\mathbf{j}\mathbf{m}\iota}(\mathbf{g})=\sum_{\mathbf{n}}C^{\mathbf{j},\iota}_{\mathbf{n}}D^{\mathbf{j}}_{\mathbf{m},\mathbf{n}}(\mathbf{g})$
(10)

form a basis of the gauge invariant Hilbert space and will be called spin
network vertex functions. The space $\mathcal{I}_{\mathbf{j}}$ consists of
intertwiners between the representations associated to the faces of the
3-simplex, and when a basis of it is expressed in terms of the Wigner
matrices, the $3j$-symbols provide the coefficients.
When viewing the simplex as dual to a 4-valent graph vertex with its edges
piercing the faces, one can associate both holonomies and representations with
the 4 edges originating at the center. This dual vertex picture will be
central in what follows - it allows us to prescribe a (classical) simplicial
complex through a dual, coloured graph. Let us also mention that the basis
$\ket{\mathbf{j}\mathbf{n}\iota}$ forms a set of eigenvectors of the area
operators
$A_{\alpha}=\sum_{\mathbf{j}}\sqrt{j_{\alpha}(j_{\alpha}+1)}\ket{\mathbf{j}\mathbf{n}\iota}\bra{\mathbf{j}\mathbf{n}\iota}$
acting on one of the edges each. There is also an analogue of a volume
operator given by a properly ordered quantisation of the triple product
$\vec{J}_{1}\cdot(\vec{J}_{2}\times\vec{J}_{3})$ that turns out to be diagonal
in the intertwiner basis. In this way, the tetrahedral quantisation and Peter-
Weyl decomposition leads to the spin basis in which areas of faces and volumes
take sharp values. The same cannot be said about edge lengths or angles,
however - not all quantities commute. Therefore, the geometry of glued quantum
simplices will not be exactly that of a simplicial complex, but it will show
definite values for areas and volumes in spin network vertex states. Angles,
edge lengths and other quantities will not be sharp.
This is a picture of a quantum geometry, in the sense that measurements of
properties of the system will not commute and present a Markov process.
Measuring an edge length of the tetrahedron will in general produce random
results. One can still, however, speak about general statistical properties of
these measurements like averages, variances etc.

#### Glueing of tetrahedra

Having established the single-tetrahedron Hilbert space, we now want to glue
them together to form larger complexes, according to a minimal procedure in
terms of matching their areas up by perfect correlation. For concreteness, we
will focus on the case of $N=2$ tetrahedra. We wish to glue the tetrahedral
states along the faces coloured $\alpha=0$. In the spin basis, where the faces
are labeled by areas $j_{\alpha}$, this involves fixing the areas of the two
faces to be the same, as well as making their orientation numbers $m$ match
oppositely.444Another alternative view is to require the normal vectors to be
opposite to each other: $\vec{J}_{x}+\vec{J}_{y}=0$, which implies the gauge
invariance shown below. As such, in the spin decomposition of a product state
of the two,

$\ket{\Psi_{1}}\otimes\ket{\Psi_{2}}=\sum_{\mathbf{j}^{x}\mathbf{m}^{x}\iota^{x}}\sum_{\mathbf{j}^{y}\mathbf{m}^{y}\iota^{y}}\Psi_{1,\mathbf{j}^{x}\mathbf{m}^{x}\iota^{x}}\Psi_{1,\mathbf{j}^{y}\mathbf{m}^{y}\iota^{y}}\ket{\mathbf{j}^{x}\mathbf{m}^{x}\iota^{x}}\ket{\mathbf{j}^{y}\mathbf{m}^{y}\iota^{y}}$
(11)

we insert a Kronecker delta
$\delta_{j^{x}_{0},j^{y}_{0}}\delta_{m^{x}_{0}+m^{y}_{0},0}$ forcing the two
faces to be glued. The same procedure can be applied to non-product states as
well.

Figure 5: Even if the two tetrahedra have initially independent parallel
transports, after glueing they only have a single group element between them.

Alternatively, we may look at this as a projection process: Consider the group
basis both before and after a glueing. While the two unglued links/faces of
the tetrahedra both possess a seperate parallel transport element, once they
are glued we expect there to be only a single link and thus only a single
parallel transport. So, a glueing in the group basis amounts to removing the
dependence on one of the two group elements in an indiscriminate fashion. In
fact, if we see the parallel transport defined by an element $g$ as outgoing,
the glued link from vertex $x$ to $y$ should have parallel transport first by
the element $g^{x}$ and then the reverse transport of $g^{y}$. The sought-for
dependence on $g^{x}(g^{y})^{-1}$ is easily produced by a right average:
Consider, for purposes of demonstration, only a function of these two
elements, $f(g^{x},g^{y})$, and perform a right group action average:

$\begin{split}\tilde{f}(g^{x},g^{y})=\int_{SU(2)}d\mu(h)f(g^{x}h,g^{y}h)\\\
=\int_{SU(2)}d\mu((g^{y})^{-1}\tilde{h})f(g^{x}(g^{y})^{-1}\tilde{h},\tilde{h})=\int_{SU(2)}d\mu(\tilde{h})f(g^{x}(g^{y})^{-1}\tilde{h},\tilde{h})\end{split}$
(12)

where in the last equality we used left invariance of the Haar measure. This
averaged function only depends on the correct combination and has full right-
invariance:

$\tilde{f}(g^{x}h,g^{y}h)=\tilde{f}(g^{x},g^{y})\,\forall h\in SU(2)$ (13)

We now have a projection operator that performs the glueing, expressed in the
group basis. In the spin basis, this same operator is rather a projection onto
a maximally entangled state of the two links:

$\ket{e_{j_{e}}}=\frac{1}{\sqrt{d_{j_{e}}}}\sum^{j_{e}}_{m_{e}=-j_{e}}(-1)^{j+m}\ket{j_{e}m_{e}}_{x}\otimes\ket{j_{e}-m_{e}}_{y}$
(14)

and the projection is given by $\ket{e}\bra{e}$. In this state, the two
subsystems share their information perfectly. It is called maximally entangled
as for each fixed spin, the reduced density matrix

$\rho_{x}=\mathrm{Tr}_{y}[\ket{e_{j}}\bra{e_{j}}]=\mathbb{I}_{j}$ (15)

gives the maximally mixed state - the entanglement entropy reaches its maximal
value

$S(\mathbb{I}_{j})=\log(d_{j})$ (16)

given by the logarithm of the dimension.
With this in mind, we can glue further simplices together by chaining these
projectors together. For a product of vertex basis states, after projection a
state of the form

$\bigotimes_{e\in
E}\bra{e}\bigotimes_{x}\ket{\mathbf{j}^{x}\mathbf{m}^{x}\iota^{x}}$ (17)

emerges. Such a state is what we will call a spin network state, as the
4-valent graph dual to the simplicial complex gives rise to a spin network
with representation labels $\vec{\textbf{j}}$, intertwiners $\vec{\iota}$ and
open edge labels $\\{m_{e}:e\in\partial\gamma\\}$. Expressed as a function of
group elements, this will in fact be a diagonally invariant function of
parallel transports along the edges of the graph, as the glueing procedure
gives each link in the graph precisely one group element.

Since they apply to different links, the projectors for different links
commute and do not interfere with each other. In this work, we will only
perform coloured glueings: Each factor in $SU(2)^{4}$ is assigned one of 4
colours, whose corresponding links may be glued together. This means that the
class of graphs dual to our simplicial complexes are 4-edge-coloured, 4(or 1
at boundary)-valent graphs. The colouring, in a very direct way by the
ordering it induces, produces an orientation of the simplices and thus of the
simplicial complexes. In this way, we study states labeled by oriented, pure
simplicial 3-complexes. This class may be motivated by starting from a smooth
oriented 3-manifold, which always admits a smooth oriented triangulation.
While not all simplicial complexes used as labels here are homeomorphic to
manifolds, they are to oriented pseudomanifolds. They therefore provide a
controllable set of labels for states by spatial topologies that extends that
of manifolds. However, due to the quantisation, it is a priori not so clear
whether these quantum geometries will behave like classical ones. Already,
even spin network states, once they are superposed, can not be expected to
behave like nice geometries.
The glueing procedure can be understood in terms of measurements as follows:
If measurements of two glued faces of tetrahedra are possible independently,
their results will show perfect correlation. It is worth emphasizing that the
question of proximity of the two tetrahedra is not answered in this way. The
only property matched between the tetrahedra is their shared area - no mention
of lengths has been made, per se. This is, however, the notion of glueing that
reproduces spin network states.

#### An aside on constraints and spacetime dynamics

Before moving on to other elements of our investigation, it is worth to give a
short remark about the relation to spacetimes. As a canonical Hamiltonian
analysis of (pure) gravity on globally hyperbolic spacetimes
$M=\mathbb{R}\times\Sigma$ shows, the Hamiltonian of classical GR consists
only of a number of classical constraints. Let $q,p_{q}$ collect whichever
other regular variables there may be, for example spatial metrics and
extrinsic curvature. Then, the transition from the Lagrangian to the
Hamiltonian formalism introduces canonical momenta $p_{q}$ for each variable
$q$ through $p_{q}=\frac{\partial L}{\partial\dot{q}}$. This may turn out to
not be a function invertible for the velocities $\dot{q}$, in which case one
instead imposes a corresponding constraint on the classical phase space,
$C(q,p_{q})=0$. A simple example may come from the point particle Lagrangian

$L=\dot{q}eA(q)$ (18)

which can be interpreted as a particle of 0 mass coupled to an electromagnetic
background $A$. The constraint here is $p=eA(q)$. Assuming that $\dot{q}$ will
be given by some function on the phase space, the Hamiltonian is.

$H(q,p)=\dot{q}p-L=\dot{q}(p-eA(q))$ (19)

In order for the constraint to hold at all times, one requires

$\\{p-eA,H\\}=(p-eA)\\{p-eA,\dot{q}\\}=0$ (20)

which is automatically true on the constraint surface. In fact, one may
achieve the same thing by introducing a new pair of conjugate variables
$N,P_{N}$ instead of seeing $\dot{q}$ as a function on phase space. If
$H=N(p-eA(q))$, then $\dot{q}=\\{q,H\\}=N$, while

$\dot{N}=\\{N,H\\}=0,\dot{P_{N}}=\\{P_{N},H\\}=-(p-eA(q)),$ (21)

which means that (1) the time evolution of $q$ is actually arbitrary and (2)
the constraint being imposed is equivalent to the conjugate momentum $P_{N}$
being constant.
In general relativity without sources, one encounters the same type of
situation. There are a number of Lagrange multipliers $N_{\alpha}$ and
constraints $C_{\alpha}$ and the Hamiltonian takes the form

$H(N_{\alpha},P^{\alpha}_{N};q,p_{q})=\int_{\Sigma}vol_{\Sigma}(x)\sum_{\alpha}N_{\alpha}(x)\,C_{\alpha}(q,p_{q})(x)$
(22)

Being a Lagrange multiplier means that the conjugate momenta $P_{N}$ must be
constant. Since they themselves do not appear, their equations of motion
$\frac{\partial H}{\partial P^{\alpha}_{N}}=\dot{N_{\alpha}}=0$ show that the
values of these multipliers are arbitrary.
This implies, through Poisson brackets, that $C_{\alpha}(x)=0$ at all times
and for all points in the spatial surface.555It is assumed that the
constraints $C_{\alpha}$ already form a closed Poisson subalgebra, so that no
further constraints are generated.
In fact, these constraints are dual (by Noether’s second theorem) to the local
symmetries of gravity: Particularly, spatial and timelike, foliation-
preserving diffeomorphisms666These symmetries map the fields to their
pushforwards under a diffeomorphism of the manifold. are a symmetry of
gravity. These local symmetries induce corresponding local constraints,
analogous to how gauge invariance of classical electromagnetism gives rise to
the electric Gauss law $\nabla\cdot E=\rho_{e}=0$.
The constraints on gravity’s naive phase space (often called kinematical phase
space) are fittingly called the spatial and timelike Diffeomorphism (or
Hamiltonian) constraints as well as a Gauss constraint in setups where spin
geometry is used and bundle automorphisms or gauge transformations of the spin
frame bundle provide another symmetry.
Given this information, we can see that the time evolution of a spatial
geometry is, in simple words, fictitious. Even after fixing some values of the
Lagrange multipliers, once a spatial geometry is found to satisfy the
constraints, the Hamiltonian will act trivially upon it. A way to reclaim the
’naive’ equations of motion is to fix values of the multipliers and compute
the equations of motion before using the constraints. In our earlier example,
this is given by

$\dot{q}=N=\text{const},\qquad\dot{p}=e\nabla_{q}A=\frac{e}{N}\frac{d}{dt}A(q)\implies\qquad
p-\frac{e}{N}A(q)=\text{const}$ (23)

The choice of a time evolution is still, however, an arbitrary one, as the
Lagrangian evolution is trivial. The first and foremost task of quantum
gravity is therefore to impose, in a meaningful, productive and controlled
way, the constraints on the kinematical phase space of the starting variables
and to reconstruct spatial geometries with the right data to produce a regime
where smooth spaces appear as a reasonable approximation.
Let us now introduce a few contexts in which spin networks naturally appear to
give a perspective on their usefulness.

####  2 Lattice gauge theories and their loop representation

The first practical context in which we would like to present spin networks is
for quantised lattice gauge theories in the loop representation. For our
purposes, a lattice gauge theory is a system with degrees of freedom given by
a connection 1-form $A_{\mu}$ analogous to Yang-Mills or Electromagnetism,
living on an oriented lattice $\Lambda$. Gauge invariance means that the only
quantities of physical interest are those lattice objects which are somehow
related to the curvature. The elementary variables in the classical system are
given by parallel transports of the gauge potential $A$ along edges or longer
paths $e$ in the lattice:

$h_{e}(A)=\mathcal{P}\exp(\int_{e}A)\in G$ (24)

These gauge group variables describe how to compare local gauges, say for
matter living on the lattice that is charged under the group $G$. As can for
example be seen most easily in the abelian case where

$\int_{e}A\mapsto\int_{e}A+\int_{e}d\chi=\int_{e}A+\chi(y)-\chi(x),$ (25)

under a continuum gauge transformation the potential changes as $A(x)\mapsto
S^{-1}(x)(A(x)+d)S(x),\,S(x)\in G$, only the endpoints experience a change:

$h_{e}(A)\mapsto S(y)h_{e}(A)S(x)^{-1}$ (26)

Clearly, if we choose the path to have the same start- and endpoint, the
resulting group element will be just conjugated under gauge transformations.
These group elements associated to loops are commonly known as holonomies. In
the loop representation of lattice gauge theories, their traces are chosen as
elementary, gauge invariant variables for quantisation.
By starting from a Segal-Bargmann type representation of the variables, gauge
invariance can be automatically imposed on the states upon quantisation.
Choose as coordinates on the (infinite dimensional) classical phase space the
traces of all the above holonomies as well as the canonically conjugate fluxes
$E$, analogous to the electric field in canonical quantisation of Maxwell
theory, satisfying $\\{A,E\\}\sim 1$ with the non-exponentiated connection.
These form an analogy to the $(z,q)$ variables presented earlier. Then, the
space of physical quantum states will be given by gauge invariant, square
integrable functionals $F[A]$ of the connection $A$. It can be shown that any
such functional may be understood as some composition of various holonomy maps
with different loops, in the following way: Let
$H:\Omega\Lambda\times\mathcal{A}\rightarrow G$ be the holonomy map from loops
and connections to group elements. Then, any gauge invariant functional will
be given by taking some combination of $H(\gamma_{i},-)$’s, turning them into
numbers using a trace, and lastly evaluating them all simultaneously on $A$.
We obtain a set of states spanning the Hilbert space:

$\ket{\\{\gamma_{i}\\}_{i}}\leftrightarrow(A\mapsto\mathrm{Tr}[\prod_{i}H_{\gamma_{i}}(A)E_{i}^{b_{i}}])$
(27)

which are the functionals associated to a set of loops $\\{\gamma_{i}\\}_{i}$
and booleans $b_{i}\in\\{0,1\\}$ indicate that fluxes may be inserted at
arbitrary points along the loops. A great feature of constructing the Hilbert
space from this set is that all states are automatically gauge invariant and
thus physical. However, it is overcomplete: There are relations between traces
of matrices called Mandelstam identities which mean that some of the above
states may be linearly related to one another44.
Thus, to obtain a basis of the gauge invariant Hilbert space, one needs to
superpose these loop states to obtain linearly independent states. By a
definite graphical procedure, one can associate to each $G$-spin network
embedded in the lattice such a superposition of loop states. And, when
collecting these states over all spin networks, one obtains a basis of the
gauge invariant Hilbert space. These basis states are appropriately named
$G$-spin network states and form a natural description of the quantum degrees
of freedom of a discretised gauge theory.

####  3 Spin network states in canonical Loop Quantum Gravity

An analogous story introduces spin networks in quantum gravity47;
rovelliQuantumGravity2004. The typical starting point of canonical
quantisation of GR is to foliate a globally hyperbolic spacetime into spatial
slices, which leads to a 3+1D split of the dynamical variables. The main
variables are the spatial 3-metrics on the slices as well as extrinsic
curvatures from the embedding. Using a sequence of canonical
transformations27, one can transform these and the Hamiltonian into
expressions dependent on connection variables, analogous to the Palatini
formulation of GR, but clearly distinct: A convenient parametrisation turns
out to be in terms of the complexified, self-dual Ashtekar-Barbero-Sen (ABS)
connection $A$ and a densitised spatial triad $E$ 3. This is a complex linear
combination of the 3D Levi-civita spin connection and the extrinsic
curvatures, most easily understood in the time gauge, where representing the
4D spin connection $\omega$, valued in $\text{spin}(1,3)$, on left-handed Weyl
spinors yields the ABS connection

$A=(\rho_{LH})_{\ast}(\omega)=(\frac{1}{2}\omega^{ij}\epsilon_{ijk}+i\omega^{0}_{k})\sigma_{k}$
(28)

by seeing the Pauli matrices as a representation of $su(2)$. So, it may be
understood as the connection for left-handed spinors on the spacetime. On the
other hand, the self-duality is to be understood when looking at the
complexified Lie algebra:

$\text{spin}(1,3)_{\mathbb{C}}\cong\text{spin}(3)_{\mathbb{C}}\oplus\text{spin}(3)_{\mathbb{C}}\cong\text{spin}(4)_{\mathbb{C}}$
(29)

in which the duality is in terms of a volume form on $\text{spin}(4)$, in a
basis described by the Levi-Civita symbol

$(\star\omega)_{ab}=\frac{1}{2}\epsilon_{abcd}\omega^{cd}.$ (30)

Using this connection variable, the constraints of the classical theory due to
diffeomorphisms take a much more amenable form. To maximise its usefulness,
one also uses as a conjugate variable the densitised spatial triad
$\tilde{E}$. Both of these variables are understood to take values in the
complexified algebra $su(2)_{\mathbb{C}}$ and will have to be restricted by
so-called reality conditions48 to recover real Lorentzian GR.777In practice,
this approach of complexification turns out to be quite involved as the
reality conditions are difficult to implement. Instead, other variants of the
Ashtekar connection are used which depend on the Immirzi parameter of the
Holst action, but for the purpose of exposition we decide to stick to this
more historical idea. With these variables, one can quantise in strong analogy
to a lattice gauge theory for $SU(2)$. Using again the Loop representation,
one can construct gauge invariant states of the system as holomorphic
functionals of $A$, again with a basis in correspondence with spin networks.44
The key differences to lattice gauge theories are the following:

  * •

Instead of being embedded in a lattice, the spin networks are equipped with an
imbedding in the spatial slice $\Sigma$ through which they can encode knotting
information among other things.

  * •

In LQG, there are additional constraints on the gauge invariant states to be
considered physical, given by the Diffeomorphism and Hamiltonian, Scalar or
Master constraints. The former set of these can be implemented directly on
spin network states, leading to states labeled by s-knots, which are akin to
isotopy classes of imbedded spin networks and depend on generalised link
invariants of the spatial slice.

In particular, the action of the Hamiltonian and Diffeomorphism constraints is
significantly simpler on this basis compared to the quantisation program in
metric variables. In fact, several solutions to the Hamiltonian constraint may
be found in this basis, providing physical states of quantum general
relativity.
Additionally, the quantisation of area and volume operators defined by analogy
from classical GR turns out to be diagonal in spin network states -
representation labels correspond to areas and intertwiners to volumes.
Therefore, these states of quantum GR have some definite geometric properties.
However, they should not be confused for exact classical geometries, either,
as lengths do not attain definite values. The situation is therefore entirely
analogous to the states labeled by simplicial complexes we introduced
earlier35; 17.
The natural occurrence of spin network states in canonically quantised general
relativity gives credence to the idea of them corresponding to states of
quantum geometry. However, the attached imbedding remains a piece of
information from the continuum, which means that even still, quantum geometry
is partially dictated by a classical one. In response to problems connected to
the continuum and, in part, due to hopes of recovering GR from purely discrete
systems, it has been suggested that one should rather start from a microscopic
model based on spin network states without imbedding data and recover a
continuum space in some limit. There have also been ideas that such data may
be incorporated in spin network states through a change of data from Lie
groups to Hopf algebras, fully deconfining the idea of quantum from classical
geometry.46

####  4 Spin foam models and Group Field Theories

Whether the spin network states one wishes to consider are from LQG or
abstract, the question of dynamics for these states remains. For example, the
Hamiltonian constraint888Rather, one might impose instead the Master
constraint $M=\int d^{3}x\frac{H(x)^{2}}{\sqrt{|h|(x)}}$ which imposes all the
Hamiltonian constraints simultaneously. may be imposed by a standard
oscillatory integral representing a projection operator:

$P=\int\mathcal{D}N\exp(iNH)$ (31)

Applied to a state, all but the kernel of $H$ will be washed out by the
integral over the ’Lapse function’ $N$. Then, the exponential operator may be
turned into a path integral by insertions of resolutions of the identity
through spin network states. The Feynman diagrams of such a path integral then
correspond to discrete 2-complexes called spin foams41. Independently from
this argument, this same structure has been proposed by Baez 5; 7 as a
categorical arrow between spin network objects. In this general definition, a
spin foam $S$ mapping spin networks $s_{1}$ into $s_{2}$ is

  1. 1.

A directed 2-complex $\Gamma$ with boundary given by the graphs of $s_{1,2}$.

  2. 2.

A labeling of its 2-cells (corresponding to time evolutions of edges in a spin
network) by representations compatibly with the bounding spin networks.

  3. 3.

A labeling of its 1-cells (corresponding to time evolutions of vertices in a
spin network) labeled by intertwiners, also chosen compatibly.

In this definition, compatibility is to be understood as choosing the dual
representation for outgoing edges of a 2-face, etc. and one may make
adjustments for spin networks with boundaries.
Given this definition, one can define a scalar product between spin network
states through a spin foam model:

$\langle s_{1},s_{2}\rangle:=\sum_{S:\partial
S=s_{1}\cup\bar{s_{2}}}\mathcal{A}(S)$ (32)

So, an amplitude is given for each spin foam mapping between the two spin
networks and a sum is performed over all such spin foams. This is entirely
analogous to the sum over Feynman diagram amplitudes in a normal relativistic
QFT. In this context, spin network states play the role of scattering states
in the relativistic QFT, which are themselves geometrically given by the
points of the point particles. Any ’time-slice’ of the Feynman diagram will
give another configuration of points. Similarly, any slice of a spin foam will
yield a spin network, as in figure 6.

Figure 6: A sequence of 3 spin networks, connected by a spin foam with
compatible labels. There are two spin foam vertices in the 2-complex
introducing new structures as the spin networks evolve discretely. A closeup
of these vertices is given on the right.
Note: Reprinted from "The Spin Foam Approach to Quantum Gravity"41, p. 29, CC
BY-4.0

However, there is no scattering process happening - this should all be
understood as in the boundary formulation of QFT, where transition amplitudes
between boundary states are all that needs to be specified for theories to
make sense34. Specifying the amplitude for any two spin network states will
therefore give the full definition of a quantum system whose Hilbert space is
spanned by these states.
Spin foam models can then also be taken as an independent, possibly covariant
alternative quantum theory of geometry, in which the spin foam takes on the
identity of a quantum pre-spacetime. In certain limits, some spin foam models
have been shown to indeed produce the semiclassical behaviour expected of a
lattice gravity path integral.41 In this setting, one has only few
restrictions on which types of boundary states one should use. Past the
specification of an amplitude for each spin foam, the question of recovering a
classical geometry remains. There are essentially two options here: The first
is to consider limits of refinement of the spin foam and networks. The other
is to construct a sum over spin foams that organises these in a controlled
fashion. One systematic way of doing the latter is through Group Field
Theories (GFT)28; 35; 37:
Interpreting spin foams as Feynman diagrams allows one to consider the
transition amplitudes given by a spin foam model in a productive way. One sees
them as contributing to a correlation function of two spin network operators
in a quantum field theory. Said QFT needs to be defined on a number of copies
of the Lie group $G$. This can be most easily seen by another analogy to
scattering diagrams: There, labels on the edges (analogies of the 2-faces in a
spin foam) are given by momenta, which are representations of the translation
group $\mathbb{R}^{4}$. The QFT’s Hilbert space can be shown to be spanned by
abstract spin network states, similar to the situation in LQG. In fact, when
the QFT is taken to be unitarily equivalent to a free one, the Hilbert space
is precisely the Fock space $\mathbb{F}(L^{2}(\frac{G^{4}}{G_{Diag}}))$, built
on the same Hilbert space of the single tetrahedron that we constructed in the
beginning. The analogue of the diagonal group action in the relativistic case
is given by the mass constraint $p^{2}=m^{2}$, reducing
$L^{2}(\mathbb{R}^{1,3})$ to $L^{2}(\mathbb{R}^{1,3}/\mathbb{R})\cong
L^{2}(\mathbb{R}^{3}_{m^{2}})$.
The dynamics of such a GFT are given, in general, by a path integral of some
form, which specifies a partition function13; 29

$Z=\int\mathcal{D}\Phi\mathcal{D}\bar{\Phi}e^{-S(\bar{\Phi},\Phi)}$ (33)

where the group fields $\Phi:G^{D}\rightarrow\mathbb{C}$ are to satisfy some
gauge invariance condition. In the free case, these fields give rise to
creation/annihilation operators associated to tetrahedra. There are many
different ways these systems relate to other quantum gravity approaches, due
to the many different representations that (wave)functions on a group can
have.8 In particular, the Peter-Weyl decomposition allows extracting a spin
foam model from any GFT’s Feynman diagrams.
The specifics of the dynamics depend largely on which class of spin foams are
to be generated in the perturbative expansion. One may include multiple fields
and several interaction terms, which all should correspond to some building
block of the spin foam. A particularly simple class of these interactions are
the simplicial ones, in which 5 tetrahedra are glued to form a 4-simplex. In
group variables, it takes the schematic form

$\mathcal{V}=\int_{G^{20}}d\mathbf{g_{0}}d\mathbf{g_{1}}d\mathbf{g_{2}}d\mathbf{g_{3}}d\mathbf{g_{4}}\Phi(\mathbf{g_{0}})\Phi(\mathbf{g_{1}})\Phi(\mathbf{g_{2}})\Phi(\mathbf{g_{4}})\Phi(\mathbf{g_{5}})V(\mathbf{g_{0}},\mathbf{g_{1}},\mathbf{g_{2}},\mathbf{g_{3}},\mathbf{g_{4}})$
(34)

where the combinatorial factor $V$ is visually encoded in a stranded diagram
as in Figure 7.

Figure 7: Schematic of how contractions happen in a 4D uncoloured simplicial
interaction. The arguments of the 5 involved group fields, four each, are
contracted with group Dirac delta functions $\delta(gh^{-1})$ according to the
green lines connecting the arguments in this schematic.
The purpose of this combinatorial scheme is that, interpreting the 4 arguments
of each field as the parallel transports along faces of a tetrahedron, this
mimicks the glueing of 3-simplices to form a 4-simplex.

Spin network states, as we have seen, can also be understood as glued or
entangled tetrahedral states. In this way, they form an analogue of entangled
many-body particle states of the GFT. This holds not just for GFT states, but
in generality for all spin network states regardless of the approach.17 In the
GFT formalism, though, the connections to entanglement are perhaps more
transparent. However, even discarding the imbedding information in LQG and
staying at the prefock (non-symmetrised) space level of the (free) GFT, the
way these states are organised in the full Hilbert space is very different
between approaches: While, for fixed number of vertices the scalar products
agree in the two theories, GFT spin networks are orthogonal if their number of
vertices is not identical36. This may be, in part, understood as a difference
in dynamics: The free GFT does not have the same physical scalar product as
LQG as it does not have the proper constraints imposed on its states. However,
the dynamics of the GFT may effectively implement a different scalar product
to make the relation29.
To understand the function of spin networks as tensor networks, we first give
a bit of background on tensor network states. Then, we review classes of
holographic codes and their natural progression towards random tensor
networks, which form the foundation of this project.

####  5 Spin network states as tensor networks

A tensor network38, for our purposes, is a multidimensional array

$T^{\mu_{1},\dots\mu_{n}}_{\nu_{1},\dots,\nu_{n}}\stackrel{{\scriptstyle
ex.}}{{=}}\sum_{\alpha_{1},\dots,\alpha_{n}}\prod_{i}V^{\mu_{i}\alpha_{i}}_{\nu_{i}}\Pi_{\alpha_{1},\dots,\alpha_{n}}$
(35)

composed from smaller arrays of various sizes by contraction according to
rules specified by a graphical calculus. Above, we have given a very simple
example where a rank (n,n)-tensor is decomposed into n tensors $V$ of rank
(2,1) and contracted using another tensor $\Pi$.
A corresponding concept in physics is that of a tensor network state: Starting
from a factorised Hilbert space, the simplest types of states one can form
from those of subsystems are tensor product states. Tensor network states form
a natural, but controlled extension of this class where the subsystem states
function as tensor blocks and are contracted or projected in a way that, when
expressed in bases of the subsystems, will produce a tensor network. A simple
class of these states is given by so-called projected entangled pair
states(PEPS)15; 38: Elementary vertex states $\ket{V_{x}}$ are projected onto
a set of maximally entangled link states (also known as EPR pairs)
$\bigotimes_{e}\ket{e}$ that encode a connectivity pattern.
Consider, for motivation, a system of up/down spins $S_{x}$ on a lattice
(V,E), say a rectangular one. The total Hilbert space factorises into

$\mathbb{H}=\bigotimes_{x\in V}\mathbb{C}^{2}_{x}$ (36)

but a typical Hamiltonian will respect the lattice structure of the system in
some form, for example by only including nearest-neighbour interactions

$\hat{H}=\sum_{e\in E}\lambda_{e}\frac{1-\hat{S_{x}}\hat{S_{y}}}{2}.$ (37)

This type of Hamiltonian is local in the sense that any site has
vanishing999More generally, a power-law or exponential decay with spatial
distance suffices for qualitatively similar statements. interaction with other
sites outside of a finite region. Sites that are two edges apart will only
couple indirectly.
In the above Hamiltonian, a ground state is fairly easy to find: if, say, all
edge couplings $\lambda_{e}>0$, then the energy is minimised by having all
spins equal. This is a simple product state

$\ket{FM}=\bigotimes_{x}\ket{\uparrow_{x}}.$ (38)

However, if the couplings are inhomogeneous and some are negative, the ground
state is not known in general. An approach to approximate it is to add
auxiliary Hilbert spaces $V_{x,\alpha}$ to the sites, two for each, left and
right. These may be intertwined with the existing site vertex Hilbert space
$\mathbb{C}^{2}_{x}$ by writing the full state in $\mathbb{H}$ as a tensor
network state in tensor blocks coming from
$V_{x}=\otimes_{\alpha}V_{x,\alpha}$. Then, the tensors have auxiliars indices
("legs") that can be contracted in accordance to the shape of the lattice or
some sublattice. The auxiliary legs of two adjacent sites are projected onto a
maximally entangled state $\ket{e}$, for example

$\ket{e_{xy}}=\frac{1}{\sqrt{D}}\sum_{m}\ket{m}_{x}\ket{m}_{y},$ (39)

which amounts to contracting the indices in the tensor network with Kronecker
deltas. For a 1-dimensional lattice with $N$ sites, placing vertex tensors
$T^{x,s^{x}}_{t^{x}_{0},t^{x}_{1}}$ on each site with auxiliary legs
$t^{x}_{0},t^{x}_{1}$, this might look like

$\ket{\Psi}=\sum_{s^{1},\dots,s^{N}\in\\{\pm
1\\}}\Psi^{s^{1},\dots,s^{N}}\ket{s^{1},\dots,s^{N}}\qquad\Psi^{s^{1},\dots,s^{N}}=\sum_{\\{t^{x}_{\alpha}\\}}\prod_{x}T^{x,s^{x}}_{t^{x}_{0},t^{x}_{1}}\prod_{i<j}\delta_{t^{i}_{1},t^{j}_{0}}$
(40)

where we have obtained this state as

$\ket{\Psi}=\bigoplus_{i<j}\bra{e}_{ij}\bigotimes_{x}\ket{T^{x}}.$ (41)

Figure 8: Tensor network contraction of tensors $T^{s}_{t_{0},t_{1}}$ into a
1D chain. Each left auxiliary leg, labeled by $0$, is contracted with the
right one of the site on its left, labeled by $1$. Free legs give the physical
indices.

The above general form of a product state projected onto a set of maximally
entangled link states is a PEPS.
By itself, the product state $\bigotimes_{x}\ket{T^{x}}$ has no entanglement
between the site spaces $\tilde{\mathbb{H}}_{x}\cong\mathbb{C}^{2}_{x}\otimes
V_{x}$. The projection, however, introduces entanglement in a geometrically
transparent, combinatorial way and can be controlled easily by adjusting the
combinatorial pattern. Practically, one will prepare an Ansatz for a
variational problem using these easily controlled elements of entanglement
between sites and approximate the ground state through it.
At this point, it should be clear that the spin network basis states we
introduced earlier are highly analogous to PEPS states. For fixed spins, the
tensors $\ket{\mathbf{j}^{x}\mathbf{m}^{x}\iota^{x}}$ correspond to
$\mathbb{H}_{x}$ in the above, and the auxiliary legs are now ’real’ legs
$V^{\mathbf{j}^{x}}$. They are contracted by projecting on maximally entangled
link states. However, a few key difference arise to standard PEPS:

  * •

The auxiliary leg spaces are of physical relevance in spin networks. Not all
of them are contracted for this reason.

  * •

The intertwiners function as the original ’site’ legs, but depend in their
dimension on the dimension of the surrounding auxiliary spaces.

  * •

In the spin network case, these states only form a basis of the full Hilbert
space. There is, however, always the option of superposing these states in a
certain form that preserves the combinatorial structure, as we will do later.

###  2 Holography and (Random) tensor networks

There have been many instances of concrete correspondences between boundary
and bulk field theories, with or without gravity, since the initial proposal
of by Maldacena31. The earliest and perhaps most well known of these is the
general $AdS_{d+1}/CFT_{d}$ conjecture. In this line of ideas, one makes use
of the asymptotic symmetry of the conformal boundary of Anti-deSitter
spacetime (concretely realised as a projective lightcone) to establish an
exact equality of partition functions of two field theories33. Typically, one
considers low-energy effective actions of string theory in the bulk, but the
duality is in principle not limited to this example. The crucial steps involve
the following: First, a set of valid boundary conditions for the bulk theory
must be found for which the solution to the classical equations of motion is
uniquely determined by their boundary values. In this step, the causal
compactness of the spacetime is crucial. Any two points on the asymptotic
boundary are connected by a geodesic of finite proper time. This feature
forces the bulk solution to heavily depend on the boundary values and in some
cases determines it uniquely.

Second, one introduces Ansätze for field theories on both ends, for example as
classical actions. The corresponding partition function then depend on the
specified boundary values, as for example in the bulk case through

$Z_{b}[\phi]=\int_{\Phi|_{\partial}=\phi}\mathcal{D}\Phi e^{-S_{b}[\Phi]},$
(42)

while the boundary CFT couples to the boundary value through some operator
$\mathcal{O}$. Requiring $Z_{b}[\phi]=Z_{\partial}[\phi]$ is the statement of
full duality and can be analysed through a semiclassical approximation. On the
level of actions, uniqueness of the bulk solution makes the correspondence
unambiguous. The two theories can then be matched by comparing their
correlators, typically starting with the 2-point function.

To characterise the geometry of this correspondence better, Ryu and Takayanagi
proposed a criterion45. When a general system is given a bipartite split into
regions $A$ and $B$, for example a time slice of the boundary CFT, one can
study its entanglement (von Neumann) entropy through path integral and other
methods. When properly regulated, this entropy is, at leading order,
proportional to the area of the surface dividing $A$ and $B$. In order to find
a bulk pendent to this quantity, they turned to a known case of entropy
scaling like a surface in gravity: That of a black hole’s horizon, acting as a
surface dividing interior from exterior. Their proposal is, in general, to
associate to each boundary CFT region $A$ a surface $\gamma_{A}$ in the bulk,
sharing the same boundary $\partial A$. This surface is further required to be
homologous to $A$ and have minimal area. Then, the CFT entanglement area of
$A$ would be given by

$S_{A}=\frac{\text{Vol}(\gamma_{A})}{4G_{N}}$ (43)

so precisely the Bekenstein-Hawking formula. In fact, this formula has been
shown to hold in fair generality in the context of $AdS/CFT$. The crucial idea
of geometry emergent from holography stems from this; There is a duality
between $(d-1)$-boundary regions and $(d-2)$-surfaces in the bulk, even with
precise areas. This suggests that, with knowledge of all entanglement
entropies in the CFT, one might be able to reconstruct the full geometry of
the bulk. Of course, not every entanglement structure of the boundary will
yield a reasonable bulk geometry. However, there are even clear cases where
the minimal surfaces never enter a region of the bulk, such as a black hole’s
interior, conventionally called the entanglement shadow. This region’s
geometry is inaccessible through the Ryu-Takayanagi (RT) formula. Without
further corrections to the formula or additional data in the boundary theory,
entanglement is then not enough to construct the interior completely. In fact,
it has been suggested by work of Susskind and Maldacena30 on maximally
extended black hole spacetimes that one needs additional data from the
boundary to recover entanglement shadowed regions and correctly relate
distances to entanglement. Rather one should see entanglement as providing
‘nontraversible wormholes’ and their length being given by some state-
dependent quantity in the CFT dual.

Additionally, there are certain tensor networks such as MERA whose application
to critical systems in low dimensions have invited their usage in the same
type of correspondence42.

Figure 9: (a),(b): A visual representation of a MERA network.
Note: Reproduced from "Exact holographic mapping and emergent space-time
geometry"42, p.2

In one of their layers, correlation functions in these networks behave as in a
CFT; they function as a boundary. Then, a state can be fixed in this layer,
which induces a corresponding one in the bulk Hilbert space. The bulk can then
be given an effective geometry through a distance function determined through
the mutual information of two bulk sites:

$d(x,y):=-\xi\log\frac{I(x,y)}{I_{0}},$ (44)

where the constant $I_{0}$ is chosen such that neighbouring maximally
entangled sites are seen as having $0$ distance between them. The correlation
length $\xi$, in turn, determines the overall scale of the metric. In a way,
this only fixes the metric up to a conformal factor. The definition is
motivated by the typical behaviour of correlations of operators $A,B$ observed
in ground states of gapped local Hamiltonians, where

$\frac{|\langle A(x)B(y)\rangle|}{||A||\cdot||B||}\leq
C_{0}e^{-\frac{d(x,y)}{\xi}}$ (45)

provides a universal bound.32; 11. It is worth noting that this exponential
decay of correlations depends crucially on properties of the ground state, as
selected by the dynamics of the system. While not all properties of $AdS/CFT$
can be directly recovered from MERA alone, it has been suggested12 that a MERA
with highly entangled boundary state may show features such as an RT formula.
This example shows that in a certain sense, the correspondence between bulk
geometries and boundary CFTs is universal and motivates looking for further
examples.

Such examples of holography are easily found in 3D and 4D asymptotically flat
gravity, where, similar to the AdS case, symmetries can be used to make
precise mappings. In 3D gravity the local triviality of the dynamics (every
solution of Einstein’s equations in vacuum is locally flat) allows a complete
solution of Einstein’s equations near asymptotic or even finite distance
boundaries24 with flat boundary conditions. Then, infinitesimal symmetries of
the boundary, given by (extensions of) the Bondi-Metzner-Sachs (BMS) algebra
may be used to study boundary quantities that correspond to bulk operators in
a very similar fashion to the AdS case, where a part of the boundary
symmetries is given by conformal transformations.

In a perhaps more physically interesting setting like 4D spacetimes, we have
other types of holographic correspondences. When self-dual boundary conditions
are put on the horizon in a Schwarzschild spacetime46, one can create a
correspondence between the space of conformal blocks of a Chern-Simons theory
on the horizon and a subspace of the physical Hilbert space of the quantum
geometry around it (in the sense that they satisfy the constraints of LQG).
Additionally, these dimensions of these Hilbert spaces satisfy, in a certain
limit, the Bekenstein-Hawking bound in the sense that

$\log(\dim(\mathbb{H}_{CB,S}))\sim A_{S},$ (46)

so, the logarithm of the dimension is given by the area in Planck units up to
a factor. If indeed one assumes that all physical quantum gravity states must
come from Hilbert spaces with such dimensionality, then there is a one-to-one
correspondence between the Chern-Simons theory and the quantum geometry
outside the horizon.
If instead asymptotically flat spacetimes are studied19, one finds the notion
of celestial holography: The asymptotic boundary is here replaced by the
celestial sphere of the spacetime, so the dual theory lives on a 2-dimensional
Euclidean sphere. Once again, the symmetries are found to be an extension of
the BMS algebra. Particle scattering in the asymptotically flat spacetime may
be expressed through correlation functions of equivalent operators on the
celestial sphere and various connections to infrared properties of general
relativity such as soft graviton theorems may be established.

What most of these examples have in common is that they are highly geometric
in their approach; they depend on a specific spacetime or at least its
asymptotic behaviour. While this is not a restriction in principle, it
presents certain practical difficulties in gaining general insights on
holographic correspondences, particularly in the non-asymptotic boundary case.
An alternative is to turn to a class of systems that are easier to control,
such as tensor networks similar to MERA. These are easier to control due to
their combinatorial buildup, can to some extent be studied numerically, often
have very simple Hilbert spaces with factorisation properties, their
entanglement can be studied using elementary means, and seem to capture a lot
of the same features of holography as full space(time)s.

Two clear steps in this direction are given by the network proposed by Harlow,
Pastawski, Preskill and Yoshida (known as the HaPPY code) as well as that by
Yang, Qi and Hayden (YQH code in the following): In the HaPPY code, the disk
model of hyperbolic space is tiled with pentagons39.

Figure 10: A graphical depiction of the pentagonal tensor network named
introduced by Harlow et al. Five of the six legs of each tensor are contracted
with surrounding legs, while the remaining leg, shown as a central dot on each
tensor, is kept as an input. The network has a finite size beyond which it
ends in a number of boundary legs.
Note: Reprinted from "Holographic quantum error-correcting codes: toy models
for the bulk/boundary correspondence"39, p.7, CC BY-4.0

Each tile is decorated with a 6-index perfect tensor, of which 5 indices are
contracted with the adjacent tiles’ tensors. The additional index functions as
an input or output. Perfectness here means that any bipartition of its indices
yields an isometric map from any one set of indices to the rest. Everywhere in
the network, the same bond dimension $v$ is chosen. This network has a number
of interesting properties:

  * •

A lattice RT formula holds; Given a region of boundary legs $A$, there is a
minimal surface in the network near it $\gamma_{A}$ such that
$S_{A}=||\gamma_{A}||\log(v)$, proportional to the number of links piercing
the minimal surface. Importantly, this suggests an interpretation of the
logarithm of the bond dimension as a quantity of area, if this is to be
directly analogous to the usual RT formula.

  * •

There is a causal wedge $C_{A}$ associated to each connected boundary region
$A$, from which any operator may be reconstructed as an operator supported in
the region $A$.

  * •

The network is quantum error-correcting: As the mapping of bulk operators to
boundary ones is not single-valued, one may remove parts of the boundary and
is still able to reconstruct said operators. Said differently, one needs only
a subset of the boundary to cover the entire bulk by its causal wedges.

Meanwhile, the YQH code uses no fixed contraction pattern, but even more
special pluperfect tensors with 4 contractible legs of bond dimension $D$ as
well as a physical one of dimension $D^{4}$51. The physical Hilbert space of
the bulk is taken as the image of the boundary under the YQH and is
constrained by a type of $SU(D)$ gauge invariance acting on each tensor. Said
gauge invariance implies that local operators in the bulk are mapped to the
trivial one on the boundary - there are no physical local observables, as in a
diffeomorphism invariant theory. However, this can be remedied by first
specifying a background bulk state (perhaps to be thought of as a vacuum), on
which a local operator can act as an excitation. The resulting state, now
consisting of the background as well as a local excitation, can be mapped to
the boundary as usual. In fact, the correlations of such excitations were
found to behave like those of a Quantum field theory on a geometric background
as long as there are not too many excitations present. In this way, the YQH
code features an effective, perturbative locality of correlations.

While both of these examples are incredibly promising insights into general
principles of holographic behaviour, their tensors are fairly constrained.
However, as explained by YQH, pluperfect tensors may be understood as an
idealised approximation of random tensors39: In the limit of large bond
dimensions, random tensors behave approximately as pluperfect ones. It is this
limit of large bond dimensions that we will see resurface over and over
throughout this work in the form of a lower cutoff on spins.
Random tensor networks, importantly, themselves also feature holographic
behaviour. As this case is more directly relevant to this work, it is worth
elaborating on in more detail. Before that, we wish to mention that all
properties mentioned for the other tensor networks hold, still: For connected
boundary regions, there are entanglement wedges with error correction
properties and RT formula. Local bulk operators are trivialised upon transport
to the boundary. Given a background bulk state, one can determine local ’code
subspaces’ of the bulk Hilbert spaces which describe QFT-like excitations.
However, in addition to these properties, there are two new insights:

  * •

Boundary 2-point correlation functions between operators $O_{A},O_{B}$ with
support resp. in $A,B$ with disjoint entanglement wedges can be shown to have
the same singular value decomposition as a corresponding bulk correlation
function. In particular, as the regions on the boundary become small, the
entanglement wedges approach the boundary. In this sense, boundary 2-point
functions are in one-to-one correspondence with bulk ones.

  * •

By placing a random entangled bulk state in the interior of the network, one
can create an entanglement shadow around the area of the bulk state, and thus
create a kind of phase transition when the rest of the network is modeled
after an Anti-deSitter space.

All of these properties will continue to hold in some capacity in the
remainder of this work, as we will work with the same techniques.
Consequently, tensor network holography provides an exciting and very concrete
setting to explore universal features of holographic correspondences that may
even extend past toy models. On the other hand, the connection of these tensor
networks to true geometry is less clear and often needs to be put in by hand,
as in MERA through the mutual information metric.

####  1 Random tensor networks

In random tensor networks26, individual tensors are randomly picked according
to a probability distribution, under which one may then compute average
quantities. In other words, with a different choice of setting and
interpretation, one can obtain the same type of tensor network holography as
for pluperfect tensors.
The setup is as follows: In a PEPS state, write each individual vertex state
as a unitary operator applied to some reference state:

$\ket{\Psi_{x}}=U_{x}\ket{\Psi_{ref}}$ (47)

The unitaries are then picked randomly from a distribution $\rho$ on the group
$\mathcal{U}(\mathbb{H}_{x})$. Conventionally, this distribution is chosen to
be just the Haar measure on the unitary group, with a constant function as
density. This is the distribution of maximal entropy on the group when no
constraints are imposed on it. However, as will be explored later in section
1, there are, in principle, other choices for this distribution.
With these random pure states, one can construct a PEPS state with a fixed
entanglement pattern. The key question of holography in this context is to
establish an isometry between a set of bulk Hilbert spaces and one of boundary
output spaces, or to calculate entropies in order to establish an RT formula.
To specify the method, we first elaborate on a specific, strong notion of
holography based on mappings of operators.

#### A notion of holography

The projected state lives in a factorised Hilbert space. In this setting of a
bi/tripartite system (bulk/boundary or a partition of the boundary space), we
can study holographic properties of the state. For this, in general, consider
a system described by a Hilbert space with tripartition
$\mathbb{H}\cong\mathbb{H}_{A}\otimes\mathbb{H}_{B}\otimes\mathbb{H}_{C}$,
where we interpret A as a subsystem from which information is read (output), C
as one where information is inserted (input) and B as the entire rest of the
system, acting as an environment, bath or background.

Figure 11: An example setup for holography particularly relevant to this work.
The Hilbert space is given by the completely factorised set of boundary link
spaces $\bigotimes_{e}\mathbb{H}_{e}$.
The tripartition in question is labeled by colours. If the full state induces
an isometric transport map $\mathcal{T}_{C,A}$, we may take any operator
acting on the boundary links in C and turn it into one acting on those of A
instead without losing information.

In this setting, consider a pure state $\ket{\Phi}$ of the system using the
natural self-duality of Hilbert spaces
$\mathbb{H}_{i}\cong(\mathbb{H}_{i})^{\ast}$. Since
$\ket{\phi}\in\mathbb{H}_{A}\otimes\mathbb{H}_{B}\otimes\mathbb{H}_{C}\cong\mathbb{H}_{A}\otimes\mathbb{H}_{B}\otimes(\mathbb{H}_{C})^{\ast}$,
we may see it (by turning kets into bras) as a map from subsystem C to
subsystems A and B. Schematically, if we consider a factorised state
$\ket{\phi}=\ket{\phi}_{AB}\otimes\ket{\phi}_{C}$,

$\ket{\phi_{AB}}\otimes\ket{\phi_{C}}\hat{=}\ket{\phi_{AB}}\otimes\bra{\phi_{C}}=\ket{\phi_{AB}}\bra{\phi_{C}}\in\hom(\mathbb{H}_{C},\mathbb{H}_{A}\otimes\mathbb{H}_{B}).$
(48)

This may straightforwardly extended into an (anti)linear map
$\mathbb{H}_{AB}\otimes\mathbb{H}_{C}\rightarrow\mathbb{H}_{AB}\otimes(\mathbb{H}_{C})^{\ast}\cong\hom(\mathbb{H}_{C},\mathbb{H}_{A}\otimes\mathbb{H}_{B})$.
We are particularly interested in a question of information transport: Given
input data on system C (a "core"), can it be recovered from system A (as for
example from a boundary region A)? This question can be answered positive if
an associated map to the one just described is isometric. Our main objective
is to investigate which random tensor network states induce isometric
mappings. These states will be called transporting or holographic in the
following.
First, begin from input data $\theta$ on the system C, described as a state
$\ket{\theta}\in\mathbb{H}_{C}$. The available data on the remainder of the
system after transport is
$\Phi\ket{\theta}=\bra{\theta}\phi\rangle\in\mathbb{H}_{A}\otimes\mathbb{H}_{B}$.
As we only consider information recoverable from A, we work with the reduced
density matrix

$\rho_{A}(\theta)=\mathrm{Tr}_{B}[\Phi\ket{\theta}\bra{\theta}\Phi^{\dagger}]$
(49)

More generally, then, for data described by a mixed state
$\rho_{c}\in\text{BL}(\mathbb{H}_{C})$, the transported state on A is

$\mathcal{T}_{C,A}(\rho_{c})=\mathrm{Tr}_{B}[\Phi\rho_{c}\Phi^{\dagger}]$ (50)

So, the transport through the system determines a superoperator
$\mathcal{T}_{C,A}=\mathrm{Tr}_{B}[\Phi(-)\Phi^{\dagger}]:BL(\mathbb{H}_{C})\rightarrow
BL(\mathbb{H}_{A})$. We can write its result in components with respect to a
basis $\\{\ket{a}\\}$ as
$\bra{a}\mathcal{T}_{C,A}(X)\ket{a^{\prime}}=\mathrm{Tr}_{C}[X\Phi^{\dagger}(\ket{a^{\prime}}\bra{a}\otimes\mathbb{I}_{B})\Phi]$
Assume now that the dimension of $\mathbb{H}_{C}$ does not exceed that of
$\mathbb{H}_{A}$ so that isometry between spaces of operators is possible in
principle. Equip the spaces with the Hilbert-Schmidt norm; If the map is
isometric, we have a transporting state. This is the case if for all operators
$X,Y\in BL(\mathbb{H}_{C})$

$\mathrm{Tr}_{A}[\mathcal{T}_{C,A}(X)^{\dagger}\mathcal{T}_{C,A}(Y)]=\mathrm{Tr}_{C}[X^{\dagger}Y].$
(51)

We can write this condition as a trace over two copies of subsystem C:

$\displaystyle\mathrm{Tr}_{C^{2}}[(X^{\dagger}\otimes
Y)\sum_{a,a^{\prime}}(\Phi^{\dagger}\otimes\Phi^{\dagger})(\ket{a^{\prime}}\bra{a}\otimes
1_{B}\otimes\ket{a}\bra{a^{\prime}}\otimes
1_{B})(\Phi\otimes\Phi)]=\mathrm{Tr}_{C^{2}}[(X^{\dagger}\otimes
Y)\mathcal{S}_{C}]$ (52)
$\displaystyle\Leftrightarrow\mathrm{Tr}_{C^{2}}[(X^{\dagger}\otimes
Y)(\Phi^{\dagger}\otimes\Phi^{\dagger})(\mathcal{S}_{A}\otimes 1_{B}^{\otimes
2})(\Phi\otimes\Phi)]=\mathrm{Tr}_{C^{2}}[(X^{\dagger}\otimes
Y)\mathcal{S}_{C}]$ (53)

where we have introduced a swap operator $\mathcal{S}_{C}$ which exchanges the
two factors in the tensor product $\mathbb{H}^{\otimes 2}_{C}$ So, we need to
require

$(\Phi^{\dagger}\otimes\Phi^{\dagger})(\mathcal{S}_{A}\otimes 1_{B}^{\otimes
2})(\Phi\otimes\Phi)=\mathcal{S}_{C}$ (54)

Reverting to the state picture makes this

$\mathrm{Tr}_{(AB)^{2}}[(\ket{\phi}\bra{\phi})^{\otimes
2}(\mathcal{S}_{A}\otimes 1_{B}^{\otimes 2})]=\mathcal{S}_{C}$ (55)

Which is the general condition to get an isometry between operator spaces.
However, we can instead start by demanding a bit more from the map
$\mathcal{T}_{C,A}$ from the beginning. In particular, we can require it to be
a quantum channel101010A completely positive, trace preserving map.: First
notice that the conjugation by $\Phi$ is completely positive, and the partial
trace is CPTP50. Therefore, the transport is a quantum channel if we require
conjugation by $\Phi$ to be trace preserving. This is the case precisely iff
$\Phi^{\dagger}\Phi=\mathbb{I}_{C}$, so if the map $\Phi$ is an isometry from
$\mathbb{H}_{C}$ to $\mathbb{H}_{AB}$. If we assume T to be a quantum channel,
the condition for T to be an isometry between B(H) simplifies to the
following:

$\displaystyle\langle
X,Y\rangle_{C}=\mathrm{Tr}_{C}[X^{\dagger}Y]=\mathrm{Tr}_{C}[X^{\dagger}\Phi^{\dagger}\Phi
Y\Phi^{\dagger}\Phi]=\mathrm{Tr}_{AB}[\Phi X^{\dagger}\Phi^{\dagger}\Phi
Y\Phi^{\dagger}]=\mathrm{Tr}_{(AB)^{2}}[(\Phi
X^{\dagger}\Phi^{\dagger}\otimes\Phi
Y\Phi^{\dagger})(\mathcal{S}_{A}\otimes\mathcal{S}_{B})]$ (56)
$\displaystyle\langle\mathcal{T}_{C,A}(X),\mathcal{T}_{C,A}(Y)\rangle_{A}=\mathrm{Tr}_{A}[\mathrm{Tr}_{B}[\Phi
X^{\dagger}\Phi^{\dagger}]\mathrm{Tr}_{B}[\Phi
Y\Phi^{\dagger}]]=\mathrm{Tr}_{(AB)^{2}}[(\Phi
X^{\dagger}\Phi^{\dagger}\otimes\Phi
Y\Phi^{\dagger})(\mathcal{S}_{A}\otimes\mathbb{I}_{B^{2}})]$ (57)

So the isometry condition for quantum channels is

$(\Phi^{\dagger}\otimes\Phi^{\dagger})(\mathcal{S}_{A}\otimes\mathbb{I}_{B^{2}})(\Phi\otimes\Phi)=(\Phi^{\dagger}\otimes\Phi^{\dagger})(\mathcal{S}_{A}\otimes\mathcal{S}_{B})(\Phi\otimes\Phi)$
(58)

In particular, notice that for $B=\emptyset$ the condition is automatically
fulfilled. In this work, we will content ourselves with establishing when the
transport superoperator is a quantum channel - so, equivalently, when $\Phi$
is an isometry. The method we use is entirely analogous to the one used in
previous works18; 16; 26; 43. Assume that the dimension of C is lower or equal
to that of AB. We rewrite

$\Phi^{\dagger}\Phi=\mathbb{I}_{C}\;\text{ as
}\;\mathrm{Tr}_{AB}[\ket{\phi}\bra{\phi}]=\mathbb{I}_{C}.$ (59)

More explicitly, the map has components $\Phi_{O,I}$ (O labeling a basis in
AB, I in C). Then

$(\Phi^{\dagger}\Phi)_{I,I^{\prime}}=\sum_{O}(\Phi^{\dagger})_{I,O}\Phi_{O,I^{\prime}}=\sum_{O}\bra{I}(\Phi^{\dagger})\ket{O}\bra{O}\Phi\ket{I^{\prime}}=\sum_{O}\bra{O}\Phi\ket{I^{\prime}}\overline{\bra{O}\Phi\ket{I}}$
(60)

And by using the defining relation
$\bra{O}\Phi\ket{I}\coloneqq\bra{O}\bra{I}\phi\rangle$:

$(\Phi^{\dagger}\Phi)_{I,I^{\prime}}=\sum_{O}\bra{O}\bra{I^{\prime}}\ket{\phi}\bra{\phi}\ket{O}\ket{I}=\bra{I^{\prime}}\left(\sum_{O}\bra{O}\ket{\phi}\bra{\phi}\ket{O}\right)\ket{I}=\bra{I^{\prime}}\mathrm{Tr}_{AB}[\ket{\phi}\bra{\phi}]\ket{I}$
(61)

We can then answer the question of isometry by calculating the purity of the
reduced state $\rho_{C}$, for example as the Rényi 2-entropy49; 6:

$e^{-S_{2}(\rho_{C})}=\frac{\mathrm{Tr}[\rho_{C}^{2}]}{\mathrm{Tr}[\rho_{C}]^{2}}$
(62)

If this expression reaches its minimum of $\dim(\mathbb{H}_{C})^{-1}$, the map
is a quantum channel. Via the replica trick111111Letting $\mathcal{S}$ be the
operator swapping two copies of a Hilbert space $\mathbb{H}$, we have
$\mathrm{Tr}[(A\otimes B)\mathcal{S}]=\mathrm{Tr}[AB]$,
while$\mathrm{Tr}[(A\otimes B)]=\mathrm{Tr}[A]\mathrm{Tr}[B]$., we can then
rewrite this as traces over two copies of the system

$\frac{\mathrm{Tr}_{\mathbb{H}_{C}^{2}}[\rho_{C}^{\otimes
2}\mathcal{S}_{C}]}{\mathrm{Tr}_{\mathbb{H}_{C}^{2}}[\rho_{C}^{\otimes
2}]}=\frac{\mathrm{Tr}_{\mathbb{H}^{2}}[(\ket{\phi}\bra{\phi})^{\otimes
2}\mathcal{S}_{C}]}{\mathrm{Tr}_{\mathbb{H}^{2}}[(\ket{\phi}\bra{\phi})^{\otimes
2}]}$ (63)

We can now apply the technique of averaging over the initial states. This
yields the average purity of the reduced state, which will allow for a general
statement about typical holographic properties. A crucial point to consider is
the variance of the quantities we wish to compute: If the variance is small,
we can not only approximate averages of functions by functions of averages,
but can also be sure that the typicity statement holds value as most states
will be close to that average. As was demonstrated by Hayden, Qi et al26, the
variance can be bounded from above to be arbitrarily small in the limit that
the auxiliary leg dimensions (also known as bond dimensions) of the tensor
network become large. This means, in the context of spin networks, that all
involved area spins must be fairly large.121212What fairly large means is
debateable and could lie between spin values of 10 and 100.  When this
assumption holds,

$\langle
e^{-S_{2}(\rho_{C})}\rangle_{U}=\langle\frac{\mathrm{Tr}_{\mathbb{H}^{2}}[(\ket{\phi}\bra{\phi})^{\otimes
2}\mathcal{S}_{C}]}{\mathrm{Tr}_{\mathbb{H}^{2}}[(\ket{\phi}\bra{\phi})^{\otimes
2}]}\rangle_{U}\approx\frac{\langle\mathrm{Tr}_{\mathbb{H}^{2}}[(\ket{\phi}\bra{\phi})^{\otimes
2}\mathcal{S}_{C}]\rangle_{U}}{\langle\mathrm{Tr}_{\mathbb{H}^{2}}[(\ket{\phi}\bra{\phi})^{\otimes
2}]\rangle_{U}}$ (64)

from where one can use linearity of the trace to replace
$(\ket{\Psi_{x}}\bra{\Psi_{x}})^{\otimes 2}$ by the average-square state

$R=\int_{SU(\mathbb{H})}d\mu(U)\rho(U)(U\ket{\Psi_{ref}}\bra{\Psi_{ref}}U^{\dagger})^{\otimes
2}$ (65)

This operator acting on two copies of the single-particle Hilbert space has
the property that it is invariant under unitary conjugation:

$V^{\otimes 2}R(V^{\dagger})^{\otimes 2}=R$ (66)

by left-invariance of the Haar measure. Crucially, this requires the group to
be a finite dimensional Lie group - this integral does not exist on the
infinite unitary group, so our Hilbert spaces must stay finite dimensional.
With this property, we can easily find what $R$ is - the only two operators
invariant under this action are the identity and the swap operator, in the
form

$R=\frac{1}{\dim(\mathbb{H})(\dim(\mathbb{H})+1)}(\mathbb{I}+\mathcal{S}).$
(67)

Since we average over each site seperately, we really replace the initial
random vertex states by

$\bigotimes_{x}\frac{1}{\dim(\mathbb{H}_{x})(\dim(\mathbb{H}_{x})+1)}(\mathbb{I}_{x}+\mathcal{S}_{x}).$
(68)

The real trick happens now: To make the tensor product above tractable, we
recognise that, when expanded as a sum, each term will have a number of swap
operators, and identity operators do not matter. Each term can then be labeled
by the set of sites with swap operators on it, a $-1$ indicating a swap.
The method by Hayden et al is to introduce on each site a $\pm 1$-valued Ising
spin $\sigma_{x}$, which indicates whether a swap is on that site or not. This
means the product turns into the sum over Ising configurations

$\prod_{x}\frac{1}{\dim(\mathbb{H}_{x})(\dim(\mathbb{H}_{x})+1)}\sum_{\vec{\sigma}}\bigotimes_{x}\mathcal{S}^{\frac{1-\sigma_{x}}{2}}_{x}$
(69)

To explain, each term in the original sum is mapped to a unique Ising
configuration such that the region of swap operators is the region of Ising
spin-downs. Then, every configuration must be summed over. This turns the
numerator and denominator of the average purity into Ising partition
functions:

$Z_{1|0}=\sum_{\vec{\sigma}}\mathrm{Tr}[\Pi_{\Gamma}^{\otimes
2}\bigotimes_{x}(\mathcal{S}^{\frac{1-\sigma_{x}}{2}}_{x})\mathcal{S}^{1|0}_{C}]=\sum_{\vec{\sigma}}e^{-H_{1|0}(\vec{\sigma})}$
(70)

and evaluation of the average purity is turned into a calculation of Ising-
like partition sums. In the case of large bond dimensions, we can approximate
the sums by their ground state values as the lowest bond dimension functions
as a notion of inverse temperature. The result is that an isometry, so minimal
purity of the reduced state, is attainable depending on the size of the local
input and output legs, as well as the graph structure.

###  3 Bulk-to-boundary holography for single spin sectors

It is the goal of this work to study a tensor network-like class of states,
derived from spin network states, with respect to their holographic
properties. The Hilbert space of $N$ tetrahedra is, due to the Peter-Weyl
decomposition,

$\mathbb{H}^{N}\cong(L^{2}(G^{D}/G))^{N}\cong\bigotimes^{N}_{x=1}(\bigoplus_{\mathbf{j}^{x}}V^{\mathbf{j}^{x}}\otimes\mathcal{I}_{\mathbf{j}^{x}})\cong\bigoplus_{{\mathbf{j}^{x}:x}}\bigotimes_{x}V^{\mathbf{j}^{x}}\otimes\mathcal{I}_{\mathbf{j}^{x}},$
(71)

which we can abbreviate by collecting all the independent representation spins
into a single label $\vec{\textbf{j}}$:

$\mathbb{H}^{N}\cong\bigoplus_{\vec{\textbf{j}}}V^{\vec{\textbf{j}}}\otimes\mathcal{I}_{\vec{\textbf{j}}}.$
(72)

Each term in this decomposition will be called a spin sector and we will
seperate the study of states here by the amount of sectors involved: First, a
single sector will be considered. The original work of this thesis is to
extend the results there to the case where spin sectors are superposed.
In this part, we will review previous studies on random spin tensor networks
with a single spin sector - so, where all edge spins have been fixed once and
for all. These states, when not randomised, are labeled by a twisted
simplicial geometry and have definite values for the areas of all faces of the
simplices. Still, they have volume information encoded in the intertwiner
degrees of freedom at each vertex of the dual graph, which may be put into
quantum superposition without changing the spins on the edges. We wish to
construct a geometry from $N$ individual simplices whose facial areas have
been fixed. The Hilbert space of these is given by

$\mathbb{H}_{\vec{\textbf{j}}}=\bigotimes_{x}\mathbb{H}_{\mathbf{j}^{x}}=\bigotimes_{x}\mathcal{I}_{\mathbf{j}^{x}}\otimes
V^{\mathbf{j}^{x}}.$ (73)

We pick a product state $\ket{\Psi}=\bigotimes_{x}\ket{\Psi_{x}}$ and wish to
glue the tetrahedra according to a simplicial complex dual to a given
4-valent, connected 4-coloured graph $\gamma$ with open edges. To do this, we
project the product state onto a maximally entangled link state for each pair
of faces to be glued. Name the total edge set of the graph $E$ and its subset
of internal links $\Gamma$, while the open boundary links will be called
$\partial\gamma$. Alternatively, the valency of the graph may be either 4 or
1, where the open edges are then capped off with a 1-valent vertex. This point
of view allows us to see our open edges as true edges of a graph. We now
project the product state onto

$\ket{\Gamma}=\bigotimes_{e\in\Gamma}\ket{e}\qquad\ket{e}=\frac{1}{\sqrt{d_{j_{e}}}}\sum_{m_{e}}(-1)^{j_{e}+m_{e}}\ket{j_{e}m_{e}}_{s(e)}\ket{j_{e},-m_{e}}_{t(e)}$
(74)

where $s(e),t(e)$ name the source and target vertices of the edge $e$,
respectively. Each individual state is normalised to $1$.
$\ket{\Gamma}\bra{\Gamma}$ is the projector which performs the glueing. If,
instead, we project out the internal links, we have a bulk/boundary state

$\ket{\phi}=\bra{\Gamma}\Psi\rangle\in\mathbb{H}_{\partial\gamma,\vec{\textbf{j}}|_{\partial\gamma}}\otimes\mathcal{I}_{\vec{\textbf{j}}}$
(75)

containing only relevant information on the intertwiners and boundary edges.
We will call this type of state, once randomised using unitaries, a Random
spin tensor network (RSTN) state. One can now see this state as defining a map
from bulk data, given by intertwiners, to the boundary space.

####  1 Bulk-to-boundary isometry

Taking as $\mathbb{H}_{C}=\mathcal{I}_{\vec{\textbf{j}}}$,
$\mathbb{H}_{B}=\emptyset$ and
$\mathbb{H}_{A}=\mathbb{H}_{\partial\gamma,\vec{\textbf{j}}|_{\partial\gamma}}$,
one can immediately apply the technique of random tensor networks.16 Assume,
that all involved spins are large enough to suppress fluctuations in the
unitary average over individual vertex states. This amounts to making the
geometry more semiclassical in the same sense as large angular momentum values
behaving approximately semiclassically and thus making variances of the
involved quantities smaller. Then,

$\langle\frac{\mathrm{Tr}_{\mathbb{H}^{2}}[(\ket{\phi}\bra{\phi})^{\otimes
2}\mathcal{S}_{C}]}{\mathrm{Tr}_{\mathbb{H}^{2}}[(\ket{\phi}\bra{\phi})^{\otimes
2}]}\rangle_{U}\approx\frac{\langle\mathrm{Tr}_{\mathbb{H}^{2}}[(\ket{\phi}\bra{\phi})^{\otimes
2}\mathcal{S}_{C}]\rangle_{U}}{\langle\mathrm{Tr}_{\mathbb{H}^{2}}[(\ket{\phi}\bra{\phi})^{\otimes
2}]\rangle_{U}}$ (76)

Writing each vertex state

$\ket{\Psi_{x}}=U_{x}\ket{Ref},$ (77)

a unitary average over each vertex gives, through Schur’s lemma,

$\langle\ket{\Psi_{x}}\bra{\Psi_{x}}^{\otimes
2}\rangle_{U_{x}}=\frac{\mathbb{I}_{\mathbb{H}_{j^{x}}^{\otimes
2}}+\mathcal{S}_{j^{x}}}{\dim(\mathbb{H}_{j^{x}})(\dim(\mathbb{H}_{j^{x}})+1)}$
(78)

which passes through the traces due to linearity. Then, we can turn the tensor
product over these into a sum by introducing bookkeeping variables
$\sigma_{x}\in\\{\pm 1\\}$ on each vertex of the graph. When collecting them
over all vertices into the vector $\vec{\sigma}$, they label a term in the sum
by giving vertices with a $\mathcal{S}_{x}$ operator a $\sigma_{x}=-1$. Then
the sum is

$\langle\bigotimes_{x}\ket{\Psi_{x}}\bra{\Psi_{x}}\rangle_{U}=\bigotimes_{x}\langle\ket{\Psi_{x}}\bra{\Psi_{x}}\rangle_{U_{x}}=\frac{1}{\prod_{x}\dim(\mathbb{H}_{x})(\dim(\mathbb{H}_{x})_{x}+1)}\sum_{\vec{\sigma}}\bigotimes_{x}\mathcal{S}_{x}^{\frac{1-\sigma_{x}}{2}}$
(79)

Luckily, the prefactor is independent of the configuration and thus drops out
in the quotient 6 and we will ignore it from now on. The average purity is
then the quotient of

$Z_{1|0}=\sum_{\vec{\sigma}}\mathrm{Tr}[(\rho_{\Gamma})^{\otimes
2}\bigotimes_{x}\mathcal{S}_{x}^{\frac{1-\sigma_{x}}{2}}(\mathcal{S}_{I})^{1|0}].$
(80)

Each term in the sum can now be understood as an amplitude in a statistical
partition function $Z_{1|0}$, simply by rewriting it as an exponential:

$Z_{1|0}=\sum_{\vec{\sigma}}e^{-\mathcal{H}_{1|0}(\vec{\sigma})}$ (81)

and where the Hamiltonians are

$\mathcal{H}_{1|0}(\vec{\sigma})=\sum_{e\in\partial\gamma}\frac{1-\sigma_{s(e)}}{2}\log(d_{j_{e}})+\sum_{e\in\Gamma}\frac{1-\sigma_{s(e)}\sigma_{t(e)}}{2}\log(d_{j_{e}})+\sum_{x}\frac{1-\sigma_{x}b_{x}}{2}\log(\mathcal{D}_{\mathbf{j}^{x}})$
(82)

where we have introduced the bulk pinning field $b_{x}$ given by 1 for $Z_{0}$
and by $-1$ on bulk vertices for $Z_{1}$. Given this setup, we can compute the
purity in the regime of high spins. The reason is that we may see the lowest
spin in the sector as a notion of temperature for the partition sum - it also
specifies the energy gap between the ground and first excited states of the
Hamiltonians. If this lowest spin is large enough, excited states of the Ising
system will not contribute much to the partition sum. In this regime, we may
work with just the ground states to approximate the purity. The ground state
of $\mathcal{H}_{0}$ is easily found to be the all-up Ising configuration. For
$\mathcal{H}_{1}$, it is less trivial: One rather typically has two regions of
the graph seperated by a domain wall where one side is Ising-up and the other
down. However, to achieve an isometry, the all-up configuration must still be
the ground state. Flipping a spin somewhere down should thus increase the
energy:

$\Delta_{z}\mathcal{H}_{1}=\sum_{e\in
E:s(e)=z}\log(d_{j_{e}})-\log(\mathcal{D}_{\mathbf{j}^{z}})\stackrel{{\scriptstyle!}}{{>}}0$
(83)

which is easily satisfied - but once on flips larger regions, this may turn
negative. In fact, if one flips a region $X$ down, then the condition of
positivity is

$\Delta_{X}\mathcal{H}_{1}=\sum_{e\in E:s(e)\in X,t(e)\notin
X}\log(d_{j_{e}})-\sum_{x\in
X}\log(\mathcal{D}_{\mathbf{j}^{x}})\stackrel{{\scriptstyle!}}{{>}}0\qquad\text{
or }\qquad\prod_{e\in E:s(e)\in X,t(e)\notin X}d_{j_{e}}>\prod_{x\in
X}\mathcal{D}_{\mathbf{j}^{x}}$ (84)

which is easily violated for some graphs, for example in the homogeneous case
where all spins are equal. Thus, while the all-up configuration is always a
local minimum of the energy, one needs it to be a global one. In particular,
there is an interplay between the graph structure and geometric quantities on
it that determines which states admit such a holography. It was found that the
larger the overall inhomogeneity of spins on the graph, the easier it is to
achieve isometric behaviour. This directly corresponds to having small
intertwiner spaces, which is easy to understand from the perspective that
information from the bulk needs to be mapped to the boundary, no matter how
small the region. More succinctly, the holography condition is that
$\dim(\mathbb{H}_{\partial X})>\dim(\mathcal{I}_{X})$, which is the minimal
condition for having an isometry from $\mathcal{I}_{X}$ to
$\mathbb{H}_{\partial X}$. So, to have isometry for the whole graph, we need
precisely that every subset of the graph fulfils the minimal condition. A
similar calculation can be done for the boundary-to boundary isometry case,
where first a bulk state must be fixed.
By the same methods, one can find a type of lattice RT formula for boundary
areas:

$S_{2}(A)\sim\sum_{e\in\Sigma_{A}}\log(d_{j_{e}})$ (85)

which depends on the domain wall $\Sigma_{A}$ associated to a Hamiltonian with
distinguished boundary region $A$. Said domain wall plays the role of the
minimal surface of an RT formula. In this way, the mapping to the Ising model
provides a precise implementation of the geometry of holographic behaviour.
It is noteworthy that whether or not a bulk-to-boundary isometry exists cannot
be calculated in the case of superpositions. This is only possible on the
fixed-spin level due to factorisation of the Hilbert space.131313There may be
a possibility of doing this factorisation through the addition of edge modes.
However, this is beyond the scope of this work and reserved for future
investigations. As such, we will instead investigate boundary-to-boundary
mappings in the following, where a projection onto a fixed bulk state
factorises the remainder of the Hilbert space.

##  Chapter 1 RSTN with multiple spin sectors

###  1 Superposition of spins

We now consider the general case of states which feature a superposition of
spins, but within the class of RSTNs. The total Hilbert space of N quanta,
$\mathbb{H}:=L^{2}(G^{D}/G_{\text{Diag}})^{N}=\bigotimes_{x}\mathbb{H}_{x}\cong\bigoplus_{\vec{\textbf{j}}}\mathbb{H}_{\vec{\textbf{j}}}$
splits into spin sectors.
Each sector factorises as
$\mathbb{H}_{\vec{\textbf{j}}}\cong\mathcal{I}_{\vec{\textbf{j}}}\otimes
V_{\vec{\textbf{j}}}=\bigotimes_{x}\mathcal{I}^{\textbf{j}^{x}}\otimes\bigotimes_{x}V^{\textbf{j}^{x}}$
with $\mathcal{I}^{\textbf{j}^{x}}$ the intertwiner space and
$V^{\textbf{j}^{v}}\cong\bigotimes_{\alpha=1}^{D}V^{j^{x}_{\alpha}}$ the space
of open semilinks associated with the tetrahedron $x$, of dimension
$d_{j^{x}_{\alpha}}=2j^{x}_{\alpha}+1$. Here, we have labeled the links with
$\alpha\in\\{1,2,\dots,D\\}$ as colours.
We once again want to take a product N-particle state and project it on some
data, then reduce the resulting state to a boundary region $C$ and calculate
its Rényi entropy. Through this, we can test whether that region is mapped
isometrically onto its boundary complement.

We will assume to work with a fixed graph and use the following definitions:

  * •

Fix the connectivity pattern $\Gamma$ of links between the $N$ vertices, for
example as an adjacency matrix. We require the graphs to be colored, meaning
in particular that vertices can not be connected by more than one link. Some
links are left open.

  * •

Let $(L)E$ be the set of (semi)links of the graph. Each of these is then
either an internal link of the form $(x,y;\alpha)$ or a boundary
($\partial\gamma$) link $(x;\alpha)$ belonging either to the outer boundary
$\partial\gamma_{out}$ or the inner boundary $\partial\gamma_{in}$, which form
a partition of the boundary links. When calculating the entropy of a region
$A\subset\partial\gamma_{out}$, we could choose it to be in the outer
boundary, while the inner one provides external information, for example from
low area regions of the spin network that we traced out. For internal links
$e\in\Gamma$, we make the distinction between a single link and its two
constituent semilinks by calling the set of internal semilinks $\Gamma_{L}$.
For boundary links, this distinction is vacuous.

  * •

Let
$\ket{e_{j}}:=\frac{1}{\sqrt{d_{j}}}\sum_{m}(-1)^{j+m}\ket{jm}_{1}\ket{j-m}_{2}$
be a maximally entangled state of 2 semilinks, in the spin-j sector. We
superpose them to
$\ket{e}=\sum_{j_{e}\in\frac{\mathbb{N}}{2}}g_{j_{e}}\ket{e_{j_{e}}}$ with
coefficients $g_{j}$ of our choice. Then the full state to project on is
$\ket{\Gamma}=\bigotimes_{e}\ket{e}$.

  * •

Let $\ket{\zeta_{\vec{\textbf{j}}}}\in\mathcal{I}_{\vec{\textbf{j}}}$ be an
intertwiner state in each spin sector, and
$\ket{\zeta}=\sum_{\vec{\textbf{j}}}\ket{\zeta_{\vec{\textbf{j}}}}$. Later on,
we will replace this with a more general, mixed state $\rho_{I}$. Do the same
for a core state $\ket{\theta}=\sum_{j_{in}}\ket{\theta_{j_{in}}}$ on which we
project the inner boundary. For the most part, this last piece of data can be
considered optional, but in the appendix it proves useful.

  * •

Finally, we will the notation $||A||$ to denote the number of boundary links
associated to a subset of the vertices of the graph, while $|A|$ denotes the
number of vertices.

Now start with the product state

$\ket{\Psi}=\bigotimes_{x}\ket{\Psi_{x}}$ (1)

and project it onto the data we have given:111Technically, this scalar product
is not defined, so this presents an abuse of notation. However, the projection
on a subspace of states with the given intertwiner and boundary data is a
well-defined operator and behaves in exactly the same way as this naive
implementation of projection.

$\ket{\Phi}=\bra{\theta}\bra{\zeta}\bra{\Gamma}\Psi\rangle\in\mathbb{H}_{\partial\gamma,out}$
(2)

Importantly, the projection forces the contributions of the intertwiners and
links to be from the same spin sectors in the following sense:

$\begin{split}\bra{\theta}\bra{\zeta}\bra{\Gamma}\Psi\rangle=\sum_{\vec{\textbf{j}}}\bra{\theta}\bra{\zeta_{\vec{\textbf{j}}}}\bra{\Gamma}\Psi\rangle=\sum_{\vec{\textbf{j}}}\bra{\theta}\bra{\zeta_{\vec{\textbf{j}}}}\bra{\Gamma}\Psi_{\vec{\textbf{j}}}\rangle\\\
=\sum_{\vec{\textbf{j}}}\bra{\theta_{\vec{\textbf{j}}|_{\partial\gamma_{in}}}}\bra{\zeta_{\vec{\textbf{j}}}}\bra{\Gamma_{\vec{\textbf{j}}|_{\Gamma}}}\Psi_{\vec{\textbf{j}}}\rangle=(\sum_{\vec{\textbf{j}}}\bra{\theta_{\vec{\textbf{j}}|_{\partial\gamma_{in}}}}\bra{\zeta_{\vec{\textbf{j}}}}\bra{\Gamma_{\vec{\textbf{j}}|_{\Gamma}}})\Psi\rangle=\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}\bra{\theta,\zeta,\Gamma}\Psi\rangle\end{split}$
(3)

so that we make no restriction on the possible data we project on by instead
using $\ket{\theta,\zeta,\Gamma}$, which is diagonal in spin sectors.
The boundary Hilbert space the projected state lives in factorises as

$\mathbb{H}_{\partial\gamma,out}\cong\bigoplus_{j_{\partial\gamma,out}}\mathbb{H}_{j_{\partial\gamma}}\cong\bigoplus_{j_{A}}\bigoplus_{j_{\bar{A}}}\mathbb{H}_{A,j_{A}}\otimes\mathbb{H}_{\bar{A},j_{\bar{A}}}\cong\bigoplus_{j_{A}}\mathbb{H}_{A,j_{A}}\otimes\bigoplus_{j_{\bar{A}}}\mathbb{H}_{\bar{A},j_{\bar{A}}}\cong\mathbb{H}_{A}\otimes\mathbb{H}_{\bar{A}}$
(4)

So we can reduce the state to the region $A$ by tracing out the complement.

####  1 Calculating average entropy

Instead of calculating the entropy for a particular state, we will make a
typicity statement about a class of states of interest with data as presented
earlier. These states have a fixed graph structure and data on intertwiners,
but are not limited to one spin sector. For them, we can write the average
entropy as a partition function of a randomised Ising model. To start, we
first average the exponential of the entropy over a distribution of starting
states $\ket{\Psi_{x}}=U_{x}\ket{\Psi_{ref}}$, where we choose some reference
state as before. This distribution is chosen to be uniform over the unitary
group relating different 1-particle-states. More explicitly, we perform an
integral

$\langle\ket{\Psi_{x}}\bra{\Psi_{x}}\rangle_{U_{x}}:=\int_{\mathcal{U}(\mathbb{H}_{x})}d\mu_{Haar}(U_{x})(U_{x}\ket{\Psi_{ref}}\bra{\Psi_{ref}}(U_{x})^{\dagger})^{\otimes
2}$ (5)

with the Haar measure on the unitary group of each vertex/tetrahedron’s
Hilbert space seperately. By linearity this average commutes with taking
traces and we denote it by $\langle-\rangle_{U}$ in the following.
Then, if we choose a large enough lower cutoff on participating spins
$\mathbb{j}$, we can suppress fluctuations in the quotient

$\langle
e^{-S_{2}(\rho_{C})}\rangle_{U}=\langle\frac{\mathrm{Tr}_{\mathbb{H}^{2}}[(\ket{\phi}\bra{\phi})^{\otimes
2}\mathcal{S}_{C}]}{\mathrm{Tr}_{\mathbb{H}^{2}}[(\ket{\phi}\bra{\phi})^{\otimes
2}]}\rangle_{U}\approx\frac{\langle\mathrm{Tr}_{\mathbb{H}^{2}}[(\ket{\phi}\bra{\phi})^{\otimes
2}\mathcal{S}_{C}]\rangle_{U}}{\langle\mathrm{Tr}_{\mathbb{H}^{2}}[(\ket{\phi}\bra{\phi})^{\otimes
2}]\rangle_{U}}=:\frac{Z_{1}}{Z_{0}}$ (6)

as has been shown in random tensor networks. Then, we expand the boundary
state in the numerator and denominator:

$Z_{1|0}=\mathrm{Tr}[(\rho_{I})^{\otimes 2}(\rho_{\Gamma})^{\otimes
2}\langle(\ket{\Psi}\bra{\Psi})^{\otimes
2}\rangle_{U}(\mathcal{S}_{C})^{1|0}]$ (7)

If we also introduce an upper cutoff on spins $J$, we can then use Schur’s
lemma to evaluate 5, as the averaging makes the quantity
$Conj(U_{x})$-invariant. The result is that

$\langle\ket{\Psi_{x}}\bra{\Psi_{x}}\rangle_{U_{x}}=\frac{\mathbb{I}_{\mathbb{H}_{x}^{2}}+\mathcal{S}_{x}}{\dim(\mathbb{H}_{x})(\dim(\mathbb{H}_{x})_{x}+1)}$
(8)

Then, we can turn the tensor product over these into a sum by introducing
bookkeeping variables $\sigma_{x}\in\\{\pm 1\\}$ on each vertex of the graph.
when collecting them over all vertices into the vector $\vec{\sigma}$, they
label a term in the sum by giving vertices with a $\mathcal{S}_{x}$ operator a
$\sigma_{x}=-1$. Then the sum is

$\langle\bigotimes_{x}\ket{\Psi_{x}}\bra{\Psi_{x}}\rangle_{U}=\bigotimes_{x}\langle\ket{\Psi_{x}}\bra{\Psi_{x}}\rangle_{U_{x}}=\frac{1}{\prod_{x}\dim(\mathbb{H}_{x})(\dim(\mathbb{H}_{x})_{x}+1)}\sum_{\vec{\sigma}}\bigotimes_{x}\mathcal{S}_{x}^{\frac{1-\sigma_{x}}{2}}$
(9)

Luckily, the prefactor is independent of the configuration and thus drops out
in the quotient 6 and we will ignore it from now on.
With this, the quantities 7 can be expanded into

$\displaystyle Z_{1|0}$
$\displaystyle=\sum_{\vec{\sigma}}\mathrm{Tr}[(\rho_{I})^{\otimes
2}(\rho_{\Gamma})^{\otimes
2}\bigotimes_{x}\mathcal{S}_{x}^{\frac{1-\sigma_{x}}{2}}(\mathcal{S}_{C})^{1|0}]$
(10)
$\displaystyle=\sum_{(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})}\mathrm{Tr}_{\mathbb{H}_{\vec{\textbf{j}}}\otimes\mathbb{H}_{\vec{\textbf{k}}}}[(\rho_{I})^{\otimes
2}(\rho_{\Gamma})^{\otimes
2}\bigotimes_{x}\mathcal{S}_{x}^{\frac{1-\sigma_{x}}{2}}(\mathcal{S}_{C})^{1|0}]$
(11)

where we used that the Hilbert spaces split into orthogonal spin sectors. The
traces in each term are now over spaces that factorise over vertices and
links, and accordingly the single-site swap operators do so, too:
$\mathcal{S}_{x}=\mathcal{S}_{\mathcal{I},x}\prod_{\alpha}\mathcal{S}_{\alpha,x}$.
The swaps here are indiscriminate of spin sector. The traces can then be
evaluated over intertwiner, link and boundary parts seperately.
To illustrate the calculations, the intertwiner factor is

$\displaystyle\mathrm{Tr}_{\mathcal{I}_{\vec{\textbf{j}}}\otimes\mathcal{I}_{\vec{\textbf{k}}}}[(\rho^{I})^{\otimes
2}\bigotimes_{x}\mathcal{S}_{I,x}^{\frac{1-\sigma_{x}}{2}}]$
$\displaystyle=\mathrm{Tr}_{\bigotimes_{S_{\downarrow}}\mathcal{I}_{\mathbf{j}^{x}}\otimes\mathcal{I}_{\mathbf{k}^{x}}}[(\mathrm{Tr}_{\bigotimes_{S_{\uparrow}}\mathcal{I}_{\mathbf{j}^{x}}}[\rho^{I}]\otimes\mathrm{Tr}_{\bigotimes_{S_{\uparrow}}\mathcal{I}_{\mathbf{j}^{x}}}[\rho^{I}])\bigotimes_{S_{\downarrow}}\mathcal{S}_{I,x}]$
(12)
$\displaystyle=\mathrm{Tr}[\bigotimes_{S_{\downarrow}}\mathbb{P}_{\mathcal{I}_{\mathbf{j}^{x}}}(\rho^{I}\bigotimes_{S_{\uparrow}}\mathbb{P}_{\mathcal{I}_{\mathbf{j}^{x}}})_{\downarrow}\bigotimes_{S_{\downarrow}}\mathbb{P}_{\mathcal{I}_{\mathbf{k}^{x}}}(\rho^{I}\bigotimes_{S_{\uparrow}}\mathbb{P}_{\mathcal{I}_{\mathbf{k}^{x}}})_{\downarrow}]$
(13)

where we use the shorthand $\mathbb{P}_{\mathcal{I}_{\mathbf{k}^{x}}}$ for the
projector onto the intertwiner space $\mathcal{I}_{\mathbf{k}^{x}}$ and
$(-)_{\downarrow}$ denotes reduction to the subspace
$\bigotimes_{x:\sigma_{x}=-1}\mathcal{I}_{x}$ associated to $S_{\downarrow}$.
So, a copy of $\rho^{I}$ is first reduced to the region
$S_{\downarrow}=\\{x:\sigma_{x}=-1\\}$, but with the spins set by
$\vec{\textbf{j}}(\vec{\textbf{k}})$ on $S_{\uparrow}$. Then, the two copies
are multiplied and traced over the remainder of vertices, but with spins fixed
again to be $j$ or $k$ between them. In particular, for certain values of
$j,k$ on $S_{\downarrow}$, this factor will vanish depending on the
intertwiner state $\rho^{I}$ chosen. For example, if it is diagonal in spin
sectors, meaning
$(\rho^{I})_{\vec{\textbf{j}},\vec{\textbf{k}}}\sim\delta_{\vec{\textbf{j}},\vec{\textbf{k}}}$,
this gives a constraint on which pairs of spin sectors have a nonvanishing
intertwiner factor, of the form

$\Delta_{I}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})=\prod_{x\in
S_{\downarrow}}\delta_{\mathbf{j}^{x},\mathbf{k}^{x}}$ (14)

We will write this contribution from intertwiners as a general boolean-valued
factor $\Delta$, a $\sigma$-independent term and an exponential as follows:

$\mathrm{Tr}_{\mathcal{I}_{\vec{\textbf{j}}}\otimes\mathcal{I}_{\vec{\textbf{k}}}}[(\rho^{I})^{\otimes
2}\bigotimes_{x}\mathcal{S}_{I,x}^{\frac{1-\sigma_{x}}{2}}]=\Delta_{I}(\vec{\sigma},\vec{\textbf{j}},\vec{\textbf{k}})\mathrm{Tr}_{\mathcal{I}_{\vec{\textbf{j}}}}[\rho^{I}]\mathrm{Tr}_{\mathcal{I}_{\vec{\textbf{k}}}}[\rho^{I}]e^{-\Sigma_{I}(\vec{\sigma},\vec{\textbf{j}},\vec{\textbf{k}})}$
(15)

where we include an entropy-like quantity

$\Sigma_{I}(\vec{\sigma},\vec{\textbf{j}},\vec{\textbf{k}})=-\log[\frac{\mathrm{Tr}_{\bigotimes_{S_{\downarrow}}\mathcal{I}_{\mathbf{j}^{x}}\otimes\mathcal{I}_{\mathbf{k}^{x}}}[(\mathrm{Tr}_{\bigotimes_{S_{\uparrow}}\mathcal{I}_{\mathbf{j}^{x}}}[\rho^{I}]\otimes\mathrm{Tr}_{\bigotimes_{S_{\uparrow}}\mathcal{I}_{\mathbf{j}^{x}}}[\rho^{I}])\bigotimes_{S_{\downarrow}}\mathcal{S}_{I,x}]}{\mathrm{Tr}_{\mathcal{I}_{\vec{\textbf{j}}}}[\rho^{I}]\mathrm{Tr}_{\mathcal{I}_{\vec{\textbf{k}}}}[\rho^{I}]}]$
(16)

and it is understood that the $\Delta$-factor forces the term to be 0 if
$\Sigma_{I}$ reaches $\infty$. Its presence serves the purpose of setting the
right hand side to be 0 whenever the left hand side is, in a transparent
manner. In sums over spin sectors later on, these types of factors serve as
constraints or indicators on which spin sectors contribute. In particular, we
note three special cases:

  * •

$\vec{\textbf{j}}=\vec{\textbf{k}}$:
$\Sigma_{I}(\vec{\sigma},\vec{\textbf{j}},\vec{\textbf{k}})=S_{2}((\rho^{I}_{\vec{\textbf{j}}})_{\downarrow})$
the Rényi 2-entropy of the reduced intertwiner state.

  * •

$\vec{\sigma}=\vec{1}$: $\Sigma_{I}=0$.

  * •

$\vec{\sigma}=\vec{-1}$:
$\Sigma_{I}=-\log[\frac{\mathrm{Tr}[(\rho^{I})_{\vec{\textbf{j}},\vec{\textbf{k}}}(\rho^{I})_{\vec{\textbf{k}},\vec{\textbf{j}}}]}{\mathrm{Tr}_{\mathcal{I}_{\vec{\textbf{j}}}}[\rho^{I}]\mathrm{Tr}_{\mathcal{I}_{\vec{\textbf{k}}}}[\rho^{I}]}]$.

This pattern continues: We will split the contributions from the other parts
of the graph in the same way into an indicator, a contribution independent of
the Ising spins and an exponential.
For the internal links, we find that the space to be traced over factorises,
as does the state of interest. For each internal link of the graph, there are
two semilinks, and so

$\mathbb{H}_{\Gamma}\cong\bigotimes_{l\in\Gamma_{L}}\mathbb{H}_{l}\;\qquad\mathbb{H}_{l}=\bigoplus_{j_{l}}V^{j_{l}}$
(17)

which means that the restriction to the $\vec{\textbf{j}}$-spin sector just
leaves each (semi)link with a single spin value:

$\mathbb{H}_{\Gamma,\vec{\textbf{j}}}\cong\bigotimes_{e\in\Gamma}\mathbb{H}_{e,j_{e}}\;\qquad\mathbb{H}_{e,j_{e}}=V^{j_{e}}_{s(e)}\otimes
V^{j_{e}}_{t(e)}$ (18)

so there is a degree of freedom of dimension $2j_{e}+1$ at each end of the
link $e$. The projection operator also factorises over internal links:

$\rho^{\Gamma}=\bigotimes_{e\in\Gamma}\rho^{\Gamma}_{e}\;\qquad\rho^{\Gamma}_{e}=\ket{e}\bra{e}\;\qquad\ket{e}=\sum_{j_{e}}\frac{g_{j_{e}}}{\sqrt{d_{j_{e}}}}\sum_{m_{e}}(-1)^{j_{e}+m_{e}}\ket{j_{e}m_{e}}_{s(e)}\ket{j_{e}m_{e}}_{t(e)}$
(19)

Internal links will either have 2 swap operators or 1 (depending on whether
they are fully contained in $S_{\uparrow,\downarrow}$ or not, respectively).
If we name the subset of internal links on the boundary of the two regions
$\partial S$, then

$\displaystyle\mathrm{Tr}_{\mathbb{H}_{\Gamma,\vec{\textbf{j}}}\otimes\mathbb{H}_{\Gamma,\vec{\textbf{k}}}}[(\rho^{\Gamma})^{\otimes
2}\bigotimes_{l\in\Gamma_{L}}\mathcal{S}_{l}^{\frac{1-\sigma_{x}}{2}}]$
$\displaystyle=\prod_{e\in\Gamma}\mathrm{Tr}_{\mathbb{H}_{e,j_{e}}\otimes\mathbb{H}_{e,k_{e}}}[(\ket{e}\bra{e})^{\otimes
2}\mathcal{S}_{e}^{\frac{1-\sigma_{s}(e)\sigma_{t}(e)}{2}}]$ (20)
$\displaystyle=\prod_{e\in\partial
S}\delta_{j_{e},k_{e}}\prod_{e\in\Gamma}d_{j_{e}}^{-\frac{1-\sigma_{s}(e)\sigma_{t}(e)}{2}}\prod_{e\in\Gamma}|g_{j_{e}}|^{2}|g_{k_{e}}|^{2}.$
(21)

For this, we state the result for both the single-and no-swap case, where for
simplicity $s(e)=x,y=t(e)$:

$\begin{split}\mathrm{Tr}_{\mathbb{H}_{e,j_{e}}\otimes\mathbb{H}_{e,k_{e}}}[(\ket{e}\bra{e})^{\otimes
2}\mathcal{S}_{e}]=\\\
\sum_{m^{x},m^{y},n^{x},n^{y}}\bra{jm^{x}}_{x}\bra{jm^{y}}_{y}\bra{kn^{x}}_{x}\bra{kn^{y}}_{y}(\ket{e}\bra{e})^{\otimes
2}\mathcal{S}_{e}\ket{jm^{x}}_{x}\ket{jm^{y}}_{y}\ket{kn^{x}}_{x}\ket{kn^{y}}_{y}\\\
=\sum_{m^{x},m^{y},n^{x},n^{y}}\bra{jm^{x}}_{x}\bra{jm^{y}}_{y}\ket{e}\bra{kn^{x}}_{x}\bra{kn^{y}}_{y}\ket{e}\bra{e}\bra{e}\mathcal{S}_{e}\ket{jm^{x}}_{x}\ket{jm^{y}}_{y}\ket{kn^{x}}_{x}\ket{kn^{y}}_{y}\\\
=\sum_{m^{x},m^{y},n^{x},n^{y}}\bra{jm^{x}}_{x}\bra{jm^{y}}_{y}\ket{e}\bra{kn^{x}}_{x}\bra{kn^{y}}_{y}\ket{e}\bra{e}\bra{e}\ket{jm^{x}}_{x}\ket{kn^{y}}_{y}\ket{kn^{x}}_{x}\ket{jm^{y}}_{y}\\\
=\sum_{m^{x},m^{y},n^{x},n^{y}}(-1)^{j+m^{x}+k+n^{x}}\frac{g_{j_{e}}}{\sqrt{d_{j_{e}}}}\delta_{m^{x}+m^{y},0}\frac{g_{k_{e}}}{\sqrt{d_{k_{e}}}}\delta_{n^{x}+n^{y},0}\bra{e}\ket{jm^{x}}_{x}\ket{kn^{y}}_{y}\bra{e}\ket{kn^{x}}_{x}\ket{jm^{y}}_{y}\\\
=\sum_{m^{x},m^{y},n^{x},n^{y}}\frac{g_{j_{e}}}{\sqrt{d_{j_{e}}}}\delta_{m^{x}+m^{y},0}\frac{g_{k_{e}}}{\sqrt{d_{k_{e}}}}\delta_{n^{x}+n^{y},0}\frac{\overline{g_{j_{e}}}}{\sqrt{d_{j_{e}}}}\frac{\overline{g_{k_{e}}}}{\sqrt{d_{k_{e}}}}\delta_{n^{x}+m^{y},0}^{2}\delta_{j_{e},k_{e}}^{2}\\\
=\delta_{j_{e},k_{e}}\frac{|g_{j_{e}}|^{4}}{d_{j_{e}}^{2}}\sum_{m^{x}}1=\delta_{j_{e},k_{e}}\frac{|g_{j_{e}}|^{2}|g_{k_{e}}|^{2}}{d_{j_{e}}}\end{split}$
(22)
$\begin{split}\mathrm{Tr}_{\mathbb{H}_{e,j_{e}}\otimes\mathbb{H}_{e,k_{e}}}[(\ket{e}\bra{e})^{\otimes
2}]\\\
=\sum_{m^{x},m^{y},n^{x},n^{y}}\bra{jm^{x}}_{x}\bra{jm^{y}}_{y}\ket{e}\bra{kn^{x}}_{x}\bra{kn^{y}}_{y}\ket{e}\bra{e}\ket{jm^{x}}_{x}\ket{jm^{y}}_{y}\bra{e}\ket{kn^{x}}_{x}\ket{kn^{y}}_{y}\\\
=\sum_{m^{x},m^{y},n^{x},n^{y}}\frac{g_{j_{e}}}{\sqrt{d_{j_{e}}}}\delta_{m^{x},m^{y}}^{2}\frac{g_{k_{e}}}{\sqrt{d_{k_{e}}}}\delta_{n^{x},n^{y}}^{2}\frac{\overline{g_{j_{e}}}}{\sqrt{d_{j_{e}}}}\frac{\overline{g_{k_{e}}}}{\sqrt{d_{k_{e}}}}=|g_{j_{e}}|^{2}|g_{k_{e}}|^{2}\end{split}$
(23)

Similarly, the boundary has its own factor: By introducing a boundary pinning
field on the boundary vertices, with $h_{x}=-1$ for $Z_{1}$ if $x$ is a
boundary vertex at $C$ and $0$ else, we find by the same kind of calculation

$\mathrm{Tr}_{\mathbb{H}_{\partial\gamma,\vec{\textbf{j}}}\otimes\mathbb{H}_{\partial\gamma,\vec{\textbf{k}}}}[\bigotimes_{e\in\partial\gamma}\mathcal{S}_{e}^{\frac{1-\sigma_{s(e)}}{2}}\mathcal{S}_{C}]=\mathrm{Tr}_{\mathbb{H}_{\partial\gamma,\vec{\textbf{j}}}\otimes\mathbb{H}_{\partial\gamma,\vec{\textbf{k}}}}[\bigotimes_{e\in\partial\gamma}\mathcal{S}_{e}^{\frac{1-\sigma_{s(e)}h_{t(e)}}{2}}]=\prod_{e:\sigma_{x}h_{e}=-1}\delta_{j_{e},k_{e}}d_{j_{e}}^{-\frac{1-\sigma_{s}(e)}{h}_{t(e)}{2}}\prod_{e\in\partial\gamma}d_{j_{e}}d_{k_{e}}.$
(24)

From this we can see that the result would be uniform across all links if we
chose the normalisation factors to be $|g_{j}|=\sqrt{d_{j}}$. As should be
clear from a quick glance, this is not normalisable if we take the cutoff $J$
to infinity. For this calculation as before, the cutoff becomes a crucial
assumption.

####  2 The random Ising model

Now, we can write 7 as

$Z_{1|0}=\sum_{(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})}\Delta_{1|0}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})e^{B(\vec{\textbf{j}})+B(\vec{\textbf{k}})}e^{-\mathcal{H}_{1|0}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})}$
(25)

where

$\mathcal{H}_{1|0}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})=\sum_{e\in\partial\gamma}(\frac{1-\sigma_{s(e)}h_{t(e)}}{2})\log(d_{j_{e}})+\sum_{e\in\Gamma}(\frac{1-\sigma_{s(e)}\sigma_{t(e)}}{2})\log(d_{j_{e}})+\Sigma_{I}(\vec{\sigma},\vec{\textbf{j}},\vec{\textbf{k}})$
(26)
$K_{\vec{\textbf{j}}}=e^{B(\vec{\textbf{j}})}=\prod_{e\in\partial\gamma}d_{j_{e}}\prod_{e\in\Gamma}|g_{j_{e}}|^{2}\mathrm{Tr}_{\mathcal{I}_{\vec{\textbf{j}}}}[\rho^{I}]$
(27)

and the (partial) constraint factor is given by

$\Delta_{1|0}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})=\prod_{e:\sigma_{x}h_{e}=-1}\delta_{j_{e},k_{e}}\prod_{e\in\partial
S}\delta_{j_{e},k_{e}}\,\Delta^{I}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})$
(28)

and constraints from intertwiners are to be incorporated in the final factor.
For example, if one chooses a product state over vertices for intertwiners,
$\rho^{I}=\otimes_{x}\rho^{I}_{x}$, we get

$\Delta^{I}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})=\prod_{x:\sigma_{x}=-1}\delta_{\mathbf{j}^{x},\mathbf{k}^{x}}$
(29)

Let us also define, for reference, the quantities

$Z^{(\vec{\textbf{j}},\vec{\textbf{k}})}_{1|0}=\sum_{\vec{\sigma}}\Delta_{1|0}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})e^{-\mathcal{H}_{1|0}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})}$
(30)

which enable us to phrase the discussion of 25 nicely. By defining the
normalised distribution over spin sectors

$P(\vec{\textbf{j}},\vec{\textbf{k}})=\frac{K_{\vec{\textbf{j}}}K_{\vec{\textbf{k}}}}{Z_{0}}Z^{(\vec{\textbf{j}},\vec{\textbf{k}})}_{0}$
(31)

we see our quantity of interest as a probability average

$\frac{Z_{1}}{Z_{0}}=\sum_{(\vec{\textbf{j}},\vec{\textbf{k}})}P(\vec{\textbf{j}},\vec{\textbf{k}})\sum_{\vec{\sigma}}\frac{e^{-\mathcal{H}_{1}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})}}{Z^{(\vec{\textbf{j}},\vec{\textbf{k}})}_{0}}\Delta_{1}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})$
(32)

In particular, we can write

$\displaystyle\mathcal{H}_{1}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})$
$\displaystyle=\mathcal{H}_{0}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})+\sum_{e\in\partial\gamma}\frac{(1-h_{e})}{2}\sigma_{x}\log(d_{j_{e}})$
(33)
$\displaystyle=\mathcal{H}_{0}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})+\sum_{e\in
C}\sigma_{x}\log(d_{j_{e}})=\mathcal{H}_{0}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})+Q(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma}).$
(34)

So the difference is a quantity supported on the boundary region $C$.
Then, the quantity at the right end of 32 takes the form of a type of
statistical average:

$\sum_{\vec{\sigma}}\frac{e^{-\mathcal{H}_{1}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})}}{Z^{(\vec{\textbf{j}},\vec{\textbf{k}})}_{0}}\Delta_{1}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})=\frac{\sum_{\vec{\sigma}}e^{-\mathcal{H}_{0}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})}e^{-Q(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})}\Delta_{1}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})}{\sum_{\vec{\sigma}}e^{-\mathcal{H}_{0}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})}\Delta_{0}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})}$
(35)

If one disregards the constrained sum due to the $\Delta$-factors, this
quotient is an average in a Gibbs-type canonical ensemble for the Hamiltonian
$\mathcal{H}_{0}$.
We adopt the following interpretation: The average exponentiated Rényi entropy
is given as a quantity associated to a random Ising model on our fixed graph
whose couplings $\log(d_{j})$ are distributed according to the function $P$.
In fact, disregarding the difference between the $\Delta$-factors, the
quantity is actually $\langle\langle e^{-Q}\rangle_{Z_{0}}\rangle_{P}$, so if
$\mathcal{H}_{0}$ is taken as the Hamiltonian, we just take the average
expectation value of $e^{-Q}=\prod_{e\in C}d_{j_{e}}^{-\sigma_{x}}$. The
individual Ising partition functions for fixed spins are given by 30.
Inspiration for this comes from SYK-type models, where a similar, but
Gaussian, average is performed over couplings and induces holographic
behaviour as well2; 14. Two things must be highlighted before proceeding
further. First, the role of the constraint factors. Second, the shape of the
distribution of couplings and what controls it.
The constraints remove certain terms from the sum if the area spins on a given
set of links is not the same in the tuple
$(\vec{\textbf{j}},\vec{\textbf{k}})$. If we take the real-space viewpoint on
Rényi k-entropy33, we can interpret them in a geometric way. Say we wish to
compute the entropy of a local QFT’s state in a certain region $A$. Thus, we
fix a state $\ket{\Psi}$ in the bipartite Hilbert space
$\mathbb{H}_{A}\otimes\mathbb{H}_{A^{c}}$. A geometric way to compute said
entropy is then to consider k copies of the domains of the fields, but glued
together at the region A in a certain way by attaching cells and arranging the
copies in a ’chain’ with independent fields in each copy, but matched using
boundary conditions along each glueing. Given this, the entropy is given as a
partition function of the field theory living on this extended domain.
For our case, we have a similar situation: 2 copies of the graph are needed
and constraints connect the values of area (spins) between the two copies on
subregions determined by the Ising spin configuration. As such, we can give
the following interpretation: In computing the entropy, we glue, for a fixed
Ising configuration, the two copies with given area spins together in a
subregion (see figure 3). However, we only glue them if the areas agree on the
given region - analogous to how we only consider tetrahedra glued when their
areas are maximally entangled. In a sense, one may only glue what fits
together geometrically. In the end, we sum over all Ising configurations, so
different regions will be glued together. The intuition to be taken from this
is that if graphs with different area values are too far apart geometrically,
the pair will not contribute to the entropy.

Figure 1: A schematic of the glueing imposed by the constraints. Along the
minimal surface $\partial S_{\downarrow}$, there is a fictitious glueing of
the two spin networks where areas are required to be equal. Conversely, any
surface where a glueing occurs needs to agree in spin labels in the two
geometries, similar to the situation in a spin foam.

Next, it is clear from 27,31 that only the normalisations of each spin sector
contributing to the states matter for the distribution of areas. If one
chooses the intertwiner state to contain only a single spin sector (so only
one fixed discrete geometry is in the state), the distribution will become
sharp on that geometry (in the sense of areas). Additionally, the unbounded
growth of the boundary factor shows that only a class of intertwiner states
obeying a ’boundary condition’ can have a sensible entropy in our setting once
the upper cutoff on spins is removed. For each boundary link,
$(d_{j_{e}}\cdot\mathrm{Tr}_{\vec{\textbf{j}}}[\rho^{I}](j_{e}))_{j_{e}}$
needs to be a summable sequence. Similarly we need also that for each internal
link,$(g_{j_{e}})_{{j_{e}}}\in\mathfrak{l}^{2}(\frac{\mathbb{N}_{0}}{2})$ and
$(\mathrm{Tr}_{\vec{\textbf{j}}}[\rho^{I}])_{\vec{\textbf{j}}}\in\mathfrak{l}^{1}((\frac{\mathbb{N}_{0}}{2})^{|}E|)$.

#### Alternative interpretation: A sum over geometries

As an alternative interpretation, one considers the set of spins
$(\vec{\textbf{j}},\vec{\textbf{k}})$ on the two copies as a background
geometry for the Ising models 30. Then, our partition function is gained by
promoting said background to a dynamical variable and summing over its
configurations with the weight $P$. In the cases where the glueing is
imperfect, not all Ising configurations are permissible - the Ising system can
only be spin-down on the subset of links and sites where the geometry is well-
defined. In this interpretation, one may see our result as a discrete path
integral over geometries for the Rényi 2-entropy. However, the interesting
point there is that a single state of a quantum geometry already gives rise to
such a sum over geometries. In a sense, the superposition provides already a
sufficient setting for summing over spatial geometries.

To generalise this idea, consider a Hilbert space
$\mathbb{H}=\bigoplus_{\vec{\textbf{j}}}\mathbb{H}^{\gamma}_{\vec{\textbf{j}}}\otimes\mathbb{H}^{M}_{\vec{\textbf{j}}}$
which splits over sectors of geometric data $\vec{\textbf{j}}$. Each sector
then has geometric and additional matter degrees of freedom. If we consider an
expectation value of some operator,

$\langle
X\rangle_{\rho}=\sum_{\vec{\textbf{j}}}(\frac{\mathrm{Tr}_{\vec{\textbf{j}}}[\rho]}{\mathrm{Tr}[\rho]})(\frac{\mathrm{Tr}_{\vec{\textbf{j}}}[\rho
X]}{\mathrm{Tr}_{\vec{\textbf{j}}}[\rho]})=\sum_{\vec{\textbf{j}}}P(\vec{\textbf{j}})\langle
X\rangle_{j}$ (36)

it can be expressed as a probability average in the same way as we had before.
Specifically, if the state has no interference between sectors, so only
$\rho_{\vec{\textbf{j}},\vec{\textbf{j}}}$ exist, then $\langle
X\rangle_{j}=\langle X\rangle_{\rho_{\vec{\textbf{j}},\vec{\textbf{j}}}}$. So,
if the sectors correspond to geometries, this formulation means that in states
which are classical superpositions of geometries averages are given by
weighted averages in the individual geometries. This gives a very
straightforward explanation to how our weighted Ising model comes to be, and
puts it into a larger picture of quantum gravity models with superposed
geometries.