#### Another alternative: spin foam models

We may even see the sums as something akin to a spin foam model. To see this,
forget about the intertwiners on a spin network on $\gamma$, which is then
just labeled by its graph and the representations $\vec{\textbf{j}}$.
Introduce a 2-complex
$W((\gamma,\vec{\textbf{j}})\mapsto(\gamma,\vec{\textbf{k}});\sigma)$
trivially connecting the two graphs in the obvious way. Then, assign to the
vertical 1-faces the Ising spins and to the 2-faces the representations of the
spin networks they end on. This may not be a unique assignment, but by
suitably restricting the labels, the weights assigned to this 2-complex will
still be unique. In fact, the role of the $\Delta$-constraints in this
formulation is precisely that of restricting the sum over all such spin foams
with fewer data to that subset for which the weight is unambiguously
calculable. Said weight is simply given by the same trace as in 10. The
calculation may therefore be understood as follows:
Fixing the graph $\gamma$, state $\rho_{I}$ and region $C$, we define two spin
foam model partition sums

$Z_{1|0}=\sum_{W_{\text{comp}}}w_{2}(\vec{\textbf{j}}\cup\vec{\textbf{k}})w_{1}(\vec{\textbf{j}}\cup\vec{\textbf{k}};\vec{\sigma})$
(37)

where the class of spin foam models is analogous to the one we introduced just
now - they trivially connect two copies of $\gamma$ and have
$\mathbb{Z}_{2}$-labels on their vertical 1-faces representing the Ising
spins, which replace the intertwiner labels. The weights of the model depend
on the intertwiner state and the region under consideration and are the same
as before up to multiplicity factors coming from counting
$(\vec{\textbf{j}},\vec{\textbf{k}})$ configurations which lead to the same
amplitude. They factorise into a part depending only on the 2-faces and one
with dependence on the 1-faces. Some of them will vanish on certain spin
foams, in which case these are excluded from the sum. So, the
$\Delta$-constraints that are always present in the partition sums are
implemented in the spin foam models by restricting to a set of compatible
foams.

####  3 Characterising holographic states

We turn to the question of holography again. Necessary conditions on states
that minimise the purity can be straightforwardly derived. First, see from 25
that the terms summed over are all nonnegative:

$\mathrm{Tr}_{\mathbb{H}_{\vec{\textbf{j}}}\otimes\mathbb{H}_{\vec{\textbf{k}}}}[(\rho_{I})^{\otimes
2}(\rho_{\Gamma})^{\otimes
2}\bigotimes_{x}\mathcal{S}_{x}^{\frac{1-\sigma_{x}}{2}}(\mathcal{S}_{C})^{1|0}]$
(38)

First trace out all the degrees of freedom without swap operators; This turns
the nonnegative $\rho_{I}\rho_{\Gamma}$ into another nonnegative operator.
Then, the swappers make the remaining trace into the trace of its square,
which is nonnegative. All projectors onto fixed-spin subspaces involved are
nonnegative operators. We thus see that every term is nonnegative in
principle. So, the only ways to reduce the sum in 32 is to make terms smaller
or leave them out entirely. For this, we first study the $\Delta$-constraints.
Note that $\Delta_{1|0}(\vec{\textbf{j}},\vec{\textbf{j}},\vec{\sigma})=1$
regardless of the configuration, as by our previous considerations. However,
we can choose our intertwiner and link data such that the remaining cases
vanish as often as possible. Let

$A_{\vec{\textbf{j}},\vec{\textbf{k}}}:=\\{e\in L:j_{e}=k_{e}\\}$ (39)

be the set of links where the areas of two spin sectors agree. Valid glueings
between the two geometries can only happen on this set. If now all involved
spin sectors in our chosen data have
$A_{\vec{\textbf{j}},\vec{\textbf{k}}}=\emptyset$ for
$\vec{\textbf{j}}\neq\vec{\textbf{k}}$, as for example in the unit matrix
$\mathbb{I}$, then there is no constraint as only glueings between copies of
the same geometry can happen. These states are classical superpositions of
definite geometries (again, only in the sense of definite areas). In this
case, the sum collapses to the diagonal pairs:

$\frac{Z_{1}}{Z_{0}}=\sum_{\vec{\textbf{j}}}P(\vec{\textbf{j}},\vec{\textbf{j}})\sum_{\vec{\sigma}}\frac{e^{-\mathcal{H}_{1}(\vec{\textbf{j}},\vec{\textbf{j}},\vec{\sigma})}}{Z^{(\vec{\textbf{j}},\vec{\textbf{j}})}_{0}}=\sum_{\vec{\textbf{j}}}P(\vec{\textbf{j}},\vec{\textbf{j}})\langle
e^{-Q}\rangle_{Z^{\vec{\textbf{j}}_{0}}}$ (40)

The geometries in these superpositions are clearly distinguished by their
areas. In our optimisation, we can restrict to this subset of states. Next, we
need to find, for a fixed spin sector $\vec{\textbf{j}}$, the value of
$\langle e^{-Q}\rangle_{Z^{\vec{\textbf{j}}_{0}}}$. This is simply a result
from previous research from the fixed-spin case18.
If all spins in a given sector are large enough, we can perform a crucial
approximation to the partition sums. In the Ising model, we may approximate
the partition function by its ground state contribution if the excited states
have very low weight. This is the case if the couplings of the model are very
large, as any spin flip will increase the energy by an amount proportional to
that coupling constant. When the spins are all large, we may approximate, in
particular,

$Z^{(\vec{\textbf{j}},\vec{\textbf{k}})}_{0}\approx 1,\qquad
Z^{(\vec{\textbf{j}},\vec{\textbf{k}})}_{1}\approx\exp(-\mathcal{H}_{1|0}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma}_{GS}))$
(41)

which massively simplifies the distribution $P$, as well:

$Z_{0}=\sum_{\vec{\textbf{j}},\vec{\textbf{k}}}K_{\vec{\textbf{j}}}K_{\vec{\textbf{k}}}Z^{(\vec{\textbf{j}},\vec{\textbf{k}})}_{0}\approx(\sum_{\vec{\textbf{j}}}K_{\vec{\textbf{j}}})^{2}\qquad
P(\vec{\textbf{j}},\vec{\textbf{k}})\approx
p_{\vec{\textbf{j}}}p_{\vec{\textbf{k}}}\qquad
p_{\vec{\textbf{j}}}=\frac{K_{\vec{\textbf{j}}}}{\sum_{\vec{\textbf{k}}}K_{\vec{\textbf{j}}}}$
(42)

In particular, given that $Z_{0}=\langle Tr[\rho]^{2}\rangle_{U}$, we can
interpret the factorisation of the partition function as the statement
$\langle Tr[\rho]^{2}\rangle_{U}=\langle Tr[\rho]\rangle^{2}_{U}$ in the high-
spin regime. The given spins serve as an external scale for the fixed-spin
Ising models defined by 30 and provide a notion of temperature to it. The
diagonal Hamiltonians take the simplified form

$\mathcal{H}_{1|0}(\vec{\textbf{j}},\vec{\textbf{j}},\vec{\sigma})=\sum_{e\in\partial\gamma}(\frac{1-\sigma_{s(e)}h_{t(e)}}{2})\log(d_{j_{e}})+\sum_{e\in\Gamma}(\frac{1-\sigma_{s(e)}\sigma_{t(e)}}{2})\log(d_{j_{e}})+S_{2}((\rho^{I}_{\vec{\textbf{j}}})_{\downarrow})$
(43)

Then let us restrict our study to those states where the minimal area spin is
large enough to suppress contributions to partition sums coming from the non-
minimal configurations. Assuming that the energy gap in the two Hamiltonians
$\mathcal{H}_{1|0}$ is equal, we can achieve a straightforward bound on the
relative error

$|\frac{\frac{Z^{j}_{1}}{Z^{j}_{0}}-e^{-E_{1}}}{e^{-E_{1}}}|\leq\frac{2^{N}-1}{e^{\Delta
E}}.$ (44)

In particular, if we assume the energy gap grows with the minimal area present
in the geometry, as it does when $S_{2}(\rho^{I})=0$, where $\Delta
E=\log(d_{j_{min}})$, then raising the minimal area will suppress all higher
configurations to a certain degree. The minimal configuration, in turn, is
built from a minimal surface $\Sigma$ in the geometry. The result from the
fixed-spin case is schematically

$e^{-E_{1}}\approx(\prod_{e\in\Sigma}d_{j_{e}})^{-1}e^{-S_{2}((\rho^{I})_{\downarrow})}$
(45)

In seeking a smaller $\frac{Z_{1}}{Z_{0}}$, we then see that including only
large geometries is beneficial to making the reduced spin network state more
mixed. We may for example adjust our probability weights $P$ to do this in
order to get a smaller value. The optimal case is then one where all areas are
equal to the upper cutoff $J$ we imposed earlier, and thus only one single
spin sector is present in the state. We emphasize that this is only the states
which have a higher mixedness than the others. For them, we can estimate

$\frac{Z^{j}_{1}}{Z^{j}_{0}}\approx
d_{J}^{-||\Sigma||}e^{-S_{2}((\rho^{I}_{J})_{\downarrow})}.$ (46)

And then, as the spin sector is homogeneous, we know that holographic
behaviour is not possible. So, the most mixed state over the set of states for
arbitrary spin sectors is not holographic.
However, as the following example shows, this is an ill-posed optimisation. We
should instead fix the spin sectors under consideration and only adjust the
remaining parameters.

####  4 Example calculation

Figure 2:  We consider a superposition of two spin sectors which are only
different on a single link. The values are chosen as to give a controllably
low intertwiner space dimension of 1 or 2.

To illustrate the complexity of the calculations involved, we shall give an
innocuous example. The graph in question is the simplest nontrivial one, and
we only consider two distinct spin sectors, which agree on all but one link.
This means that there are only four Ising configurations. We choose the area
spins such that only one intertwiner space is nontrivial. We will also refer
to the vertices just by L(eft) or R(ight) for convenience. In spin sector
$\vec{\textbf{k}}$, all intertwiner spaces are 1-dimensional, while in
$\vec{\textbf{j}}$, the right one is 2-dimensional. We can then give the
intertwiner state

$\rho^{I}=\begin{bmatrix}\rho^{I}_{\vec{\textbf{j}},\vec{\textbf{j}}}&\rho^{I}_{\vec{\textbf{j}},\vec{\textbf{k}}}\\\
\rho^{I}_{\vec{\textbf{k}},\vec{\textbf{j}}}&\rho^{I}_{\vec{\textbf{k}},\vec{\textbf{k}}}\end{bmatrix}\qquad\rho^{I}_{\vec{\textbf{j}},\vec{\textbf{j}}}=\begin{bmatrix}a&b\\\
\bar{b}&d\end{bmatrix}\qquad\rho^{I}_{\vec{\textbf{j}},\vec{\textbf{k}}}=\begin{bmatrix}u\\\
v\end{bmatrix}\qquad\rho^{I}_{\vec{\textbf{k}},\vec{\textbf{k}}}=w=1-(a+d)$
(47)

From this, we can derive the reduced entropies

$S_{2}((\rho^{I}_{\vec{\textbf{j}},\vec{\textbf{j}}})_{R})=S_{2}((\rho^{I}_{\vec{\textbf{j}},\vec{\textbf{j}}}))=-\log\left[\frac{a^{2}+d^{2}+|b|^{2}}{(a+d)^{2}}\right]\qquad
S_{2}((\rho^{I}_{\vec{\textbf{j}},\vec{\textbf{j}}})_{L})=0$ (48)

while the ones for $\vec{\textbf{k}}$ vanish.
We choose the area $C$ to be the rightmost link, where the two sectors
disagree. The individual sectors induce, at least for large spins, an
isometry, making the only question what parameters need to be chosen to make
their superposition induce an isometry.
Explicit calculation as in Appendix C gives us an expression for the purity,
in the large-spin limit:

$\frac{Z_{1}}{Z_{0}}=\frac{2w^{2}-2w+1}{6s}+\frac{(1-2w)^{3}}{36s^{2}}+O\left(\left(\frac{1}{s}\right)^{3}\right)$
(49)

Particularly, the lower bound of $\frac{1}{12s}$ is achieved in the regime of
large $s$ for $(a,d,w)\approx(\frac{1}{4},\frac{1}{4},\frac{1}{2})$. The
dimension of the boundary link space is precisely $(6s+1)+(6s-1)=12s$, so an
isometry can be achieved with these parameters.
From this simple example, we already gain valuable information. First, the set
of spin sectors under consideration must be fixed to have a more tractable, as
well as interesting, minimisation problem. Additionally, it is clear that the
intertwiner state has to be adjusted depending on the boundary region chosen
and that different spin sectors may contribute with different weights.
Therefore, the question of characterising holographic states becomes a fairly
involved question, as the number of parameters for the intertwiner states
grows very fast with number of vertices, spin sectors and spin values. To
illustrate this, consider only $K$ spin sectors which are all homogeneous. The
intertwiner spaces for each sector have dimensions
$\\{D_{k}=(2j_{k}+1)^{N}\\}^{K}_{k=1}$, such that the intertwiner state has
$(\sum_{k}D_{k})^{2}-1$ real parameters before imposing positivity
constraints. For large enough $N$ or large spins, this scales as
$4^{N}(\sum_{k}s^{N}_{k})^{2}$. The curse of dimensionality binds us to the
regime of very low $N$ if we wish to do numerical investigations without heavy
approximations.

Figure 3:  For the second example, consider two spin sectors of the same form,
differing only in spin. Once again, intertwiner space dimensions are kept at 1
or 2.

We will consider another instructive example. Consider the same graph
structure and boundary region as before, but make the two spin sectors of the
same form, differing only in their average area. Then, as no two edge spins
agree anywhere, the constraints make all off-diagonal partition sums vanish.
The sum simplifies to

$\frac{Z_{1}}{Z_{0}}=P(j,j)\frac{Z_{1}^{j}}{Z_{0}^{j}}+P(k,k)\frac{Z_{1}^{k}}{Z_{0}^{k}}$
(50)

Take both of these spins to be fairly large, with fraction
$\nu\coloneqq\frac{d_{t}}{d_{s}}=\frac{2t+1}{2s+1}\approx\frac{t}{s}$. Then,
we may approximate the quotients as before by their ground state value - e.g.,
for choosing $C$ to be the right link as before,

$\frac{Z_{1}^{j}}{Z_{0}^{j}}=d_{3s-1}^{-1}$ (51)

and same for the other sector, replacing s with t. We can also easily write
the probability weights for the two sectors as

$P(j,j)=\left[1+(\frac{|g_{t}|^{2}}{|g_{s}|^{2}}\frac{d_{t}^{4}d_{3s}d_{3s-1}}{d_{s}^{4}d_{3t}d_{3t-1}}\frac{c_{k}}{c_{j}})\right]^{-2},\qquad
P(k,k)=\left[1+(\frac{|g_{t}|^{2}}{|g_{s}|^{2}}\frac{d_{t}^{4}d_{3s}d_{3s-1}}{d_{s}^{4}d_{3t}d_{3t-1}}\frac{c_{k}}{c_{j}})^{-1}\right]^{-2}$
(52)

In the large spin limit and taking $\frac{|g_{t}|^{2}}{|g_{s}|^{2}}\approx 1$,
this is approximately

$P(j,j)=\left[1+(\nu^{6}\frac{c_{k}}{c_{j}})\right]^{-2},\qquad
P(k,k)=\left[1+(\nu^{6}\frac{c_{k}}{c_{j}})^{-1}\right]^{-2}.$ (53)

Note that this is the same as approximating the geometries by homogeneous
ones. After factorising, this yields

$\frac{Z_{1}}{Z_{0}}=d_{3s-1}^{-1}(\frac{1}{(1+\nu^{6}a)^{2}}+\frac{\nu^{-1}}{(1+\nu^{-6}a^{-1})^{2}})$
(54)

where $a=c_{k}/c_{j}$. We can now characterise which states are holographic:
We wish the previous expression to be

$(\prod_{e}d_{j_{e}}+d_{k_{e}})^{-1}\approx(\prod_{e}d_{j_{e}}(1+\nu))^{-1}=(1+\nu)^{-1}(d_{3s-1})^{-1}.$
(55)

So the isometry constraint is

$(1+\nu)(\frac{1}{(1+\nu^{6}a)^{2}}+\frac{\nu^{-1}}{(1+\nu^{-6}a^{-1})^{2}})\approx
1$ (56)

which has solution $a\approx\nu^{-5}$, or equivalently
$(c_{j},c_{k})=(\frac{1}{1+\nu^{-5}},\frac{1}{1+\nu^{5}})$. This is
independent of the intertwiner state chosen, as it was for the individual
sectors. It is thus easy to see that if one geometry (j) is large in terms of
areas and the other (k) small, we must give a lot of weight to the smaller
geometry to make their combination induce an isometry. As they approach each
other (corresponding to $\nu\rightarrow 1$), the two sectors require equal
weight, as one might expect from intuition.
Let us generalise this setting: Assume that we work with $K$ spin sectors
sharing no two spins, labeled $s_{k}$, and fix a boundary region such that for
each sector, the fixed-spin purity is the minimal value, which makes the
sector induce an isometric map. By our arguments in the appendix, this should
generically leave a lot of parameters free. So,
$\frac{Z_{1}^{k}}{Z_{0}^{k}}=\dim(\mathbb{H}_{C,n})^{-1}=(\prod_{e\in
C}d_{s_{k},e})^{-1}$. For this, we assume intertwiner data has been adjusted
as necessary. Then, we may still adjust the $c_{n}$. The condition for
isometry becomes

$\begin{split}\frac{Z_{1}}{Z_{0}}\dim(\mathbb{H}_{C})=\sum^{K}_{n=1}P(n,n)\prod_{e\in
C}\sum_{j\in\\{s_{k,e}:k\in[1:K]\\}}\frac{d_{j}}{d_{s_{n,e}}}=\\\
\sum^{K}_{n=1}P(n,n)\prod_{e\in
C}\left(\frac{d_{s_{n,e}}}{{\sum^{K}_{m=1}}d_{s_{m,e}}}\right)^{-1}=\sum^{K}_{n=1}P(n,n)\prod_{e\in
C}\rho_{e}^{-1}\stackrel{{\scriptstyle!}}{{=}}1.\end{split}$ (57)

In the end, this condition is a polynomial one: Writing, with
$\tilde{K}_{n}=\prod_{e\in\partial\gamma}d_{j_{n,e}}\prod_{e\in\Gamma}|g_{j_{n,e}}|^{2}$,
$P(n,n)=\frac{c_{n}^{2}\tilde{K}_{n}^{2}}{\sum_{m}c_{m}^{2}\tilde{K}_{m}^{2}}$
and $u_{n}=\prod_{e\in C}\rho^{-1}_{e}$, this is equivalent to

$\sum_{n}\tilde{K}_{n}^{2}c_{n}^{2}(u_{n}-1)=0,$ (58)

which is just a multinomial of degree 2 in the variables $\\{c_{n}\\}_{n}$. As
long as some $u_{n}<1$, this should admit solutions in the region of interest
$0<c_{n}<1,\sum_{n}c_{n}=1$. For these $u_{n}$, the value of $c_{n}$ can then
be large, while the other sectors will need small $c_{n}$ weights to make the
sum 0. Additionally, large spin sectors will have their weights be suppressed
by $\tilde{K}_{n}^{-2}$. The existence of holographic states is thus not much
of an issue - as long as there are enough parameters and the spin sectors are
not too uncooperative, we should be able to reach the level set of holographic
states by adjusting a few parameters.
The next question becomes the dimensionality of this set. As shown in the
appendix and used analogously here, we expect the set of holographic state
parameters to be of codimension one in the full parameter space. So, even
though we fixed some parameters per spin sector already, we are technically
free to change some more once we consider the superposition - there is not
even a requirement for each individual sector to be holographic, per se.
Let us attempt the above in yet slightly larger generality. To achieve
holography, we require $\frac{Z_{1}}{Z_{0}}\dim(\mathbb{H}_{C})=1$, or, in the
regime of large spins

$0=\sum_{m,n}\tilde{K}_{n}c_{n}\tilde{K}_{m}c_{m}(\frac{Z^{m,n}_{1}}{Z^{m,n}_{0}}-\frac{1}{\dim(\mathbb{H}_{C})})=\sum_{n}\tilde{K}^{2}_{n}c^{2}_{n}(\frac{Z^{n,n}_{1}}{Z^{n,n}_{0}}-\frac{1}{\dim(\mathbb{H}_{C})})+2\sum_{m\neq
n}\tilde{K}_{n}c_{n}\tilde{K}_{m}c_{m}(\frac{Z^{m,n}_{1}}{Z^{m,n}_{0}}-\frac{1}{\dim(\mathbb{H}_{C})})$
(59)

which we can easily analyse: In the diagonal part of the sum,
$\frac{Z^{n,n}_{1}}{Z^{n,n}_{0}}$ is the purity of the part of the full state
contained in sector $n$. This means that the usual entropy bound applies:
$1\geq\frac{Z^{n,n}_{1}}{Z^{n,n}_{0}}\geq\frac{1}{\dim(\mathbb{H}_{C,n})}\geq\frac{1}{\dim(\mathbb{H}_{C})}$,
meaning all terms in the diagonal part are nonnegative. So, to achieve
holography, the off-diagonal terms are crucial. In the best case, where
$Z^{m,n}_{1}=0$ off of the diagonal, the entire off-diagonal sum is negative
and holography might be achieved by adjusting the $c_{n}$.

As our final generalisation, take
$\beta_{m,n}=\frac{Z^{m,n}_{1}}{Z^{m,n}_{0}}\dim(\mathbb{H}_{C})-1$ and
$\alpha_{n}=\tilde{K}_{n}c_{n}$, then the condition can be written as
$\langle\alpha,\beta\alpha\rangle=0$. We then can find solutions iff the
$K\times K$ matrix $\beta$ has negative or $0-$eigenspaces. In the simplest
case, $\beta_{m,n}=\delta_{m,n}\frac{1}{\rho_{n}}-1$, where
$\rho_{n}=\frac{\dim(\mathbb{H}_{C,n})}{\dim(\mathbb{H}_{C})}$ sums to 1.
This matrix has the general determinant
$\det(\frac{1}{\rho})(1-\mathrm{Tr}(\rho))=0$, so there is a 0-eigenspace
which admits a solution. This space is in fact spanned by the solution
$\alpha_{n}=\dim(\mathbb{H}_{C,n})$. Take an $\alpha$ to be from this
eigenspace and recover the corresponding
$c_{n}=\frac{\dim(\mathbb{H}_{C,n})}{\tilde{K}_{n}}$. Their sum is not 0, so
we may rescale $\alpha$ within this eigenspace to normalise the sum to
$\sum_{n}c_{n}=1$, yielding holography for the state. The final result for the
weights of the individual sectors for superpositions where one can take
$Z_{1}^{m,n}\sim\delta_{m,n}$ and the individual spin sectors holographic, is
then

$c_{n}=\left(\sum^{K}_{m=1}\frac{\prod_{e\in
C^{c}}d_{j_{n,e}}\prod_{e\in\Gamma}|g_{j_{n,e}}|^{2}}{\prod_{e\in
C^{c}}d_{j_{m,e}}\prod_{e\in\Gamma}|g_{j_{m,e}}|^{2}}\right)^{-1}$ (60)

in the high-spin regime. In particular, this result extends to cases where the
off-diagonal partition sums may not be zero, but numerically negligible. Also,
the general condition for a 0-eigenspace to exist is, in the same vein as
earlier, that

$\sum_{m,n}(Q^{-1})_{m,n}=1$ (61)

(where $Q_{m,n}=\frac{Z^{m,n}_{1}}{Z^{m,n}_{0}}\dim(\mathbb{H}_{C})$). For
diagonal $Q$, this is precisely the statement that
$\sum_{n}(\frac{Z^{n,n}_{1}}{Z^{n,n}_{0}})^{-1}=\dim(\mathbb{H}_{C})$ \- in
other words, that each sector is by itself holographic. In this way, we have
found a general sufficient and necessary criterion for holography for the case
where the off-diagonal sums are negligible.
A particularly simple case is when each internal link has the same weight for
the possible values of spins, $g_{j_{m,e}}=\frac{1}{\sqrt{K}}$: Then the
weight is given by

$\tilde{K}_{n}c_{n}=(\frac{\dim(\mathbb{H}_{C^{c},n})}{\dim(\mathbb{H}_{C^{c}})})^{-1}$
(62)

Heuristically, this confirms what we saw before: For spin sectors with larger
geometry, the weight must be smaller than for those with small geometries. In
fact, our earlier example gets precisely the same values of
$c_{1}=\frac{1}{1+\nu^{5}}$ as before.

###  2 Averages of observables

We can also study boundary observables in this setting of typical spin network
states. For this, we use the following analogue of the replica trick:

$\langle X\rangle_{\rho}\langle Y\rangle_{\rho}=\frac{\mathrm{Tr}(\rho
X)\mathrm{Tr}(\rho Y)}{\mathrm{Tr}(\rho)^{2}}=\frac{\mathrm{Tr}(\rho^{\otimes
2}(X\otimes Y))}{\mathrm{Tr}(\rho\otimes\rho)}=$ (63)

which allows us to use the same averaging method as before. The reason we
introduce two observables $X,Y$ at this point is the following fact:

$\langle\langle
XY\rangle_{\rho}\langle\mathbb{I}\rangle_{\rho}\rangle_{U}=\langle\langle
XY\rangle_{\rho}\rangle_{U}\neq\langle\langle X\rangle_{\rho}\langle
Y\rangle_{\rho}\rangle_{U}$ (64)

which hints that the measure of correlation of the two observables is not
entirely erased by the average over the vertex wavefunctions. We can thus not
only calculate average expectation values, but also average correlations.
Importantly, we can derive the same type of random Ising model as before, with
the boundary part replaced by a quantity related to $X,Y$: The
$(\vec{\textbf{j}},\vec{\textbf{k}})$-boundary contribution in the numerator
is now

$\mathrm{Tr}_{\mathbb{H}_{\partial\gamma,\vec{\textbf{j}}}\otimes\mathbb{H}_{\partial\gamma,\vec{\textbf{k}}}}[\bigotimes_{e\in\partial\gamma}\mathcal{S}_{e}^{\frac{1-\sigma_{s(e)}}{2}}(X\otimes
Y)]=\mathrm{Tr}_{\mathbb{H}_{\partial\gamma,\vec{\textbf{j}}}}[\mathbb{I}]\mathrm{Tr}_{\mathbb{H}_{\partial\gamma,\vec{\textbf{k}}}}[\mathbb{I}]e^{-c_{XY}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})}$
(65)

where

$c_{XY}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})=-\log\left[\frac{\mathrm{Tr}[\Pi_{j_{\downarrow}}\Pi_{j_{\uparrow}}X\Pi_{j_{\uparrow}}\Pi_{k_{\downarrow}}\Pi_{k_{\uparrow}}Y\Pi_{k_{\downarrow}}\Pi_{j_{\downarrow}}]}{\mathrm{Tr}_{\mathbb{H}_{\partial\gamma,\vec{\textbf{j}}}}[\mathbb{I}]\mathrm{Tr}_{\mathbb{H}_{\partial\gamma,\vec{\textbf{k}}}}[\mathbb{I}]}\right].$
(66)

Here, we have labeled for simplicity $\Pi_{j_{\uparrow}}$ the projector onto
the boundary subspace of links in $S_{\uparrow}$ with spins from sector
$\vec{\textbf{j}}$, others analogous. In particular, for the all-up
configuration, this quantity turns into a sum, signalling again the glueing on
the spin-down region. The Hamiltonian for the numerators
$Z^{\vec{\textbf{j}},\vec{\textbf{k}}}_{1}$ is then given as

$\mathcal{H}_{1}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})=\sum_{e\in\Gamma}\frac{1-\sigma_{s(e)}\sigma_{t(e)}}{2}\log(d_{j_{e}})+c_{XY}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})+\Sigma_{I}(\vec{\sigma},\vec{\textbf{j}},\vec{\textbf{k}})$
(67)

and new boundary constraints may arise from the form of the quantities $X,Y$.
However, the distribution of couplings $P(\vec{\textbf{j}},\vec{\textbf{k}})$
in the randomised Ising model will remain the same by design. In this way, we
may use the results of entropy calculations as a starting point for
characterising other boundary quantities.
We will calculate the expectation value of the area222Here, we use a
simplified approximation to usual models of the area operator where all
eigenvalues are given by $j$ instead of $\sqrt{j(j+1)}$. As we deal with large
spins, this should not be an issue. of the boundary region $C$ when the
geometry is in a holographic state. For this, we calculate

$\langle\langle
A\rangle_{\rho}\langle\mathbb{I}\rangle_{\rho}\rangle_{U}=\langle\langle
A\rangle_{\rho}\rangle_{U}\qquad\text{for }A=\sum_{e\in C}A_{e}=\sum_{e\in
C}\sum_{j_{e}}j_{e}\mathbb{I}_{j_{e}}.$ (68)

Observables of the form

$X=\bigotimes_{e\in L}X_{e}$ (69)

always factorise $X=X_{\uparrow}\otimes X_{\downarrow}$. Here we consider
operators of the form $X_{e}=\sum_{j_{e}}\lambda_{e,j}\ket{jm}\bra{jm}$, which
are diagonal in spins. For these, the Hamiltonian contribution is particularly
simple. There is a constraint
$\Delta_{\partial\gamma}=\prod_{e\in\partial\gamma\cap
S_{\downarrow}}\delta_{j_{e},k_{e}}$, and the factor for boundaries turns out
to be

$e^{-c_{X\mathbb{I}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})}}=\frac{\prod_{e\in
L}\lambda_{e,j}\prod_{e\in\partial\gamma\cap
S_{\uparrow}}d_{j_{e}}\prod_{e\in\partial\gamma\cap
S_{\downarrow}}d_{j_{e}}\prod_{e\in\partial\gamma\cap
S_{\uparrow}}d_{k_{e}}}{\prod_{e\in\partial\gamma\cap
S_{\uparrow}}d_{j_{e}}\prod_{e\in\partial\gamma\cap
S_{\downarrow}}d_{j_{e}}\prod_{e\in\partial\gamma\cap
S_{\uparrow}}d_{k_{e}}\prod_{e\in\partial\gamma\cap
S_{\downarrow}}d_{k_{e}}}=\frac{\prod_{e\in
L}\lambda_{e,j}}{\prod_{e\in\partial\gamma\cap S_{\downarrow}}d_{j_{e}}}.$
(70)

For the area operator, we put $\lambda_{e,j}=1$ for all but one $e$, where
$\lambda_{e,j}=j_{e}$. Then, we sum this over all edges of $C$. The numerator,
even after sum, is independent of the Ising configuration - therefore, it can
be factored out from the fixed-spin partition sums. Remarkably, then,

$Z^{\vec{\textbf{j}},\vec{\textbf{k}}}_{1}=A_{C,\vec{\textbf{j}}}\sum_{\vec{\sigma}}\Delta_{1}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})e^{-\mathcal{H}_{0}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})}$
(71)

which indeed includes the same Hamiltonian as in $Z_{0}$ after all,
$(\prod_{e\in\partial\gamma\cap
S_{\downarrow}}d_{j_{e}})^{-1}=e^{-\sum_{e\in\partial\gamma}\frac{1-\sigma_{s(e)}h_{t(e)}}{2}\log(d_{j_{e}})}$
if we take $h_{x}=1$ everywhere. This means that the only difference between
numerator and denominator is in the $\Delta$-factors. In particular, for
$\vec{\textbf{k}}=\vec{\textbf{j}}$ the quotient is always exactly 1. In the
regime of high spins, we will thus approximate both numerator and denominator
by the ground state value $1$. Then, say for a superposition of $K$ spin
sectors,

$\langle
A\rangle=\sum^{K}_{n=1}P(m,n)\frac{Z^{m,n}_{1}}{Z^{m,n}_{0}}=\sum^{K}_{m,n=1}p_{m}p_{n}A_{C,n}=\sum^{K}_{n=1}p_{n}A_{C,n},$
(72)

which means that we may use our earlier result for the distribution. In the
same assumptions as before:

$p_{n}=\frac{\dim(\mathbb{H}_{C,n})}{\dim(\mathbb{H}_{C})}=\frac{\prod_{e\in
C}\frac{d_{j_{n,e}}}{2}}{\sum_{m}\prod_{e\in C}\frac{d_{j_{m,e}}}{2}}$ (73)

So that for large enough spins we may approximate

$\langle A\rangle=\sum_{n}p_{n}A_{C,n}\approx\frac{\sum_{n}A_{C,n}\prod_{e\in
C}j_{n,e}}{\sum_{m}\prod_{e\in
C}j_{m,e}}=\frac{\sum_{n}A_{C,n}^{2}}{(\sum_{m}A_{C,m})^{2}}\sum_{m}A_{C,m}=e^{-S_{2}((A_{C,n}))}\sum_{m}A_{C,m}$
(74)

which expresses that the expected area is proportional to the sum of the
individuals, up to a factor between $1$ and $\frac{1}{K}$. The more spin
sectors are superposed, the smaller the actual area may become. However, none
of the areas can be equal by our earlier assumption - at best there is a
difference of $||C||$ between the values of $A_{C,n}$. If, however, we let
$A_{C,n}=A_{1}+n||C||$, the prefactor still scales like $\frac{4}{3K}$ as K
grows larger. So in practice, the effective area of a boundary segment that
gets holographically mapped onto the boundary complement may be between the
mean or the sum of the individual areas in the superposition.
We can even make the same calculation for the variance $\langle
A^{2}\rangle-\langle A\rangle^{2}$ of this same area in a similar way. Since
the Hamiltonian factors for each of the two involved averages are

$e^{-c_{A^{2}\mathbb{I}}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})}=\frac{\prod_{e\in
L}\lambda_{e,j}^{2}}{\prod_{e\in\partial\gamma\cap
S_{\downarrow}}d_{j_{e}}}\text{ and }\qquad
e^{-c_{AA}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})}=\frac{\prod_{e\in
L}\lambda_{e,j}\lambda_{e,k}}{\prod_{e\in\partial\gamma\cap
S_{\downarrow}}d_{j_{e}}},$ (75)

the fixed-spin partition sums are given by

$\frac{Z^{m,n}_{1}}{Z^{m,n}_{0}}_{A^{2},1}=A_{C,n}^{2}\qquad\frac{Z^{m,n}_{1}}{Z^{m,n}_{0}}_{A,A}=A_{C,m}A_{C,n}$
(76)

so that the unitary average of the expectations turns out to be precisely the
variance in the distribution $p$:

$\displaystyle\langle\langle A^{2}\rangle_{\rho}-\langle
A\rangle_{\rho}^{2}\rangle_{U}=\sum_{n}p_{n}A_{C,n}^{2}-(\sum_{n}p_{n}A_{C,n})^{2}=\langle
A^{2}\rangle_{p}-\langle A\rangle_{p}^{2}$ (77)
$\displaystyle=\left(\frac{\sum_{n}A_{C,n}^{3}}{(\sum_{n}A_{C,n})^{3}}-\left(\frac{\sum_{n}A_{C,n}^{2}}{(\sum_{n}A_{C,n})^{2}}\right)^{2}\right)(\sum_{m}A_{C,m})^{2}=(e^{-S_{3}((A_{C,n}))}-$
$\displaystyle e^{-2S_{2}((A_{C,n}))})(\sum_{m}A_{C,m})^{2}$ (78)

For the large-K limit and the sequence of areas as before, the prefactor
behaves approximately as $\frac{2}{9K^{2}}$. On the other hand, if one takes
only 2 geometries, one of which is much larger than the other, we find the
opposite result: the prefactor of the area gets arbitrarily close to $1$
($\sim 1-\frac{2A_{2}}{A_{1}}$), while the variance’s still becomes very small
($\sim\frac{4A_{2}}{A_{1}}$).
We can finally see that not only do geometric properties of even simple types
of quantum gravity states like the ones we studied depend sensitively on the
way they are superposed, but also the relative sizes of geometries in the
superposition. However, in holographic states, these dependencies are easily
controlled and explicitly studied.

##  Chapter 2 Extensions and Limitations

We will now highlight a few ways our analysis can be extended to other
interesting questions, while highlighting which conceptual or practical
problems arise.

###  1 Alternative distributions

When picking tetrahedral states at random, we must specify a distribution.
This choice crucially influences the typicity result - for example, the
support of the distribution may be different and produce a more ’coarse-
grained’ result, say by averaging over all tetrahedra simultaneously, as is
presented in the appendix.
So, the choice of distribution matters for the statement we wish to make. The
previously used constant distribution (with respect to the Haar measure) is
characterised by having maximal entropy among all distributions: The
Lagrangian

$L[\rho]=-\int_{SU(\mathbb{H})}d\mu(U)\rho(U)\log(\rho(U))-\alpha(\int_{SU(\mathbb{H})}d\mu(U)\rho(U)-1)$
(1)

has equation of motion $\log(\rho(U))+1+\alpha=0$, so the distribution is
constant.
We take this as the ’most conservative’ guess for a distribution, as maximal
entropy distributions are typically understood as having the least amount of
useful information ’put in’. For the constant distribution, it implies that we
choose our tetrahedra completely indiscriminately.
We can now characterise interesting distributions by the same principle: Say,
perhaps, we wish to only pick tetrahedra with a certain mean area

$\bar{A}=\sum_{\mathbf{j}}(\frac{1}{D}\sum_{\alpha}j_{\alpha})\;\mathbb{I}_{\mathbf{j}},$
(2)

which means we impose a further constraint on the distribution:

$L[\rho]=-\int_{SU(\mathbb{H})}d\mu(U)\rho(U)\log(\rho(U))-\alpha(\int_{SU(\mathbb{H})}d\mu(U)\rho(U)-1)-\beta(\int_{SU(\mathbb{H})}d\mu(U)\rho(U)\mathrm{Tr}[\bar{A}U\rho_{0}U^{\dagger}]-\mathcal{A})$
(3)

This Lagrangian has the similarly simple equation of motion with corresponding
solution

$\log(\rho(U))+1+\alpha+\beta\mathrm{Tr}[\bar{A}U\rho_{0}U^{\dagger}]=0\qquad\implies\qquad\rho(U)=\frac{e^{-\beta\mathrm{Tr}[\bar{A}U\rho_{0}U^{\dagger}]}}{Z(\beta)}$
(4)

which is just an analogue to the standard Boltzmann weight. This is the ’least
biased’ distribution still only giving only tetrahedra with the sought-for
average surface area. In principle, we could have chosen other quantities such
as the volume. More importantly, though, we have to begin from some reference
state $\rho_{0}$. To consider its influence, notice that while the
distribution depends on it, the only thing really playing a part is its
conjugacy class. In other words, the only meaningfully distinct $\rho_{0}$ are
not connected by conjugation with a unitary. So, our choices are reduced to
$\mathcal{D}(\mathbb{H})/SU(\mathbb{H})$, or, in lieu of a true quotient, the
set of path components of the orbit space. Luckily, in finite dimensions, this
is easy to describe:
If the matrix $\rho_{0}$ is diagonalisable, it is so by an
$SU(\mathbb{H})$-matrix conjugation. So any diagonalisable matrix, modulo such
conjugations,is specified by its eigenvalues, which, for density matrices on
$\mathbb{C}^{m}$, form a standard $(m-1)$-simplex. In this simplex, we have to
remove the corner vertices, since these correspond to pure states, which
require a seperate treatment. In fact, the set of pure states is transitively
acted upon by $SU(\mathbb{H})$, so any two pure states are connected through
it. Collapsing the orbits then yields a single point. So, the full set of
choices $\rho_{0}$ is given by an $(m-1)$-simplex whose corner vertices are
identified. This means that choosing $\rho_{0}$ pure, the distribution does
not have further parameters coming from it.
However, the distribution constructed this way has a reduced symmetry, as
well. This can be seen especially well in the average state

$\hat{K}_{A}=\int_{SU(\mathbb{H})}d\mu(U)\rho(U)\hat{U}\hat{\rho_{0}}\hat{U}^{\dagger}$
(5)

which satisfies

$VK_{A}V^{\dagger}=K_{VAV\dagger}\qquad\forall V\in SU(\mathbb{H})$ (6)

so, is only invariant under the stabiliser, or commutant, of $A$ in the
unitary group. We can now see the success of the random tensor technique in a
new light: It produces an average state with a particularly high symmetry in
which evaluation of quantities is fairly easy to do. By introducing further
constraints, we reduce this effective symmetry to the largest one that
respects them. In the case above, the symmetry
$SU(\bigoplus_{\mathbf{j}}\mathbb{H}_{\mathbf{j}})$ is broken down to
$\bigtimes_{\mathbf{j}}SU(\mathbb{H}_{\mathbf{j}})$. It seems fitting to then
instead consider an averaging only over this smaller group, in which the
distribution is again invariant under all transformations. In practice, such
finer averages make the analysis more involved. To see this, an average over
all tetrahedra simultaneously has been performed in the appendix for
demonstration, where the result turns out to be much easier to calculate.

###  2 Superposition of graph structures

We can leave the class of classical spin network states further by replacing
the link states by more arbitrary ones. Instead of a maximally entangled
state, consider for each open semilink of a vertex the $(N-1)$ other possible
vertices it can be connected to with the same colour $\alpha$. The most
general coloured state of these semilinks is a mixed state
$\rho^{\Gamma}_{\alpha}\in\bigoplus_{\\{j^{x}_{\alpha}\\}}\bigotimes_{x}V^{j^{x}_{\alpha}}$,
and then the most general coloured link state is simply
$\rho^{\Gamma}=\bigotimes_{\alpha}\rho^{\Gamma}_{\alpha}$. However, this
simple looking generalisation opens up a new, more pressing question. How is
the boundary space identified? We can think of two solutions to this.

The first is to identify ’sufficiently non-entangled’ semilinks in the state,
which are then taken as a boundary. This might be done by considering
arbitrary bipartitions of the set of links and computing their entanglement. A
maximal set of semilinks under the proposed entanglement threshold will be
then a good boundary of the state. This, however, is only algorithmically
determined. To actually work with this selected boundary, and ask questions,
this choice must be characterised in a closed form expression. The second
solution is to reverse the problem: Instead of starting from a given link
state, we first fix a boundary and then select only link states in the
complement of the boundary semilinks.

An interesting point to make is that the operator on the links in $Z_{1|0}$
used so far is

$\rho_{L}=\bigotimes_{e_{xy}\in\Gamma}\mathbb{I}_{e_{xy}}\otimes\bigotimes_{e_{x}\in\partial\gamma}\mathbb{I}_{e_{x}}.$
(7)

So in words, it is a state where every link is uncorrelated with any other
link and each individual one is maximally mixed. From this, we can think of a
few generalisations. The first is to consider convex superpositions of
different geometries, each with their own boundary. The second is to replace
the maximally mixed states on each link by something more general. The third
is to allow correlations between different links. While the first two options
are fairly easy to implement, the third makes the analysis considerably less
explicit by not letting us use local per-link terms in the Ising model. All of
these are valid generalisations of the states considered so far, so a choice
of study must be motivated by some application. Consider a generic coloured
state of $N$ tetrahedra with fixed spins and intertwiner data. One way to
apply our results so far is to ask whether it is possible to produce a spin
tensor network from the existing state. Then, our problem is the following:
Given a generic mixed state, approximate it by a convex linear combination of
spin tensor network states with different combinatorial patterns.
Additionally, we might consider a superposition of different node numbers.
This confronts us with the nontrivial choice of how to embed the fixed-node-
number spaces in a larger one. The simplest choice is to make these spaces
orthogonal - this choice is made in free GFTs. Another choice is to impose a
kind of consistency condition that makes "similar looking" graphs close in the
scalar product, even if they have a different number of nodes - this would be
similar to the cylindrical consistency conditions of LQG. The GFT choice, of
course, allows one to apply a different set of tools to study the state space.
Notably, the same issue with identifying boundaries arises in this case.

###  3 GFT Dynamics in holographic calculations

Let us consider the influence of dynamics of a GFT model on the previous
calculations. The type of dynamics we have in mind is that defined by a
coherent state path integral, implementing, in a weak sense, constraints that
we wish to place on the degrees of freedom of the microscopic tetrahedra. The
first thing we wish to explore is how we can recover the previous results from
a ’free theory’ type, quadratic action. For this, it is instructive to first
consider an analogy with free scalar field theories on $\mathbb{R}^{n}$.

####  1 The free field theory as weak imposition of constraints

Let us consider the unconstrained, ’off-shell’ Hilbert space
$\mathbb{H}=L^{2}(\mathbb{R}^{n})$of a free particle, here for simplicity in
Euclidean spacetime. The on-shell constraint of the particle is given by
$P^{2}=m^{2}$ for the momentum of the particle. This defines a projection
$\Pi$ as well as a constraint operator $C=\frac{P^{2}}{m^{2}}-1$ and makes the
on-shell space of states its kernel. In terms of the off-shell scalar product
$<,>$ on $\mathbb{H}$, we can write a scalar product that respects this
constraint as $<a,b>_{D}=<a,\Pi b>$. We can also achieve a projection onto the
constrained subspace in different ways: First, we can take
$<a,b>_{D}=\lim_{\beta\rightarrow\infty}<a,e^{-\beta C}b>$, as states for
which $C\neq 0$ will be exponentially suppressed. A different way of achieving
the same thing is through a highly oscillatory integral: If again the
constraint is not satisfied, then $e^{isC}$ will integrate to 0 over $s$. So,
an integral $\int_{0}^{\infty}e^{is(C+i\epsilon)}ds$ with a regulator
$\epsilon>0$ will ’wash away’ the parts that have $C\neq 0$. Then, a sensible
scalar product is

$<a,b>_{D}=\lim_{\epsilon\rightarrow
0}<a,b>_{D,\epsilon}=\lim_{\epsilon\rightarrow
0}\int_{0}^{\infty}<a,e^{is(C+i\epsilon)}b>ds=\lim_{\epsilon\rightarrow
0}<a,\mathcal{G}_{\epsilon}b>$ (8)

where we define the regularised Green 2-point operator

$\mathcal{G}_{\epsilon}=\int_{0}^{\infty}e^{is(C+i\epsilon)}=i(C+i\epsilon)^{-1}$
(9)

where the second equality is by Schwinger’s identity. Writing out the
constraint yields

$\mathcal{G}_{\epsilon}=\frac{i}{P^{2}-m^{2}+i\epsilon}$ (10)

and inserting a resolution of the identity by momentum states to get the
scalar product of coordinate eigenstates

$<a,b>_{D,\epsilon}=\mathcal{G}_{\epsilon}(x,y)=\bra{x}\mathcal{G}_{\epsilon}\ket{y}=\int_{\mathbb{R}^{n}}d\mu(p)\frac{i\bra{x}p\rangle\bra{p}y\rangle}{p^{2}-m^{2}+i\epsilon}=\int_{\mathbb{R}^{n}}d\mu(p)e^{-ip(x-y)}\frac{i}{p^{2}-m^{2}+i\epsilon}$
(11)

which is just the usual 2-point function of free scalar field theory. From the
unconstrained Hilbert space point of view, we can thus see the on-shell field
theory either as the result of implementing the on-shell constraint or as
imposed dynamics on an unconstrained field. The partition function

$Z_{\epsilon}[J]=\exp\left(\int
J(x)\mathcal{G}_{\epsilon}(x,y)J(y)dxdy\right)$ (12)

produces this scalar product and can be obtained from a coherent state path
integral

$\det(\frac{\Delta}{\pi})^{\frac{1}{2}}\int\mathcal{D}\phi
e^{-(\phi,\Delta\phi)+(\phi,J)}$ (13)

where $\Delta$ is the formal inverse of the Green’s function. However, as we
have seen, that formal inverse is just the constraint itself. Thus, flipping
the definition around, we can take the dynamics of the free theory to be given
through the path integral

$Z_{C}[J]=\det(\frac{C}{\pi})^{\frac{1}{2}}\int\mathcal{D}\phi
e^{-(\phi,C\phi)+(\phi,J)}$ (14)

whose 2-point function will reproduce the scalar product in the constrained
space. 1-particle states will be created through functional differentials with
respect to J or equivalently through insertions of the coherent field.
Arbitrary 1-particle states can then be put into the scalar product by
superposition. This procedure extends to n-particle states straightforwardly
and allows for a direct generalisation to arbitrary dynamics. In the above
integral, for $J=0$ the extremising set of coherent field configurations is
given by those that satisfy the constraint. For more general actions, then,
the interpretation is that one implements a constraint given by the equations
of motion, but weakly so.
We might use this approach of implementing dynamics in various ways to
incorporate GFT dynamics. The way which we choose here is to rephrase the
imposition of the gauge invariance constraint as a Schwinger-Dyson equation
arising from the GFT path integral. The action

$S(\phi)=\int[dg]\phi(g)[\vec{J}_{Diag}^{2}-S(S+1)]\phi^{\dagger}(g)$ (15)

will enforce every vertex’s intertwiner space to be a representation of spin
$S$ under the action of the diagonal subgroup of $SU(2)^{D}$. The choice $S=0$
then retrieves the standard case of gauge invariant intertwiner spaces. The
Schwinger-Dyson equation then asserts

$[\vec{J}_{Diag}^{2}(g)-S(S+1)]\langle\phi(g)\phi^{\dagger}(h)\rangle=-i\hbar\delta(g,h)$
(16)

and many others. If we rephrase the entropy calculation in terms of GFT
correlation functions, we can then impose the dynamics ’straightforwardly’. A
test of this would be to recover the gauge invariant calculation starting from
the unconstrained one.

####  2 Expressing the trace as a GFT integral

We will actually take a slightly different approach to proceed, which is
however inspired by the above argument. We draw inspiration from condensed
matter physics and write the trace

$\mathrm{Tr}[e^{-\beta H}X]$ (17)

as a coherent state path integral in periodic, Euclidean time. Here we use
$H=C$ both as a constraint and a Hamiltonian operator. This allows us to see
the GFT constraint as the ground state space of the Hamiltonian $H$ \- so the
GFT becomes the study of ground state properties of $H$. The system on which
the Hamiltonian will live is the same as before - A many-particle system of
tetrahedra. If, in addition to the constraint, we have additional data on the
tetrahedra, say from a scalar field $\phi$, then the Hamiltonian should not
vanish completely and provides actual time evolution. In that case, we may see
the system as those additional data with its own Hamiltonian $H_{Matter}$, but
coupled to a background system in its ground state. One way to analyse this
trace is, as said, to go to a path integral representation. Then, by the usual
way of coherent state bosonic path integrals,

$\mathrm{Tr}[e^{-\beta
H}X]=\int_{\phi(0)=\phi(\tau)}\mathcal{D}[\phi(\tau)\bar{\phi}(\tau)]e^{-S(\bar{\phi},\phi)}\bra{\phi(\beta)}X\ket{\phi(0)}$
(18)
$S(\bar{\phi},\phi)=\int_{0}^{\beta}d\tau[\bar{\phi}\partial_{\tau}\phi+H(\bar{\phi},\phi)]$
(19)

So we integrate over fields which have an additional periodic time variable.
In the limit of $\beta\rightarrow\infty$, the action becomes an integral over
$\mathbb{R}$, and we lose the periodicity requirement.
Our purpose here is to use this method, roughly, to analyse traces of the form
$\mathrm{Tr}_{\mathbb{H}^{\otimes 2}}[(e^{-\beta H})^{\otimes 2}X]$, so we
will have 2 independent field theories of $\phi,\chi$ which mix only through
matrix elements of $X$. The Hilbert spaces in question are once again the
(unconstrained/constrained) Fock ones. Let $\mathbf{m},\mathbf{n}$ label a
basis of the one-particle Hilbert space. Then, if X is given by some first-
quantised 1-particle operator, we can write it as

$X=\sum_{\mathbf{m_{1}},\mathbf{n_{1}},\mathbf{m_{2}},\mathbf{n_{2}}}X_{\mathbf{m_{1}},\mathbf{n_{1}};\mathbf{m_{2}},\mathbf{n_{2}}}\ket{\mathbf{m_{1}}}_{1}\bra{\mathbf{n_{1}}}_{1}\otimes\ket{\mathbf{m_{2}}}_{2}\bra{\mathbf{n_{2}}}_{2}$
(20)

And its second quantised form will simply be

$X=\sum_{\mathbf{m_{1}},\mathbf{n_{1}},\mathbf{m_{2}},\mathbf{n_{2}}}X_{\mathbf{m_{1}},\mathbf{n_{1}};\mathbf{m_{2}},\mathbf{n_{2}}}\hat{\phi}_{\mathbf{m_{1}}}^{\dagger}\hat{\phi}_{\mathbf{n_{1}}}\otimes\hat{\chi}_{\mathbf{m_{2}}}^{\dagger}\hat{\chi}_{\mathbf{n_{2}}}$
(21)

Similar extensions from 1st to 2nd quantisation apply for higher-body
operators. In our case, we are interested in $X$ being a density matrix for
$N\otimes N$ particles times a swap operator. As such, the expression for $X$
will involve $N$ fields of $\phi,\phi^{\dagger}$ and $\chi,\chi^{\dagger}$
each. The density matrix part will factorise over the two independent field
operators while the swap operator mixes them. In our 1-body example, the trace
is then a combination of thermal correlation functions:

$\mathrm{Tr}[(e^{-\beta H})^{\otimes
2}X]=\sum_{\mathbf{m_{1}},\mathbf{n_{1}},\mathbf{m_{2}},\mathbf{n_{2}}}X_{\mathbf{m_{1}},\mathbf{n_{1}};\mathbf{m_{2}},\mathbf{n_{2}}}\mathrm{Tr}[e^{-\beta
H}\hat{\phi}_{\mathbf{m_{1}}}^{\dagger}\hat{\phi}_{\mathbf{n_{1}}}]\mathrm{Tr}[e^{-\beta
H}\hat{\chi}_{\mathbf{m_{2}}}^{\dagger}\hat{\chi}_{\mathbf{n_{2}}}]$ (22)
$\langle
X\rangle_{\beta}=\sum_{\mathbf{m_{1}},\mathbf{n_{1}},\mathbf{m_{2}},\mathbf{n_{2}}}X_{\mathbf{m_{1}},\mathbf{n_{1}};\mathbf{m_{2}},\mathbf{n_{2}}}\langle\hat{\phi}_{\mathbf{m_{1}}}^{\dagger}\hat{\phi}_{\mathbf{n_{1}}}\rangle_{\beta}\langle\hat{\chi}_{\mathbf{m_{2}}}^{\dagger}\hat{\chi}_{\mathbf{n_{2}}}\rangle_{\beta}$
(23)

We should then get the expectation value we want by calculating the
correlation functions in the zero temperature limit and contracting the tensor
$X_{\mathbf{m_{1}},\mathbf{n_{1}};\mathbf{m_{2}},\mathbf{n_{2}}}$ with them.
The image is that the tensor network given by $X$ is capped off by the Feynman
diagrams of the GFT, which are spin foams connecting the two ends of a
cobordism between N particles/tetrahedra. In this sense, the spin foams are
glued together into a closed discrete spacetime by putting the tensor network
on the ends.

Figure 1: Schematic of how contractions happen in the setting of the traced
two copies of a GFT. Two copies of the 2N-point functions of the GFT are
contracted with density matrices. Preliminary image.

Importantly, one need not use the condensed matter path integral for this. Any
definition of correlation functions may function as a way to specify the GFT
dynamics that influences the entropy calculation. In particular, one might ask
how the above is related to the timeless GFT path integral: One gets the usual
GFT path integral by integrating over fields which have trivial time
dependence $\phi(\tau)=\phi$. They may still depend on group parameters or
modes, but all nonzero Matsubara frequency modes of the periodic time path
integral are dropped. Then, the integral is simply

$\mathrm{Tr}[e^{-\beta H}X]=\int\mathcal{D}\phi\mathcal{D}\bar{\phi}e^{-\beta
H(\bar{\phi},\phi)}\bra{\phi}X\ket{\phi}$ (24)

so one recovers the typical form of GFT path integrals. This reduction can be
equivalently seen as replacing $e^{-\beta H}$ by $:e^{-\beta H}:$ in the
initial trace. Of course, this means one is in principle calculating a very
different object and sufficient care may be necessary to see which definition
is more appropriate.
The simplest example of this is in the free GFT case, where we can choose our
basis of field modes such that the correlation function
$\langle\hat{\phi}_{\mathbf{m}}^{\dagger}\hat{\phi}_{\mathbf{n}}\rangle_{\infty}=\delta_{\mathbf{m},\mathbf{n}}$.
For example, we can reduce go from the unconstrained to the Gauge constrained
Hilbert space by switching the modes from $(\mathbf{j},\mathbf{m},\mathbf{n})$
to $(\mathbf{j},\mathbf{m};S,\iota_{S})$, where the label S specifies the
representation under the diagonal subgroup of $SU(2)^{4}$ and $\iota_{S}$ a
basis of that space. One receives the constrained Hilbert space as the
subspace for which $S=0$. However, in a theory which places a nontrivial spin
on each tetrahedron, say through torsion in the classical spacetime induced by
fermions, it might be necessary to dynamically set this constraint to
something weaker. Then, the unconstrained Fock space might be necessary again.
Either way, in the limit of $\beta\rightarrow\infty$ the trace reduces to the
already known contraction of tensor networks, with an added symmetry in the
indices due to the permutation symmetry of the bosonic fields. Thus the
calculation is in principle the same as before, but with a bosonic or
unlabeled tensor network state.
Next we can consider typical simplicial GFT interactions. In 4D, these are of
the form $\hat{\phi}^{5}+h.c.$ Regardless of colouring, we can see that due to
our operator consisting of $2N$ fields $\hat{\phi}$, interactions will change
the relevant correlation function at order $\mathcal{O}(\lambda^{2})$ in the
GFT coupling.
For a tensorial/bubble interaction with bipartite field set, this is different
and will contribute at lower order as well. However, from this point on the
conceptual issues are cleared up and the calculation involves once again
’just’ contractions of tensors. Perturbative corrections to the entropy
calculation will have to be renormalised and an analysis of the
renormalisation group flow might be necessary. At each order in perturbation
theory, then, we will have a sum over contracted spin foams.
Instead of calculating the correlation functions for a given GFT, we can
instead ask once again which conditions we might impose on them to achieve
holography. So, we see the CFs as another element in the tensor contraction
which, along with intertwiner and core states, may be tweaked to achieve
maximal entropy. In this way, we might find conditions for more generic spin
tensor network states to admit holography, or see how a different scalar
product, closer to that of LQG, could change the effective traces and
entropies. Alternatively, one might select a dynamics by requiring as many
states as possible to be holographic in some form. Such questions should be
explored in future work to answer whether holography could be a guiding
principle for finding simple spin foam models or GFTs that produce the right
behaviour for a theory of quantum gravity.

####  3 Extension of the spin foam picture

As we saw in section 1, one may see the calculation of the two Ising partition
functions 25 as a spin foam partition function. More specifically, the only
spin foams being summed over have the form of being the trivial ones
connecting two copies of the graph $\gamma$. From here, many generalisations
through the GFT approach are more obvious. For one, these trivial spin foams
are the only ones available in a free GFT’s expansion. So, when calculating
transition amplitudes between spin network states in such a GFT, only these
will contribute to the sum, which is represented as a spin foam model. So, the
partition functions used to calculate the entropy become once again numbers
calculated from correlation functions in the GFT. By replacing the GFT with an
interacting one, yet leaving the boundary as-is, we will obtain corrections
though nontrivial spin foams connecting the two copies of the graph. The
difference to the approach presented above is that we have a direct path to
obtain the free GFT generating the spin foam model from which we calculate the
entropy, and that there is only one copy of the GFT involved instead of two.
Said GFT will have additional variables on each vertex given by the
$\mathbb{Z}_{2}$ Ising spin.
Sums over graphs, even with varying numbers of vertices are also more
straightforward in this setting, as this just involves a sum over larger class
of Feynman diagrams of the GFT. However, the interpretation of these
quantities is still as conceptually unclear as in the approach outlined
before.

####  4 Implementing dynamically distinguishable quanta

As another problem to consider, we need to write a tensor network state of
indistinguishable tetrahedra in a 2nd quantised setting somehow to apply GFT
dynamics. We propose a simple way to implement this on a technical level,
along with some criteria for when this works.
The issue is that, while GFT dynamics are specified in a second-quantised
form, we wish to study a tensor network state with fixed connectivity under
the new scalar product. While we can technically implement this by starting
from a first-quantised tensor network, then producing the equivalent second-
quantised one, this produces different results from the ones we had before,
even for the free theory, by virtue of symmetrising the network. This means
that somehow, a labeling of the 2nd quantised state must be introduced to
compare the results properly. From an operational point of view, an effective
labeling of vertices can arise through additional degrees of freedom. Let us
consider an $N$-particle state in the symmetric Fock space
$\mathbb{F}_{s}(L^{2}(G^{D}\times\mathbb{R}^{D}))$ with the special form

$\ket{\Psi}=\int d\mu(g_{1})\dots
d\mu(g_{N})f(g_{1},\dots,g_{N})\ket{(g_{1},x_{1}),\dots,(g_{N},x_{N})},$ (25)

with some fully symmetrised function $f$. We may understand the set of these
states for fixed $x_{i}\in\mathbb{R}^{D}$ as one living in the slightly
different Fock space

$\mathbb{F}_{s}(L^{2}(G^{D}\times\mathbb{R}^{D};d\mu_{Haar}\times
d\mu_{Count})),$ (26)

in which only the measure on the second set of data has changed: Instead of
integrating over all of $\mathbb{R}^{D}$, only those fixed $x_{i}$ are
relevant now. Actually, we may see this $L^{2}$-space as

$L^{2}(G^{D}\times\mathbb{R}^{D};d\mu_{Haar}\times d\mu_{Count})\cong
L^{2}(\sqcup_{i}G^{D}_{i};\sum_{i}d\mu_{Haar,i})\cong\bigoplus_{i}L^{2}(G^{D}_{i};d\mu_{Haar,i})$
(27)

which is entirely analogous to a mode splitting as on compact spaces:
$L^{2}(S_{1})\cong\bigoplus_{n}\text{span}(e^{i2\pi nx})$. A key property of
the Fock functor is that it has adjoints - it turns direct sums into tensor
products, in particular.

$\mathbb{F}_{s}(L^{2}(G^{D}\times\mathbb{R}^{D};d\mu_{Haar}\times
d\mu_{Count}))\cong\bigotimes^{N}_{i=1}\mathbb{F}_{s}(L^{2}(G^{D}_{i};d\mu_{Haar,i}))$
(28)

which is a simple factorisation that we can make use of: In the state above,
none of the modes labeled by $x_{i}$ have higher occupation numbers than 1.
Therefore, when identifying tetrahedra by this mode label in some way, it
makes sense to speak of the tetrahedron with label $x_{i}$. Moreover, the
factorisation allows for a straightforward usage of reduced states and similar
operations. In other words, we can see the starting state as one in the space
of $N$ distinguishable quanta, inducing a map

$\bigotimes^{N}_{i=1}\mathbb{F}_{s}(L^{2}(G^{D}_{i};d\mu_{Haar,i}))\rightarrow\bigotimes^{N}_{i=1}L^{2}(G^{D}_{i};d\mu_{Haar,i})$
(29)

mapping the 0-particle sectors to 0.
The remaining natural questions are when such states exist and how stable
these identifications are under constraints or other dynamics. We think of the
following criteria for this procedure of effective distinguishability to work:

  1. 1.

The system of indistinguishable must have a sufficiently large set of quantum
numbers to identify them.

  2. 2.

Observables or similar quantities must exist that allow identifying values of
these quantum numbers in a state.

  3. 3.

A state must be sharp in the values of these labels and said values must be
distinguishable (on an operational level).

  4. 4.

The dynamics of the theory must make these values evolve continuously and
without having them intersect too often. Possibly, the evolution of these
labels should even be slow compared to some typical timescales.

A typical setting where these criteria are fulfilled is in low-energy solid
state systems: A priori indistinguishable atoms arrange in a lattice, where
their spatial positions are nearly conserved in time. This allows using these
positions as labels to distinguish the atoms. From this example, one can
already see that this effective distinguishability is made possible both by
the states and the dynamics of the system only in a certain regime.

##  Chapter 3 Conclusion

Notions of holography help in many different ways to understand and simplify
quantum field theories with high degrees of complexity. In the case of
gravity, there are several suggestions that degrees of freedom of QG might be
equally well described by boundary theories. However, most known cases of
holography study dualities or recovery of information in a quasi-classical
setting: There are always definite geometric quantities or even a
semiclassical spacetime. Even in the tensor network case, one finds behaviour
more reminiscent of a single metric or geometry. In this thesis, we made a few
first steps in exploring what properties holographic quantum states must have,
if they are true superpositions of distinct geometries. For this purpose, we
applied techniques developed in the context of random tensor networks and
combined them with ideas from systems with disorder to understand the specific
case of RSTNs with multiple, but finitely many spin sectors. We uncovered a
few key insights that likely have a more general analogue beyond our setting:

  * •

The Hilbert space of states does not factor into bulk and boundary parts, so
one can only make typicity statements about classes of states with fixed bulk
data.

  * •

The purity of a reduced boundary state as well as averages of boundary
observables can be mapped to a random Ising model with a distribution of
couplings related to the relative weight the geometries have in the state.

  * •

Such a random Ising model may also be understood as a kind of sum over
background geometries for the Ising model to live on, thus providing a notion
of quantum gravity path integral for it.

  * •

For holographic states, for which the map from a boundary region $C$ to its
complement $C^{c}$ is an isometry, said relative weight of geometries is
required to be inversely proportional to the size of the spin sectors, in
terms of their boundary area in $C^{c}$.

  * •

The expectation value of the area of $C$ in such a state was found to be
related in a very simple way to the sequence of areas of $C$ in the involved
spin sectors. In particular, the value is at least the $mean$ and does not
exceed the $sum$ of the individual areas.

It is easy to envision many extensions of our work, particularly as we did not
use most of our assumptions on the graph structure, like valency, colouring,
etc. In principle, also the dimensions of the representation spaces are not
essential and one might replace the $SU(2)$-data by that of other Lie groups
or other structures. We also highlighted possible extensions to consider
superpositions of different graphs labeling the states, as well as the
inclusion of dynamics in the form of a GFT path integral. We expect that, with
more work in all of these directions, a more coherent picture of general
properties of states in many QG approaches will emerge. In fact, as spin
networks are used beyond discrete quantum gravity approaches, any context in
which many-body quantum states and lattice gauge theory-like methods might
make use of the extensions of RSTN holography that we have started to explore
here. We expect the results of this work to be of use in investigating
holographic properties of a large number of superposed spin sectors and
entanglement patterns, which will play a key role in the description of
semiclassical geometries in quantum gravity.

##  Chapter 4 Appendix

###  1 Average over all tetrahedra

We can perform, to contrast our main approach, an average over all tetrahedral
states as a whole, as opposed to averaging over individual tetrahedra’s
states. While this simplifies the calculations significantly, we lose the
correspondence to a random Ising model and the bulk and boundary decouple as
far as the boundary entropy is concerned. Still, we find that there is a
leftover transition of a minimal surface that is not seen explicitly. We
choose as the state to be projected a general

$\begin{split}\ket{\Psi}=U\ket{\Psi_{0}}=\bigoplus_{\vec{\textbf{j}}}\ket{\Psi_{\vec{\textbf{j}}}}=\bigoplus_{\vec{\textbf{j}}}\sum_{\vec{\textbf{k}}}U_{\vec{\textbf{j}},\vec{\textbf{k}}}\ket{\Psi_{0,\vec{\textbf{k}}}}\end{split}$
(1)

where $U$ is a general unitary $\mathbb{H}\mapsto\mathbb{H}$ and
$\ket{\Psi_{0}}$ is a reference state. Now, we can define the boundary state

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

as before.

We are interested in the second Rényi entropy of this reduced state in
dependence on the connectivity $\Gamma$ and the presence of an interior
$\rho_{\text{core}}$ and intertwiner data $\rho_{I}$, from now on assumed to
be a mixed state. For this, we need to use the replica trick again.

$e^{-S_{2}(A)}=\frac{\text{Tr}_{A}[\rho_{A}^{2}]}{\text{Tr}_{A}[\rho_{A}]^{2}}=\frac{\text{Tr}_{\mathbb{H}_{\partial\gamma}^{\otimes
2}}[(\rho\otimes\rho)\mathcal{S}_{A}]}{\text{Tr}_{\mathbb{H}_{\partial\gamma}^{\otimes
2}}[\rho\otimes\rho]}=\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}\frac{Z_{1}}{Z_{0}}$
(3)

where the swap operator mixes all spin sectors. By using the definition 2, we
can write the objects $Z_{1|0}$ as

$\begin{split}Z_{1|0}&=\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma,in}\otimes\mathbb{H}_{\partial\gamma,out}\otimes\mathbb{H}_{b})^{\otimes
2}}\left[\rho_{\text{core}}^{\otimes 2}\rho_{I}^{\otimes
2}\left(\ket{\Gamma}\bra{\Gamma}\right)^{\otimes
2}\left(\ket{\Psi}\bra{\Psi}\right)^{\otimes
2}(\mathcal{S}_{A}|\mathbb{I})\right]\end{split}$ (4)

where we note that the trace is now performed not over the Hilbert space
$\mathbb{H}$, but
$\mathbb{H}^{\partial\gamma}\otimes\mathbb{H}^{I}\otimes\mathbb{H}^{\Gamma}$.
However, because the state $\Psi$ is from $\mathbb{H}$ and thus ’diagonal’ in
$\mathbb{H}_{\partial\gamma}\otimes\mathbb{H}_{b}$, we can switch to
$\mathbb{H}$ instead.

$Z_{1|0}=\mathrm{Tr}_{\mathbb{H}^{\otimes 2}}\left[\rho_{\text{core}}^{\otimes
2}\rho_{I}^{\otimes 2}\left(\ket{\Gamma}\bra{\Gamma}\right)^{\otimes
2}\left(\ket{\Psi}\bra{\Psi}\right)^{\otimes
2}(\mathcal{S}_{A}|\mathbb{I}_{\mathbb{H}})\right].$ (5)

We now need to perform an average over some distribution of unitaries $U$ to
proceed. The simplest way to do this is for a uniform choice of states $\Psi$,
corresponding to the Haar measure on the unitary group
$\mathcal{U}(\mathcal{H}_{J})$, where

$\mathcal{H}_{J}\coloneqq\bigoplus_{\vec{\textbf{j}},\mathbb{j}\leq
j^{x}_{\alpha}\leq J}\mathbb{H}_{\vec{\textbf{j}}}\quad\text{with dimension
}\mathcal{D}_{J}$ (6)

For this case, we may use the same application of Schur’s lemma as before. If
$\langle f\rangle_{U}$ denotes the average of a function $f$ on
$U(\mathcal{H}_{J})$ with respect to the Haar measure, we have that

$\langle\left(\ket{\Psi}\bra{\Psi}\right)^{\otimes
2}\rangle_{U}=\frac{\mathbb{I}_{\mathcal{H}_{J}\otimes\mathcal{H}_{J}}+\mathcal{S}_{\mathcal{H}_{J}\otimes\mathcal{H}_{J}}}{\mathcal{D}_{J}(\mathcal{D}_{J}+1)}$
(7)

where S swaps the two copies of $\mathcal{H}_{J}$. We note two things. First,
it is not possible to take the limit $J\rightarrow\infty$ here, and do this
argument in the full, untruncated Hilbert space. One can see this from the
normalisation making the RHS vanish, while the LHS stays of trace 1 in that
limit. Second, contrast this to the previously considered case of a seperate
average over each spin sector seperately. In that case, we had
$\langle\ket{f_{\vec{\textbf{j}}}}\bra{f_{\vec{\textbf{j}}}}\otimes\ket{f_{\vec{\textbf{j}}}}\bra{f_{\vec{\textbf{j}}}}\rangle_{U}=\frac{\mathbb{I}_{\vec{\textbf{j}}}+\mathcal{S}_{\vec{\textbf{j}}}}{\mathcal{D}_{\vec{\textbf{j}}}(\mathcal{D}_{\vec{\textbf{j}}}+1)}$,
so that the current $\langle\left(\ket{\Psi}\bra{\Psi}\right)^{\otimes
2}\rangle_{U}$ would be equivalent to
$\bigotimes_{x}\bigoplus_{\textbf{j}^{x}}\frac{\mathbb{I}_{\vec{\textbf{j}}}+\mathcal{S}_{\vec{\textbf{j}}}}{\mathcal{D}_{\vec{\textbf{j}}}(\mathcal{D}_{\vec{\textbf{j}}}+1)}$.
This, inserted into the trace 5 would lead to a much more fine grained,
vertex-dependent expression, which we do not have here. Now, we insert this
into 5 and proceed as usual.

$\begin{split}\langle Z_{1|0}\rangle_{U}=\mathrm{Tr}_{\mathbb{H}^{\otimes
2}}\left[\rho_{\text{core}}^{\otimes
2}\left(\ket{\zeta}\bra{\zeta}\right)^{\otimes
2}\left(\ket{\Gamma}\bra{\Gamma}\right)^{\otimes
2}\overline{\left(\ket{\Psi}\bra{\Psi}\right)^{\otimes
2}}(\mathcal{S}_{A}|\mathbb{I})\right]\\\ =\mathrm{Tr}_{\mathbb{H}^{\otimes
2}}\left[\rho_{\text{core}}^{\otimes
2}\left(\ket{\zeta}\bra{\zeta}\right)^{\otimes
2}\left(\ket{\Gamma}\bra{\Gamma}\right)^{\otimes
2}\frac{\mathbb{I}_{\mathcal{H}_{J}\otimes\mathcal{H}_{J}}+\mathcal{S}_{\mathcal{H}_{J}\otimes\mathcal{H}_{J}}}{\mathcal{D}_{J}(\mathcal{D}_{J}+1)}(\mathcal{S}_{A}|\mathbb{I})\right]\\\
=\frac{1}{\mathcal{D}_{J}(\mathcal{D}_{J}+1)}\mathrm{Tr}_{\mathbb{H}^{\otimes
2}}\left[\rho_{\text{core}}^{\otimes
2}\left(\ket{\zeta}\bra{\zeta}\right)^{\otimes
2}\left(\ket{\Gamma}\bra{\Gamma}\right)^{\otimes
2}\left(\mathbb{I}_{\mathcal{H}_{J}\otimes\mathcal{H}_{J}}+\mathcal{S}_{\mathcal{H}_{J}\otimes\mathcal{H}_{J}}\right)(\mathcal{S}_{A}|\mathbb{I})\right]\\\
\end{split}$ (8)

If we also neglect fluctuations in the low spin regime111This requires that
the lower cutoff must scale polynomially in the number of vertices of the
graph, as in $\mathbb{j}>>N^{k}$ for some $k>\frac{2}{\Delta E}$ with the
spectral gap of the later Ising model., we may discard the $J$-dependent
prefactor in the quotient and write

$\langle
e^{-S_{2}(A)}\rangle_{U}=\langle\frac{Z_{1}}{Z_{0}}\rangle_{U}\approx\frac{\langle
Z_{1}\rangle_{U}}{\langle Z_{0}\rangle_{U}}$ (9)

However, we can no longer perform the conversion to an Ising model as before.
The reason is that for that conversion, we need a tensor product
$\bigotimes_{x}\left(\mathbb{I}_{\mathcal{H}_{x}\otimes\mathcal{H}_{x}}+\mathcal{S}_{\mathcal{H}_{x}\otimes\mathcal{H}_{x}}\right)$
of operators acting on the tetrahedra individually. Working with completely
generic classes of states $\Psi$ has removed the local structure from the
problem entirely, and thus averaging over it will not allow us to recover that
local data.
Note that this is to be expected: An average removes information from a
distribution or random variable. The larger, or coarser, the average we
perform, the more data we remove in the process. On the other hand, removing
said data can pinpoint typical behaviour and allow for simpler calculations.
In our case, the removal of local data clearly makes the calculation simpler -
so simple, in fact, that we will not be able to talk about holographic
surfaces or entanglement wedges or similar concepts, as those objects are not
needed for the entropy calculation.
In fact, we can perform the calculation as-is. For this, it is actually
convenient to work in the form of 4, where the traces over bulk and boundary
can be performed seperately.

$Y_{1|0}=\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma}\otimes\mathbb{H}_{b})^{\otimes
2}}\left[\rho_{\text{core}}^{\otimes 2}\rho_{I}^{\otimes
2}\left(\ket{\Gamma}\bra{\Gamma}\right)^{\otimes
2}\left(\mathbb{I}_{\mathbb{H}_{\partial\gamma}\otimes\mathbb{H}_{\partial\gamma}}\mathbb{I}_{\mathbb{H}_{b}\otimes\mathbb{H}_{b}}+\mathcal{S}_{\mathbb{H}_{\partial\gamma}\otimes\mathbb{H}_{\partial\gamma}}\mathcal{S}_{\mathbb{H}_{b}\otimes\mathbb{H}_{b}}\right)(\mathcal{S}_{A}|\mathbb{I})\right]$
(10)

The key fact is that the bulk state has no support on the boundary, so it does
not matter for the trace over the boundary space at all. We study both parts
seperately and find:

$\displaystyle Y_{0}$
$\displaystyle=\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma}\otimes\mathbb{H}_{b})^{\otimes
2}}\left[\rho_{\text{core}}^{\otimes 2}\rho_{I}^{\otimes
2}\left(\ket{\Gamma}\bra{\Gamma}\right)^{\otimes
2}\left(\mathbb{I}_{\mathbb{H}_{\partial\gamma}\otimes\mathbb{H}_{\partial\gamma}}\mathbb{I}_{\mathbb{H}_{b}\otimes\mathbb{H}_{b}}+\mathcal{S}_{\mathbb{H}_{\partial\gamma}\otimes\mathbb{H}_{\partial\gamma}}\mathcal{S}_{\mathbb{H}_{b}\otimes\mathbb{H}_{b}}\right)\right]$
(11) $\displaystyle=\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma})^{\otimes
2}}\left[\rho_{\text{core}}^{\otimes
2}\mathbb{I}_{\mathbb{H}_{\partial\gamma}\otimes\mathbb{H}_{\partial\gamma}}\right]\mathrm{Tr}_{(\mathbb{H}_{b})^{\otimes
2}}\left[\rho_{I}^{\otimes 2}\left(\ket{\Gamma}\bra{\Gamma}\right)^{\otimes
2}\right]$ (12)
$\displaystyle+\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma})^{\otimes
2}}\left[\rho_{\text{core}}^{\otimes
2}\mathcal{S}_{\mathbb{H}_{\partial\gamma}\otimes\mathbb{H}_{\partial\gamma}}\right]\mathrm{Tr}_{(\mathbb{H}_{b})^{\otimes
2}}\left[\rho_{I}^{\otimes 2}\left(\ket{\Gamma}\bra{\Gamma}\right)^{\otimes
2}\mathcal{S}_{\mathbb{H}_{b}\otimes\mathbb{H}_{b}}\right]$ (13)
$\displaystyle=\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma})^{\otimes
2}}\left[\rho_{\text{core}}^{\otimes
2}\right]\mathrm{Tr}_{(\mathbb{H}_{b})^{\otimes 2}}\left[\rho_{I}^{\otimes
2}\left(\ket{\Gamma}\bra{\Gamma}\right)^{\otimes
2}\right]+\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma})^{\otimes
2}}\left[\rho_{\text{core}}^{\otimes
2}\mathcal{S}_{\mathbb{H}_{\partial\gamma}\otimes\mathbb{H}_{\partial\gamma}}\right]\mathrm{Tr}_{(\mathbb{H}_{b})^{\otimes
2}}\left[\rho_{I}^{\otimes 2}\left(\ket{\Gamma}\bra{\Gamma}\right)^{\otimes
2}\right]$ (14)
$\displaystyle=\left(\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma})^{\otimes
2}}\left[\rho_{\text{core}}^{\otimes
2}\right]+\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma})^{\otimes
2}}\left[\rho_{\text{core}}^{\otimes
2}\mathcal{S}_{\mathbb{H}_{\partial\gamma}\otimes\mathbb{H}_{\partial\gamma}}\right]\right)\mathrm{Tr}_{\mathbb{H}_{b}}\left[\rho_{I}\ket{\Gamma}\bra{\Gamma}\right]^{2}$
(15)

While

$\begin{split}Y_{1}=\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma}\otimes\mathbb{H}_{b})^{\otimes
2}}\left[\rho_{\text{core}}^{\otimes 2}\rho_{I}^{\otimes
2}\left(\ket{\Gamma}\bra{\Gamma}\right)^{\otimes
2}\left(\mathbb{I}_{\mathbb{H}_{\partial\gamma}\otimes\mathbb{H}_{\partial\gamma}}\mathbb{I}_{\mathbb{H}_{b}\otimes\mathbb{H}_{b}}+\mathcal{S}_{\mathbb{H}_{\partial\gamma}\otimes\mathbb{H}_{\partial\gamma}}\mathcal{S}_{\mathbb{H}_{b}\otimes\mathbb{H}_{b}}\right)\mathcal{S}_{A}\right]\\\
=\left(\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma})^{\otimes
2}}\left[\rho_{\text{core}}^{\otimes
2}\mathcal{S}_{A}\right]+\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma})^{\otimes
2}}\left[\rho_{\text{core}}^{\otimes
2}\mathcal{S}_{\mathbb{H}_{\partial\gamma}\otimes\mathbb{H}_{\partial\gamma}}\mathcal{S}_{A}\right]\right)\mathrm{Tr}_{\mathbb{H}_{b}}\left[\rho_{I}\ket{\Gamma}\bra{\Gamma}\right]^{2}\end{split}$
(16)

Which leads to the interesting result that the entropy does not depend on the
bulk at all - in fact, the global average completely erased the information
about the bulk combinatorics:

$\langle
e^{-S_{2}(A)}\rangle_{U}=\frac{Y_{1}}{Y_{0}}=\frac{\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma})^{\otimes
2}}\left[\rho_{\text{core}}^{\otimes
2}\mathcal{S}_{A}\right]+\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma})^{\otimes
2}}\left[\rho_{\text{core}}^{\otimes
2}\mathcal{S}_{\mathbb{H}_{\partial\gamma}\otimes\mathbb{H}_{\partial\gamma}}\mathcal{S}_{A}\right]}{\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma})^{\otimes
2}}\left[\rho_{\text{core}}^{\otimes
2}\right]+\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma})^{\otimes
2}}\left[\rho_{\text{core}}^{\otimes
2}\mathcal{S}_{\mathbb{H}_{\partial\gamma}\otimes\mathbb{H}_{\partial\gamma}}\right]}$
(17)

We can simplify this further using the fact that $\rho_{\text{core}}$ only has
support on the inner boundary, and $\mathcal{S}_{A}$ only on the outer one.
For example,

$\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma})^{\otimes
2}}\left[\rho_{\text{core}}^{\otimes
2}\mathcal{S}_{\mathbb{H}_{\partial\gamma}\otimes\mathbb{H}_{\partial\gamma}}\mathcal{S}_{A}\right]=\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma,out})^{\otimes
2}}\left[\mathcal{S}_{\mathbb{H}_{\partial\gamma,out}}\mathcal{S}_{A}\right]\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma,in})^{\otimes
2}}\left[\rho_{\text{core}}^{\otimes
2}\mathcal{S}_{\mathbb{H}_{\partial\gamma,in}}\right]$ (18)

which, when used in the above, yields

$\begin{split}\langle
e^{-S_{2}(A)}\rangle_{U}=\frac{Y_{1}}{Y_{0}}=\frac{\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma,out})^{\otimes
2}}\left[\mathcal{S}_{A}\right]+e^{-S_{2}(\rho_{\text{core}})}\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma},out)^{\otimes
2}}\left[\mathcal{S}_{\mathbb{H}_{\partial\gamma,out}}\mathcal{S}_{A}\right]}{\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma,out})^{\otimes
2}}\left[\mathbb{I}\right]+e^{-S_{2}(\rho_{\text{core}})}\mathrm{Tr}_{(\mathbb{H}_{\partial\gamma,out})^{\otimes
2}}\left[\mathcal{S}_{\mathbb{H}_{\partial\gamma,out}}\right]}\\\
=\frac{\dim(\mathbb{H}_{A})\dim(\mathbb{H}_{\bar{A}})^{2}+e^{-S_{2}(\rho_{\text{core}})}\dim(\mathbb{H}_{A})^{2}\dim(\mathbb{H}_{\bar{A}})}{\dim(\mathbb{H}_{\partial\gamma,out})^{2}+e^{-S_{2}(\rho_{\text{core}})}\dim(\mathbb{H}_{\partial\gamma,out})}=\frac{h^{||\bar{A}||}+e^{-S_{2}(\rho_{\text{core}})}h^{||A||}}{h^{||\partial\gamma_{out}||}+e^{-S_{2}(\rho_{\text{core}})}}\end{split}$
(19)

where we defined $h=\dim(\bigoplus_{\mathbb{j}\leq j^{x}_{\alpha}\leq
J}V^{j^{x}_{\alpha}})=\frac{d_{J}(d_{J}+1)-d_{\mathbb{j}}(d_{\mathbb{j}}+1)}{2}$.
So,

$-\log(\langle
e^{-S_{2}(A)}\rangle_{U})=-\log(\frac{h^{||\bar{A}||}+e^{-S_{2}(\rho_{\text{core}})}h^{||A||}}{h^{||\partial\gamma_{out}||}+e^{-S_{2}(\rho_{\text{core}})}})$
(20)

which, for large enough $h$, has limiting behaviour:

$\langle S_{2}(A)\rangle_{U}\approx-\log(\langle
e^{-S_{2}(A)}\rangle_{U})\approx\min\\{S_{2}(\rho_{\text{core}})+||\bar{A}||\ln(h),||A||\ln(h)\\}$
(21)

and in particular $\langle
S_{2}(\partial\gamma)\rangle_{U}=\min\\{||\partial\gamma_{out}||\ln(h),S_{2}(\rho_{\text{core}})\\}$.
(This approximation is very good for even low values of $h$.) It appears that
the entropy will only depend on the size of the region A, the outer boundary
and the entropy of the core. One might read this, in fact, as a form of a
general Ryu-Takayanagi formula due to statistical considerations alone. Low
spin fluctuations may change this result. The crucial fact is that after total
randomisation, no connection between bulk and boundary exists anymore.

###  2 Fixed-spin isometry

We give a sufficient criterion for the fixed-spin case of geometries to induce
isometries as well as an argument for the dimensionality of the set of
holographic states.
For the single-spin sector case, we need to find the minimal configuration of
$\mathcal{H}_{1}(\vec{\textbf{j}},\vec{\textbf{j}})$ and adjust it such that
the ground state value becomes $\sum_{e\in C}\log(d_{j_{e}})$. The most
obvious way to achieve this is when the all-up configuration is the ground
state: Then, only boundary links in $C$ contribute and we recover precisely
the value we need. So, in a way similar to [BulkEntropyRef], we can tell by
certain conditions on the spin sector or graph combinatorics whether the state
is holographic. Let us assume that we change the configuration around the all-
up one. Then, the change in energy from flipping a single spin $z$ with
distance $|z-C|>1$ is always positive. If instead, we flip a spin with
distance $1$ from $C$, so adjacent to that boundary region, we instead have
the change

$\sum_{\alpha:(z,\alpha)\notin
C}\log(d_{j^{z}_{\alpha}})-\sum_{\alpha:(z,\alpha)\in
C}\log(d_{j^{z}_{\alpha}})+S_{2}((\rho^{I})_{z}).$ (22)

If this quantity is positive for all vertices with distance 1 from $C$, the
all-up configuration is a local minimum and the state is, if spins are large
enough, approximately holographic. This condition translates to

$\prod_{\alpha:(z,\alpha)\notin
C}d_{j^{z}_{\alpha}}\prod_{\alpha:(z,\alpha)\in
C}d_{j^{z}_{\alpha}}^{-1}>e^{-S_{2}((\rho^{I})_{z})}$ (23)

or more generally for spin-down regions $X$:

$\prod_{e\in E:s(e)\in X,t(e)\notin
X}d_{j_{e}}^{h_{t(e)}}>e^{-S_{2}((\rho^{I})_{X})}$ (24)

which should hold for all regions $X$, with the understanding that h is 1 on
all vertices except on the boundary region C. This gives a condition on the
entropy of the intertwiner state; If the left hand side is smaller than $1$,
this constrains the intertwiners to be not too pure. As reducing a general
random state to a subsystem like $X$ typically produces high entropies, this
should not be too much of a constraint in many cases. However, for this to
work, there is still the necessity that the left hand side exceeds the lower
bound on the purity of the reduced density matrix on the right:

$\prod_{e\in E:s(e)\in X,t(e)\notin X}d_{j_{e}}^{h_{t(e)}}>\prod_{x\in
X}\mathcal{D}_{\mathbf{j}^{x}}$ (25)

Otherwise, the holography condition cannot be achieved. Like the bulk-to-
boundary case, this condition depends on the graph and spins. For homogeneous
graphs, this translates to

$||\partial X\setminus C||-||\partial X\cap C||>|X|$ (26)

for all subsets of the graph, which can be violated on many graphs easily. One
of the smallest type of region for homogeneous spins that can do this is the
"once-fine-grained vertex",

Figure 1: The once-fine-grained vertex graph. When choosing the regions A,C as
outlined, flipping all spins down violates the condition 26 for homogeneous
spins, as $3-1=2<5$. The spin-down region $X$ has been enclosed in green here.

which has 5 vertices but 4 boundary edges. However, if the region $X$ is close
to $C$, there will be even smaller geometries. It is easy to see that the
condition for holography presented here is most easily satisfied for
inhomogeneous geometries, where the intertwiner dimensions are as small as
possible. Like in the bulk-to-boundary question, the interpretation here is
that the minimal condition for isometry from $\mathbb{H}_{\partial X\cap C}$
to $\mathbb{H}_{\partial X\setminus C}$ is modified slightly by the presence
of a bulk state $\rho^{I}$ through which information can dissipate. Most
succinctly,

$\dim(\mathbb{H}_{\partial X\setminus
C})>e^{-S_{2}((\rho^{I})_{X})}\dim(\mathbb{H}_{\partial X\cap C})$ (27)

demonstrates that each region in the bulk must have an impure enough reduced
bulk state to make the effective input dimension on the right smaller than the
output one. If equality is reached, though, one has a degeneracy in the ground
state as flipping that spin $z$ will not change the energy. Any degeneracies
will persist into the limit of large spins and destroy holographic behaviour.

Given this simple criterion, which we may see as a region of parameters
$\rho^{I}$ to be tuned, we turn to the general question of characterising the
space of holographic states. Our claim is the following: Fixing a graph
structure and a spin sector, the set of intertwiner states yielding, on
average, a holographic state, is either empty or a subspace of
$\mathcal{D}(\mathcal{I}_{\vec{\textbf{j}}})$, the space of density matrices,
with codimension 1. This means that, while the space is not dense in the full
set of states, it is still of relatively high dimension - a lot of parameters
may be tuned when starting from a holographic state. In particular, the
dimension of the space of holographic states still scales exponentially with
$N$, the number of vertices. However, it is still of measure zero in the full
space.
To illustrate this idea, consider our holography question as a differential
geometric problem222We stealthily ignore the complication that the space of
density matrices is not globally a smooth manifold of constant dimension - it
is rather a stratified space made of smooth components. When restricting to
the interiors of the strata, however, we can apply standard arguments.: Let
the map

$\displaystyle F:\mathcal{D}$
$\displaystyle(\mathcal{I}_{\vec{\textbf{j}}})\longrightarrow\mathbb{R}$ (28)
$\displaystyle\rho_{I}\mapsto\sum_{\vec{\sigma}}e^{-\mathcal{H}_{1}(\vec{\textbf{j}},\vec{\textbf{j}},\vec{\sigma})}$
(29)

define the purity - our interest is the level set
$Hol(\gamma,\vec{\textbf{j}})\coloneqq
F^{-1}(\dim(\mathbb{H}_{C,\vec{\textbf{j}}})^{-1})$. By showing that the
differential of this map is of rank 1 in the interior, we can use standard
regular level set arguments to see that the level set is locally a smooth
submanifold of codimension 1. For this purpose, first factor the map $F$ into

$\displaystyle\mathcal{D}$ $\displaystyle(\mathcal{I}_{\vec{\textbf{j}}})$
$\displaystyle\stackrel{{\scriptstyle a}}{{\longrightarrow}}$
$\displaystyle[0,1]^{2^{N}}$
$\displaystyle[0,1]^{2^{N}}\stackrel{{\scriptstyle b}}{{\longrightarrow}}$
$\displaystyle\mathbb{R}$ (30) $\displaystyle\rho^{I}$
$\displaystyle\longmapsto$
$\displaystyle(e^{-S_{2}((\rho^{I})_{S})})_{S\subset\gamma}$
$\displaystyle,(q_{S})_{S\subset\gamma}$ $\displaystyle\longmapsto$
$\displaystyle\sum_{S\subset\gamma}\prod_{e\in\partial S\Delta
C}d_{j_{e}}^{-1}\cdot q_{S}$ (31)

So we can study the linearisation of $a$ as

$T_{\rho}a(\epsilon)=e^{-S_{2}((\rho+\epsilon)_{\downarrow})}-e^{-S_{2}(\rho_{\downarrow})}=\frac{2}{\mathrm{Tr}(\rho)^{2}}\mathrm{Tr}\left[(\epsilon\otimes\rho)(\mathcal{S}_{S}-e^{-S_{2}(\rho)}\mathbb{I})\right]+\mathcal{O}(\epsilon^{2})=\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}\langle
Q_{S}(\rho),\epsilon\rangle_{HS}.$ (32)

Similarly, the linearisation of $b$ is also a scalar product, as b itself is:

$T_{q}b(t)=\langle\alpha,t\rangle=\sum_{S\subset\gamma}\alpha_{S}\cdot t_{S}$
(33)

where $\alpha_{S}=\prod_{e\in\partial S\Delta C}d_{j_{e}}^{-1}$. Then the
composition of the two is simply

$T_{\rho}F(X)=\langle\alpha,{Q(\rho),X}_{HS}\rangle=\langle\sum_{S\subset\gamma}\alpha_{S}Q_{S}(\rho),X\rangle_{HS}=\langle\tilde{Q}(\rho),X\rangle_{HS}$
(34)

which can be written as

$\frac{2}{\mathrm{Tr}(\rho)^{2}}\sum_{S\subset\gamma}\alpha_{S}\langle\rho_{S},(X-\frac{\mathrm{Tr}(X)}{\mathrm{Tr}(\rho)}\rho)_{S}\rangle.$
(35)

In particular, $T_{\rho}F(\rho)=0$. However, we can perturb this around $\rho$
to see that this map is nonzero, so of rank 1. Let $\mathrm{Tr}(\epsilon)=0$,
then

$T_{\rho}F(\rho+\epsilon)=\frac{2}{\mathrm{Tr}(\rho)^{2}}\sum_{S\subset\gamma}\alpha_{S}\langle\rho_{S},\epsilon_{S}\rangle\approx\frac{1}{\mathrm{Tr}(\rho)^{2}}\sum_{S\subset\gamma}\alpha_{S}\left[\langle(\rho+\epsilon)_{S},(\rho+\epsilon)_{S}\rangle-\langle\rho_{S},\rho_{S}\rangle\right].$
(36)

This can be, in turn, written as

$T_{\rho}F(\rho+\epsilon)=\sum_{S\subset\gamma}\alpha_{S}\left[e^{-S_{2}((\rho+\epsilon)_{S})}-e^{-S_{2}(\rho_{S})}\right].$
(37)

So, starting from a given $\rho$, we need to establish that the map is not
identically 0. This is easily seen if $e^{-S_{2}(\rho_{S})}\neq 1$, as then it
is easy to perturb the state with $\epsilon$ to increase the entropy. Then,
each of the terms should become nonnegative and the sum will be, as well. As
long as there is at least one positive term then, the map will be nonzero. The
only complication arises from pure states, which incidentally have a lower
dimensionality in the space of states and live in a different stratum from the
rest. However, for nonpure states, this yields the desired result: as a map
from nonpure intertwiner states, the map $F$ has constant rank 1 and its
regular level sets will have codimension 1. The intent of this argument is not
to provide full rigour, but to highlight a crucial aspect of our endeavour to
characterise holographic states: The space of parameters for which we achieve
holography is relatively large in the sense of dimensionality.

###  3 Example calculations

As we will see, even a simple example can become quite involved.
We first discuss the constraints on configurations. The case chosen here has
intertwiner constraints which can be ignored, so we have the following:

  1. 1.

$\Delta_{0}(\vec{\textbf{j}},\vec{\textbf{j}},\vec{\sigma})=\Delta_{0}(\vec{\textbf{j}},\vec{\textbf{k}},\vec{\sigma})=1$

  2. 2.

$\Delta_{0}(\vec{\textbf{j}},\vec{\textbf{k}},(+_{L},+_{R}))=1$

  3. 3.

$\Delta_{0}(\vec{\textbf{j}},\vec{\textbf{k}},(-_{L},-_{R}))=\prod_{e:\sigma_{x}=-1}\delta_{j_{e},k_{e}}=0$

  4. 4.

$\Delta_{0}(\vec{\textbf{j}},\vec{\textbf{k}},(+_{L},-_{R}))=\prod_{e:\sigma_{x}=-1}\delta_{j_{e},k_{e}}=0$

  5. 5.

$\Delta_{0}(\vec{\textbf{j}},\vec{\textbf{k}},(-_{L},+_{R}))=1$

Essentially, as the spins on the right vertex do not agree between the two
sectors, the Ising spin may not be down on it. The only difference in the
numerator $\Delta$-factors is that the boundary pinning field flips around
where spins have to agree. We take the pinning field to be $-1$ on the
rightmost link where the areas disagree:

  1. 1.

$\Delta_{1}(\vec{\textbf{j}},\vec{\textbf{k}},(+_{L},+_{R}))=0$

  2. 2.

$\Delta_{1}(\vec{\textbf{j}},\vec{\textbf{k}},(-_{L},-_{R}))=1$

  3. 3.

$\Delta_{1}(\vec{\textbf{j}},\vec{\textbf{k}},(+_{L},-_{R}))=1$

  4. 4.

$\Delta_{1}(\vec{\textbf{j}},\vec{\textbf{k}},(-_{L},+_{R}))=0$

In other words, the right Ising spin must be down. This is a manifestation of
the general rule that configurations where the value of the Hamiltonian would
be ambiguous need to be excluded. Overall, even in this innocuous example we
see that there can be a fair reduction of possible configurations contributing
to the mixed partition sums. Then, we can calculate the Hamiltonian values
individually.

Hamiltonian values for the 4 configurations
---
|  (+,+) |  (+,-) |  (-,+) |  (-,-)
$\mathcal{H}_{0}(\vec{\textbf{j}},\vec{\textbf{j}})$ |  $0$ |  $3\ln(2s+1)+\ln(6s-1)+S_{2}$ |  $3\ln(2s+1)+\ln(6s-1)$ |  $4\ln(2s+1)+\ln(6s-1)+\ln(6s+1)+S_{2}$
$\mathcal{H}_{0}(\vec{\textbf{k}},\vec{\textbf{k}})$ |  0 |  $3\ln(2s+1)+\ln(6s+1)$ |  $3\ln(2s+1)+\ln(6s+1)$ |  $4\ln(2s+1)+2\ln(6s+1)$
$\mathcal{H}_{0}(\vec{\textbf{j}},\vec{\textbf{k}})$ |  0 |  disallowed |  $2\ln(2s+1)+\ln(6s+1)+\Sigma_{(-,+)}$ |  disallowed
$\mathcal{H}_{1}(\vec{\textbf{j}},\vec{\textbf{j}})$ |  $\ln(6s-1)$ |  $3\ln(2s+1)+S_{2}$ |  $3\ln(2s+1)+\ln(6s+1)+\ln(6s-1)$ |  $4\ln(2s+1)+\ln(6s+1)+S_{2}$
$\mathcal{H}_{1}(\vec{\textbf{k}},\vec{\textbf{k}})$ |  $\ln(6s+1)$ |  $3\ln(2s+1)$ |  $3\ln(2s+1)+2\ln(6s+1)$ |  $4\ln(2s+1)+\ln(6s+1)$
$\mathcal{H}_{1}(\vec{\textbf{j}},\vec{\textbf{k}})$ |  disallowed |  $3\ln(2s+1)+\Sigma_{(+,-)}$ |  disallowed |  $3\ln(2s+1)+\ln(6s+1)$+$\Sigma_{(-,-)}$

We have used the shorthands

$\displaystyle\Sigma_{(\sigma_{L},\sigma_{R})}=\Sigma(\vec{\textbf{j}},\vec{\textbf{k}};(\sigma_{L},\sigma_{R}))\qquad
S_{2}=S_{2}(\rho^{I}_{\vec{\textbf{j}},\vec{\textbf{j}}}).$ (38)

We thus find the six partition sums

  1. 1.

$Z^{\vec{\textbf{j}},\vec{\textbf{j}}}_{0}=1+(2s+1)^{-3}(6s-1)^{-1}e^{-S_{2}}+(2s+1)^{-3}(6s-1)^{-1}+(2s+1)^{-4}(6s-1)^{-1}(6s+1)^{-1}e^{-S_{2}}$

  2. 2.

$Z^{\vec{\textbf{k}},\vec{\textbf{k}}}_{0}=1+(2s+1)^{-3}(6s+1)^{-1}+(2s+1)^{-3}(6s+1)^{-1}+(2s+1)^{-4}(6s+1)^{-2}$

  3. 3.

$Z^{\vec{\textbf{j}},\vec{\textbf{k}}}_{0}=1+(2s+1)^{-2}(6s+1)^{-1}e^{-\Sigma_{(-,+)}}$

  4. 4.

$Z^{\vec{\textbf{j}},\vec{\textbf{j}}}_{1}=(6s-1)^{-1}+(2s+1)^{-3}e^{-S_{2}}+(2s+1)^{-3}(6s-1)^{-1}(6s+1)^{-1}+(2s+1)^{-4}(6s+1)^{-1}e^{-S_{2}}$

  5. 5.

$Z^{\vec{\textbf{k}},\vec{\textbf{k}}}_{1}=(6s+1)^{-1}+(2s+1)^{-3}+(2s+1)^{-3}(6s+1)^{-2}+(2s+1)^{-4}(6s+1)^{-1}$

  6. 6.

$Z^{\vec{\textbf{j}},\vec{\textbf{k}}}_{1}=(2s+1)^{-3}e^{-\Sigma_{(+,-)}}+(2s+1)^{-3}(6s+1)^{-1}e^{-\Sigma_{(-,-)}}$

and we note that as expected, the values of the $Z_{0}$ partition sums
approach 1 as we increase the areas. Each term in the $Z_{1}$ sums also decays
with some power of the area. Additionally, the decay in the mixed sums is
stronger than the others due to suppression of certain configurations.
It is instructive to see the single-sector results at this point. For the
first sector, the purity is

$\displaystyle\frac{Z^{\vec{\textbf{j}},\vec{\textbf{j}}}_{1}}{Z^{\vec{\textbf{j}},\vec{\textbf{j}}}_{0}}$
$\displaystyle=\frac{(6s-1)^{-1}+(2s+1)^{-3}e^{-S_{2}}+(2s+1)^{-3}(6s-1)^{-1}(6s+1)^{-1}+(2s+1)^{-4}(6s+1)^{-1}e^{-S_{2}}}{1+(2s+1)^{-3}(6s-1)^{-1}e^{-S_{2}}+(2s+1)^{-3}(6s-1)^{-1}+(2s+1)^{-4}(6s-1)^{-1}(6s+1)^{-1}e^{-S_{2}}}$
(39) $\displaystyle\approx(6s-1)^{-1}$ (40)

which is the reciprocal dimension of the boundary input space in the large-
spin limit. The same thing happens with the other sector, where
$\frac{Z^{\vec{\textbf{k}},\vec{\textbf{k}}}_{1}}{Z^{\vec{\textbf{k}},\vec{\textbf{k}}}_{0}}\approx(6s+1)^{-1}$.
Thus both sectors individually yield an isometric map when spins are large
enough. Now, with the two weights

$e^{B(\vec{\textbf{j}})}=(6s+1)(6s-1)(2s+1)^{4}(a+d)\qquad
e^{B(\vec{\textbf{k}})}=(6s+1)^{2}(2s+1)^{4}w$ (41)

and the entropies of the intertwiner state

$e^{-S_{2}}=t\coloneqq\frac{a^{2}+d^{2}+|b|^{2}}{(a+d)^{2}}\qquad
e^{-\Sigma_{(-,-)}}=e^{-\Sigma_{(+,-)}}=q\coloneqq\frac{|u^{2}|+|v|^{2}}{w(a+d)}\qquad
e^{-\Sigma_{(+,+)}}=e^{-\Sigma_{(-,+)}}=1$ (42)

this gives us $\frac{Z_{1}}{Z_{0}}$. The full expression is rather
unilluminating, but we give a few special cases of interest. When the spin is
taken to be asymptotically large:

$\displaystyle\frac{Z_{1}}{Z_{0}}$
$\displaystyle=\frac{2w^{2}-2w+1}{6s}+\frac{(1-2w)^{3}}{36s^{2}}$ (43)
$\displaystyle+\frac{27(w-1)^{2}e^{-S_{2}}+(61-54q)w^{2}+(54q-10)w+24w^{4}-48w^{3}+1}{216s^{3}}+O\left(\left(\frac{1}{s}\right)^{4}\right).$
(44)

This is particularly simple in that the $b$ and $u,v$ parameters do not
contribute up to second order. Thus, in the high spin limit, these tend to not
matter as much. Studying only the leading 2 orders, we can ask which
parameters give us the most mixing. These are easily found to be, for fixed s,
$w\in\\{\frac{1}{2},\frac{2s+1}{2}\\}$, which means that only $w=\frac{1}{2}$
is a valid minimum.
If we take the third order into account, we instead have as minima

$(w,q)=(1,\frac{2}{27}\left(18s^{2}-9s+16\right))\approx(1,\frac{4}{3}s^{2})$
(45)

irrespective of the value of the entropy $e^{-S_{2}}$. Further orders preserve
this minimum. A numerical study with Mathematica of the minimum of the purity
for given $s$ reveals that the overall value decreases with spin inversely as
expected:

$G(s):=\min_{a,d,w,t,q}(\frac{Z_{1}}{Z_{0}}(a,d,w,t,q,s))\approx\frac{1}{12s}.$
(46)

So, the maximal achievable entropy for fixed spin $s$ is then

$S_{2}\approx\ln(12s)$ (47)

We can then estimate the maximal dimension for a holographic subspace by
$G^{-1}\approx 12s$, which is precisely the dimension of $\mathbb{H}_{C}$ in
the studied case. Numerically, we find that this is achieved when
$(a,d,w)\approx(\frac{1}{4},\frac{1}{4},\frac{1}{2})$ in the large-spin limit,
with not much dependence on $q$ or $t$. This, in particular, implies
$c_{j}=c_{k}=\frac{1}{2}$. The result agrees qualitatively with the second
order result from 43, confirming that the large-spin regime is well
approximated by it.
If we instead choose the region $C$ to be the upper right link, we get the
same type of result: There is a state of maximal entropy which makes the
induced map into an isometry, with $G(s)=\frac{1}{2s+1}$, and which is the
minimum of $\frac{Z_{1}}{Z_{0}}$ for fixed spin sectors. The parameters of the
minimum are, however, different:
$(a,d,w)\approx(\frac{1}{3},\frac{1}{3},\frac{1}{3})$. Here, $c_{j}=2c_{k}$.
This shows that whether an isometry exists or not can depend sensitively on
the region under consideration.

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Declaration of Academic Integrity

Hereby, I declare that I have written this thesis independently on my own and
without any other sources than the ones mentioned.

Simon Langenscheidt Munich, August 27, 2024