1. Introduction
The AdS/CFT correspondence is described as a quantum theory (more precisely, a
conformal field theory) lying on the boundary of an
anti-de Sitter spacetime geometry [
1]. Many particular features of this correspondence remain mysterious, in particular the link with quantum information theory and
entanglement. It was shown in [
2] that for a fixed time slice, the entanglement behavior of a given region of the boundary quantum theory is proportional to the minimal hypersurface bulk area homologous to the region of interest (known as the Ryu–Takayanagi entanglement entropy). In the context of AdS/CFT, the Ryu-Takayanagi formula demonstrates a crucial link between the entanglement behavior of an intrinsic quantum theory and its connection with the bulk gravitational field. These results open up a new perspective on the understanding of quantum gravity in the AdS/CFT framework from the viewpoint of entanglement and quantum information theory. For a complete introduction, see [
3] and references therein.
The difficulty of computing the entanglement properties of boundary quantum theories has led to the development of attractable simple models, particularly the
tensor network and
random tensor network frameworks. Initially, the tensor network framework started with “good” models approximating ground states in condensed matter physics. In the context of condensed matter physics, tensor networks represent ground states of a class of gapped Hamiltonians [
4]. Moreover, tensor networks have paved the way to understanding different physical properties such as the classification of topological phases of matter. An extensive review of all the different applications can be found in [
4]. Recently, other extensions of tensor networks to random tensor networks for studying random matrix product states or projected entangled pairs of states have been introduced in [
5,
6,
7]. The random tensor network (or simply RTN) was initiated in [
8] as a toy model reproducing the key properties of entanglement behavior in the AdS/CFT context [
9,
10,
11,
12,
13,
14,
15]. Moreover, the random tensor network framework has appeared in different active areas of research ranging from condensed matter physics to random quantum circuits and measurement frameworks [
16,
17,
18,
19,
20,
21,
22,
23,
24].
In general, a random tensor network (or simply RTN) is defined as a random quantum state via a given fixed graph structure, as will be described below. The main problem is to compute the
average entanglement between two complementary regions of the graph as
, where
D denotes the dimension of the Hilbert space of the model. Various results have been established, allowing the entanglement entropy of RTN models to be understood as toy models that mimic the entanglement behavior in quantum gravity. The scaling of the entanglement entropy (in the
limit) as a function of the size and
number of minimal cuts needed to separate the region of interest from the rest of the graph has been explored in different works [
8,
25]. Moreover, several directions have been explored that move beyond the toy model picture of (random) tensor networks [
26,
27].
The focus of this work is a
maximal flow approach to the analysis of general random tensor networks. The use of the maximal flow approach was already explored in [
10] to compute the entanglement negativity, in [
28] to derive the Ryu–Takayanagi entanglement entropy in the continuum setting, and in Appendix C of [
8] in a moment computation similar to the ones discussed in this work; see also [
9,
25]. As described in the previous paragraph, the model consists of defining a random quantum state from a given fixed graph structure. This model consider a graph with edges (bulk edges) and half-edges (boundary edges). The use of bulk and boundary edges will become clear from the definition of the model. A finite-dimensional Hilbert space
is associated with each half-edge and a tensor product Hilbert space
with each edge. The edge Hilbert spaces generate a local Hilbert space associated with each vertex of the graph. In order to define an RTN, each component of the graph is associated with a quantum state generated at random. For this, a random Gaussian state is generated for each vertex and a
maximally entangled state is associated with each of the edges. A
random tensor network is defined by projecting all of the maximally entangled states associated with the graph onto the total random states generated in each vertex. The obtained random tensor network lies in the full-boundary Hilbert space. The main goal of this work is to consider a sub-boundary region
A of the graph and evaluate the entanglement behavior of the associated residual state as
. The first computation is to estimate the moment computation of the state associated with the region
A as
. With the help of the
maximal flow, which is developed in detail in this work, the moments are estimated
without any assumption regarding cuts, and it is shown that they converge to the moment of a graph-dependent measure. It is shown that if the obtained
partial order is
series-parallel, the measure associated with the graph can be explicitly constructed
without any cut assumption through the use of
free probability theory. Moreover, the
existence of higher-order correction terms of the entanglement entropy (provided by a graph-dependent measure) is demonstrated, and these can be explicitly provided if the partial order is series-parallel. In different examples, it is shown how the measures associated with the initial graph can be computed explicitly in the case where the obtained partial order is series-parallel. The link between quantum information theory, free probability, and random tensor networks was already explored in [
25]. There, a general link state was used to represent the effect of the bulk matter field in the AdS/CFT context, extending the semiclassical regime with correction terms of the entanglement structure. Moreover, the obtained results in [
25]
assumed the existence of two disjoint minimal cuts separating region
A from the rest. In this work, only maximally entangled states in the bulk edge of the model are used; higher-order correction terms are obtained without any cut assumption, which may be interpreted as
intrinsic fluctuations. In the context of AdS/CFT, these are intrinsic to the quantum spacetime nature of bulk gravitational field without any bulk matter field.
The main differences between this work and the previous literature [
8,
25] are highlighted below. Through the use of the maximum flow approach, we emphasize the importance of the partial order defined by disjoint paths achieving the maximum flow to the computation of the correction term for the entropy of entanglement of the random tensor network states. Indeed, both the minimum cut and the maximum flow perspectives provide the dominating term (
). However, performing the analysis with the maximum flow allows for a combinatorial characterization of the moments of the reduced state. Moreover, in the case where the flow partial order is
series-parallel, an explicit formula for the correction term is provided in terms of a probability measure constructed from the network structure using both the free and classical multiplicative convolutions from (free) probability theory.
The rest of this work is organized as follows.
Section 2 provides a summary of the main results.
Section 3 introduces the random tensor network framework.
Section 4 presents the moment computation of a given state
associated with a given sub-boundary region of the graph
A.
Section 5 uses the maximal flow approach to compute the asymptotic scaling of the moments and demonstrates convergence to a graph-dependent measure.
Section 6 introduces the notion of series-parallel partial order and uses the free probability to explicitly show how a graph-dependent measure can be constructed with the free product convolution and classical measurement product.
Section 7 provides various examples of random tensor networks and explicitly shows the associated measure obtained in the case of an obtained partial order that is series-parallel.
Section 8 presents the main technical results, which use concentration inequalities to show that the obtained higher-order entanglement correction terms are graph-dependent without any cut assumption; moreover, the graph-dependent measure can be explicitly constructed if the partial order is series-parallel.
2. Main Results
This section introduces the main definitions and results obtained in this work. The focus is placed on the computation of the entanglement entropy of a given random tensor network. The most general framework of random tensor networks is considered, and the entanglement structure of the random tensor is studied with respect to a fixed bipartition of the total Hilbert space. By addressing the problem through a
network flow approach, the leading term can be computed along with the higher-order correction terms of the entanglement entropy, which are
graph-dependent. The higher-order correction terms are found to play a crucial role in various areas, particularly in the context of AdS/CFT [
8,
25,
27,
29], as will be discussed following the presentation of the main results. The key results of this work can be informally summarized as follows.
In the limit of large local Hilbert space dimension D, the average Rényi entanglement entropy of an RTN G across a given bipartition has the following:
In the case where is a series-parallel graph, the distribution of the entanglement spectrum (and consequently the finite entropy correction) can be computed as an iterative classical and free convolution of Marc̆henko–Pastur distributions.
A random tensor network is associated with a corresponding random quantum state
that encodes the structure of a graph
G. To describe this, the graph
G and relevant terminology are introduced below. Further technical details and definitions of the model are provided in
Section 3. Let
denote a connected undirected finite graph with (full) edges and half-edges (sometimes referred to as an
open graph); the former encode the internal entanglement structure of the quantum state
, while the latter represent the physical systems (Hilbert spaces) on which
is supported. The sets of edges (bulk edges) and half-edges (boundary edges) are denoted by
and
, respectively. Formally, the sets of edges and half-edges are provided by
and
, where
. The corresponding
random tensor is then defined as
where
are taken to be random Gaussian states defined in the local Hilbert space of each vertex
x. Moreover, a maximally entangled state
is associated with each (full) edge
, which serves to contract the internal degrees of freedom of the tensor network. A representation of a random tensor network is provided in
Figure 1, which is discussed in detail to illustrate the main results of this work. Further details are provided in Definition 3.
As mentioned earlier, the objective of this work is to evaluate the
entanglement entropy of the random quantum state
along a bipartition
of the boundary edges
. This analysis is carried out in the limit of the large local Hilbert space dimension
. To evaluate the entanglement entropy of the pure state
, the asymptotic
entanglement spectrum along the bipartition
is computed, that is, the limiting spectrum of the density matrix
. From this spectral information, the average Rényi entanglement and von Neumann entropies for the approximate normalized state
can be deduced, respectively provided by:
Above, the expectation is taken with respect to the Gaussian distribution of the independent random tensors
present at each vertex of the graph. The use of approximate normalized state instead of a correctly normalized state
will become clear from
Section 8.
First, the moments of the random matrix are computed exactly, then the main contributing terms at large dimensions are analyzed by relating the problem to a maximum flow question in a related graph. By employing the maximal flow and tools from free probability theory, the leading and fluctuating terms of the Rényi entropy are derived and the behavior of the von Neumann entanglement entropy is subsequently deduced.
Moment computation. First, the un-normalized state
is considered and its moments are computed. As a first step, the
graphical Wick formula from [
30] is used to obtain
where
can be understood as the Hamiltonian of a classical “spin system” in which each spin variable takes a value from the permutation group
:
In the above, the identity permutation
is associated with the region
B (corresponding to taking the partial trace over
B), while the full-cycle permutation
is associated with the region
A (corresponding to the trace of the
n-th power of
). A more precise statement and proof can be found in Proposition 1 in
Section 4. It should also be noted that the contribution of the normalization term of
is provided by
where
Note that in the above formula the quantity
is simply
with
; see Proposition 2 for more details. Note that in the particular case with
, the authors of [
8] provided an exact mapping to the partition function of a classical Ising model. Notice the frustrated boundary conditions of the Hamiltonian above; vertices connected to region
A prefer the configuration
, while vertices connected to region
B prefer the low energy state
. The connection between the combinatorics of the dominating terms in the random matrix model considered here and an effective spin system Hamiltonian have already been extensively studied in the quantum statistical mechanics literature; see, e.g., [
22,
31,
32].
Maximal flow. The
(max)-flow approach consists of identifying the leading terms from the moment formula above as
. For this, a
network is introduced. This network is derived from the original graph
G by connecting all of the half-edges in
A to an extra vertex
(sink) and all of the half-edges in
B to id (source). In
, the vertices are valued in the permutation group
, with all the half-edges connected either to the source id or to the sink
. The flow approach consists of looking at the different paths starting from the source id to the sink
. The different paths in the flow approach induce an ordering structure (more precisely a
poset structure) in the network
. Intuitively, the maximal flow will consist of searching the maximal number of these paths such that the source and sink will be disconnected upon removing the paths from the network. More precisely, by
Menger’s theorem, the maximum flow in this graph is equal to the number of edge-disjoint
augmenting paths that start from the source id and end in the sink
.
Figure 7 represents the different paths achieving maximal flow in the network
from the original graph
G as represented in
Figure 1. This procedure allows us to find a
lower bound to the Hamiltonian
that can be attained by some choice of the variables
.
Theorem 1. For all , one haswhere denotes the extended Hamiltonian in the network . After all augmenting paths achieving the maximum flow in have been removed, a clustered graph
remains, which is formed by clustering all the remaining connected components (see Figure 8). Importantly, the maximality of the flow implies that the cluster-vertices and in this clustered graph are disjoint. Further details and the proof of the above result are provided in Proposition 5. As a direct consequence of the above result, the moment convergence as can be deduced; further details of the following result are provided in Theorem 4.
Theorem 2. In the limit , for all one haswhere are the moments of a probability measure and . Moreover, we can show that the normalization term converges to 1, as shown in Corollary 1. The previous maximum flow computation provides the first order in the formula for the average entanglement entropy of random tensor network states:
Free probability theory and entanglement. The main contribution of this work is to show that the
second order (or the finite corrections) of the Rényi and von Neumann entanglement entropy can be found by carefully analyzing the set of augmenting paths achieving the maximum flow in the graph
. When the different paths achieve the maximal flow in the graph
, a
partial order is obtained after the clustering operation, where the vertices are the different permutation clusters formed from the clustered graph
.
Figure 9 shows the obtained partial order from the original graph
G in
Figure 1. The results are general, and become explicit in the setting where the partial order
is
series-parallel. With the help of
free probability theory, the second-order correction terms of each of the Rényi and von Neumann entropy can be deduced in this setting.
Definition 1. A graph G is called series-parallel if it can be constructed recursively using the following two operations:
Series concatenation: is obtained by identifying the sink of with the source of .
Parallel concatenation: obtained by identifying the sources and sinks of and .
Definition 2. A probability measure is associated with a series-parallel graph G, defined recursively as follows:
The Dirac mass at 1 is associated with the trivial graph , as follows: .
Series concatenation: . Here is the Marc̆henko–Pastur distribution and ⊠
is the free convolution product. For more details, see Appendix A. Parallel concatenation:
Theorem 3. In the limit , the average Rényi entanglement entropy and von Neumann entropy of an approximate normalized state respectively behave as follows: For further details and the proof of the above statements, the reader is referred to Corollary 2. In particular, if the obtained partial order
is series-parallel, then the measure
can be explicitly constructed, as detailed in Theorem 5. The use of the approximate normalized state rather than the normalized state
is justified by the concentration result for
presented in
Section 8.1.
It was previously argued in [
8,
27] that if one wants to encode the
quantum fluctuations, then instead of a maximally entangled state it is necessary to use a general “link state”
defined by
It was recently shown in [
25] that the quantum fluctuations beyond the semiclassical regime in AdS/CFT are obtained for the non-flat spectra of the link state under the existence
assumption of
two minimal cuts. The use of a generic link state in the context of AdS/CFT represents the bulk matter field contribution.In this work, the
maximal flow approach has been used to demonstrate the existence of quantum fluctuations without any
minimal cut assumption and with the
maximally entangled state serving as the link state. The higher-order correction terms obtained in this context can be interpreted as the “intrinsic” quantum fluctuations of spacetime geometry in the absence of any bulk matter field represented by a general link state.
For example, in the case of the graph represented in
Figure 1, the resulting partial order
is series-parallel (see
Figure 9), where
As represented in
Figure 10, the graphs
and
are trivial; hence,
. The graph
can be factored as a parallel composition of two other graphs, as represented in
Figure 11:
The graph
, as represented in
Figure 12, factorizes as follows:
where we use the fact that
and
are series compositions of two trivial graphs; thus,
, while
.
Moreover, the graph
, as represented in
Figure 13, factorizes as follows:
with the associated measure
where the series composition for
and
has been used iteratively, with their respective measures provided by
and
. In the case of random tensor network represented in
Figure 1, the partial order is series-parallel, with the associated measure
with
which is obtained by combining all the results stated above. The minimal cuts associated with the network
(see
Figure 7) can also be considered, as represented in
Figure 14. It should be noted that several minimal cuts exist; for simplicity, four ways of achieving the minimal cut in the network are represented. Note that some of the cuts share common edges.
4. Moment Computation
Given a random tensor network, we next investigate the behavior of entanglement between a specified subregion of the network and the remainder. To this end, we first address the moment computation of the quantum state for a subregion . This initial computation will subsequently enable our analysis of the Rényi and von Neumann entropies in the following sections.
Let be a sub-boundary region of the graph G. The complementary region of A is denoted by . The Hilbert spaces associated with the boundary regions A and B are denoted by and , respectively.
In this work, we computed the
average entanglement entropy at large bond dimensions:
where
is the normalized quantum state obtained by tracing out the region
B, i.e.,
, with the partial trace over the Hilbert space
. In the above expression, the average is taken over all of the random Gaussian tensors at the vertices.
The first computation to be addressed here is the moment computation, as described in the following proposition. This computation will subsequently enable our calculation of the average entanglement entropy (Rényi and von Neumann entropies) in the limit
, as analyzed in detail in the following sections. The result has previously been obtained in a very similar setting by Hastings [
33] (Theorem 3,
ensemble); see also
Section 5 of [
8] for a closely related derivation.
Proposition 1. For any , it holds thatwhere can be understood as the Hamiltonian of a classical “spin system” in which each spin variable takes a value from the permutation group : Before providing the proof of the above proposition, we recall some properties of the permutation group
and fixed the relevant notation. Here, the symbol
is used to denote the
total cycle in the permutation group
evaluated at
:
Recall that a notion of distance in
, known as the
Cayley distance, can be defined by
where
stands for the number of cycles in
. The distance
provides the minimum number of transpositions required to turn
into
. In general, the distance in
satisfies the triangle inequality, where
In particular,
is said to be a
geodesic between
and
in
if
. The following notation for the distance is adopted instead of
, where
Proof. To prove the result announced in the proposition, it should first be remarked that we can use the well-known replica trick to write the trace on the left-hand side of Equation (
9) as follows:
The trace on the left-hand side (on
) may be rewritten as a full trace on
n copies of the complete (bulk and boundary) Hilbert space, as shown on the right-hand side of the equation above. The notation
is used to denote the tensor product of unitary representations of the permutation
for each half-edge
By expanding and taking the average over random Gaussian tensors, we obtain
where the shorthand notation
in the last equation above is used in place of
We recall the following property of random Gaussian states (see [
34]):
where
is the unitary representation of
. Each permutation
acts on each vertex in Hilbert space; hence, each acts implicitly on each half-edge incident to the vertex
. Thus, the moment formula becomes
where the number of loops obtained by contracting the maximally entangled states (edges) upon taking the trace is counted by the formula above. The factor
arises from the normalization associated with the contraction of the bulk edges (see Equation (
2), where each edge contributes a factor of
). By employing the relation between the Cayley distance and the number of loops, the result stated in the proposition is obtained. □
Graphically, the formula can be interpreted using
Figure 4, where the case with
is illustrated. By employing the graphical integration technique for Wick integrals as presented in [
30,
35], loops are obtained. Consequently, Cayley distances of three types arise: (a) between
and elements directly connected to it, originating from the region
B; (b) between
and elements directly connected to it, originating from the region
A; and (c) between elements neither directly connected to id nor
, originating from the bulk. Based on this, the Hamiltonian can be rewritten in terms of Cayley distances, as follows:
where
represents the half-edges in
A,
, represents the half-edges in
B, and
represents the edges in the bulk of the tensor network.
In the above proposition, only the numerator term of the normalized quantum state
have been addressed. However, to compute the von Neumann and Rényi entropies (see Equations (
4) and (
5)), the state must be normalized and the moment computed.
The following proposition provides the moment computation of the normalization term in .
Proposition 2. For any , it is found thatwhere the Hamiltonian is provided by Proof. The proof of this proposition follows directly from Proposition 1 by taking such that is obtained as the special case of with . □
5. Asymptotic Behavior of Moments
In this section, the leading contributing terms of the moment as are described using the (maximal) flow approach. First, the (maximal) flow approach is introduced, which allows for the estimation of the leading terms of the moments as ; further details can be found in Proposition 5. Based on this result, the convergence of the moment as to the moments of a graph-dependent measure can be deduced, as detailed in Theorem 4.
First, the results from the previous section are recalled. In Proposition 1, it was shown that the moments are provided by
where the spin-valued Hamiltonian in the permutation group
is provided by
In particular, the contribution of the normalization term in
(see Equation (
8)) is the extended Hamiltonian
, as shown in Proposition 2 when one takes
in
.
The main goal of this section is to analyze the main contributing terms of the moment as
. The leading terms will consist of solving the following minimization problem:
In particular,
is minimized, yielding the leading contributing term as
for the normalization term of
. The minimization problem described above enables deduction of the moment convergence as
to the moment of the graph-dependent measure
in Theorem 4.
The minimization problem above is addressed using the (maximal)-flow approach. This approach consists of first constructing a network from the original graph G. This network is constructed by first adding two extra vertices and id to G in such a way that all the half-edges associated with A are connected to the total cycle and that half-edges in B are connected to id. The network has the same bulk structure of G, with the difference that all the vertices in are valued in the permutation group .
The flow approach is based on the identification of different augmenting paths in the network , each starting from id and ending at . These various paths induce an order structure in . By removing all augmenting paths in , a lower bound for the extended Hamiltonian can be established; further details are provided in Proposition 3. Furthermore, it is demonstrated that the minimum is attained when the maximal flow from id to is achieved, as discussed in Proposition 5. In particular, it is shown that the minimum of the extended Hamiltonian is zero; see Proposition 6 for additional details.
Before the flow approach is introduced, it should be noted that the contributing terms of the moments at large dimensions were previously analyzed in [
25] using the
(minimal) cut approach. In that work, the existence of two
disjoint minimal cuts in the graph (separating the region of interest from the remainder of the graph) was assumed to determine the contributions at large bond dimensions. In contrast, in the maximal flow approach presented here, no assumption regarding the existence of (minimal) cuts is made. By identifying various augmenting paths that achieve the maximal flow and applying the well-known maximal-flow minimal-cut theorem (see, e.g., [
36], Theorem 8.6), the different minimal cuts can be deduced without any prior assumption.
Definition 4. Let the network be defined from the initial graph such thatwhere the regions and is defined as follows:where and respectively denote all of the vertices associated with the boundary region A and B. Moreover the vertices are valued in the permutation group , where It should be remarked that in the above definition, the graph is constructed in such a way that all of the half-edges are connected to and that the half-edges are connected to id. Note also that there are no half-edges in ; the bulk region in the network remains the same as the one in graph G.
The extended Hamiltonian
of
in the network
is provided by
where each term in the new Hamiltonian is valued in the network
. Moreover, the sums in the above formula are over the vertices
,
, and
, which are the vertices with the respective half-edges in the region
A,
B, and
.
As mentioned earlier, the flow approach consists of analyzing different paths that start from id and end in . This induces a natural orientation of the network (more precisely, a poset structure). In the following, the sets of different (edge-disjoint) paths in is defined.
Definition 5. Let be the set of all possible paths from the source to the sink in , where the source and sink are id
and γ, respectively. Formally, the set of paths is defined as where are all the paths connecting id
to γ.
Definition 6. Let be the set of all families consisting of edge-disjoint paths
in : Remark 1. It is clear from this definition that .
The process of searching for different paths that start from id and end at induces an ordering, more precisely a poset structure, in the network . In the following, a definition of a poset structure is provided which will later be utilized in the maximal flow approach to minimize .
Definition 7. The poset structure is a homogeneous relation denoted by ≤ and satisfying the following conditions for all :
Reflexivity: ;
Antisymmetry: and , implying ;
Transitivity: and , implying .
Definition 8. The natural ordering
is defined as for a path provided by Another useful notion in (maximal) flow analysis is the permutation cluster. A permutation cluster of a given permutation is defined as all of the edge-connected permutations to .
Definition 9. A permutation cluster is defined as all the edge-connected permutations to .
Remark 2. With the poset structure in , a naturally-induced ordering is obtained in the cluster structures for each permutation connected to the permutation elements , where all the properties of the above definition can be extended to the cluster of a given permutation . More precisely, the following hold: Definition 10. The max flow in is the maximum of all of the edge-disjoint paths in : The following proposition provides a lower bound of the extended Hamiltonian
that is saturated when the maximal flow in
is achieved, as shown in Proposition 5. Similar ideas have been used by Hastings in Lemma 4 of [
33] to lower-bound the moments of a random tensor network map.
Proposition 3. Let and be an arbitrary set of edge-disjoint paths in and set ; then, the following inequality holds:where , defined as follows: It should be mentioned that the sums in the above proposition are over for , which make up the set of the different boundary and bulk regions when removing all the different edges and vertices that will contribute in different paths in .
Proof. Consider a set of edge disjoint paths
. We can fix a path
for a given
where
is a path that starts from id, explores
vertices, and ends in
. Using Equation (
14) with the path defined above, we obtain
where we have used the triangle inequality for the Cayley distance together with the fact that
. The Hamiltonian
is the contribution when the path
from
is used.
By iteration over all the edge-disjoint paths , the desired result is obtained. The second inequality is obtained by observing that , which ends the proof of this proposition. □
Proposition 4. Given a graph G, there exist a tuple of permutations α such that .
Proof. By the celebrated max-flow min-cut theorem, the maximum flow in the network is equal to its minimal cut. Recall that a
cut of a network is a partition of its set of vertices into two subsets
and
, with the size of the cut being the number of
edges. In the current setting, the max-flow min-cut theorem (see, e.g., [
36], Theorem 8.6) implies that there exists a partition of the vertex set
of
(see Definition 4) into two subsets
with
and
, such that
For
, we define
Because
and
, we have
where in the last claim we have used the fact that there are no edges between id and
in
; see
Figure 5. □
Proposition 5. For all , it holds that Proof. This proof follows from the two previous propositions. □
When all the augmenting paths in the network achieving the maximal flow have been identified and removed, a clustered graph is obtained by identifying different remaining connected permutations.
The following example provides an illustration of the different steps described above to analyze the maximum flow problem in the case of the tensor network represented in
Figure 1.
Example 2. Figure 7 represents the network associated with the random tensor network from Figure 1. The vertices in the network are valued in the permutation group . The network is constructed by adding two extra vertices γ and id
by connecting all of the half-edges in A to γ and the half-edges in B to id
. The flow approach induces a flow from id
to γ; here, the maximum flow in Figure 7 is 4, where the augmenting paths achieving it are shown in color. By removing the four edge-disjoint augmenting paths, we obtain the clustered graph in Figure 8 by identifying the remaining connected edges as a single permutation cluster (i.e., id
with ) to form the cluster . Theorem 4. In the limit , for all we have the following:where is the number of permutations achieving the minimum of the network Hamiltonian . These numbers are the moments of a probability measure which is compactly supported on . Proof. For fixed
n, the convergence to
(the number of minimizers of the Hamiltonian
) follows from Proposition 1 and Proposition 5. The claim that the numbers
are the moments of a compactly supported probability measure follows basically from Prokhorov’s theorem ([
37],
Section 5; see also [
33], Footnote 2). Indeed, we note that at fixed
D, the quantity
is the
n-th moment of the empirical eigenvalue distribution of the random matrix
, restricted to a subspace of dimension
containing its support (this follows from the fact that
is an upper bound on the rank of
; see Equation (
6)). These measures have finite second moments, meaning that the sequence (index by
D) is tight. The limiting moments satisfy Carleman’s condition, since
, proving that the limit measure
has compact support; recall that
is the
n-th
Catalan number (see
Appendix A). Because the matrix
is positive semi-definite,
must be supported on
. □
Remark 3. The obtained moments are provided by a graph-dependent measure. It will be shown in the following sections that such measures can be explicitly constructed if the partial order is series-parallel; see Section 6 and Theorem 5 for more details. In the above, the contribution terms at a large bond dimension of are those that minimize . As shown in Proposition 5, the quantity is minimized when the maximal flow is attained in .
For later purposes, if the moment of
is to be analyzed, then the contribution of the normalization term of
at large bond dimension should also be considered. Recall from Proposition 2 that the contribution of the normalization term is provided by
where
At a large dimension
, the contributed terms are provided by the one that will minimize the extended Hamiltonian
in
:
where the first is over all the vertices
with boundary edges and where
are the bulk vertices.
Proposition 6. Let be the extended Hamiltonian in ; then, for all we havewhich is achieved by identifying all the permutations with id
. Proof. To minimize the Hamiltonian , the same approach is followed; all half-edges A are connected to and all half-edges B to id. However, in all of the boundary terms are connected to id, meaning that no path starts from id and ends in . Per the bulk connectivity of G, the minimum is achieved by identifying all of the permutations with id; thus, the result follows from Proposition 5. □
Remark 4. In Proposition 6, the Hamiltonian is obtained by taking in
(see Equation (
10)
). It should be mentioned that if , then the same form of the Hamiltonian is obtained, where instead of all the half-edges being connected to id
, they are all connected to γ. Therefore, it can be deduced that there are no paths that start from id and end at γ; hence, the minimum is 0, which is achieved by identifying all the permutations with γ. Corollary 1. For any moments of the normalization term, it converges to 1
; more precisely, Proof. By taking the average, as shown in Proposition 2, we obtain the Hamiltonian . Per the maximal flow, the Hamiltonian is minimized by identifying all of the permutations to id; therefore, we have , as shown in Proposition 6. By removing all of the augmenting paths that achieve maximal flow, the obtained residual graph is trivial, with only two disjoint vertices and the identity cluster . Hence, by Theorem 4 we obtain the desired result. □