# Remote detectability from entanglement bootstrap I: Kirby’s torus trick

Bowen Shi, Jin-Long Huang, John McGreevy

###### Abstract

Remote detectability is often taken as a physical assumption in the study of
topologically ordered systems, and it is a central axiom of mathematical
frameworks of topological quantum field theories. We show under the
entanglement bootstrap approach that remote detectability is a necessary
property; that is, we derive it as a theorem. Starting from a single wave
function on a topologically-trivial region satisfying the entanglement
bootstrap axioms, we can construct states on closed manifolds. The crucial
technique is to immerse the punctured manifold into the topologically trivial
region and then heal the puncture. This is analogous to Kirby’s torus trick.
We then analyze a special class of such manifolds, which we call _pairing
manifolds_. For each pairing manifold, which pairs two classes of excitations,
we identify an analog of the topological $S$-matrix. This pairing matrix is
unitary, which implies remote detectability between two classes of
excitations. These matrices are in general not associated with the mapping
class group of the manifold. As a by-product, we can count excitation types
(e.g., _graph excitations_ in 3+1d). The pairing phenomenon occurs in many
physical contexts, including systems in different dimensions, with or without
gapped boundaries. We provide a variety of examples to illustrate its scope.

Department of Physics, University of California at San Diego, La Jolla, CA
92093, USA

###### Contents

  1. 1 Introduction
    1. 1.1 Reader’s guide
  2. 2 Central pillars of entanglement bootstrap
    1. 2.1 Axioms and existing tools
    2. 2.2 Immersed regions
    3. 2.3 Constrained fusion space
  3. 3 Regions dual to excitations
    1. 3.1 2d bulk and 3d bulk regions
    2. 3.2 Systems with gapped boundaries in 2d and 3d
  4. 4 Closed manifolds completed from immersed regions
    1. 4.1 Vacuum block completion
    2. 4.2 Examples of vacuum block completion
    3. 4.3 Other examples of completion in ${\cal N}$
  5. 5 Pairing manifolds
    1. 5.1 Motivating questions and Kirby’s torus trick
    2. 5.2 Pairing manifolds, the definition
    3. 5.3 Consequences of pairing
  6. 6 Examples of pairing manifolds
    1. 6.1 Pairing manifold examples in the bulk
    2. 6.2 Pairing manifold examples with gapped boundaries
    3. 6.3 Natural partitions used in examples
    4. 6.4 Non-examples of pairing manifolds
  7. 7 Pairing matrix and remote detectability
  8. 8 Discussion
  9. A Notation glossary
  10. B Consistency Relations with Quantum Dimensions
  11. C Information convex set for handlebodies
    1. C.1 Relations between quantum dimensions and fusion probabilities of graph excitations
    2. C.2 Quantum double examples
  12. D Quantum double illustration of pairing manifold relations

##  1 Introduction

Topological quantum field theory (TQFT) [1, 2, 3] is a machine that eats
arbitrary manifolds and produces invariants. The entanglement bootstrap [4, 5,
6] begins with a single state on a topologically-trivial region, e.g., a ball
or a sphere. How can it incorporate the data on nontrivial closed manifolds?
In this work, we provide a method to construct quantum states on various
manifolds from the reference state on a ball. With this technique at hand, we
further give a general proof of the remote detectability for topologically
ordered systems.

Remote detectability [7, 8] is the statement that, in topologically ordered
systems, each nontrivial topological excitation (which can be a particle or a
loop or something else) must be detectable by a remote process involving
another (possibly different) class of excitations. It is a broad phenomenon
that is believed to occur in many physical setups, including topologically
ordered systems in arbitrary dimensions ($d\geq 2$) [7, 9, 10, 11] and systems
with gapped boundaries [12, 13, 14]. Remote detectability is a vast
generalization of the braiding non-degeneracy of anyons; see Ref. [15]
Appendix E. In these previous approaches, remote detectability was an axiom
(or principle). In entanglement bootstrap, we shall derive this property as a
theorem.

The main idea in our derivation of remote detectability is as follows. We
start with a reference state $\sigma$ on a topologically trivial region, which
can be either a ball $\mathbf{B}^{n}$ or a sphere $\mathbf{S}^{n}$.111We use
boldface letters $\mathbf{B}^{n}$ and $\mathbf{S}^{n}$ (instead of the more
standard topological notation $B^{n}$ and $S^{n}$) to indicate that these
regions are those on which we define the reference state. We immerse (i.e.,
locally embed) a closed manifold ${\cal M}$ of interest into the ball or
sphere upon removing a ball from it, that is, we consider ${\cal W}={\cal
M}\setminus B^{n}$ and an immersion ${\cal W}\looparrowright\mathbf{B}^{n}$ or
${\cal W}\looparrowright\mathbf{S}^{n}$. Then with a trick to heal the
puncture, we construct a state on the manifold ${\cal M}$. We identify a
specific class of manifolds that is important for the study of remote
detectability; we refer to them as _pairing manifolds_. Each pairing manifold
(${\cal M}=X\bar{X}=Y\bar{Y}$222Here $\bar{X}$ is the complement of $X$ in
${\cal M}$, that is $\bar{X}={\cal M}\setminus X$.) pairs two excitation
types, namely the excitation types characterized by the information convex
sets of regions $X$ and $Y$ respectively. For each pairing manifold, we
identify a “pairing matrix”, which is a finite-dimensional unitary matrix
analogous to the topological $S$ matrix in the anyon theory. The remote
detectability of the excitations follows from the unitarity of the pairing
matrix. (See Ref. [16] for an entanglement bootstrap study of the topological
$S$ matrix in the anyon theory, which contains a trick we adopt.)

Our method can be thought of as a quantum analog of Kirby’s torus trick [17]:
it uses immersion and pulls back structures from a topologically trivial
region to closed manifolds (e.g., the torus). Then, because the structure on
the closed manifold (i.e., the stability in Kirby’s case) is better
understood, insight into the topologically trivial region (i.e.,
$\mathbb{R}^{n}$ in Kirby’s case) is gained by the consistency. In our method,
we pull back quantum states to punctured manifolds and then heal the puncture.
The structures, i.e., universal data, associated with the quantum state on the
topologically trivial region understood by our method include the pairing
matrix, best defined on the pairing manifolds, as well as various consistency
relations derived by making use of the pairing manifolds. Intriguingly,
Hastings considered an application of Kirby’s torus trick for gapped
invertible phases [18], where the local Hamiltonian is pulled back. In
comparison, our approach works for gapped invertible phases as well as
intrinsic topological orders, and we only make use of the quantum states. One
innovation is the use of _building blocks_ , which carry the instruction for
picking the right state on the punctured manifold. Such states allow a
“smooth” healing of the puncture.

Why are we interested in the entanglement bootstrap approach to topological
orders and TQFT? The entanglement bootstrap is an independent theoretical
framework: it requires an input reference state that satisfies two axioms on
bounded-sized regions (see Eq. (2.1)). From there, various rules are derived
as information-theoretic consistency relations of the quantum theory. In the
process of deriving such consistency relations, mathematical objects are
identified that capture the universal data of the gapped phase. These data are
encoded in the reference state, which is physically the ground state. Below
are some additional perspectives:

  1. 1.

TQFT and its categorical description, while standing as the best-known
candidate of the underlying mathematical theory for topological field theory,
are not without ambiguities. In mathematics, it is possible to add various
adjectives to TQFTs and tensor category theories. Therefore, there has always
been the question of which adjective is the right one for a given physical
setup. Furthermore, the study of TQFT and topological orders in higher
dimensions ($d\geq 3$) (see e.g. [7, 8, 11, 19]) is an ongoing research
direction.

  2. 2.

The entanglement bootstrap is a bridge between the program of classifying
gapped quantum phases and the classification of quantum states. This is
because any data we identify (such as the pairing matrix), becomes a label of
the reference state which satisfies an entanglement area law captured by the
axioms. In this way, entanglement bootstrap puts labels on quantum states
satisfying the axioms and thus classifies them.

  3. 3.

One expects to discover new connections between the ground states and
universal properties. For instance, in this work, we identify excitations that
have not been studied, e.g., the graph excitations in 3d topological order.
They are excitations occupying a thin handlebody. Handlebodies in 3d are
classified by genus, and for each genus, we identify a class of “graph
excitations”. The remote detectability for this new class of graph excitation
is one of the many examples we give. As a byproduct, we provide a counting of
the number of graph excitations for an arbitrary genus. As we shall explain,
the counting manifests the fact that the coherence in the fusion space in the
particle cluster is needed in detecting the graph excitations.

Recently, some of the tools developed in entanglement bootstrap have been
useful guidelines for finding new topological invariants calculable from a
ground state wave function. (See [20] for a proposed formula for the chiral
central charge, in terms of the modular commutator. See [21] for a proposed
formula for the Hall conductance, in the context of which the reference state
is symmetric under a $U(1)$ onsite symmetry.) These studies suggest that the
ground state wave function on a topologically-trivial region contains data
beyond those captured by a tensor category. One may wonder if this line of
ideas generalizes into higher dimensions.

  4. 4.

Lastly, entanglement bootstrap, rooted in earlier related works, suggests a
new philosophical relation between the universal data and the ground state(s).
It suggests, in a concrete manner, that many (possibly all) data of the gapped
phase of matter (or TQFT) can be extracted from a topologically trivial patch
of the ground state wave function. In other words, the universal data is
encoded in the seed or its “DNA” (reference state on a patch larger than the
correlation length) of the “plant” (topologically ordered system). Our
approach is to first grow a plant from the seed and then study the morphology
of the plant; therefore, we do not need to start with the whole plant. See
Fig. 1 for an illustration.

Figure 1: Illustrated is an analog to the growth of a plant from a seed.

The pairing matrices we identify are analogs of the topological $S$ matrix in
the anyon theory. A surprise is that, while each of these pairing matrices is
associated with a closed manifold, the matrix is not associated with its
mapping class group (MCG), unless the two classes of excitations characterized
by $X$ and $Y$ are identical. This distinguishes them from a previous
generalization of the topological $S$ matrix in 3d, associated with the 3-tori
[22, 10]. Some of the matrices considered in [10, 23, 24] are examples of
pairing matrices. Pairing matrices fall into three kinds, depending on whether
$X$ and $Y$ encode a nontrivial fusion space; see Table 1 for the examples we
consider.

###  1.1 Reader’s guide

This section is a reader’s guide. It includes a figure that summarizes the key
concept developed (or reviewed) in each section and the relations between the
sections (see Fig. 2), and a table that summarizes the examples of pairing
manifolds (see Table 1). Below are some more details.

####  1.1.1 Content of the sections

§2: Backgrounds & Toolsaxioms, information convex setsstructures, vacuum,
associativityimmersion§3: Regions $\leftrightarrow$ Excitationsbulk and gapped
boundaries:§4 Manifolds completed from immersed regionscompletion $({\cal
M},{\cal W}\looparrowright\mathbf{S}^{n}\textrm{ or
}\underline{\underline{\mathbf{B}}}^{n})$building blocks $\\{{\cal V},{\cal
V}^{\Diamond},{\cal V}^{\triangle}\\}$vacuum block completion
$(\mathcal{M},\\{{\cal V}_{i}\\},{\cal W}\looparrowright\mathbf{S}^{n}\textrm{
or }\underline{\underline{\mathbf{B}}}^{n})$completion trick“torus trick”§5
and §6 Pairing manifolds & examplespairing:
$(\mathcal{M},\\{X,\bar{X}\\},\\{Y,\bar{Y}\\},{\cal
W}\looparrowright\mathbf{S}^{n}\textrm{ or
}\underline{\underline{\mathbf{B}}}^{n})$consequences of
pairing:$\sum_{a\in{\mathcal{C}}_{\partial
X}}N_{a}(X)^{2}=\sum_{\alpha\in{\mathcal{C}}_{\partial
Y}}N_{\alpha}(Y)^{2}=\dim\mathbb{V}({\cal M})$capacity matching:
$\mathcal{Q}(X)=\mathcal{Q}(Y)$, TEE $=\ln\mathcal{Q}$.§7 Pairing
matrices$S_{(a,i,j)(\alpha,I,J)}=\langle
a_{X}^{(i,j)}|\alpha_{Y}^{(I,J)}\rangle/\sim$three types depending on $X$, $Y$
Figure 2: A summary of the content of the main text. The focus is on the
concepts developed and the relations between the sections.

In §2, we collect useful tools of entanglement bootstrap, that are developed
in previous works. We start by reviewing the axioms in general dimensions. The
structure theorems of the information convex sets enable us to talk about
superselection sectors and fusion spaces. The merging theorem allows us to
glue regions on either an entire entanglement boundary or part of the
entanglement boundary. The associativity theorem tells us how the dimensions
of the fusion spaces match upon gluing an entire entanglement boundary. We
introduce the idea of constrained information convex set, which allows a nice
reformulation of the associativity theorem. We recall the definition of
quantum dimensions and the properties of the vacuum. Finally, we recall the
concept of immersed regions333As is explained in Ref. [6], the idea is
essentially the same as topological immersion: a continuous map from a
topological manifold where every point in the source has a neighborhood on
which the map restricts to an embedding. We only need immersion maps between
manifolds of the same dimension, and in this case, the immersion in question
is also a submersion., which will play an important role in this work.

In §3, we explain the duality between excitations and regions. In the bulk, we
shall consider excitations (which can be one excitation or a cluster of
excitations) that live on a sphere $\mathbf{S}^{n}$. The region dual to the
excitations is homeomorphic to the subsystem of a sphere that occupies the
complement of the excitations. We also discuss the immersed version of such
regions. These regions will be used as “building blocks” for constructing
closed manifolds. We further discuss similar dualities in the physical context
of gapped boundaries. We also discuss excitation types that are new to our
knowledge. For instance, in the 3d bulk, we identify _graph excitations_ which
are located in “graphs”.444An accurate statement is the excitations are
supported on thin handlebodies. Handlebodies in 3d are solid genus-$g$
surfaces. They are more general than the familiar loop excitations. The
regions that detect them are also handlebodies. These graph excitations are
classified into subclasses labeled by the genus.

In §4, we discuss the main conceptual progress of this work: that is the idea
of making closed manifolds ${\cal M}$ from immersed regions ${\cal
W}\looparrowright\mathbf{S}^{n}$, where ${\cal W}={\cal M}\setminus B^{n}$.
The concepts and techniques to achieve this goal include completion
(Definition 4.1), completion trick (Lemma 4.4), building blocks (Definition
4.5), and vacuum block completion (Definition 4.9). Intuitively, if a closed
manifold ${\cal M}$ allows a completion, then there exist quantum states on
${\cal M}$, which are assembled from the reduced density matrices of the
reference state on small patches. The completion trick is a general trick to
achieve that. Vacuum block completion is a special type of completion; it
carries an instruction to build the closed manifold from a set of building
blocks. Each manifold so constructed has a canonical state with respect to the
choice of building blocks. This canonical state is obtained by assembling the
“vacuum” states of these building blocks. Importantly, the canonical state
satisfies the entanglement bootstrap axioms on all balls contained in ${\cal
M}$. We provide many examples to illustrate these concepts and techniques.

In §5, we introduce the concept of _pairing manifold_ (Definition 5.3). It is
a special class of manifolds that allows _a pair_ of vacuum block completions,
as

${\cal M}=X\bar{X}=Y\bar{Y},$ (1.1)

in addition to a couple of extra conditions. The most important condition is
that $X$ and $Y$ cut each other into balls and thus hide information from each
other completely. From the definition, we prove a set of properties of pairing
manifolds. For instance, the Hilbert space dimension on ${\cal M}$ is
determined by the information convex set of either $X$ or $Y$. In short, this
is a manifestation that a pairing manifold is a machine that pairs two classes
of excitations.555When either $X$ or $Y$ is not sectorizable, it is useful to
think of these as two clusters of excitations. One class is that detected by
the region $X$ and another is that detected by the region $Y$.

In §6, we provide many examples of pairing manifolds in various spatial
dimensions and systems with or without gapped boundaries. See Table 1 for the
examples in 2, 3, and 4 spatial dimensions. As a by-product of this analysis,
we count the total number of graph excitations for any given genus. The result
is expressed in terms of the fusion multiplicity of point particles. We show
that some manifolds can be a pairing manifold in multiple ways.

In §7, we introduce the pairing matrix ($S$), a generalization of the well-
known topological $S$-matrix. This is a different generalization that has been
considered in [22], and the pairing matrices are not necessarily associated
with the mapping class group. A pairing matrix generates a unitary
transformation between two bases specified by the two cuts, and it is a
unitary matrix. The pairing matrices fall into three types, depending on
whether a fusion space needs to be involved in the remote detection process;
see Table 1 for some examples. A pairing matrix of the first type can be
thought of as the braiding matrix between two classes of excitations; these
excitations detect each other without making use of fusion spaces, and the
number of excitations in each class must be identical. A pairing matrix of the
second type provides an example of remote detection involving nontrivial
fusion spaces. The physical picture to keep in mind is that a cluster of
coherently-created excitations detects the excitations in another class. A
pairing matrix of the third type generically requires the fusion spaces of
both excitation clusters (one detected by $X$ and another detected by $Y$) to
participate in the remote detection process.

§8 discusses open questions. In Appendix A, we summarize the notations and
provide a glossary. In Appendix B, we prove a set of consistency conditions on
quantum dimensions and fusion rules, generalizing those found in [6]. This is
used in Appendix C, which gives an exposition of graph excitations, and
exemplifies them using quantum double models. Appendix D illustrates some of
the consequences of the fact that $(S^{2}\times S^{1})\\#(S^{2}\times S^{1})$
is a pairing manifold in two ways, in the family of 3d quantum double models.

####  1.1.2 Why Kirby’s torus trick?

Finally, we explain the somewhat mysterious statement that our method is an
analog of Kirby’s torus trick. The first question is _what is the torus
trick?_ The general idea of using such an immersion to pull back structure
from $\mathbb{R}^{n}$ to another topological manifold is sometimes called the
Kirby’s torus trick [17]. In Kirby’s work, this idea is used to pull back
smooth or piecewise-linear structures. In [18] this idea was used to pull back
local Hamiltonians for invertible phases. Hastings [18] and Freedman [25, 26]
also used this idea to pull back Quantum Cellular Automata.

We pull back the information about the universal property of the gapped phases
encoded in the quantum (ground) states, and therefore, it can be thought of as
a quantum analog of Kirby’s torus trick [17]. We use immersion to pull back
structures from a topologically trivial region to closed manifolds, e.g., the
torus. More specifically, we first construct quantum states on immersed
punctured manifolds and then heal the puncture. The structures associated with
the topologically trivial region understood this way include the pairing
matrix, best defined on the pairing manifolds, as well as various consistency
relations derived by making use of the pairing manifolds.

The second question is: _which sections of this work contain the analog of the
torus trick?_ An answer is indicated in Fig. 2. It says that the trick is most
relevant to §4 and §5. The reason is as follows. Section 4 solves the problem
of how to pull back quantum states from a ball (or a sphere) to a closed
manifold. Section 5 identifies a class of closed, connected manifolds that can
provide insights into some mathematical structures, e.g., consistency
relations and the braiding nondegeneracy. (The discussion of pairing matrices
in Section 7 is a continuation of this analysis.)

That a canonical state on a closed manifold can be constructed with the
additional instruction carried by the building blocks is a new feature. This
enables us to construct a state on the punctured manifold ${\cal W}$, for
which a “smooth” healing of puncture is possible even for intrinsic
topological orders (i.e., non-invertible phases). This provides one way to
overcome a difficulty about intrinsic topological orders, emphasized by
Hastings [18] (we note also that we are dealing with quantum states rather
than Hamiltonians).

context | pairing manifold, $\mathcal{M}$ | $X$ | $Y$ |  excitation detected by $X$ |  excitation detected by $Y$
---|---|---|---|---|---
2d bulk | $T^{2}$ | annulus | annulus | particle pair | particle pair
2d bulk | genus-$g>1$ Riemann surface |  $g$-hole disk |  $g$-hole disk |  $g+1$-particle cluster |  $g+1$-particle cluster
2d gapped boundary |  cylinder with two identical gapped boundaries |  half-annulus | boundary annulus  |  boundary particle pair |  bulk anyon condensed on boundary
3d bulk | $S^{2}\times S^{1}$ | solid torus | sphere shell | loop | particle pair
3d bulk | $\underbrace{(S^{2}\times S^{1})\\#\cdots\\#(S^{2}\times S^{1})}_{g\text{ times},~{}g>1}$ |  secret
genus-$g$
handlebody
| $\underbrace{\text{\color[rgb]{0.90234375,0.37890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.90234375,0.37890625,0}{ball minus $g$ balls}}}_{\equiv{\cal B}_{g}}$ |  genus-$g$ graph excitation  |  $g+1$-particle cluster
3d bulk | $(S^{2}\times S^{1})\\#(S^{2}\times S^{1})$ | ball minus torus | ball minus torus |  loop-particle cluster  |  loop-particle cluster
3d gapped boundary |  sphere shell with
two gapped boundaries |  solid cylinder |  boundary sphere shell |  boundary string excitation  |  bulk anyon condensed on boundary
3d gapped boundary |  solid torus with
gapped boundary |  bundt cake |  half sphere shell |  string excitation ending on the boundary  |  boundary particle pair
3d gapped boundary |  genus-$g$ handlebody
with gapped boundary |  genus-$g$
bundt cake |  ${\cal B}_{g}$ cut in half  |  $g$-arch bridge excitation  |  $g+1$ boundary particles
4d bulk | $S^{3}\times S^{1}$ |  secret
3-sphere shell,
$B^{1}\times S^{3}$
| $B^{3}\times S^{1}$ | particle | 2-brane
4d bulk | $S^{2}\times S^{2}$ | $S^{2}\times B^{2}$ | $B^{2}\times S^{2}$ | loop | loop
Table 1: This table summarizes some examples of the pairing manifolds we
study. The examples of $X$ or $Y$ that are not sectorizable are this color,
while those that are sectorizable are this color. In the cartoons, gapped
boundaries are depicted in thick black; entanglement boundaries are in blue
here. In the two rightmost columns, the support of the excitation is in red;
in grey is a possible support of the flexible operator that creates the
excitations. The icing on the bundt cake is decorative.

##  2 Central pillars of entanglement bootstrap

In this section, we summarize the essential working tools of the entanglement
bootstrap [4, 5], as extended to three spatial dimensions in [6]. These tools
are available in a general space dimension, as the previous works give a clear
clue to such generalizations. As the proofs were given in previous works, we
shall focus on the statements and physical explanations. In §2.2, we give an
in-depth discussion of the topology of immersed regions. In §2.3, we introduce
the concept of constrained fusion spaces, which will be useful later.

In appendix A, we review the notations and terminology that we use often. In
particular, because we shall also discuss physical systems with gapped
boundaries, we need to distinguish between gapped boundaries and _entanglement
boundaries_ : an entanglement boundary is a component of the boundary of a
region that is not part of the gapped boundaries.

###  2.1 Axioms and existing tools

The entanglement bootstrap axioms can be stated in an arbitrary dimension. In
the bulk, we always have two axioms. For concreteness, we start by stating the
axioms for the entanglement bootstrap in three dimensions (Eq. (2.1)). We
assume given a reference state $\sigma$ supported on a large ball
$\mathbf{B}^{3}$. (Note the distinction between $\mathbf{B}^{n}$ and an
arbitrary ball $B^{n}$. We shall denote _the_ large ball with the reference
state as $\mathbf{B}^{n}$ for $n$-dimensional entanglement bootstrap.) We
assume that the reference state $\sigma$ lives on a many-body quantum system
whose Hilbert space is the tensor product of local onsite Hilbert spaces
${\mathcal{H}}_{\rm total}=\otimes_{i}{\mathcal{H}}_{i}$. This assumption
means that we restrict our attention to systems made of bosons.666Fermionic
systems have a $\mathbb{Z}_{2}$-graded Hilbert space instead. We further
assume that each local Hilbert space ${\mathcal{H}}_{i}$ is finite-
dimensional. This is for the purpose of avoiding possible exotic cases that
arise only in infinite dimensions.

We assume that the following two axioms hold on ball-shaped subsystems
contained in $\mathbf{B}^{3}$, for the reference state $\sigma$:

$C$$B$${\bf A0}:~{}~{}(S_{BC}+S_{C}-S_{B})_{\sigma}=0$$C$$B$$D$${\bf
A1}:~{}~{}(S_{BC}+S_{CD}-S_{B}-S_{D})_{\sigma}=0$ (2.1)

where $(S_{X})_{\sigma}=-{{\rm Tr}}(\sigma_{X}\ln\sigma_{X})$ is the von
Neumann entropy of the state $\sigma$ reduced to the region $X$. Each
partition of the balls shown in Eq. (2.1) is topologically equivalent to the
volume of revolution of the indicated 2d region.

We shall denote the entropy combinations appearing in the axioms as

$\Delta(B,C)\equiv S_{BC}+S_{C}-S_{B},~{}~{}~{}\Delta(B,C,D)\equiv
S_{BC}+S_{CD}-S_{B}-S_{D}.$ (2.2)

We shall refer to the two axioms as A0 and A1. As in the 2d case [4], the
strong subadditivity (SSA) [27] implies that if the axioms hold for balls of a
certain length scale,777We remark that, the balls are topological balls. They
do not need to be round. We only require that the region is topologically
equivalent to the one shown in Eq. (2.1). Physically speaking, the thickness
of $B$ and $D$ needs to be thicker than the correlation length. We introduced
the volume of revolution in figures only for the visualization of the
topology, and rotational symmetry is not required. the same conditions must
hold for all larger balls. In other words, there is no problem zooming out to
a larger length scale. Because of this, we shall assume that we have a fine
enough lattice, and we only consider large enough regions consisting of enough
(but finite) lattice sites and having sufficient distance separation between
each other. This continuum limit allows us to borrow topology concepts such as
ball, annulus, and sphere.888The mathematical theory for going from the
lattice to the continuum topology is nontrivial. It is unknown if an existing
branch of mathematics can formulate this idea with full rigor. This is a
meaningful topic for future studies. (Concrete intuitions can, however, be
obtained by choosing a particular coarse-grained lattice. Interested readers
are encouraged to look at the exercises presented in Chapter 9 of [28], the
context of which is the 2d bulk.)  999In realistic settings, and in particular
for chiral phases, these axioms may not be exact. Nonetheless, we believe that
they are satisfied in a renormalization group sense for a large class of
physical systems. That is, the violation of the axioms decay towards zero (at
a fast enough speed) as we coarse grain the lattice further. Justification of
this conjecture is an interesting future problem.

The axioms A0 and A1 can be defined in an arbitrary space dimension $n$. For
axiom A0, $B$ will be the sphere shell, and for axiom A1, $BD$ will be the
sphere shell, where $B$ and $D$ are hemisphere shells. We shall study the
generalizations of the axioms to systems with gapped boundaries as well, see
§3.2 for the axioms in that setup.

The axioms A0 and A1, are closely related to two well-known quantities,
respectively, the mutual information $I(A:C)\equiv S_{A}+S_{C}-S_{AC}$ and the
conditional mutual information $I(A:C|B)\equiv S_{AB}+S_{BC}-S_{B}-S_{ABC}$.
SSA refers to the statement that $I(A:C|B)\geq 0$ for any tripartite mixed
state. If a tripartite state $\rho_{ABC}$ satisfies $I(A:C|B)_{\rho}=0$, i.e.,
if it saturates the strong subadditivity, we say it is a quantum Markov state
(with respect to this partition).

One important object that can characterize various nontrivial structures is
the information convex set, denoted as $\Sigma(\Omega)$. It depends on the
region $\Omega$ and on the reference state $\sigma$. We suppress the
dependence of $\sigma$ in the notation $\Sigma(\Omega)$ since the reference
state is fixed at the beginning. There are equivalent ways to define
$\Sigma(\Omega)$. One intuitive definition is: _$\Sigma(\Omega)$ is the set of
density matrices on $\Omega$ which can be smoothly extended to any larger
regions $\Omega^{\prime}$ ($\Omega^{\prime}$ regular homotopic to $\Omega$ and
$\Omega^{\prime}\supset\Omega$), where the state on $\Omega^{\prime}$ is
locally indistinguishable with the reference state._101010A state on an
immersed region is locally indistinguishable with the reference state if the
reduced density matrix on any (small) embedded ball is identical to that of
the reference state. This is equivalent to two other definitions that appeared
in [4] by the isomorphism theorem. As the name suggests, $\Sigma(\Omega)$ is a
convex set of density matrices. The region $\Omega$ can be immersed, as we
shall explain.

From the axioms A0 and A1, properties of information convex sets can be proved
as theorems. Results appearing in previous literature [4, 5, 6] include:

  * •

Merging technique: It includes the _merging lemma_ [29] and the _merging
theorem_ [4]. When we use the word merging as a physical process, we always
refer to the process described by the merging lemma [29]. That is, it is
possible to construct a unique quantum Markov state $\tau_{ABCD}$ (with
$I(A:D|BC)_{\tau}=0$) from two quantum Markov states $\rho_{ABC}$ (with
$I(A:C|B)_{\rho}=0$) and $\lambda_{BCD}$ (with $I(B:D|C)_{\lambda}=0$), as
long as $\rho_{BC}=\lambda_{BC}$. The resulting “merged” state $\tau_{ABCD}$
is identical with $\rho_{ABC}$ and $\lambda_{BCD}$ on marginals $ABC$ and
$BCD$.

In entanglement bootstrap, merging theorem [4, 5] says that whenever two
quantum Markov states in two information convex sets can merge,111111There is
one extra mild requirement that the regions in question need to be wide
enough. This requirement is satisfied in all our applications. the resulting
state must be an element of a third information convex set.

  * •

Immersed region: The concept of immersed region [6, 16] is a natural
generalization of subsystems. An immersed region is locally embedded and has
the same dimension as the physical system. For each immersed region one can
consider the information convex set. Since immersed regions will be crucial in
this work, we shall introduce them further in §2.2.

  * •

The isomorphism theorem: If two immersed regions are connected by a _path_ ,
then the information convex sets are isomorphic. In other words,

$\Omega_{0}\sim\Omega_{1}\quad\Rightarrow\quad\Sigma(\Omega^{0})\cong\Sigma(\Omega^{1}).$
(2.3)

Here, a path between $\Omega_{0}$ and $\Omega_{1}$ is a finite sequence of
immersed regions $\\{\Omega^{t}\\}$, $t\in\\{0,1/N,2/N,\cdots,1\\}$ such that
adjacent regions in the sequence are related by adding (removing) a small ball
in a topologically trivial manner. This relation is an elementary step; see
Fig. 3 for an illustration. In the remaining of the paper, when two regions
are connected by a path, we say one region can be “smoothly” deformed into
another. The isomorphism $\cong$ refers to the isomorphism of two information
convex sets as convex sets, as well as the preservation of distance measures
and the entropy difference. (We do not require the entropy to be preserved,
only the entropy difference.) The isomorphism $\cong$ between the two
information convex sets can depend on the path. Nonetheless, it can be shown
that only the topological class of the path matters.

Figure 3: An illustration of an elementary step. Here the ball $C$ and added
to $AB$ in a topologically trivial manner. $A$ can be large and has an
arbitrary topology. $BC$ is part of a small ball $BCD$ on which the axioms are
imposed. $A$ is contained in the complement of $BCD$. $AB\to ABC$ is an
elementary step of extension, whereas $ABC\to AB$ is an elementary step of
restriction. Axiom A1, on $BCD$ (with $D$ not shown such that $BD$ separate
$C$ from the “outside”) implies that $I(A:C|B)$ must vanish for any element of
information convex set $\Sigma(ABC)$.

  * •

Structure theorems: The geometry of information convex sets must be of a
certain form. While the isomorphism theorem says that we only need to consider
topological classes of regions, the structure theorems describe powerful
statements on the topological dependence:

    1. 1.

_Simplex theorem:_ If an immersed region $S$ is _sectorizable_ 121212A region
$X$ is said to be sectorizable if it contains two disjoint pieces $X^{\prime}$
and $X^{\prime\prime}$ such that each can be deformed to $X$ via a sequence of
extensions., the information convex set is a simplex:

$\Sigma(S)=\left\\{\sum_{I\in{\cal
C}_{S}}p_{I}\rho_{S}^{I}\right|\left.\sum_{I}p_{I}=1,p_{I}\geq 0\right\\}.$
(2.4)

Here ${\cal C}_{S}$ is the (finite) set of labels, where each label represents
a superselection sector. The set of density matrices $\\{\rho^{I}_{S}\\}$ are
the extreme points, and they are mutually orthogonal.

    2. 2.

_Hilbert space theorem:_ For any immersed region $\Omega$, the information
convex set $\Sigma(\Omega)$ is a convex hull of mutually orthogonal convex
subsets $\Sigma_{I}(\Omega)$, where $I\in{\cal C}_{\partial\Omega}$. (Here
$\partial\Omega$ is the thickened entanglement boundary131313A thickened
entanglement boundary is always of the form $m\times\mathbb{I}$, where $m$ is
a manifold and $\mathbb{I}$ is an interval. In this work, we always consider
thickened entanglement boundaries that are thick enough so that the interval
$\mathbb{I}$, though being a lattice analog, can be partitioned into smaller
intervals. of $\Omega$, and it is always sectorizable.) Each subset is
isomorphic to the state space (i.e., the set of density matrices) of a finite-
dimensional Hilbert space $\mathbb{V}_{I}(\Omega)$. We denote this by
$\Sigma_{I}(\Omega)\cong\mathcal{S}(\mathbb{V}_{I}(\Omega))$, where
$\mathcal{S}(\mathbb{V})$ denotes the state space of $\mathbb{V}$. The
dimension of the Hilbert spaces, denoted as
$\\{N_{I}(\Omega)\equiv\dim\mathbb{V}_{I}(\Omega)\\}$, are the fusion
multiplicities.

  * •

Associativity theorem: If an immersed region $\Omega$ can be cut into halves
by a hypersurface that does not touch the entanglement boundary of $\Omega$,
then the fusion space dimensions associated with $\Omega$ are completely
characterized by the multiplicities of its subsets ($\Omega_{L}$ and
$\Omega_{R}$) and the way the hypersurface connects them:

$\dim\mathbb{V}_{a_{L}}^{a_{R}}(\Omega)=\sum_{i\in{\mathcal{C}}_{S}}\dim\mathbb{V}_{a_{L}}^{i}(\Omega_{L})\cdot\dim\mathbb{V}_{i}^{a_{R}}(\Omega_{R}),$
(2.5)

where $S$ is the thickened hypersurface. It is guaranteed that $S$ is
sectorizable and ${\cal C}_{S}$ denotes the set of superselection sectors on
$S$. (Alternatively, one can write:
$N_{a_{L}}^{a_{R}}(\Omega)=\sum_{i\in{\mathcal{C}}_{S}}N_{a_{L}}^{i}(\Omega_{L})N_{i}^{a_{R}}(\Omega_{R})$.)
This theorem [6] is proved by considering the merging of whole boundary
components and making use of the Hilbert space theorem.

  * •

Vacuum and sphere completion: Let $\Omega\subset\mathbf{B}^{n}$ be a subsystem
of the ball. The vacuum state on $\Omega$ is defined as $\sigma_{\Omega}$,
i.e., the reduced density matrix of the reference state. It is shown that the
vacuum state is an isolated extreme point of $\Sigma(\Omega)$ [6]. If $S$ is a
sectorizable region embedded in $\mathbf{B}^{n}$, then
$\rho^{1}_{S}\equiv\sigma_{S}$ corresponds to a very special label in ${\cal
C}_{S}$, i.e., the _vacuum sector_ , denoted as $1\in{\cal C}_{S}$.

Moreover, a reference state on a sphere $\mathbf{S}^{n}$ always has a vacuum,
that is the unique pure state in $\Sigma(\mathbf{S}^{n})$ [4]. A vacuum state
on a sphere can be obtained from one on the ball by the “sphere completion”
[6]. We shall generalize the completion trick in later sections.

To what extent can the definition of vacuum generalize to immersed regions
that are not embedded? This largely remains an open problem, but we report
some progress in §4.1.

Figure 4: Partitions of sphere shells relevant to the quantum dimension of
point excitations.

  * •

Quantum dimension: For a sectorizable region $S$ embedded in $\mathbf{B}^{n}$,
the quantum dimension of a sector $I$ can be defined as

$d_{I}=\exp{\left({S(\rho_{S}^{I})-S(\sigma_{S})\over 2}\right)},\quad\forall
I\in{\cal C}_{S}.$ (2.6)

We shall need another definition of quantum dimension for point-like
excitations, which are characterized by the information convex set of a sphere
shell. Let the sphere shell be $S^{n-1}\times\mathbb{I}=BCD$, where
$\mathbb{I}=[0,1]$ is an interval. $S^{n-1}=NS$, where $N$ is the northern
hemisphere and $S$ is the southern hemisphere. Let $C=S^{n-1}\times[1/3,2/3]$,
$B=(N\times\mathbb{I})\setminus C$ and $D=(S\times\mathbb{I})\setminus C$.
This partition for the case $n=3$ is shown in Fig. 4(a). Then the quantum
dimension for point excitation $a$ is

$d_{a}=\exp{\left({\Delta(B,C,D)_{\rho^{a}}\over 4}\right)},\quad a\in{\cal
C}_{S^{n-1}\times\mathbb{I}},\quad\textrm{for Fig.~{}\ref{fig:quantum-dim-
point}(a).}$ (2.7)

If instead, $C=S^{n-1}\times[1/3,2/3]$, $B=N\times[2/3,1]$ (or
$B=(N\times[0,1/3]$ ) and $D$ is the rest, the quantum dimension is141414The
technically nontrivial part of the claim in Eq. (2.8) is that the partitions
in Fig. 4(b) and (c) are related by turning the sphere shell inside-out. The
proof is written by one of us and will appear elsewhere. (Note however, in 3d,
this is intuitively related to the fact that any sphere shell immersed in a
3-dimensional ball can be turned inside out.)

$d_{a}=\exp{\left({\Delta(B,C,D)_{\rho^{a}}\over 2}\right)},\quad a\in{\cal
C}_{S^{n-1}\times\mathbb{I}},\quad\textrm{for Fig.~{}\ref{fig:quantum-dim-
point}(b) and (c).}$ (2.8)

If we know $\Delta(B,C,D)_{\rho^{a}}$ for any one of the three partitions, we
have a definition of quantum dimension for point particles. This alternative
definition has an advantage. We only need the state $\rho^{a}$ on the sphere
shell, and there is no need to compare it with the vacuum. This makes Eq.
(2.7) and Eq. (2.8) work for immersed sphere shells. Moreover, there are
generalizations of them for many other excitation types [6]. Moreover, from
Eq. (2.7) and Eq. (2.8) it is manifest that $d_{a}\geq 1$. This is because
$\Delta(B,C,D)$ is nonnegative.

###  2.2 Immersed regions

The concept of _immersed region_ is a natural generalization of subsystems. A
subsystem is an embedded region that has the same dimension as the physical
system. An immersed region is a region that is _locally_ embedded and has the
same dimension as the physical system. By this definition, embedding is a
special case of immersion.

Figure 5: Illustration of the three kinds of immersed regions.

Immersed regions come in three kinds (see Fig. 5):

  1. 1.

Regions of the first kind can always be smoothly deformed to embedded regions.
An example is a ball in any dimension; see Fig. 5(a).

  2. 2.

Regions of the second kind have inequivalent1414footnotetext: We say two
immersions are inequivalent if no path can connect the pair of regions. In the
continuous limit, this is the statement that the two immersions are not
regular homotopy equivalent. immersions into the physical system. One is an
embedding, and at least one other is a nontrivial immersion. One such example
is the annulus in 2d; the figure eight at right in Fig. 5(b) cannot be
deformed by regular homotopy to the embedded annulus at left because their
boundaries have different winding numbers.

  3. 3.

Regions of the third kind cannot be embedded in a ball. One example is a torus
with a ball removed; see Fig. 5(c).

In particular, because of the existence of the third kind, immersed regions
are strictly more diverse than embedded regions.

Below, we give a few examples of immersed regions, focusing on the topology
that cannot be obtained from regions embedded within a ball. Let
$\mathcal{W}\equiv\mathcal{M}\setminus B^{n}$. Here ${\cal M}$ is a
$n$-dimensional closed manifold and $B^{n}$ is a solid ball. In other words,
$\mathcal{W}$ is a closed manifold with a ball removed. Here are some choices
of such ${\cal W}$, which can be immersed in a ball ${\cal W}\looparrowright
B^{n}$, for $n=2,3$:

  1. 1.

In 2d, any closed connected orientable surface (classified by genus-$g$) with
a ball removed is a choice of ${\cal W}$. It is not hard to visualize these
immersions [30].

  2. 2.

In 3d, it is shown recently that every closed connected orientable 3-manifold
can be immersed in a ball upon removing a ball [31]. In particular, the
immersions of the following manifolds151515We shall extensively use the
standard notation for manifold topology, e.g., $S^{n}$ refers to $n$-sphere,
$T^{n}$ refers to $n$-dimensional torus; see Appendix A for other notations we
frequently use in this work. are known explicitly.

    1. (a)

$(S^{2}\times S^{1})\setminus B^{3}$.

    2. (b)

$\underbrace{(S^{2}\times S^{1})\\#\cdots\\#(S^{2}\times S^{1})}_{\text{$g$
times}}\setminus B^{3}\equiv\\#g(S^{2}\times S^{1})\setminus B^{3}$. Here
$\\#$ means connected sum.161616A connected sum of two $n$-dimensional
manifolds is also a $n$-dimensional manifold. It is formed by first deleting a
ball in the interior of each manifold, then gluing together the resulting
boundary spheres.

    3. (c)

$T^{3}\setminus B^{3}$, where $T^{3}$ is the three-dimensional torus. This
case is nontrivial to visualize. In the math literature, the existence proof
of such immersion appeared first [32], and was generalized to arbitrary
dimensions later on [33, 34]. Several explicit constructions followed, see
[35, 36] for example.

We shall describe other examples of immersion in §3; some of these examples
are related to the physical setup of gapped boundaries. Furthermore, in §4, we
shall find ways to obtain reference states on various closed manifolds with
the combination of two powerful techniques: immersion and the completion trick
we shall discuss. For later convenience, we shall always consider immersed
regions that leave enough space so that we can thicken them while keeping them
immersed.

###  2.3 Constrained fusion space

We have discussed information convex set $\Sigma(\Omega)$ and the subsets
$\Sigma_{I}(\Omega)$ in which the sectors ($I$) on the thickened entanglement
boundaries are specified. One motivation for studying these sets is to
characterize how quantum information is distributed in subsystems of a quantum
state.

For this purpose, it is sometimes useful to specify a constraint that is not
on the thickened entanglement boundary. This motivates us to define
information convex sets with constraints, and their constrained fusion spaces
(or constrained Hilbert space).

######  Definition 2.1 (Constrained information convex sets and constrained
fusion spaces).

Let $\Omega=AE$ and choose $\rho^{\kappa}_{A}\in\textrm{ext}(\Sigma(A))$, the
set of extreme points of $\Sigma(A)$. We define the constrained information
convex set as

$\displaystyle\Sigma_{[\kappa_{A}]}(\Omega)$
$\displaystyle\equiv\\{\rho_{\Omega}\in\Sigma(\Omega)|{{\rm
Tr}}_{E}\,\rho_{\Omega}=\rho_{A}^{\kappa}\\},\quad\textrm{and }$ (2.9)
$\displaystyle\Sigma_{I[\kappa_{A}]}(\Omega)$
$\displaystyle\equiv\\{\rho_{\Omega}\in\Sigma_{I}(\Omega)|{{\rm
Tr}}_{E}\,\rho_{\Omega}=\rho_{A}^{\kappa}\\}.$

According to Lemma 2.2 below,
$\Sigma_{I[\kappa_{A}]}(\Omega)\cong\mathcal{S}(\mathbb{V}_{I[\kappa_{A}]})$,
for some fusion space $\mathbb{V}_{I[\kappa_{A}]}(\Omega)$. We shall refer to
$\mathbb{V}_{I[\kappa_{A}]}(\Omega)$ as a constrained fusion space.

######  Remark.

We further allow the usage of $[\rho^{\kappa}_{A}]$ as an alternative for
$[\kappa_{A}]$, where $\rho^{\kappa}_{A}$ is the extreme point of $\Sigma(A)$
in question. Constrained fusion spaces are particularly convenient for the
study of closed manifolds. When $\mathcal{M}$ is a closed manifold the index
$I$ is dropped, we have constrained fusion spaces
$\mathbb{V}_{[\kappa_{A}]}({\cal M})$.

######  Lemma 2.2.

The various constrained sets defined in Definition 2.1 satisfy:

  1. 1.

$\Sigma_{[\kappa_{A}]}(\Omega)$ and $\Sigma_{I[\kappa_{A}]}(\Omega)$ are
compact convex sets.

  2. 2.

$\Sigma_{I[\kappa_{A}]}(\Omega)\cong\mathcal{S}(\mathbb{V}_{I[\kappa_{A}]}(\Omega))$,
where $\mathbb{V}_{I[\kappa_{A}]}(\Omega)$ is a finite-dimensional Hilbert
space, and it is a subspace of $\mathbb{V}_{I}(\Omega)$.

  3. 3.

If $A$ is sectorizable, we can relabel $\kappa$ as $J\in{\mathcal{C}}_{A}$. In
this case,

$\mathbb{V}_{I}(\Omega)=\oplus_{J}\mathbb{V}_{I[J_{A}]}(\Omega),$ (2.10)

is a direct sum.171717A Hilbert space is a direct sum when the subspaces in
the sum are mutually orthogonal.

###### Proof.

The proof of statement 1 is as follows. It is evident that
$\Sigma_{[\kappa_{A}]}(\Omega)$ and $\Sigma_{I[\kappa_{A}]}(\Omega)$ are
convex sets. The compactness follows from that of $\Sigma(\Omega)$.

To prove statement 2, we can shrink $A$ a little bit, without changing its
topology. Let $\widetilde{A}=A\setminus\partial A$ and
$\widetilde{E}=E\cup\partial A$. Then $\widetilde{A}$ and $\widetilde{E}$
share an entire boundary. The extreme point label $\kappa$ on $A$ induces a
label $\Phi(\kappa)$ of the extreme points of $\Sigma(\partial A)$. Note that
$\partial A$ is a sectorizable region. While $\kappa$ may be a continuous
label, $\Phi(\kappa)$ must be a discrete finite set. Thus, any element of
$\Sigma_{I[\kappa_{A}]}(\Omega)$ is obtained by merging some element of
$\Sigma_{I\Phi(\kappa)}(\widetilde{E})$ with _the same_ state
$\rho^{\kappa}_{A}$. Thus,
$\Sigma_{I[\kappa_{A}]}(\Omega)\cong\mathcal{S}(\mathbb{V}_{I\Phi(\kappa)}(\widetilde{E}))$.
Thus, statement 2 is true, and furthermore
$\dim\mathbb{V}_{I[\kappa_{A}]}(\Omega)=\dim\mathbb{V}_{I\Phi(\kappa)}(\widetilde{E})$.

Statement 3 is a corollary of statement 2. To see this, we observe that
density matrices in $\Sigma_{I}(\Omega)$, which reduces to the extreme points
of $\Sigma(A)$ associated with $J,J^{\prime}\in{\mathcal{C}}_{A}$ must be
orthogonal, for $J\neq J^{\prime}$. This follows from the monotonicity of
fidelity and that $F(\rho_{A}^{J},\rho_{A}^{J^{\prime}})=0$ for $J\neq
J^{\prime}$. Second, if we take all different choices of $J$, the right-hand
side of Eq. (2.10) gives the Hilbert space dimension that matches that for the
left-hand side. ∎

With the language of constrained Hilbert space, we can usefully refine the
Associativity Theorem as:

######  Theorem 2.3 (Associativity theorem, new form).

Let $S$ be the thickened hypersurface that appeared in the setup of the
Associativity Theorem (see around Eq. (2.5)), then

$\dim\mathbb{V}_{a_{L}[i_{S}]}^{a_{R}}(\Omega)=\dim\mathbb{V}_{a_{L}}^{i}(\Omega_{L})\cdot\dim\mathbb{V}_{i}^{a_{R}}(\Omega_{R}),\quad\forall
i\in{\cal C}_{S}.$ (2.11)

##  3 Regions dual to excitations

In this section, we discuss various regions that can detect either an
excitation or a cluster of excitations. As reviewed in §2.1, immersed regions
can be classified into sectorizable regions and non-sectorizable regions. A
sectorizable region has an information convex set isomorphic to a simplex; the
extreme points correspond to the superselection sectors. For non-sectorizable
regions, information about fusion spaces can be detected; in the case of a
nontrivial fusion space, the information is quantum.

Sometimes, we say a region can detect the superselection sector of some
excitation or the fusion space associated with a cluster of excitations. When
does this happen?181818The question becomes nontrivial when the region is not
embedded in a ball or a sphere. The thickened Klein bottle in 3d is one
example where we don’t know what are the excitations it characterizes, or
whether these excitations can be identified as excitations living in a sphere.
The purpose of this section is to explain this. We further provide many
examples, the physical setups of which differ in the dimensions and the
presence (absence) of a gapped boundary.

The basic intuition is as follows. Imagine a ground state $|\psi\rangle$ of a
topologically ordered system, on a sphere. Consider an excited state
$|\varphi\rangle$ with a few excitations on the sphere. The state
$|\varphi\rangle$ on a region $\Omega$ separated from the excitation(s) by a
few correlation lengths must be locally indistinguishable from the ground
state. Roughly speaking, the region $\Omega$ is the complement of the
excitations. If the excitations are nontrivial, that is, if the excitations
cannot be created by local operators, the region $\Omega$ must be able to
detect them.

In entanglement bootstrap, this intuition is guaranteed. The reference state
plays the role of the ground state. The reduced density matrix of
$|\varphi\rangle$ on $\Omega$ must be an element of the information convex set
$\Sigma(\Omega)$. From the “sphere completion lemma” of Ref. [6], we can
always construct a reference state on the sphere ($\mathbf{S}^{n}$), where the
axioms A0 and A1 are satisfied everywhere. (We shall not review the sphere
completion technique here. Instead, we review it after proving a more powerful
version Lemma 4.4; see Fig. 21.) In the context of gapped boundaries, as we
discuss in §3.2, the analog of sphere completion is the fact that we can
always define a reference state on a ball with an entire gapped boundary,
which we denote as $\underline{\underline{\mathbf{{B}}}}^{n}$. String
operators and membrane operators creating the excitations can also be
constructed in entanglement bootstrap, and the properties of these operators
can be useful in proving things; see [16]. In this work, we will not rely on
string or membrane operators to prove any statement, but as they provide
complementary intuition, we sometimes draw them in figures. They will be
discussed more explicitly in [37].

All regions considered in this section _can_ be embedded in a ball
$\mathbf{B}^{n}$ (or $\mathbf{S}^{n}$, or
$\underline{\underline{\mathbf{{B}}}}^{n}$ in the presence of a gapped
boundary). However, we also consider the immersed version of these regions.
They will be useful later, as building blocks (see §4.1). We start with the 2d
bulk and 3d bulk. Then we discuss the boundary generalizations.

###  3.1 2d bulk and 3d bulk regions

The setup and axioms of the 2d bulk and 3d bulk have been discussed in §2.1.
The intuition to keep in mind is that the combination of the two axioms
(especially axiom A1) makes it possible to deform the regions smoothly: if two
regions can be connected by a path, i.e., if two regions are related by a
regular homotopy, then the information convex sets associated with the pair of
regions are isomorphic. Thus, we shall only be interested in the topological
class of immersed regions.

####  3.1.1 2d bulk regions

The 2d bulk is the physical context of anyons and topological orders, and it
is the context in which 2+1d TQFT [2, 3, 38] is expected to apply. As studied
in detail in [4], there is a finite set of basic regions: the disk, the
annulus, and the pair of pants.

(3.1)

These regions are closely related to anyons and the string operators creating
them; see Eq. (3.1). First, the regions are homeomorphic to the complement of
the anyon excitations on $S^{2}$. The string operators pass through the
regions and cut them into disks. Note that a single excitation on the sphere
always carries a trivial superselection sector.

In the topology literature, it is well-known that any compact orientable
surface can be constructed by gluing these basic topology types along closed
curves. This is known as the pants decomposition [39]. This intuition is known
in the framework of topological quantum field theory (TQFT), see e.g. [3] and
[38]. In §4.1, we will construct reference states on more interesting
manifolds by building them from pieces of a state on a topologically trivial
region. To achieve that, we shall make use of immersed versions of these
regions. To gently prepare the reader for this discussion, we illustrate
immersed versions of regions in Fig. 6.

Figure 6: Immersed versions of embedded regions. For every example shown here,
there exists a smooth deformation that connects the two configurations. The
deformation is done on $\mathbf{S}^{2}$. Note that the deformation is more
flexible on $\mathbf{S}^{2}$ compared with $\mathbf{B}^{2}$.

Even for the simple class of embedded regions, e.g., a sphere minus $k$-balls,
nontrivial immersed versions exist. If we only allow deformations within
$\mathbf{B}^{2}$, the immersed regions described in Fig. 6 are not regular
homotopic to embedded regions. This is denoted as
$\Omega\overset{\mathbf{B}^{2}}{\nsim}\Omega^{\prime}$. Nonetheless, they
_are_ regular homotopic to embedded regions on the background manifold
$\mathbf{S}^{2}$, (denoted as
$\Omega\overset{\mathbf{S}^{2}}{\sim}\Omega^{\prime}$). In other words, the
deformation is more flexible on $\mathbf{S}^{2}$ compared with
$\mathbf{B}^{2}$. This is one of the reasons why the notion of manifold
completion, detailed in §4, is useful.

####  3.1.2 3d bulk regions

The 3d bulk provides more diverse topology types. One simple class of regions
is genus-$g$ handlebodies; see Fig. 7. Note that it is already an infinite
series. While there are other interesting topologies, we remark that the
seemingly simple list of handlebodies is surprisingly fundamental: every
closed compact orientable 3-manifold can be divided into two handlebodies;
this is known as Heegaard splitting [40]; see Fig. 8 for an illustration. (The
Heegaard genus, which is the smallest number of handles of a Heegaard
splitting, is additive under connected sums of 3-manifolds. Therefore, all
genus-$g$ handlebodies are needed to construct an arbitrary closed compact
orientable 3-manifold in this approach.)

Figure 7: Genus-$g$ handlebodies, where $g=0,1,2,\cdots$.

Figure 8: A Heegaard splitting of a 3-sphere, inspired by [40].

Figure 9: Two ways to fuse flux loops: (left) along the whole loop or (right)
along a segment. The latter produces a genus-two graph excitation.

Interestingly, all these handlebodies are sectorizable. The excitations
detected by these regions are supported on thin handlebodies of the same
genus. Thin handlebodies look like graphs, and for this reason, we call these
excitations _graph excitations_. For genus $g=1$, the graph excitations are
detected by the solid torus; they are familiar in the literature, and known as
pure flux loops; see, e.g., [7, 8, 11, 6].

How about graph excitations with a higher genus? Are they intrinsically new?
Shouldn’t they be labeled by two flux loops as is suggested by the fusion
process of loops, illustrated in Fig. 9? As it turns out, the answer is ‘no’
in general. For instance, the $G=S_{3}$ quantum double model has $3$ flux
sectors and $11>3\times 3$ graph excitations191919Among the 11 types, 6 are
_genuine_ graph excitations, meaning that they cannot be reduced to a single
loop. This number is greater than the number ($9-5=4$) predicted by the naive
approach. for genus $g=2$. Here $S_{3}$ is the permutation group of three
elements. This is illustrated in the example of the 3-dimensional $S_{3}$
quantum double model below.

Here are the various superselection sector labels for 3d topological orders,
which we shall use in Example 3.1 and later. We use the same notation as Ref.
[6]. ${\mathcal{C}}_{\rm point}=\\{1,a,b,\cdots\\}$ refers to the
superselection sectors of point particles detected by the sphere shell.
${\mathcal{C}}_{\rm flux}=\\{1,\mu,\nu\\}$ is the set of pure flux loops,
i.e., graph excitations supported on $g=1$ graphs. Below, we shall use
${\mathcal{C}}_{g}$ to denote the superselection sectors for the graph
excitations on an unknotted genus $g$ graph; they are detected by a genus $g$
handlebody.

######  Example 3.1 (3d $S_{3}$ quantum double).

The finite group $S_{3}$ is the smallest non-abelian group:
$S_{3}=\\{1,r,r^{2},s,sr,sr^{2}|r^{3}=s^{2}=1,sr=r^{2}s\\}$. The quantum
double with $S_{3}$ group has the following superselection sectors:

  1. 1.

${\mathcal{C}}_{\rm{point}}$ contains 3 labels, with
$\\{d_{a}\\}=\\{1,1,2\\}$.

  2. 2.

${\mathcal{C}}_{\rm{flux}}$ contains 3 labels, with
$\\{d_{\mu}\\}=\\{1,\sqrt{2},\sqrt{3}\\}$.

  3. 3.

${\mathcal{C}}_{g}$ contains $6^{g-1}+3^{g-1}+2^{g-1}$ labels. In particular,
${\mathcal{C}}_{g=2}$ contains 11 labels.

######  Proposition 3.2 (Graph sectors for 3d quantum double).

For 3d quantum double with finite group $G$, the set (${\mathcal{C}}_{g}$) of
superselection sectors of graph excitations characterized by the information
convex set of genus-$g$ handlebody has

$|{\mathcal{C}}_{g}|={1\over|G|}\sum_{h\in G}|E(h)|^{g}$ (3.2)

elements, where $E(h)$ is the centralizer group of $h:E(h)\equiv\\{k\in
G|kh=hk\\}$. $|G|$ denotes the order of finite group $G$. In particular when
$G=S_{3},|{\cal C}_{g}|=6^{g-1}+3^{g-1}+2^{g-1}$.

The proof of Proposition 3.2 is given in Appendix C.

A few other basic topologies are shown in Fig. 10. The first one detects a
particle,202020On a 3-sphere, the complement of a sphere shell is two disjoint
balls. For this reason, one can say the sphere shell detects a particle-
antiparticle pair. the second one detects a three-particle cluster and the
third one detects a particle-loop cluster. Only the sphere shell is
sectorizable. These regions are studied in [6].

Figure 10: (a) A sphere shell. (b) A ball with two balls removed. The two
removed balls are smaller, disjoint, and are within the interior of the large
ball. (c) A ball with a solid (unknotted) torus removed.

The sphere shell and $B^{3}$ with two balls removed, shown in Fig. 10(a) and
(b) are direct analogs of the 2d region we discussed in §3.1.1. In fact, every
region in Fig. 10 is the volume of revolution of a 2d region along an axis,
and therefore, the dimensional reduction consideration in [6] applies.
Clearly, this list is incomplete. For instance, there are regions with knotted
or linked torus entanglement boundaries; some of them are studied in [6].

As with the 2d bulk, we can make immersed versions of these basic regions.
Intuitively speaking, immersion can let the region “flip” along an
entanglement boundary; see Fig. 11.

Figure 11: Examples of embedded regions in $\mathbf{S}^{3}$ and immersed
regions obtained by a “smooth” deformation starting from embedded ones on
$\mathbf{S^{3}}$. Whether the rightmost region can be obtained by the
indicated deformation is unclear to us, and this is indicated by the question
mark.

Connected components of thickened entanglement boundaries provide another
basic class of 3d region. The topology of any such region is of the form
$m\times\mathbb{I}$, where $m$ is a genus-$g$ surface and $\mathbb{I}$ is an
interval. In other words, such regions are thickened genus-$g$ surfaces.
Therefore, it is always possible to embed them in $\mathbf{S}^{3}$.
Interestingly, when $g\geq 1$, it is also possible to immerse such a region in
$\mathbf{S}^{3}$ nontrivially, in a way not regular homotopic to any
embedding; see, e.g., Corollary 1.3 of Ref. [41].

###  3.2 Systems with gapped boundaries in 2d and 3d

We shall first describe the setup and the axioms of entanglement bootstrap for
the gapped boundary problems [5]. They are direct analogs of the axioms in
other previously studied contexts [4, 6]. After that, we describe the basic
choices of regions and the excitations or fusion spaces they characterize.
Although we focus here on gapped boundaries, the same technology applies to
the more general case of gapped domain walls between topological phases [5].

####  3.2.1 Axioms for gapped boundaries in 2d and 3d

2d setup: The entanglement bootstrap setup of the 2d gapped boundary problem
is a reference state ($\sigma$) on a half disk ($\underline{\mathbf{B}}^{2}$)
adjacent to the gapped boundary. The total Hilbert space is the tensor product
of finite dimensional Hilbert spaces on lattice sites. The number of such
lattice sites is a finite number but is large enough. These lattice sites make
a sensible discretization of a topological manifold. One can, for example,
obtain such a lattice by coarse-graining a realistic many-body system made of
qudits on a manifold.212121The same considerations as in footnote 9 apply
here. Each site has a finite-dimensional Hilbert space. The region with the
gapped boundary is depicted in Fig. 12, where the thick black line is the
gapped boundary. We assume that nontrivial Hilbert space only exists on one
side of the boundary, and the other side is empty. Two boundary axioms are
assumed in addition to the bulk axioms. These axioms are of similar forms, in
the sense that we always partition a small region into two or three pieces,
labeled by $BC$ or $BCD$:

  * •

If we partition the region into two pieces, we call the interior $C$ and call
the thickened entanglement boundary $B$, then we require
$\Delta(B,C)_{\sigma}=0$ on the reference state, for the indicated region
(yellow disk or half-disk in Fig. 12).

  * •

If we partition the region in three pieces, we call the interior $C$ and call
the thickened entanglement boundary $BD$, where the topology of $B$ and $D$
are indicated in Fig. 12 (green regions). We require that
$\Delta(B,C,D)_{\sigma}=0$.

Figure 12: The setup of 2d entanglement bootstrap with a gapped boundary. One
may start from a reference state on either: (a) a “half disk” adjacent to the
boundary ($\underline{\mathbf{B}}^{2}$), or (b) a disk with an entire boundary
($\underline{\underline{\mathbf{B}}}^{2}$). Axioms are imposed on bounded
radius disks both within the bulk and on the gapped boundary. The gapped
boundary is represented by a thick black line.

An equally simple setup is a reference state on a disk with an _entire_
boundary, with analogous axioms imposed; see Fig. 12(b) for an illustration.
The justification is a direct analog of the sphere completion lemma (Lemma 3.1
of Ref. [6]). See e.g. [42, 43] for some solvable models of gapped boundaries
in 2+1d and see [44] for explicit computation of information convex sets in a
related context. We note that there are systems that admit only gapless
boundaries, see [45, 46]; we expect that the boundary axioms are violated for
these systems.

We clarify the distinction between terminologies. The _gapped boundary_ should
not be confused with the _entanglement boundary_. The entanglement boundary is
a boundary that has nontrivial physical degrees of freedom lying on both
sides, and for the quantum states we are interested in, these physical degrees
of freedom have entanglement across the entanglement boundary. Another
perspective is that it is possible to pass the entanglement boundary with the
extensions allowed by the (generalized) isomorphism theorem. When we write
$\partial\Omega$ for an immersed region $\Omega$, we always mean the thickened
entanglement boundary. (In the remainder of the paper, we sometimes refer to
an entanglement boundary as a boundary for short, but we always say gapped
boundary.)

Figure 13:  The setup of 3d entanglement bootstrap with a gapped boundary. We
impose bulk and boundary versions of axioms A0 and A1 on bounded radius balls.
Here the reference state is given on a 3-dimensional ball with an entire
gapped boundary, denoted as $\underline{\underline{\mathbf{B}}}^{3}$, where
the reference state is pure. (As in 2d, another reasonable starting point is a
reference state on a half ball $\underline{\mathbf{B}}^{3}$ adjacent to a
gapped boundary. The “completion trick” shows that these two starting points
are equally simple.)

3d setup: Below we describe the entanglement bootstrap setup for the gapped
boundary of a 3d gapped system. It is very similar to the 2d setup, as can be
seen in Fig. 13. We refer to the half-ball adjacent to the boundary as
$\underline{\mathbf{B}}^{3}$. An alternative starting point is the (pure)
vacuum state on a ball with an entire boundary
$\underline{\underline{\mathbf{B}}}^{3}$.

######  Remark.

A topologically ordered system in 2d and 3d usually has multiple gapped
boundary types. In 3d, models of gapped boundaries have been studied by
several authors (see e.g. [47, 48] and references therein), where we expect
our axioms to apply. A complete classification of gapped boundaries of 3d
topologically ordered systems is an open problem. These gapped boundary types
cannot be converted to each other by finite depth circuit, and they should be
thought of as different “boundary phases” separated by a phase transition.
These different boundary types have different universal data and a “defect” on
the boundary is necessary to separate two boundary phases. (Here the defect is
codimension 1 to the gapped boundary.) Axiom A1 is expected to be violated on
such defects. Each reference state we consider in Fig. 12 or Fig. 13 is
associated with a particular boundary type.

####  3.2.2 2d: regions adjacent to a gapped boundary

Figure 14 shows a few basic topology types of regions, for a 2d system with a
gapped boundary. We draw them either on ${\underline{\mathbf{B}}}^{2}$ or
$\underline{\underline{\mathbf{B}}}^{2}$, whichever is more convenient.

Figure 14: A few basic topologies of regions adjacent to a gapped boundary.

For some of the topology types, one can make immersed versions of the regions.
Two examples are shown in Fig. 15. In particular, in the second example, the
“clock” region cannot be smoothly deformed into the “mushroom” region within
$\underline{\underline{\mathbf{B}}}^{2}$. This is a new feature, and because
of this, it is not obvious whether the two regions have isomorphic information
convex sets. Nevertheless, it is possible to show that the information convex
sets of the “clock” and the “mushroom” are isomorphic.

Figure 15: Immersed versions of embedded regions. (a) The immersed region is
regular homotopic to the embedded one on
$\underline{\underline{\mathbf{B}}}^{2}$. (b) The immersed region is not
regular homotopic to the embedded one on
$\underline{\underline{\mathbf{B}}}^{2}$, although the two regions are
homeomorphic.

We also note that the gapped boundary of a 2d system only contributes one type
of thickened entanglement boundary. That is the half annulus, i.e., the second
region in Fig. 14(a). On $\underline{\underline{\mathbf{B}}}^{2}$, any
immersion of the half annulus is regular homotopic to the embedded one.

####  3.2.3 3d: regions adjacent to a gapped boundary

In the context of 3d systems with a gapped boundary, we discuss two classes of
basic topologies. One class is the boundary analog of genus-$g$ handlebodies;
see Fig. 16. There are three subclasses. Each region is sectorizable. Another
class can be thought of as the boundary analog of ball minus $k$-ball; see
Fig. 17.

Figure 16: Boundary analogs of genus-$g$ handlebodies. There are three
subclasses.

Figure 17: Boundary analogs of ball minus $k$ balls.

Connected components of thickened entanglement boundaries adjacent to the
gapped boundary provide another class of basic regions. They are of the form
genus-$g$ handlebody shell cut by circles. We note that there can be
inequivalent ways to “fill in” the shell and obtain connected sectorizable
regions. In fact, the orange regions and the blue regions in Fig. 16 are
related in this way. To see this, we observe that the orange regions and blue
regions in Fig. 16 complement each other in
$\underline{\underline{\mathbf{B}}}^{3}$.

##  4 Closed manifolds completed from immersed regions

In this section, we shall be interested in finding reference states on closed
manifolds. If a gapped many-body system is described by topological quantum
field theory (TQFT) [2, 3, 38], it is natural to put the system on various
space manifolds (and spacetime manifolds). If an entanglement bootstrap
reference state secretly is a state that obeys the TQFT description, as is
suggested by all the progress up to now, one should expect that a reference
state can be put on various orientable space manifolds. The justification of
this for general space dimensions remains an important open problem. Below, we
solve the 2d case and make concrete progress on the 3d case.

Our progress rests on the idea of completing immersed regions to compact
manifolds. As before, we use ${\cal A}\looparrowright{\cal B}$ to denote
${\cal A}$ immersed in ${\cal B}$. When it is necessary to specify the
immersion map, we use ${\cal A}\overset{\varphi}{\looparrowright}{\cal B}$. We
start with a general definition.

######  Definition 4.1 (Completion in ${\cal N}$).

Let ${\cal N}$ be a connected manifold, possibly with entanglement boundaries,
equipped with a reference state $\sigma_{{\cal N}}$ that satisfies the
entanglement bootstrap axioms. We say a closed manifold $\mathcal{M}$ allows a
completion in ${\cal N}$, $({\cal M},{\cal
W}\overset{\varphi}{\looparrowright}{\cal N})$, if

  1. 1.

$\mathcal{W}\subset\mathcal{M}$ is obtained by removing a finite number of
separated balls from ${\cal M}$.

  2. 2.

$\mathcal{W}\overset{\varphi}{\looparrowright}{\cal N}$ is an immersion for
which $\Sigma(\mathcal{W})$ is non-empty.

If ${\cal N}=\mathbf{B}^{n}$ or $\mathbf{S}^{n}$, we call $({\cal M},{\cal
W}\overset{\varphi}{\looparrowright}{\cal N})$ a _completion_ for short.

${\cal M}=T^{2}$${\cal W}=T^{2}\setminus\Diamond$${\cal
W}\looparrowright\mathbf{S}^{2}$ Figure 18: The _completion_
$(T^{2},T^{2}\setminus B^{2}\looparrowright\mathbf{S}^{2})$ as an illustration
of Definition 4.1. When ${\cal M}$ is a torus $T^{2}$, we remove a small ball
$B^{2}$ (which we refer to as a _completion point_ , represented in the figure
as a “diamond” $\Diamond$) to obtain ${\cal W}$. The immersion ${\cal
W}\looparrowright\mathbf{S}^{2}$ is explicitly shown in the figure. In many of
the later figures, we omit the drawing of the black circle indicating
$\mathbf{S}^{n}$ whenever this is convenient and not confusing.

The reason we are interested in completion is that it provides a state on
manifold ${\cal M}$ which possesses a few nice properties (Proposition 4.2).
The state satisfies axioms A0 and A1 except possibly at a number of
“completion points”. Moreover, on $B^{n}\subset{\cal W}$ the state is locally
indistinguishable from the reference state $\sigma$. The reason we are
interested in the specific choice ${\cal N}=\mathbf{B}^{n}$ is that it is the
“cheapest” choice: a ball is a subsystem of any manifold. While we require
${\cal N}$ to be connected, ${\cal M}$ does not have to be connected. An
equally simple choice is a sphere $\mathbf{S}^{n}$; it is equally simple
because of the sphere completion lemma, which we shall review; see Fig. 21.
(We shall also consider versions of completion in the context related to
gapped boundaries, where ${\cal M}$ is a compact222222In entanglement
bootstrap, we always consider regions consisting of a finite number of sites.
We shall refer to a manifold without entanglement boundaries as compact, and
if entanglement boundaries and gapped boundaries are both absent, we say the
manifold is closed. manifold with boundaries and the simplest choices of
${\cal N}$ are $\underline{\mathbf{B}}^{n}$ and
$\underline{\underline{\mathbf{B}}}^{n}$.)

######  Proposition 4.2.

Let $({\cal M},{\cal W}\overset{\varphi}{\looparrowright}{\cal N})$ be a
completion in ${\cal N}$ of the closed manifold ${\cal M}$, where the
thickened boundary $\partial{\cal W}$ has $k$ connected components. Then,
there exists a choice of superselection sector labels $\\{a_{i}\\}_{i=1}^{k}$
and a state $|\phi^{\\{a_{i}\\}}\rangle$ on ${\cal M}$ such that

${{\rm Tr}}_{{\cal M}\setminus{\cal
W}}\,\,|\phi^{\\{a_{i}\\}}\rangle\langle\phi^{\\{a_{i}\\}}|\in{\rm
ext}(\Sigma_{a_{1}\cdots a_{k}}({\cal W})).$ (4.1)

Furthermore, A0 and A1 hold everywhere on $|\phi^{\\{a_{i}\\}}\rangle$ expect
for possible violations of A1 at the completion points: for a ball centered at
the $i$th completion point, $\Delta(B,C,D)_{|\phi^{\\{a_{i}\\}}\rangle}=2\ln
d_{a_{i}}$. Here, $d_{a_{i}}$ is the quantum dimension of $a_{i}$. The
information convex set $\Sigma_{a_{1}\cdots a_{k}}({\cal W})$ is determined by
the immersion ${\cal W}\overset{\varphi}{\looparrowright}{\cal N}$ as well as
the reference state $\sigma_{{\cal N}}$.

###### Proof.

The proof follows from the completion trick (Lemma 4.4). ∎

Intuitively speaking, Proposition 4.2 says that, if a manifold ${\cal M}$
allows a completion, we can obtain a state on ${\cal M}$ that satisfies the
entanglement bootstrap axioms, up to possible violations at a finite number of
isolated completion points. These violations are well-controlled and are
attributed to non-Abelian superselection sectors of point particles232323For
topologically nontrivial choices of ${\cal N}$, such non-Abelian
superselection sectors may also come from topological defects. See Example
4.27 in §4.3 for a discussion. at those completion points. Furthermore, the
state is locally “vacuum-like” since it is locally indistinguishable from the
reference state on ${\cal N}$ for all small balls contained in ${\cal W}$.

######  Proposition 4.3.

If $({\cal M},{\cal W}\overset{\varphi}{\looparrowright}{\cal N})$ is a
completion in ${\cal N}$ of ${\cal M}$, and ${\cal N}$ is orientable, then
${\cal M}$ is orientable.

###### Proof.

We provide proof by contradiction. Suppose that the manifold $\mathcal{M}$ is
nonorientable. Then there exists a small ball in ${\cal W}$, which can be
transported around and back to the original place and gets its orientation
flipped; the whole process happens within ${\cal W}$. This happens no matter
how small the ball is. However, because ${\cal W}\looparrowright{\cal N}$, we
can always choose the ball small enough so that it remains embedded in ${\cal
N}$ during the whole process. Such a process cannot flip the orientation of
the small ball because ${\cal N}$ is orientable. This leads to a contradiction
and accomplishes the proof. ∎

######  Remark.

Here is the more general observation in differential geometry terms. An
immersion ${\cal W}\looparrowright{\cal N}$ implies that all nontrivial
characteristic classes of the tangent bundle $T{\cal W}$ must agree with those
of $T{\cal N}$. This includes the first Stieffel-Whitney class $w_{1}$ whose
vanishing is required for orientability. Therefore, ${\cal M}$ is orientable
if ${\cal N}$ is orientable, and ${\cal M}$ is spin if ${\cal N}$ is spin (see
e.g. [49] Corollary 6.2.4).

Next, we describe the completion trick (Lemma 4.4). It is a direct
generalization of the idea of sphere completion introduced in [6]; see Lemma
3.1 therein. The completion trick allows us to collapse an arbitrary subset of
spherical entanglement boundaries, each of which becomes a completion point.
See Fig. 19 for an illustration.

######  Lemma 4.4 (Completion trick).

Let $\widetilde{{\cal W}}$ be an $n$-dimensional manifold with $l$ boundary
components. ${\cal W}\subset\widetilde{{\cal W}}$ is obtained by deleting $k$
internal balls (see Fig. 19 for an illustration). ${\cal W}$ is an immersed
region, which carries a non-empty information convex set $\Sigma({\cal W})$.
Let $\partial{\cal W}=(X_{1}\cup\cdots\cup X_{k})\cup(Y_{1}\cup\cdots\cup
Y_{l})$, where $X_{i}$ and $Y_{j}$ each is a connected component. $X_{i}$s are
spherical boundaries (i.e., $X_{i}=S^{n-1}\times\mathbb{I}$) obtained by
deleting balls from $\widetilde{{\cal W}}$. For each $\rho_{{\cal
W}}\in\Sigma_{a_{1}\cdots a_{k}b_{1}\cdots b_{l}}({\cal W})$, there is a state
$\widetilde{\rho}_{\widetilde{{\cal W}}_{+}}$ (where $\widetilde{{\cal
W}}_{+}$ is a thickening of $\widetilde{{\cal W}}$) such that

  1. 1.

${{\rm Tr}}_{\widetilde{{\cal W}}_{+}\setminus{\cal
W}}\,\widetilde{\rho}_{\widetilde{{\cal W}}_{+}}=\rho_{{\cal W}}$.

  2. 2.

If $l=0$, and $\rho_{{\cal W}}\in{\rm ext}(\Sigma_{a_{1}\cdots a_{k}}({\cal
W}))$, then $\widetilde{{\cal W}}_{+}=\widetilde{{\cal W}}$ and
$\widetilde{\rho}_{\widetilde{{\cal W}}}$ is pure.

  3. 3.

Axioms A0 and A1 hold on $\widetilde{{\cal W}}_{+}$ except for possible
violation of A1 on the “completion points”. On a ball centered at the $i$th
completion point,

$\Delta(B,C,D)_{\widetilde{\rho}}=2\ln d_{a_{i}},\quad a_{i}\in{\cal
C}_{X_{i}},\quad\textrm{for the partition in Fig.~{}\ref{fig:completion-
da}(b)}.$ (4.2)

In particular, A1 holds on the $i$-th completion point if $a_{i}$ is Abelian.

Figure 19: Illustration of the completion trick (Lemma 4.4). For illustration
purposes, we consider a region ${\cal W}$ (gray) in the 3d bulk, which has 2
spherical entanglement boundaries and 3 torus entanglement boundaries. We wish
to collapse both spherical entanglement boundaries, and therefore this
corresponds to the case with $k=2$ and $l=3$ in Lemma 4.4. The dark gray area
in the leftmost figure is $\cup_{i=1}^{k}X_{i}\cup_{j=1}^{l}Y_{j}$, the
thickened entanglement boundary of ${\cal W}$. (1) Expand ${\cal W}$ passing
its entanglement boundary and obtain its thickening ${\cal W}_{+}$.
$\partial{\cal W}_{+}={\cal W}_{+}\setminus{\cal
W}=\cup_{i=1}^{k}\widetilde{X}_{i}\cup_{j=1}^{l}\widetilde{Y}_{j}$. The orange
sphere shells are $\cup_{i=1}^{k}\widetilde{X}_{i}$. The green torus shells
are $\cup_{j=1}^{l}\widetilde{Y}_{j}$. (2) Purify each spherical entanglement
boundary of ${\cal W}_{+}$ separately, where each purple box represents a
purifying system $E_{i}$. $E=\cup_{i=1}^{k}E_{i}$. (3) Collapsing each
spherical entanglement boundary (labeled by $i\in\\{1,\dots,k\\}$) to a single
site. In terms of quantum states, this is done by identifying each purifying
region $E_{i}$ and the thickened spherical entanglement boundary
$\widetilde{X}_{i}$ (which we choose to collapse) as a site. Each site so
obtained is a _completion point_. The resulting region is $\widetilde{{\cal
W}}_{+}$.

Note that we do not need to assume that the spherical entanglement boundaries
are embedded. This trick will be useful in “filling the holes” for various
immersed spherical entanglement boundaries. The key idea, as is illustrated in
Fig. 19, is (1) for an extreme point, each connected component of the
entanglement boundary can be purified separately, and (2) each spherical
boundary, together with its associated purifying region, can be collapsed to a
completion point.

######  Remark.

For most applications, it is sufficient to thicken the subset of (spherical)
entanglement boundaries that we wish to purify. Nonetheless, we choose to
thicken all entanglement boundaries, both orange and green ones in Fig. 19.
(For this reason, the final region we obtain is $\widetilde{{\cal W}}_{+}$,
which contains $\widetilde{{\cal W}}$.) This is convenient in some
applications: the extra layer can sometimes be useful when considering
information convex sets.

###### Proof of Lemma 4.4, items 1 and 2.

First, we expand ${\cal W}$ passing every entanglement boundary to obtain
${\cal W}_{+}$, so that $\partial{\cal W}_{+}={\cal W}_{+}\setminus{\cal W}$.
See Fig. 19 for an illustration. We further write ${\cal W}_{+}\setminus{\cal
W}=(\widetilde{X}_{1}\cup\cdots\cup\widetilde{X}_{k})\cup(\widetilde{Y}_{1}\cup\cdots\cup\widetilde{Y}_{l})$,
where the labels of the connected components match that of $\partial{\cal W}$.
By the Hilbert space theorem (in particular, Proposition D.4 of Ref. [5]), any
state $\rho_{{\cal W}_{+}}\in\Sigma_{a_{1}\cdots a_{k}b_{1}\cdots b_{l}}({\cal
W}_{+})$ allows a simple factorization along the thickened boundaries:

$\rho_{{\cal
W}_{+}}=\left(\bigotimes_{i=1}^{k}\rho^{a_{i}}_{{A_{i}B_{i}^{L}}}\right)\otimes\left(\bigotimes_{j=1}^{l}\rho^{b_{j}}_{A_{k+j}B_{k+j}^{L}}\right)\otimes\lambda_{(\cup_{i=1}^{k+l}B_{i}^{R})\cup\widehat{C}},$
(4.3)

where $\widehat{C}$ is ${\cal W}_{+}\setminus\cup_{i}A_{i}B_{i}$. Here, for
$i\in\\{1,\cdots,k\\}$, $A_{i}B_{i}\subset\widetilde{X}_{i}X_{i}$,
$A_{i}\supset\widetilde{X}_{i}$, and $B_{i}$ can deform to $X_{i}$ by
extensions; similarly, for $i\in\\{k+1,\cdots,k+l\\}$,
$A_{k+i}B_{k+i}\subset\widetilde{Y}_{i}Y_{i}$,
$A_{k+i}\supset\widetilde{Y}_{i}$ and $B_{k+i}$ can deform to $Y_{i}$ by
extensions.
$\mathcal{H}_{B_{i}}=(\mathcal{H}_{B_{i}^{L}}\otimes\mathcal{H}_{B_{i}^{R}})\oplus\cdots$.
Note that $B_{i}^{L}$ and $B_{i}^{R}$ do not represent subsystems in general.
The density matrices $\\{\rho^{a_{i}}_{A_{i}B_{i}^{L}}\\}$ are supported on
$\mathcal{H}_{A_{i}}\otimes\mathcal{H}_{B_{i}^{L}}$,
$\\{\rho^{b_{i}}_{A_{k+i}B_{k+i}^{L}}\\}$ are supported on
$\mathcal{H}_{A_{k+i}}\otimes\mathcal{H}_{B_{k+i}^{L}}$, and the state
$\lambda_{(\cup_{i=1}^{k+l}B_{i}^{R})\cup\widehat{C}}$ is supported on
$(\otimes_{i=1}^{k+l}\mathcal{H}_{B_{i}^{R}})\otimes\mathcal{H}_{\widehat{C}}$.
$\\{a_{i}\\}_{i=1}^{k}$ and $\\{b_{j}\\}_{j=1}^{l}$ are labels of
superselection sectors, and the states on $\rho^{a_{i}}_{{A_{i}B_{i}^{L}}}$
and $\rho^{b_{j}}_{A_{k+j}B_{k+j}^{L}}$ only depend on the choice of them;
other information of $\rho_{{\cal W}_{+}}\in\Sigma_{a_{1}\cdots
a_{k}b_{1}\cdots b_{l}}({\cal W}_{+})$ are in
$\lambda_{(\cup_{i=1}^{k+l}B_{i}^{R})\cup\widehat{C}}$. To summarize the
intuition behind Eq. (4.3), we say a “fuzzy cut” is contained in $B_{i}$ for
any $i$.

Next, we introduce a purifying system supported on $E=\cup_{i=1}^{k}E_{i}$.
Each $E_{i}$ carries a Hilbert space $\mathcal{H}_{E_{i}}$, whose dimension is
finite but large enough. For $i\in\\{1,\cdots,k\\}$ we purify
$\rho^{a_{i}}_{A_{i}B_{i}^{L}}$ with $E_{i}$ and let the resulting state be
$|\phi^{a_{i}}_{A_{i}B_{i}^{L}E_{i}}\rangle$. We topologically collapse
$\widetilde{X}_{i}E_{i}$ to a single site (i.e., a completion point). We call
the region obtained by collapsing all the $k$ sphere shells as
$\widetilde{{\cal W}}_{+}$, which is equipped with a topology associated with
topologically collapsing the entanglement boundaries into points (i.e.,
“healing the punctures”). The state we obtain can be written as

$\widetilde{\rho}_{\widetilde{{\cal
W}}_{+}}=\left(\bigotimes_{i=1}^{k}|\phi^{a_{i}}_{A_{i}B_{i}^{L}E_{i}}\rangle\langle\phi^{a_{i}}_{A_{i}B_{i}^{L}E_{i}}|\right)\otimes\left(\bigotimes_{j=1}^{l}\rho^{b_{j}}_{A_{k+j}B_{k+j}^{L}}\right)\otimes\lambda_{(\cup_{i=1}^{k+l}B_{i}^{R})\cup\widehat{C}},$
(4.4)

From this expression, we can immediately verify statement 1. Because
$\lambda_{(\cup_{i=1}^{k+l}B_{i}^{R})\cup\widehat{C}}$ is pure for extreme
points (Eq. D.9 of Ref. [5]), statement 2 holds. This completes the
verification of statements 1 and 2 of Lemma 4.4. ∎

Figure 20: About axiom A0 and the potential violation of axiom A1 centered at
a completion point. Note that $\widetilde{X}_{i}E_{i}$ is within the
completion point.

###### Proof of Lemma 4.4, item 3.

For axiom A0 on $i$-th completion point in statement 2, consider the partition
of a ball centered around the $i$-th completion point as $BC$, shown in Fig.
20(a). We can choose $BC$ so that $X_{i}\subset C$. Here the state considered
is $\widetilde{\rho}$ in Eq. (4.4). To prove
$\Delta(B,C)_{\widetilde{\rho}}=0$, we can first prove
$\Delta(X_{i},\widetilde{X}_{i}E_{i})_{\widetilde{\rho}}=0$. Then
$\Delta(B,C)_{\widetilde{\rho}}=0$ follows from the extension of the axioms
[4].

The reduced density matrices on $X_{i}\widetilde{X}_{i}E_{i}$ for
$\widetilde{\rho}_{\widetilde{{\cal W}}_{+}}$ can be directly computed from
Eq. (4.4) as
$\widetilde{\rho}_{X_{i}\widetilde{X}_{i}E_{i}}=|\phi^{a_{i}}_{A_{i}B_{i}^{L}E_{i}}\rangle\langle\phi^{a_{i}}_{A_{i}B_{i}^{L}E_{i}}|\otimes\lambda_{R}$,
where $A_{i},B_{i}$ were defined below Eq. (4.3), and $\lambda_{R}$ is
$\lambda$ reduced to the Hilbert space ${\cal H}_{B_{i}^{R}}\otimes{\cal
H}_{X_{i}\tilde{X}_{i}\setminus A_{i}B_{i}}$. Using the tensor product
structure of $\widetilde{\rho}_{X_{i}\widetilde{X}_{i}E_{i}}$ and
$\widetilde{\rho}_{X_{i}}$, it is straightforward to see that
$\Delta(X_{i},\widetilde{X}_{i}E_{i})_{\widetilde{\rho}}=0$.

Next we show that axiom A1 is violated by an amount $2\ln d_{a_{i}}$ on the
$i$-th completion point, where $d_{a_{i}}$ is the quantum dimension of the
point excitation $a_{i}\in{\cal C}_{X_{i}}$. (Note that the sphere shell may
not be embedded and therefore we use the definition of the quantum dimension
in Eq. (2.8).) The proof idea is shown in Fig. 20(b) and is similar to the use
of the Decoupling Lemma in Appendix D of [6]. The first step is to show

$\Delta(B,C,D)_{\widetilde{\rho}}=\Delta(B,C^{\prime},DD^{\prime\prime})_{\widetilde{\rho}}$
(4.5)

for the partitions illustrated in Fig. 20(b). This is true because
$\Delta(C^{\prime},D^{\prime\prime})_{\widetilde{\rho}}=0$ and
$\Delta(BC^{\prime},D^{\prime\prime})_{\widetilde{\rho}}=0$, which are
extended axiom A0 at the $i$-th completion point, as verified in the earlier
part of the proof. The second step is to prove

$\Delta(B,C^{\prime},DD^{\prime\prime})_{\widetilde{\rho}}=\Delta(B,C^{\prime},DD^{\prime})_{\widetilde{\rho}}$
(4.6)

This follows from $\Delta(D^{\prime},D^{\prime\prime}\setminus
D^{\prime})_{\widetilde{\rho}}=0$ and
$\Delta(C^{\prime}DD^{\prime},D^{\prime\prime}\setminus
D^{\prime})_{\widetilde{\rho}}=0$, where we applied axiom A0 twice again.

From the second definition of the quantum dimension of a point particle Eq.
(2.8), we see that the correction is
$\Delta(B,C^{\prime},DD^{\prime})_{\widetilde{\rho}}=2\ln d_{a_{i}}$, and it
is nonzero only when the point particle is non-Abelian. This completes the
proof. ∎

Immediate corollaries of the completion trick (Lemma 4.4) are: (1) It is
possible to complete to a sphere $\mathbf{S}^{n}$ with a reference state on a
ball $\mathbf{B}^{n}$. This is discussed in Ref. [6]. (2) It is possible to
obtain a reference state on $\underline{\underline{\mathbf{B}}}^{n}$ from a
reference state on ${\underline{\mathbf{B}}}^{n}$. In both cases, the
reference state on the compact manifolds ($\mathbf{S}^{n}$ or
$\underline{\underline{\mathbf{B}}}^{n}$) satisfies the axioms everywhere,
including the ball containing the completion point. (This also means that the
Hilbert space dimensions on compact manifolds $\mathbf{S}^{n}$ and
$\underline{\underline{\mathbf{B}}}^{n}$ are one dimensional, i.e.,
$\dim\mathbb{V}(\mathbf{S}^{n})=\dim\mathbb{V}(\underline{\underline{\mathbf{B}}}^{n})=1$.)
We illustrate the two cases in Fig. 21.

Figure 21: Sphere completion and its analog for systems with a gapped
boundary. (Explicitly shown here are the 2d cases.) They are immediate
corollaries of the completion trick. Completion points in the bulk are
illustrated as a ‘diamond’ $\Diamond$ and completion points adjacent to a
gapped boundary are illustrated as a ‘triangle’ $\triangle$. This convention
will be adopted throughout the paper.

For a given closed manifold ${\cal M}$, is it possible to determine if allows
a completion $({\cal M},{\cal
W}\overset{\varphi}{\looparrowright}\mathbf{S}^{n})$? There are a couple of
challenges to answering this question generally. The first challenge is to
understand what closed manifold ${\cal M}$ can immerse in a sphere upon
removing a ball. In space dimensions $d\leq 4$, some interesting topology
results have been obtained recently [31]. The second challenge is that even if
${\cal M}$ can immerse in a sphere, we still need to know if the information
convex set $\Sigma({\cal W})$ is non-empty. In the presence of topological
order, where there are nontrivial superselection sectors, we need to
understand if the density matrices on pieces of the immersed regions can be
consistently merged. In addition, we wish to know if there is always a special
_abelian_ superselection sector allowed on the punctures, so that they may be
healed. The general answer to this question will be explored elsewhere.
Nonetheless, the vacuum block completion, developed in §4.1, gives a
constructive answer to many interesting cases, including all orientable
manifolds in two dimensions.

Let us point out that for invertible phases (i.e., systems with only trivial
superselection sectors), the second challenge mentioned in the previous
paragraph disappears, and the task is simpler. Intriguingly, for invertible
phases, Kitaev has developed a different approach for putting the ground state
on a class of closed manifold; see [50] at around 1 hour and 35 minutes, where
the class of manifolds is referred to as normally framed manifolds. We note
that this is the same condition242424Normally framed implies that the tangent
bundle can be made trivial by adding trivial bundles. Thus the characteristic
classes must vanish. that allows a punctured manifold to be immersed in a
sphere, at least up to dimension four [31]. It is unclear to us, however, if
our approach is related to Kitaev’s approach.

###  4.1 Vacuum block completion

In this section, we consider a specific type of completion, which we shall
refer to as _vacuum block completion_. It comes with two important features.
First, the immersed region ${\cal W}\subset{\cal M}$ decomposes as ${\cal
W}=\cup_{i}{\cal V}_{i}^{\Diamond}$, where $\\{{\cal V}^{\Diamond}_{i}\\}$ is
a set of _building blocks_ , and the “gluing” between any two building blocks
is on whole entanglement boundaries. Second, on each building block ${\cal
V}^{\Diamond}_{i}$, a “vacuum” can be specified, even though ${\cal
V}^{\Diamond}_{i}$ may not be embedded in the space on which the reference
state lives.

In the rest of the paper, we will mainly be interested in the immersion of
${\cal W}$ in $\mathbf{S^{n}}$ in the study of bulk phases and the immersion
${\cal W}$ in $\underline{\underline{\mathbf{B}}}^{n}$ in cases related to the
gapped boundaries. The reason, as explained in Fig. 21, is that it is always
possible to make a reference state on these compact manifolds starting from a
reference state on a ball ($\mathbf{B}^{n}$ or $\underline{\mathbf{B}}^{n}$).
Nevertheless, we define building blocks in such a way that they have the
freedom to be immersed in balls.

####  4.1.1 Building blocks for vacuum block completion

We first give a precise definition of building blocks.

######  Definition 4.5 (Building blocks, bulk).

We say a connected region ${\cal V}^{\Diamond}\looparrowright\mathbf{S}^{n}$
(or $\mathbf{B}^{n}$) is a building block if it satisfies:

  1. 1.

$\partial{\cal V}^{\Diamond}=(\cup_{i=1}^{k}E_{i})\cup F$, where $E_{i},F$ are
connected components of $\partial{\cal V}^{\Diamond}$. $E_{i}$s are embedded,
$F$ is either empty or an immersed sphere shell ($S^{n-1}\times\mathbb{I}$);
in the former case, we may denote ${\cal V}^{\Diamond}$ as ${\cal V}$.

  2. 2.

$\Sigma_{[1_{E}]}({\cal V}^{\Diamond})=\\{\hat{\sigma}_{{\cal
V}^{\Diamond}}\\}$, has a unique element, where $E=\cup_{i=1}^{k}E_{i}$.
Furthermore, $\hat{\sigma}_{{\cal V}^{\Diamond}}$ reduces to an Abelian sector
on $F$. (We denote this Abelian sector252525Note that $\hat{1}\in{\cal C}_{F}$
is a special Abelian sector and it is an analog of the vacuum. We use
$\hat{1}$ instead of $1$ because if $F$ is not embedded, there will be no
reference state (on $F$) to compare with. We do not know if $\hat{1}$ depends
on ${\cal V}^{\diamond}$ or not for a given $F$. Below (in Example 6.5) we
will give an example where different vacuum block completions of the same
manifold produce the same abelian state on $F$. as $\hat{1}\in{\cal C}_{F}$.)

######  Remark.

Note that, in writing $\Sigma_{[1_{E}]}({\cal V}^{\Diamond})$, we are using
the notation of constrained information convex set (Definition 2.1). In other
words, we want the convex subset of $\Sigma({\cal V}^{\Diamond})$, whose
elements reduce to the vacuum on the embedded region $E=\cup_{i}E_{i}$. We
shall often drop the distinction between the abstract region ${\cal
V}^{\Diamond}$ with its immersion in the physical system, which is equipped
with a Hilbert space and needs the immersion map to specify.

The generalization of building blocks to gapped boundaries is straightforward.
The difference is that we immerse the region in
$\underline{\underline{\mathbf{B}}}^{n}$ or $\underline{\mathbf{B}}^{n}$
(instead of $\mathbf{S}^{n}$ or $\mathbf{B}^{n}$) and allow a broader sense of
immersed “spherical” entanglement boundary. These entanglement boundaries may
either lie in the bulk or adjacent to the gapped boundary.

######  Definition 4.6 (Building blocks, gapped boundary).

We say a connected region ${\cal
V}^{\Diamond/\triangle}\looparrowright\underline{\underline{\mathbf{B}}}^{n}$
(or $\underline{\mathbf{B}}^{n}$) is a building block if it satisfies:

  1. 1.

$\partial{\cal V}^{\Diamond/\triangle}=(\cup_{i=1}^{k}E_{i})\cup F$, where
$E_{i},F$ are connected components of $\partial{\cal V}^{\Diamond/\triangle}$.
$E_{i}$s are embedded. $F$ may be empty, an immersed bulk sphere shell in the
bulk or an immersed half-sphere shell262626In $n$ space dimensions, half-
sphere shell $B^{n-1}\times\mathbb{I}$ attaches to the gapped boundary at
$S^{n-2}\times\mathbb{I}$. attaches to the boundary. In these three cases, we
may alternatively denote ${\cal V}^{\Diamond/\triangle}$ as ${\cal V}$, ${\cal
V}^{\Diamond}$, or ${\cal V}^{\triangle}$.

  2. 2.

$\Sigma_{[1_{E}]}({\cal V}^{\Diamond/\triangle})=\\{\hat{\sigma}_{{\cal
V}^{\Diamond/\triangle}}\\}$, has a unique element, where
$E=\cup_{i=1}^{k}E_{i}$. Furthermore, $\hat{\sigma}_{{\cal
V}^{\Diamond/\triangle}}$ reduces to an Abelian sector
($\hat{1}\in{\mathcal{C}}_{F}$) on $F$.

At the moment, it should not be clear how hard it is to find such building
blocks. While the first requirement can be verified by looking at the
topology, the requirement on the constrained information convex set
$\Sigma_{[1_{E}]}({\cal V}^{\Diamond/\triangle})=\\{\hat{\sigma}_{{\cal
V}^{\Diamond/\triangle}}\\}$ may not be easy to check. Nonetheless, it turns
out that a large class of immersed regions can be shown to be building blocks.
Here are some examples.

######  Example 4.7 (Building blocks, bulk).

A list of examples of building blocks ${\cal V}^{\Diamond}$ satisfying
Definition 4.5:

  1. 1.

Any dimension: any connected embedded region.

  2. 2.

2d bulk: $S^{2}$ with $(k+1)$ balls removed. Here, $k$ of the spherical
boundaries are embedded, and the remaining one is not. See Eq. (4.7) for some
examples:

(4.7)

To see why these regions are building blocks, we let the thickening of the
immersed spherical boundary be $F$. Then the first condition in Definition 4.5
is verified. Some thought goes into the verification of the second condition.
We provide two methods.

     * •

We convert the immersed region to an _embedded_ $k$-hole disk by a smooth
deformation, the same as the strategy illustrated in Fig. 6. Here, we want a
deformation that keeps the $k$ embedded boundaries embedded (in
$\mathbf{S}^{n}$) in the whole process. Therefore, the sectors must still be
the vacuum when they reach the final configuration. The remaining boundary
must also carry the vacuum, according to the structure theorem of the
information convex set of embedded $k$-hole disks.

     * •

Alternatively, we use the completion trick (Lemma 4.4). The strategy is
outlined in Eq. (4.8) below.

(4.8)

First, $\Sigma_{[1_{E}]}({\cal V}^{\Diamond})$ is non-empty. This is because
we can always obtain the immersed region in question by merging a state on an
immersed disk to the vacuum of the $k$ disjoint embedded annuli ($E_{i}$).
This gives a construction of an element of $\Sigma_{[1_{E}]}({\cal
V}^{\Diamond})$. With the completion trick, we collapse the $k$ embedded
spherical boundaries and get a disk $B^{2}$ with $k$ completion points. (Shown
in Eq. (4.8) is the case $k=3$.) Because the sector is the vacuum at all these
completion points, the axioms are satisfied on them. Finally, the information
convex set on a disk has a unique element, where the boundary carries an
Abelian sector. We identify this Abelian sector as $\hat{1}$, and this
completes the argument.

  3. 3.

3d bulk: $S^{3}$ with $k+1$ balls removed. Here, $k$ of the spherical
boundaries are embedded, and the remaining one is not. This is true for the
same reason as case 2.

  4. 4.

3d bulk: some choices of regions with torus entanglement boundaries, e.g.:

(4.9)

The nontrivial feature of these regions is the existence of a torus
entanglement boundary, which cannot be collapsed to a point. We also do not
know if it is possible to deform the regions smoothly into an embedded region
and keep the torus boundary embedded in the whole process. Alternative
justification is needed. One justification comes from the dimensional
reduction method developed in Appendix E of [6]. This method maps the problem
to a problem in a 2d system with a gapped boundary. Then, according to Example
4.8 below, these 3d regions must be valid building blocks.

######  Example 4.8 (Building blocks, gapped boundary).

Here are a few examples of building blocks adjacent to the gapped boundary. In
this example, all embeddings are in $\underline{\underline{\mathbf{B}}}^{n}$,
and in the figures, we only draw part of the gapped boundary of
$\underline{\underline{\mathbf{B}}}^{n}$.

  1. 1.

Any dimension: connected embedded regions adjacent to the gapped boundary.

  2. 2.

2d with a gapped boundary: a few non-embedded examples as shown:

(4.10)

These are building blocks as can be checked with the “collapsing” argument
explained around Eq. (4.8). The crucial observation is that, after we collapse
all the embedded entanglement boundaries, the region so obtained (with the
completion points included) is a half-disk adjacent to the gapped boundary.
Thus, the remaining entanglement boundary must be in an Abelian sector.

  3. 3.

3d with a gapped boundary: a few non-embedded examples as shown:

(4.11)

It is evident that the topology of the regions satisfies the definition of
building blocks. We also need to verify the property related to the quantum
state. The first example can be shown to be a building block by the
“collapsing” argument explained around Eq. (4.8). However, the same trick does
not apply to the remaining two examples, due to the existence of an embedded
entanglement boundary that is not spherical. Nevertheless, these are building
blocks; we shall establish this fact only after we understand pairing
manifolds. Explicitly, the first purple region is precisely the
$\bar{X}^{\triangle}$ of Example 6.7 in §6.2. The second purple region is
related to the first one by the collapsing of the spherical entanglement
boundary in the bulk.

####  4.1.2 Vacuum block completion: definition and properties

We give the precise definition of vacuum block completion and then discuss its
properties. Examples will be presented in the next few sections.

######  Definition 4.9 (Vacuum block completion, bulk).

We say a manifold $\mathcal{M}$ allows a vacuum block completion $({\cal
M},\\{{\cal V}_{i}\\},{\cal
W}\overset{\varphi}{\looparrowright}\mathbf{S}^{n})$ if ${\cal
M}=\cup_{i}{\cal V}_{i}$, glued along entire boundaries, ${\cal W}\subset{\cal
M}$, and

  1. 1.

${\cal V}_{i}\cap({\cal M}\setminus{\cal W})$ is either an internal ball of
${\cal V}_{i}$ or empty.

  2. 2.

${\cal V}_{i}^{\Diamond}\equiv{\cal V}_{i}\cap{\cal W}$, equipped with the
immersion $\varphi$ restricted onto it, is a building block (by Definition
4.5), where $\partial{\cal V}_{i}^{\Diamond}=E\cup F$ with $E=\partial{\cal
V}_{i}$ embedded in $\mathbf{S}^{n}$ and $F={\cal V}_{i}\cap\partial{\cal W}$.

######  Remark.

We will also use a boundary version of vacuum block completion. We omit the
precise definition since it is obtained by replacing $\mathbf{S}^{n}$ with
$\underline{\underline{\mathbf{B}}}^{n}$ and adopting the boundary version of
building blocks (Definition 4.6).

######  Proposition 4.10.

Any vacuum block completion $({\cal M},\\{{\cal V}_{i}\\},{\cal
W}\overset{\varphi}{\looparrowright}\mathbf{S}^{n})$ gives a completion of
${\cal M}$. Furthermore, there is a special element $\hat{\sigma}_{{\cal W}}$
of $\Sigma({\cal W})$. It is the unique state in $\Sigma({\cal W})$ that
reduces to $\hat{\sigma}_{{\cal V}_{i}^{\Diamond}}$ for any building block
${\cal V}_{i}^{\Diamond}$. We call $\hat{\sigma}_{{\cal W}}$ the canonical
state on ${\cal W}$.

###### Proof.

First, since ${\cal V}_{i}^{\Diamond}$ is a building block, it has a state
$\hat{\sigma}_{{\cal V}_{i}^{\Diamond}}$ in its information convex set. We can
merge these states, after expanding the regions (${\cal V}_{i}^{\Diamond}$)
slightly on their shared embedded hypersurfaces (i.e., embedded entanglement
boundaries). This merging is possible because these states all carry the
vacuum sector on these shared embedded hypersurfaces, and they also have the
quantum Markov state structure required in the merging theorem. The merged
state on ${\cal W}$ is an extreme point $\hat{\sigma}_{{\cal
W}}\in\Sigma_{\hat{1}}({\cal W})$. Here $\hat{1}$ refers to the fact that the
state carries an Abelian sector $\hat{1}$ on every (spherical) entanglement
boundary of ${\cal W}$. ∎

######  Definition 4.11 (Canonical state on ${\cal M}$).

For a given vacuum block completion $({\cal M},\\{{\cal V}_{i}\\},{\cal
W}\overset{\varphi}{\looparrowright}\mathbf{S}^{n})$, call the completion of
the canonical state $\hat{\sigma}_{{\cal W}}$ of ${\cal W}$ (defined in
Proposition 4.10) on ${\cal M}$ (by the completion trick, Lemma 4.4) as the
canonical state on ${\cal M}$. We denote this pure canonical state as
$|1_{\\{{\cal V}_{i}\\}}\rangle$.

The immediate consequences of this definition are (1) $|1_{\\{{\cal
V}_{i}\\}}\rangle$ reduces to $\hat{\sigma}_{{\cal V}_{i}^{\Diamond}}$ for any
building block ${\cal V}_{i}^{\Diamond}$, and (2) axioms A0 and A1 holds
everywhere on ${\cal M}$ for $|1_{\\{{\cal V}_{i}\\}}\rangle$.

Note that the state $|1_{\\{{\cal V}_{i}\\}}\rangle$ is not unique.
Nevertheless, it is unique up to tensoring in extra product degrees of freedom
and applying local unitaries $\\{U_{i}\\}$ on the completion points. Taking
these operations as equivalence relations, we say $|1_{\\{{\cal
V}_{i}\\}}\rangle$ is canonical.

###  4.2 Examples of vacuum block completion

####  4.2.1 Vacuum block completion in the 2d bulk

Every 2d orientable surface admits a vacuum block completion:

######  Proposition 4.12.

Every 2d orientable manifold allows a vacuum block completion.

We construct each case as an example; they comprise a constructive proof of
the Proposition. We leave the generalization of this statement to
nonorientable manifolds to the interested reader. The trick is to consider the
completion in a non-orientable ${\cal N}$.

######  Example 4.13 (Sphere $S^{2}$).

We already have a reference state on $S^{2}$, and therefore not much needs to
be done. For pedagogical reasons, we discuss two methods. The first method is
to pick one building block ${\cal V}=\mathbf{S}^{2}$. There is no completion
point, and ${\cal W}={\cal M}=\mathbf{S}^{2}$. The second method is to pick
two building blocks $\\{{\cal V}_{1},{\cal V}_{2}\\}$, both of which are balls
obtained by cutting the sphere in halves. In this construction, again, we do
not need any completion points. As the reader should expect, the state
completed on ${\cal M}$ is precisely the reference state on $\mathbf{S}^{2}$,
for both constructions.

######  Example 4.14 (Torus $T^{2}$).

For the torus ${\cal M}=T^{2}$, we need two building blocks $\\{{\cal
V}_{1},{\cal V}_{2}^{\Diamond}\\}$ and one completion point. ${\cal V}_{1}$ is
an embedded annulus. ${\cal V}_{2}^{\Diamond}$ is a 2-hole disk, immersed in
such a way that two of the three entanglement boundaries are embedded. Here is
an illustration:

(4.12)

The essence of vacuum block completion is that we can choose the vacuum state
on ${\cal V}_{1}$ and merge it with a unique state on ${\cal
V}_{2}^{\Diamond}$, so that the state has a special Abelian sector ($\hat{1}$)
on $\partial{\cal W}$. Then we heal the puncture to obtain a state on $T^{2}$
that satisfies the axioms everywhere.

Figure 22: The vacuum block completion for a genus-$g$ surface $\\#g\,T^{2}$.
The case $g$=2 is shown explicitly. ${\cal V}_{1}$ is a $g$-hole disk. ${\cal
V}_{2}^{\Diamond}$ is an immersed $(g+1)$-hole disk.

######  Example 4.15 (genus-$g$ surface $\\#g\,T^{2}$).

The genus-$g$ surface is the connected sum of $g$ tori, and therefore we
denote it as $\\#g\,T^{2}$. The vacuum block completion for

${\cal M}=\\#g\,T^{2}=\underbrace{T^{2}\\#\cdots\\#T^{2}}_{\text{$g$ times}},$
(4.13)

can be done in multiple ways. Consider the way shown in Fig. 22. This
construction needs two building blocks $\\{{\cal V}_{1},{\cal
V}_{2}^{\Diamond}\\}$, and there is one completion point. ${\cal V}_{1}$ is an
embedded $g$-hole disk, and ${\cal V}_{2}^{\Diamond}$ is an $(g+1)$-hole disk
with $g$ of its entanglement boundaries embedded.

####  4.2.2 Vacuum block completion in the 3d bulk

In this section, we provide examples of vacuum block completions in the 3d
bulk.

######  Example 4.16 (3-sphere $S^{3}$).

This can be done in a way analogous to the completion of $S^{2}$ in Example
4.13. Essentially, nothing needs to be done, given that we have a reference
state on the 3-sphere already.

######  Example 4.17 ($S^{2}\times S^{1}$).

The vacuum block completion of ${\cal M}=S^{2}\times S^{1}$ can be constructed
as follows.

(4.14)

This construction needs two building blocks $\\{{\cal V}_{1},{\cal
V}_{2}^{\Diamond}\\}$, and there is one completion point. Here ${\cal
V}_{1}=S^{2}\times\mathbb{I}$ is an embedded sphere shell, and ${\cal
V}_{2}^{\Diamond}$ is a ball minus two balls. ${\cal W}={\cal M}\setminus
B^{3}$. (In Eq. (4.14), ${\cal M}$ is illustrated in such a way that a pair of
$S^{2}$ are identified as indicated.) Note that this is a close analog of
vacuum block completion of the torus in 2d, considered in Example 4.14.

######  Example 4.18 ($\\#g(S^{2}\times S^{1})$).

Here $g\geq 2$ is an integer. It is possible to choose two building blocks
$\\{{\cal V}_{1},{\cal V}_{2}^{\Diamond}\\}$, and there is one completion
point, as the illustration below shows.

(4.15)

Here, ${\cal V}_{1}$ is embedded and it is homeomorphic to a ball with $g$
balls removed. ${\cal V}_{2}^{\Diamond}$ is a ball with $g+1$ balls removed,
immersed in such a way that $g$ of its spherical entanglement boundaries are
embedded.

######  Remark.

In many of the examples above, the immersed building block can be deformed to
an embedded region keeping all (but one) entanglement boundary embedded in the
whole deformation process. If we keep track of the deformation, we can further
use the isomorphism theorem to compute the Hilbert space dimensions
$\dim\mathbb{V}_{a}({\cal W})$ and $\dim\mathbb{V}({\cal M})$. We omit the
details. As we shall explain later, the pairing manifold provides another way
to compute the Hilbert space dimensions. Examples will be given in §6.

####  4.2.3 Vacuum block completion in 2d with gapped boundaries

Entanglement bootstrap works for gapped boundaries, the setup and axioms are
reviewed in §3.2. Below, we provide examples of vacuum block completion in
this physical context.

######  Example 4.19 (Disk with an entire gapped boundary
$\underline{\underline{\mathbf{B}}}^{2}$).

Let ${\cal M}=\underline{\underline{\mathbf{B}}}^{2}$. For vacuum block
completion, nothing needs to be done here, given that we already have a
reference state on $\underline{\underline{\mathbf{B}}}^{2}$; see Example 4.13
for an analog. (Recall that the reference state on
$\underline{\underline{\mathbf{B}}}^{2}$ can come from the boundary analog of
the sphere completion; see Fig. 21.)

######  Example 4.20 (A finite cylinder with a pair of gapped boundaries).

Let ${\cal M}$ be the finite cylinder $S^{1}\times\mathbb{I}$, with a pair of
gapped boundaries. Here $\mathbb{I}=[0,1]$ is a finite interval. (Note that
the cylinder with gapped boundaries is not a sectorizable region because we
cannot smoothly deform the region back to ${\cal M}$ if we trace out the
region along the boundary. This is in contrast with the annulus in the bulk.)
The vacuum block completion of ${\cal M}$ needs two building blocks $\\{{\cal
V}_{1},{\cal V}_{2}^{\triangle}\\}$, and a completion point, illustrated as
follows.

(4.16)

Here ${\cal V}_{1}$ is a half-annulus embedded in the disk
$\underline{\underline{\mathbf{B}}}^{2}$. ${\cal V}_{2}^{\triangle}$ has two
embedded entanglement boundaries and its third entanglement boundary is not
embedded. From the construction, we see that the two gapped boundaries are of
the same type because they are copies of the same boundary of the disk
$\underline{\underline{\mathbf{B}}}^{2}$.

Another vacuum block completion of the cylinder still uses two building blocks
$\\{\widetilde{{\cal V}}_{1},\widetilde{{\cal V}}_{2}^{\triangle}\\}$ and has
one completion point. This is illustrated as follows.

(4.17)

In this construction, which is different from the one above, $\widetilde{{\cal
V}}_{1}$ is the annulus that covers the entire gapped boundary, and it is
embedded in $\underline{\underline{\mathbf{B}}}^{2}$. $\widetilde{{\cal
V}}_{2}^{\triangle}$ is a “mushroom”, which has an embedded entanglement
boundary in the bulk and an immersed entanglement boundary that touches the
gapped boundary.

######  Example 4.21 (A sphere with $k$ disks removed, with $k$ gapped
boundaries).

Let ${\cal M}$ be a sphere with $k$ disks removed, where all the boundaries
are gapped boundaries. The vacuum block completion of ${\cal M}$ can be done
with $k+1$ building blocks, $\\{{\cal V}_{j}\\}_{j=1}^{k}\cup\\{{\cal
V}^{\Diamond}_{k+1}\\}$ and a completion point $\Diamond$ within the bulk, as
follows:

(4.18)

Here $\\{{\cal V}_{j}\\}_{j=1}^{k}$ are “identical copies” of the annulus that
covers the entire boundary, embedded in
$\underline{\underline{\mathbf{B}}}^{2}$. ${\cal V}_{k+1}^{\Diamond}$ is an
immersed region within the bulk, homeomorphic to a $(k+1)$-hole disk, and it
is immersed in such a way that $k$ of its $k+1$ entanglement boundaries are
embedded.

Since the construction in Eq. (4.18) is sufficiently nontrivial, we recall the
basic intuition hidden in the abstract formulation of vacuum block completion
for this example. At an intuitive level, how do we obtain ${\cal M}$ by gluing
the building blocks? First, we need to deform the $k$ copies of the boundary
annulus $\\{{\cal V}_{j}\\}_{j=1}^{k}$ such that their entanglement boundaries
match the positions of the embedded entanglement boundaries of ${\cal
V}^{\Diamond}_{k+1}$. Then we glue these entanglement boundaries. The
remaining entanglement boundary is in the bulk, and it is then filled using
the completion trick, resulting in the completion point in the bulk.

######  Remark.

It is also possible to construct vacuum block completions for nonplanar
regions with gapped boundaries. These are 2d regions that cannot be embedded
in a sphere, which have only gapped boundaries but no entanglement boundaries.
One example is illustrated below, and we leave the explicit construction to
interested readers.

(4.19)

####  4.2.4 Vacuum block completion in 3d with gapped boundaries

The setup of the following examples is the 3d entanglement bootstrap with
gapped boundaries, as stated in §3.2.

######  Example 4.22 ($\underline{\underline{\mathbf{B}}}^{3}$, the ball with
an entire gapped boundary).

As a region equipped with a reference state, we can simply take one building
block ${\cal V}=\underline{\underline{\mathbf{B}}}^{3}$. No completion point
is needed here..

######  Example 4.23 (Solid torus with an entire gapped boundary).

Let ${\cal M}$ be a solid torus with an entire gapped boundary. It is possible
to have a vacuum block completion of ${\cal M}$ with two building blocks
$\\{{\cal V}_{1},{\cal V}_{2}^{\triangle}\\}$, and it has a completion point
lying adjacent to the boundary, as illustrated below.

(4.20)

Here, ${\cal V}_{1}$ is a half-sphere shell adjacent to the gapped boundary.
${\cal V}_{2}^{\triangle}$ is a half-sphere shell with a $\underline{B}^{3}$
removed, and it is immersed in $\underline{\underline{\mathbf{B}}}^{3}$ in
such a way that two of its three entanglement boundaries are embedded. In
fact, the topologies of these regions are closely related to those that
appeared in Example 4.17, the vacuum block completion of $S^{2}\times S^{1}$
in the 3d bulk. To see the relation, we notice that every region (and the
completion point) that appeared in this example is some region in Eq. (4.14)
cut in half by the “plane of reflection symmetry”.

######  Remark.

Following the same idea, we can have a vacuum block completion for the
genus-$g$ handlebody, whose gapped boundary is a genus-$g$ surface. We left
the details as an exercise for the interested reader. An alternative (and more
efficient) way of constructing these manifolds follows from the “sequential
completion” technique discussed in §4.3. The idea is to use regions
constructed from simpler manifolds to assemble more interesting topologies.

######  Example 4.24 (Sphere shell $S^{2}\times\mathbb{I}$ with two gapped
boundaries).

The vacuum block completion of $S^{2}\times\mathbb{I}$, a sphere shell in 3d
with two spherical gapped boundaries, is shown below.

(4.21)

In this construction, we used three building blocks $\\{{\cal V}_{1},{\cal
V}_{2},{\cal V}_{3}^{\Diamond}\\}$. ${\cal V}_{1}$ and ${\cal V}_{2}$ are
sphere shells that cover an entire gapped boundary. They are embedded in
$\underline{\underline{\mathbf{B}}}^{3}$, and they are identical copies up to
deformations. Therefore, the two gapped boundaries must match in their type.
The third building blocks ${\cal V}_{3}^{\Diamond}$ is a ball with two balls
removed, within the bulk, and it is immersed in such a way that two of its
three entanglement boundaries are embedded.

How do we obtain ${\cal M}$ topologically? The idea is that we “glue” the
entanglement boundary of ${\cal V}_{1}$ (${\cal V}_{2}$) to an embedded
entanglement boundary of ${\cal V}_{3}^{\Diamond}$. An important detail is
that we need to deform the regions slightly to get the position right! The
completion point in the bulk is obtained by collapsing the immersed
entanglement boundary of ${\cal V}_{3}^{\Diamond}$.

######  Remark.

Following the same idea that we used in Example 4.21, we can obtain a vacuum
block completion for the connected sum of $k$ solid balls. It has $k$ disjoint
spherical gapped boundaries of the same type. (The $k=2$ case recovers Example
4.24.) We simply need $k$ copies ${\cal V}_{1}$ shown in Eq. (4.21) and
generate the building block in the bulk (${\cal V}_{3}^{\Diamond}$ of Eq.
(4.21)) in an obvious way.

###  4.3 Other examples of completion in ${\cal N}$

Here we provide a few examples of completions in ${\cal N}$, which are either
not vacuum block completion or not obviously so.

######  Example 4.25 (Exotic immersion of punctured torus).

Let ${\cal M}=T^{2}$ be a torus. Let ${\cal W}=T^{2}\setminus B^{2}$ be a
torus with a disk removed, i.e., a punctured torus. As we have discussed,
${\cal W}$ can be immersed in $\mathbf{S}^{2}$; see Fig. 18. Moreover, ${\cal
W}$ cannot be embedded in $\mathbf{S}^{2}$ since every embedded region is
homeomorphic to a $k$-hole disk. Here we consider another immersion ${\cal
W}\looparrowright\mathbf{S}^{2}$, as depicted in Fig. 23.

Figure 23: Another immersion of the punctured torus to $\mathbf{S}^{2}$. We
think it is not regular homotopic to the one shown in the rightmost of Fig.
18.

It is an inequivalent immersion compared with the one shown in Fig. 18. This
immersion has a non-empty information convex set $\Sigma({\cal W})$, and
therefore $(T^{2},{\cal W}\overset{\varphi}{\looparrowright}\mathbf{S}^{2})$
is a completion. However, we do not know if the immersion ${\cal
W}\looparrowright\mathbf{S}^{2}$ in Fig. 23 allows a decomposition of ${\cal
W}$ into building blocks. (The other immersion does, as is discussed in
Example 4.14 and Eq. (4.12).)

######  Example 4.26 (Torus with an anyon).

Let ${\cal M}=T^{2}$ and ${\cal W}=T^{2}\setminus B^{2}$. We take the
immersion ${\cal W}\looparrowright\mathbf{S}^{2}$ shown in Fig. 18, which is
the immersion that allows a vacuum block completion (4.12). However, for some
topologically ordered systems, there are states $\rho^{a}_{{\cal W}}\in{\rm
ext}(\Sigma_{a}({\cal W}))$, where $a$ is a superselection sector of anyon
that can exist on the puncture $\partial{\cal W}$, which is different from
$\hat{1}$, the sector obtained in the vacuum block completion. By the
completion trick, we can then obtain a state $|\varphi^{a}\rangle$ on ${\cal
M}$.

The most interesting case is when $a$ is Abelian (and different from
$\hat{1}$). In this case, the state $|\varphi^{a}\rangle$ on ${\cal M}$
satisfies axioms A0 and A1 everywhere on ${\cal M}$. A simple model where this
happens is the Ising anyon model. ${\cal C}=\\{1,\sigma,\psi\\}$, where $\psi$
is Abelian and $\sigma$ is non-Abelian with $d_{\sigma}=\sqrt{2}$ (which
should not be confused with the reference state $\sigma$). If we take
$|\varphi^{\psi}\rangle$ as the reference state on $T^{2}$, one can verify

$\dim\mathbb{V}(T^{2})=\dim\mathbb{V}_{\psi}({\cal
W})=1,\quad|\varphi^{\psi}\rangle\textrm{ as the reference state on $T^{2}$.}$
(4.22)

This is different from the answer we obtain from the vacuum block completion,
which gives $\dim\mathbb{V}(T^{2})=|{\mathcal{C}}|=3$, where the reference
state on $T^{2}$ is taken to be the canonical state associated with the vacuum
block completion; we postpone the explanation of this fact in Example 6.1 in
§6.1.

######  Remark.

The Hilbert space dimension in both cases has a nice TQFT interpretation; see,
e.g., Appendix E of [15]. In the Ising anyon model, we omitted the subtlety
that the model is chiral. Its chiral central charge is $c_{-}=1/2$. It is
likely that the axioms can only hold approximately, although we expect the
errors of the axioms to decay towards zero at large length scales in realistic
models with finite correlation length, e.g., [15]. This phenomenon described
in Example 4.26 also happens in zero correlation length models; see, e.g.,
Appendix E of [51] and [52].

######  Example 4.27 (When ${\cal N}$ is an annulus with a defect line passes
through).

Let ${\cal N}$ be an annulus, and suppose that a defect line passes through
it; see the gray annulus in Fig. 24. Let ${\cal M}=T^{2}$ be the torus and
${\cal W}={\cal M}\setminus B^{2}$. with the immersion ${\cal
W}\looparrowright{\cal N}$ in Fig. 24(a) and (b) by $\varphi_{1}$ and
$\varphi_{2}$ respectively.

Figure 24: Completion in ${\cal N}$ of a torus. Here ${\cal N}$ is an annulus
with a defect line passes through. ${\cal W}=T^{2}\setminus B^{2}$. (a) ${\cal
W}\overset{\varphi_{1}}{\looparrowright}{\cal N}$, and the decomposition of
${\cal W}$ into two building blocks. (b) ${\cal
W}\overset{\varphi_{2}}{\looparrowright}{\cal N}$, and the decomposition of
${\cal W}$ into two building blocks.

When a defect exists, ${\cal N}$ cannot be identified with any annulus
subsystem of the sphere $\mathbf{S}^{2}$. (Recall that when we call a sphere
$\mathbf{S}^{2}$ rather than $S^{2}$, we assume that there is a reference
state on it, which satisfies A0 and A1 everywhere.) This also prevents the
identification of the punctured tori ${\cal W}$ in Fig. 24(a) and (b) as
regions immersed in $\mathbf{S}^{2}$. While the punctured tori ${\cal W}$ in
the figures are divided into two building blocks, this example does not
qualify as a vacuum block completion.

We explore the analog and distinction further. Interestingly, each subset of
${\cal W}$ (i.e., ${\cal V}_{1}$ or ${\cal V}_{2}^{\Diamond}$) shown in Fig.
24(a) and (b) is still a building block. The reason is that each piece is
contained in a disk. (While the 2-hole disk ${\cal V}_{2}^{\Diamond}$ is not
immersed in any embedded disk, it is immersed in an immersed disk!) The reason
this construction is not a vacuum block completion is that the union ${\cal
W}$ is not immersed in a disk or sphere. Nonetheless, following the technique
behind vacuum block completion, we can learn a few things about this special
type of completion in ${\cal N}$.

  1. 1.

The vacuum state on the building block can merge. The resulting state has an
Abelian sector $\hat{1}$ on $\partial{\cal W}$.

  2. 2.

After healing the puncture, we have a state on $T^{2}$ which satisfies the
axioms everywhere. The Hilbert space dimension $\dim\mathbb{V}(T^{2})$ for
this reference state can be different from that of models without defects. For
instance, if we take the defect to be the “duality wall” of the toric code
[53], we should expect $\dim\mathbb{V}(T^{2})=2$, not 4.

The immersions in Fig. 24 generalize into a class of immersions that differ
from one another by the “winding number” around the defect annulus; those
shown in Fig. 24(a) and (b) correspond to the winding number equal to 1 and 2.

######  Example 4.28 (Sequential completion).

Here is another way to obtain states on a nontrivial manifold that satisfies
the axioms everywhere. The idea is that we first start from a ball
$\mathbf{S}^{n}$ or $\underline{\underline{\mathbf{B}}}^{n}$ and construct a
state on a relatively simple manifold ${\cal M}_{1}$, by vacuum block
completion. Then, we make use of pieces of density matrices on ${\cal M}_{1}$
to construct more interesting manifolds ${\cal M}_{2}$. We can do it this
repeatedly and obtain states on ${\cal M}_{3},{\cal M}_{4},\cdots$. We call
this approach _sequential completion_.

Sequential completion is interesting for a few reasons. First, it gives a
sequence of completions of ${\cal M}_{i+1}$ in ${\cal M}_{i}$. Second, the
states so obtained satisfy the axioms everywhere on closed manifolds ${\cal
M}_{i+1}$. More interestingly, as long as we avoid the completion points in
${\cal M}_{i}$ in constructing the state in ${\cal M}_{i+1}$, the construction
can be translated into an immersion of ${\cal M}_{i+1}$ with punctures into
$\mathbf{S}^{n}$ or $\underline{\underline{\mathbf{B}}}^{n}$. For this reason,
sequential completion can produce completions.

Below we explain the details of constructing a state on ${\cal M}_{i+1}$ from
${\cal M}_{i}$, using the idea of sequential completion. We are able to do the
connected sum in the bulk and adjacent to the gapped boundary.

Case 1, connected sums in the bulk: Here we obtain $\\#g{\cal M}$ from ${\cal
M}$, where ${\cal M}$ is compact, connected, and may or may not have
boundaries. (The way to obtain $\\#gT^{2}$ from $T^{2}$ is illustrated in Fig.
25.) The idea is to remove a ball from ${\cal M}$ and call the resulting
region $\mathcal{P}$. We take $g$ copies of $\mathcal{P}$ and denote the them
separately as $\\{{\cal P}_{i}\\}_{i=1}^{g}$. We pick ${\cal V}$, a building
block immersed in the ball, which is topologically a ball with $g$ balls
removed, and it is immersed in such a way that $g$ of its $g+1$ entanglement
boundaries are embedded. We merge the reduced density matrix of a state on
${\cal M}$ to the regions $\\{{\cal P}_{i}\\}_{i=1}^{g}$ and the state
$\hat{\sigma}_{{\cal V}}$, such that the merging glues whole entanglement
boundaries. The end result is a state on $\\#g{\cal M}$ with a puncture in the
bulk. The sector on the spherical entanglement boundary is Abelian (in fact, a
special Abelian sector, which we referred to as $\hat{1}$). Therefore, we can
heal the puncture as usual and obtain a state on $\\#g{\cal M}$.

Figure 25: Sequential completion to obtain $\\#gT^{2}$ from $T^{2}$. The case
$g=2$ is illustrated explicitly.

Case 2, connected sums near the boundary: We can also obtain $\natural g{\cal
M}$ from ${\cal M}$, where ${\cal M}$ is compact, connected, and with a
(connected) gapped boundary. (The case of obtaining a genus-$g$ handlebody
with a gapped boundary from a solid torus with a gapped boundary is
illustrated in Fig. 26.) Similarly, we have a set of regions $\\{{\cal
P}_{i}\\}_{i=1}^{g}$, which are “identical copies” obtained by removing a ball
$\underline{B}^{3}$ from the solid torus. We further need a region ${\cal V}$,
which is a building block adjacent to the gapped boundary. ${\cal V}$ is a
ball minus $g$ balls cut in half by the gapped boundary: it has $g+1$ half-
sphere entanglement boundaries; it is immersed in
$\underline{{B}}^{n}\subset{\cal M}$ in such a way that $g$ of these
entanglement boundaries are embedded. We can therefore attach entire
entanglement boundaries of $\mathcal{P}_{i}$ to ${\cal V}$ and obtain
$\natural g{\cal M}\setminus\underline{B}^{n}$. Then we heal the puncture to
obtain the closed manifold $\natural g{\cal M}$.

Figure 26: Sequential completion, to obtain a genus-$g$ handlebody with a
gapped boundary. The case $g=2$ is illustrated.

It is interesting to notice that the partition of all regions in Fig. 26,
restricted to the neighborhood of the gapped boundary, look similar to regions
in Fig. 25.

##  5 Pairing manifolds

Compact manifolds without entanglement boundaries are interesting in many
ways. For instance, groundstates on the torus in 2d encode the mutual braiding
of anyons [54, 55] (see [56] and [16] for other thoughts). In higher
dimensions, one may hope that certain closed manifolds can provide insight
into why one class of excitations should be paired with another class of
excitations that braid nontrivially with it; this is termed as _remote
detectibility_ [9, 7, 11] and assumed for physical reasons. Moreover, as the
existence of graph excitations (Fig. 9) shows, the understanding of remote
detectability should require _a class of_ compact manifolds. An important
motivation we have in mind is to justify remote detectability in entanglement
bootstrap for general dimensions.

For these purposes, we introduce a class of connected compact manifolds called
_pairing manifolds_. Each of these pairing manifolds can be constructed using
vacuum block completion, discussed in §4.1. Pairing manifolds satisfy a few
extra conditions, from which we derive concrete information-theoretic
constraints. We shall find many examples, in which the physical meaning of