Remote detectability from entanglement bootstrap I: Kirby's torus trick

Dear Editor, we have updated our draft, and we believe this version addresses all the questions raised by the referee. We are grateful to the referee for the detailed reading of the draft and the excellent questions and suggestions sent in the referee report. Please see below for the reply to all the referee comments.

\subsection{Strengths} \begin{it} 1- The paper gives an entanglement-based understanding of the idea of remote detectability of excitations in quantum phases exhibiting topological order, in 2 and 3 spatial dimensions. \end{it}

\begin{it} 2- The paper provides a systematic construction of quantum states on space manifolds with nontrivial topologies starting from a reference state on a topologically trivial space. The case of 2 spatial dimensions is dealt with conclusively, while a lot of progress is made towards the case of 3 spatial dimensions. This lends concrete evidence to the longstanding lore in topological order literature that says that all topological data (including, e.g., modular $S,T$ matrices that encode anyon braiding in 2+1d) should be recoverable from a single ground state wave function or density matrix, defined on a sphere. \end{it}

\begin{it} 3- The paper introduces the notion of "pairing matrices" which are closely related to modular S and T matrices of 2+1d topological orders. In 3+1d, these are morally similar to the mapping class group representations of 3-manifolds discussed in the literature, although there are key differences which are highlighted in the paper. \end{it}

\begin{it} 4- Figures are thoughtfully included throughout the paper, which greatly aid in the understanding of many of the topological manipulations. \end{it}

We thank the referee for the careful reading of our paper and for the positive assessment.

\subsection{Weaknesses} Below we reply in detail to each of the points raised by the referee.

\begin{it} 1- The authors have made clear efforts to make their paper largely self-contained. However, in some parts, the text is still not very welcoming to readers who are not already familiar with their past work on entanglement bootstrap. Some examples are identified in the Report below, along with some questions that may aid in clarifying some conceptual jumps. \end{it}

We are grateful for the helpful suggestions, which we address below.

\begin{it} 2- A significant chunk of the paper [section 3.2 (pp 30-34), definition 4.6 (pg 44), example 4.8 (pg 46), sections 4.2.3-4.2.4 (pp 51-54), section 6.2 (pp 85-89), non-examples 6.10 (pg 93) and 6.16 (pp 97-98)] discusses systems with gapped boundaries. However, it seems that almost all of the discussions in these sections are straightforward generalizations of the case with no gapped boundaries. The overall presentation would be clearer if these cases could instead be put in an appendix that is devoted to all issues associated with gapped boundaries. \end{it}

We thank the referee for the helpful suggestion, which we considered carefully. In the end, we felt that hiding all of the discussion of gapped boundaries in an appendix would not significantly improve the paper or the reader's experience. The examples involving gapped are intended to illustrate concepts appearing in various places in the body of the paper, and putting them all in one appendix would undermine that purpose. Instead, we added an explanation in the introduction indicating which parts of the paper may be skipped by a reader who might not want to read discussions about gapped boundaries. % for some reason.
Further, we added a label ($\star$) to the subsections about gapped boundaries to indicate that they may be skipped.

\begin{it} 3- Even though the authors argue that the unitarity of the pairing matrix implies remote detectability, this connection is not made clear. In particular, they write in section 7 (pp 98-99), "[...] the pairing matrix, denoted as $S$, is always unitary, which shows that the pair of excitations remotely detect each other". It is not clear in this description what remote process achieves such detactability. Further clarification on this point is crucial to provide clarity to one of the most important results (as the authors themselves note) obtained in the paper. \end{it}

We agree with the referee that the connection between remote detectability of topological excitations and the unitarity of the pairing matrix is not made completely clear in this paper.
In fact, this is precisely the reason for the symbol I" in the title. The main purpose of parts II and III is to explain this connection more completely. In part II we give an alternative point of view on the information in the information convex set in terms of a certain algebra of operators. These operators may regarded as transporting the topological excitations around closed paths. In part III we learn tocut open" these operators, so that they actually create the topological excitations. In this way, we can directly relate the braiding of the topological excitations to the pairing matrix.
Unfortunately the story is rather long, due to the generality we aimed at, and we felt that it would make it more digestible to break it into several parts.

To clarify this situation, we have amplified the part of the discussion where we raise this point, to include a digested version of the main idea behind parts II and III. This now appears as subsection 7.2.

\subsection{Report, Part 1}

\begin{it} The primary technique developed in this paper is a recipe to construct ground states on topologically nontrivial spatial manifolds (in 2 and 3 dimensions) starting from a single reference state on a topologically trivial region, either a ball or a sphere, by making use of immersions (i.e. local embeddings). The paper also discusses the case where the system has gapped boundaries in which case the reference state is defined on a ball with a gapped boundary. They provide a recipe for stitching together different "blocks" that makes use of various theorems proven earlier by (some or all of) the authors. A special class of topologically nontrivial manifolds, "pairing manifolds", are presented. Two sets of basis states can be constructed on each pairing manifold, and (roughly) the overlap between these two bases defines the "pairing matrix" for the topological order on that manifold. This matrix is unitary, which the authors argue indicates remote detectability of topological excitations that are dual to certain "building blocks" that construct those manifolds.

The paper builds on the authors' program of entanglement bootstrap, which is of great conceptual value, as noted in Strength $#2$. However, as noted in Weakness $#1$, there are some conceptual gaps at various places that we would like to request clarification on: \end{it}

1.1) \emph{On pp 18-19, the authors discuss fusion multiplicity and fusion spaces. Could they describe what these are? These concepts are not explained anywhere prior to this in the paper and they make many appearances throughout the rest of the paper.}

These terms are defined in section 2, though in the previous version the origin of the name was probably opaque.
We have added a sentence of explanation there: ``The origin of this name is the case where $\Omega$ is the 2-hole disk, and these numbers are associated with the fusion of two anyons into a third.''

1.2) \emph{1.2.a) In the proof of Lemma 2.2 (statement 1), the authors state "The compactness follows from that of $\Sigma(\Omega)$." Is this because these subsets of $\Sigma(\Omega)$ are closed? If so, it's not obvious why that is the case. Could the authors please clarify?}

The referee's reading is correct. We added a sentence clarifying what we meant: ``In more detail, the convex set $\Sigma(\Omega)$ is compact, and the subsets are defined by a set of constraints that are linear equalities."

\emph{1.2.b) In the proof of Lemma 2.2 (statement 2), it is not obvious why "The extreme point label $\kappa$ on $A$ induces a label $\Phi(\kappa)$ of the extreme points of $\Sigma(\partial A)$ " and why "any element of $\Sigma_{I[\kappa_A]}(\partial A)$ is obtained by merging some element of $\Sigma_{I\Phi(\kappa)}(\tilde{E})$ with the same state $\rho^\kappa_A$": could the authors please clarify these points further?}

Thanks for helping to identify where our proof can be clarified.
We've added a sentence of explanation for the first statement: This follows from the structure theorem for the information convex set. " and also for the second statement:The state is a Markov state on $\widetilde A, \partial{A}, E$ whose marginal on $ A$ is $\rho^\kappa_A$."

1.3) \emph{The authors refer to "topological defects" in footnote 23 (page 36) as well as under Example 4.27. Could they clarify what exactly these defects are? Are they symmetry defects, or string/membrane operators, or something else entirely?}

Indeed, the term topological defects" is used in too many different ways by different people. Bytopological defects" in this context, we meant extrinsic defects, such as the point defect in the toric code on which the duality wall ends. We have added a few sentences of explanation where this term first appears.

1.4) \emph{On page 45, the last sentence of the first bullet point states "The remaining boundary must also carry the vacuum, according to the structure theorem of the information convex set of embedded k-hole disks."-- does the "structure theorem" refer to one of the two structure theorems described in section 2 (pp 18-19)?}

Yes, ``structure theorem" refers to the structure theorem described in section 2 (pp 18-19). We also used the fusion rules of particles (anyons). We made it more explicit.

1.5) \emph{ In the proof of Proposition 4.10, how does one see that the $\hat{\sigma}_{\mathcal{V}_i}$ states "have the quantum Markov state structure"? }

Indeed it follows from the first part of the sentence, which wasn't clear from our wording. We have improved the sentence to ``This merging is possible because these states all carry the vacuum sector on these shared embedded hypersurfaces, and therefore, they also have the quantum Markov state structure required in the merging theorem."

\emph{Also, why is the "merged state on $\mathcal{W}$ [is] an extreme point" of $\Sigma_{\hat{1}}(\Omega)$?}

Indeed, this follows from Lemma 2.20 (used to prove the Associativity Theorem) of our reference [6]. We added a sentence to direct the reader's attention there.

1.6) \emph{In the proof of lemma 5.7, the reasoning behind the statements "This implies that $\rho_Y^{|\Psi\rangle}$ is the maximum-entropy state of all states consistent with the state $|\Psi\rangle$ on balls $AB$ and $BC_Y$." and "[...] $\rho_Y^{|\Psi\rangle}$ must be the maximum-entropy state in $\Sigma(Y)$" are not clear. Could the authors please elaborate on these?}

We have added a footnote to remind the reader that: The basic statement we use here is that any quantum Markov state $\lambda_{XYZ}$ is the maximum entropy state among all states that agrees with $\lambda$ on marginals $XY$ and $YZ$.

1.7) \emph{Under the last bullet point on page 67, the statements "This is sufficient to guarantee that the superselection sector on the spherical boundary of $X \cup Y$ is Abelian. The minimization of entropy follows." Could the authors please elaborate on their reasoning behind these? It does not seem entirely obvious.}

We agree that that paragraph was not a model of clarity.
We have changed the end of the paragraph to explain the following:

Axiom {\bf A0} applied to the ball $\CM \setminus (X \cup Y)$ implies the extreme point condition for the state on $X \cup Y$. An extreme point is a minimum entropy state.

Moreover, this is sufficient to guarantee that the superselection sector on the spherical boundary of $X\cup Y$ is Abelian, because the information convex set of the ball has only one element. Therefore, the entropy is the absolute minimum.

1.8) \emph{In the proof of Lemma 5.8, how exactly does the natural partition condition for the state $|\Psi_X(\mathcal{M})\rangle$ imply eq (5.4)?}

Only the equality is the natural partition condition, the inequality is SSA. We rewrote the sentence to make this clear: The natural partition condition (Def.~5.2) implies \begin{equation} 0 = \Delta(B',X, D'){\lambda} \geq I(A' :X | B')~ \end{equation} where the inequality is a form of SSA.

1.9) \emph{In the remark preceding Proposition 5.10, the authors note that "we can use Eq. (2.6) because $\partial X$ is embedded in $\mathbf{S}^n$". Do we assume that all the connected components of $\partial X$ are embedded? This certainly is not always true for building blocks, according to Definition 4.5. Could the authors please clarify what they mean in the quoted sentence? A similar statement is also made in the discussion preceding eq (7.4); does that have the same explanation?}

Indeed, for a building block there can be an immersed boundary. But in the given context, $X$ is participates in a pairing manifold, and therefore is part of a vacuum block completion, which is a building block with the immersed boundary filled in. So indeed all components of its boundary are embedded.
Indeed the same explanation applies to the discussion around equation (7.4). We added a comment to clarify.

1.10) \emph{ Below eq (6.4), the authors state that the multiplicities in eq (5.8) are either 0 or 1 since X and Y are sectorizable. Could they please elaborate on why this is the case? }

By "multiplicities" here we mean fusion multiplicities. By definition, sectorizable regions can only be associated with fusion multiplicities 0 or 1.
We added a sentence in section 2 to review this point.

\subsection{Report, Part 2}

{\it There are a few other points of confusion which we would request the authors' comments on:}

2.1) \emph{ In footnote 9, the authors state that the axioms A0 and A1 may not be exact for chiral phases. In what way do they fail? Is it related to correlation lengths diverging?}

For chiral groundstates, it is generally believed that there is a tension between zero correlation length and a tensor product Hilbert space with finite-dimensional factors.
We added a comment to this effect.

2.2) \emph{What are the labels $a_L$ and $a_R$ in eq (2.5)? Do they take values in $\mathcal{C}{\partial\Omega_L}$ and $\mathcal{C}$?}

The referee is nearly correct. $a_L, a_R$ are the labels on the components of the boundary of $\Omega$ before it is cut in half.
We have attempted to clarify our use of the labels $a_L$ and $a_R$.

2.3) \emph{In the Remark above Lemma 2.2, the notation $\Omega$ seems to be replaced by $\mathcal{M}$. Is that intended? Since the authors refer to the subregion $A$, perhaps they could clarify how $\mathcal{M}$ is related to $A$ and $\Omega$.}

Indeed, that discussion is about the special case of a region $\Omega = \CM$ which is a closed manifold. We added a reminder.

2.4) \emph{On page 24, first paragraph of section 3, the authors state "For non-sectorizable regions, information about fusion spaces can be detected; in the case of a nontrivial fusion space, the information is quantum.": could the authors please explain what they mean by this? In particular, what exactly does the information being "quantum" mean in this context.}

By saying that "the information [in nontrivial fusion spaces of the information convex set of non-sectorizable regions] is quantum" we mean that it cannot be copied by any linear operation, in contrast to the classical information that is found in the information convex set of sectorizable regions. We have added some words to make our meaning more concrete.

2.5) \emph{In the description of the 3d $S_3$ quantum double (Example 3.1), the quantum dimension of the pure flux loops are stated to be $1, \sqrt{2}, \sqrt{3}$. This seems a bit confusing since the quantum dimensions of the counterparts of these excitations in 2d are $1,2,3$, respectively (equal to the sizes of the corresponding conjugacy classes). Is this difference due to some difference in definitions?}

Yes, the discrepancy is due to an annoying clash of definitions of "quantum dimension". It arises because for a sectorizable region we can associate a quantum dimension either to the region itself or to its thickened boundary.
We have added a sentence making a note about our definition.

2.6) \emph{ In statement 2 of Lemma 4.4, what does $\widetilde{\mathcal{W}}_+= \widetilde{\mathcal{W}}$ mean? How are they equal if one is obtained from the other by thickening of the boundary.}

It's a trick question! Statement 2 refers to the case $l=0$, where there is no entanglement boundary at all, so no thickening. We rephrased the statement to avoid this seeming paradox.

2.7) \emph{What does $\sigma_Y$ mean in condition 2(b) of Lemma 5.5? Is this the reduced density matrix on Y obtained from the reference state $\sigma$?}

The referee's reading is correct, $\sigma_Y$ is the reduced density matrix on $Y$ of the reference state. This is well-defined because, by the assumption of Lemma 5.5 [see 1(d)], $Y$ is embedded. We have added a note there.

2.8) \emph{What does the notation $\dim \mathbb{V}(\mathcal{M})$, used e.g. in eq (5.8) mean? Does it refer to the Hilbert space of ground states on the manifold $\mathcal{M}$?}

Consistent with our use of the symbol $\mathbb{V}(\Omega)$ for other regions $\Omega$, $\dim \mathbb{V}(\mathcal{M})$ denotes the dimension of the Hilbert space of the information convex set of $\mathcal{M}$. (Usually we say $\dim \mathbb{V}_I(\Omega)$. Here, we omitted the lower index $I$ because it only allows to be the vacuum sector due to $\mathcal{M}$ being a closed manifold.) Indeed, we expect that this number is equal to the number of groundstates of the given topological order on $\mathcal{M}$, given a suitable reconstructed (parent) Hamiltonian.

2.9) \emph{The notation in eq (6.4) is somewhat unclear. Do $\mathcal{C}{\rm point}$ and $\mathcal{C}$ refer to ${\cal C}X$ and ${\cal C}_Y$ or ${\cal C}$ and ${\cal C}_{\partial Y}$?}

Indeed there is a possible ambiguity in the notation ${\cal C}{\rm point}, {\cal C}$.
In the revision, we specify near equation (6.4) that they refer to ${\cal C}_X$ and ${\cal C}_Y$, respectively.

%\emph{I believe the results obtained in the paper are interesting. The paper is suitable for publication in SciPost Physics, provided the authors address questions above and the requested changes below.}

\subsection{Requested changes}

\begin{it} Major changes: \begin{enumerate} \item I suggest the authors move the discussions associated with gapped boundaries to an appendix. Section 6.3 may also be better suited to an appendix. I request the authors to consider that possibility.

{\rm We seriously considered this possibility. In the end, we decided that for the benefit of the reader completely uninterested in gapped boundaries, it would suffice to label the relevant sections with an asterisk. Moving all mention of gapped boundaries would have the drawback that the context of each part of the discussion would be lost. }

\item Please address the requests for clarification in the Report above.

{\rm We strongly believe we have implemented all the required clarifications in the updated version. }

\item Please elaborate on the relation between the unitarity of pairing matrix and remote detectability (more details provided in Weakness $#3$).

{\rm We hope the added subsection summarizing the rest of the proof is a sufficient appetizer to hold the reader over until we can finish parts II and III.} \end{enumerate}

Minor changes: \begin{enumerate} \item Some additional figures would be really helpful to better understand the second bullet point under the second example of bulk building blocks described under Example 4.7. In particular, an illustration of the following would be very illuminating: "[...] obtain the immersed region in question by merging a state on an immersed disk to the vacuum of the k disjoint embedded annuli".

We updated the figure in Eq.~(4.9). Now, it shows the step of merging. Indeed, we find this makes the discussion illustrated better.

\item The sentence above Proposition 4.12 is redundant and should be removed.

{\rm Done.}

\item Please use a darker green font for the phrase "this color" in the discussion following eq (6.1). The current one makes the words rather hard to read on a white background.

{\rm Done.}

\item In the last sentence of the Remark above Example 6.2, "the relation to" is redundant.

{\rm Removed.}

\end{enumerate} \end{it}

About gapped boundaries: We added an explanation in the introduction indicating which parts of the paper may be skipped by a reader who might not want to read discussions about gapped boundaries. Further, we added a label ($\star$) to the subsections about gapped boundaries to indicate that they may be skipped.

Below are explicit changes in each section:

In section 2:

we have added a sentence of explanation about fusion multiplicity and fusion spaces: ``The origin of this name is the case where $\Omega$ is the 2-hole disk, and these numbers are associated with the fusion of two anyons into a third.''

In the proof of Lemma 2.2, we added a sentence clarifying a consequence related to compactness: ``In more detail, the convex set $\Sigma(\Omega)$ is compact, and the subsets are defined by a set of constraints that are linear equalities."

In the proof of Lemma 2.2, we've added a sentence of explanation for the first statement:
``This follows from the structure theorem for the information convex set. " and also for the second statement:
``The state is a Markov state on $\widetilde A, \partial{A}, E$ whose marginal on $ A$ is $\rho^\kappa_A$."

In section 2, we made more explicit what we mean by fusion rules near the structure theory.

We added a sentence in section 2 to review the meaning of multiplicities. We published footnote 9 to make it more clear.

Below Eq.(2.5), we added an explanation:
$a_L, a_R$ are the labels on the components of the boundary of $\Omega$ before it is cut in half.

In section 3:

At the beginning of section 3, we made footnote 18 as a concrete statement on what we mean by ``the information is quantum".

In section 4:

We amplified footnote 24 with an explanation of what we mean by ``topological defects".

On page 48, We improved a sentence to ``This merging is possible because these states all carry the vacuum sector on these shared embedded hypersurfaces, and therefore, they also have the quantum Markov state structure required in the merging theorem."

We rewrite item 2 of Lemma 4.4 in such a way that it takes less effort to decode its meaning.

We updated the figure in Eq.~(4.9). Now, it shows the step of merging.

In section 5:

We added a footnote on Page 67. See Footnote 29.

We added a clarification to the last bullet point on page 68.

Eq.(5.4) is edited for clarification purposes.

The remark before Proposition 5.10 is polished.

In Section 6:

We specify near equation (6.4) that they refer to ${\cal C}_X$ and ${\cal C}_Y$, respectively.

In Eq.(6.1) we made the green darker.

In Section 7:

In subsection 7.2, we now amplify the discussion of remote detectability to include a digested version of the main idea behind parts II and III.