Entanglement Classification in the Graph States: The generalization to $n$-Qubits States using the Entanglement Matrix

Sameer Sharma

REVIEW 4 major objections 4 minor 1 cited by

Entanglement Classification in the Graph States: The generalization to $n$-Qubits States using the Entanglement Matrix

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Maximum entanglement of complete $n$-qubit graph states is exactly piecewise quadratic, with a boost when $n$ is a multiple of 12.

desk verdict The Entanglement Matrix is a bookkeeping device, not a quantum measure, and it overcounts the 3-qubit GHZ state by a factor of two. read the letter →

arxiv 2507.11458 v1 pith:P2SHEIBO submitted 2025-07-15 quant-ph

Sameer Sharma This is my paper · ORCID

classification quant-ph MSC 81P4081P68

keywords graphstatesentanglementmatrixvonNeumannentropymultipartiteclassificationcontrolled-Zgatemidpointdegreequadraticscaling

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Graph states are the resource states behind quantum error correction, quantum communication, and measurement-based computation, so a closed-form maximum entanglement gives a concrete target for state preparation. The paper introduces the Entanglement Matrix, a symmetric array whose entries record bipartite entanglement between edge midpoints of a graph state, and uses it to classify how much entanglement a fully connected $n$-qubit graph state can hold. Its central claim is that the maximum entanglement, counted in units of $\log 2$ per contributing line through a midpoint, is exactly quadratic in $n$: $E_{\max}=n^2-n$ for odd $n$, $5n^2/4-3n/2$ for even $n$ with $C/2$ an integer, $5n^2/4-2n$ for even $n$ with $C/2$ non-integer, and $5n^2/4$ for even $n$ that are multiples of 12, where $C=n/2-1$ is the number of concentric circles of midpoints. If correct, this gives a complete analytical classification of the most entangled class of $n$-qubit graph states and identifies $n$ divisible by 12 as a special family with enhanced entanglement. The author notes that the reason for the multiples-of-12 anomaly remains an open question.

What carries the argument

The Entanglement Matrix is an $M\times M$ symmetric matrix, with $M=\binom{n}{2}$ for odd $n$ and $M=n(n/2-1)+1$ for even $n$, whose entries are entanglement contributions between edge midpoints. Primary midpoints (edges between adjacent qubits) have degree 2 and fill the first $n\times n$ block, while secondary midpoints (edges between non-adjacent qubits) have degree $d\ge2$ and fill only diagonal entries, each line through a midpoint contributing one unit of $\log 2$. The total entanglement is the sum of the diagonal plus one triangular half, so the whole maximum-entanglement calculation reduces to counting midpoints by degree on concentric circles, with the central midpoint of degree $n$ in even graphs.

What would settle it

Take an even $n$ not covered by the paper's tables, such as $n=14$ or $n=20$, draw the complete graph with vertices in convex position, and list every edge midpoint with the number of edges passing through it; if the alternating 2/4 degree pattern, the central degree-$n$ midpoint, or the single exceptional midpoint fails, the piecewise quadratic formulas and the mod-12 anomaly collapse.

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Extended reading notes

Core claim

The discovery the author claims is that maximum entanglement of complete $n$-qubit graph states, expressed through the Entanglement Matrix with von Neumann entropy, is fixed by a simple degree-counting rule: each midpoint of an edge contributes $\log 2$ per line passing through it, and the total is the sum of the matrix diagonal and one triangular half. Because odd-$n$ complete graphs produce only degree-2 midpoints, this sum evaluates to $n^2-n$. Even-$n$ complete graphs add concentric rings of midpoints whose degrees alternate between 2 and 4 plus a central midpoint of degree $n$, producing larger quadratic values; when $n$ is a multiple of 12, one ring midpoint acquires degree 6 or 8 and the formula becomes exactly $5n^2/4$. The four cases together form the claimed maximum-entanglement classification.

Load-bearing premise

The even-$n$ formulas assume an unproved geometric extrapolation: in a complete graph drawn with vertices on a circle, edge midpoints lie on concentric circles with degrees alternating between 2 and 4 from the outside inward, a central midpoint of degree $n$, and, for $n$ a multiple of 12, exactly one exceptional midpoint of degree 6 or 8.

Editorial extensions

If this is right

If the formulas are correct, every $n$-qubit graph state has entanglement at most the piecewise quadratic bound, and the complete graph state attains it.
Even-qubit complete graph states carry more entanglement than odd-qubit ones at the same $n$, with the gap growing roughly as $n^2/4$.
Graph states with $n$ a multiple of 12 form a distinct maximum-entanglement class, reaching $5n^2/4$; the paper's Table IV predicts degree-6 replacement for odd multiples and degree-8 replacement for even multiples.
The Entanglement Matrix organizes small non-isomorphic graph states into separable, bi-separable, entangled, and fully entangled classes, as tabulated for three and four qubits.
Because the paper cites an atomic-ensemble experimental realization of graph states, the midpoint-degree counting could be translated into a certificate for maximum entanglement in prepared states.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The Entanglement Matrix total adds contributions from many bipartitions, so its quadratic scaling is a classification score rather than the von Neumann entropy of any single bipartition; comparing it to standard entropy bounds requires care.
The mod-12 pattern is a number-theoretic prediction that can be tested independently of quantum mechanics by enumerating edge midpoints of complete graphs on $n=60$ and $n=72$ vertices in convex position.
The same midpoint-degree counting could extend to non-complete graphs or to the Renyi and Tsallis entropies the paper lists but does not apply, though the paper does not develop these extensions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear Colleague,

The short version: this paper's central result does not hold. The 'Entanglement Matrix' is a bookkeeping device, not a quantum-mechanical calculation, and the paper's own numbers fail on the smallest nontrivial case. For the three-qubit complete graph (the GHZ-class triangle), the correct von Neumann entanglement, summed over the three 1-vs-2 cuts, is 3 (each reduced state has eigenvalues 1/2,1/2). Table I classifies this as 'fully entangled' with total entanglement 6 log2 – a factor of two overcount. Since Eq. (2) and every even-N formula are built from the same rule (every edge contributes log2, every secondary midpoint of degree d contributes (d/2) log2), the quadratic growth and the mod-12 enhancement are consequences of definitions, not of the quantum state.

What is genuinely new: the Entanglement Matrix construction, the primary/secondary midpoint distinction, and the explicit piecewise formulas do not appear in the cited literature. The paper is organized, the tables and figures are clear, and the author honestly notes that the mod-12 replacement 'reason is still not clear.' That honesty is welcome, but it does not rescue the derivation. The 'Bipartite Entanglement Calculation' in Section IV rests on an unproven geometric claim about concentric circles and alternating midpoint degrees 2 and 4, extrapolated from Table III to all even N. No partial trace is ever performed. The special multiples-of-12 case is an ad hoc patch, and the concluding experimental paragraph cites a specific Nature Physics result without establishing any concrete connection to the Entanglement Matrix.

The known four-qubit graph-state classification (Hein et al., cited in the paper) should have been a check; instead the paper defines its own categories without comparing spectra or entropies. That citation pattern is a red flag, though the paper does cite the standard references.

Bottom line: this is not a paper a serious journal should send to referees. It is a self-contained formalism that is internally coherent but ungrounded, and it is numerically wrong for N=3. I would desk-reject it, with a note pointing to the K3 check. If the author ever derives the matrix entries from reduced density matrices, the mod-12 result might become interesting, but as it stands it is numerology.

I don't want to spend group time on it. My vote: reject.

Referee Report

4 major / 4 minor

Summary. The paper introduces an 'Entanglement Matrix' formalism for n-qubit graph states. Midpoints of graph edges are classified as primary or secondary, each matrix entry is assigned a von Neumann entropy contribution (log2 2 for primary midpoints and (d/2) log2 2 for secondary midpoints of degree d), and total entanglement is defined as the sum of the diagonal and one triangular part of this matrix. The paper claims that maximum entanglement is N^2 - N for odd N, and for even N is given by piecewise quadratic formulas, with an enhanced value 5N^2/4 when N is a multiple of 12. A classification of three- and four-qubit non-isomorphic graph states is also presented.

Significance. If correct, the paper would provide a simple closed-form quantification of maximum entanglement in graph states, with interesting number-theoretic structure. The paper does provide a clear organization of non-isomorphic classes for small N and tabulates midpoint-degree data for larger N. However, the central claim is not derived from the quantum state: no reduced density matrix is computed for any graph state beyond N=3, and the paper's own admission that the multiples-of-12 feature has 'reason... still not clear' indicates the absence of a physical derivation. The main quantitative result is already contradicted by the simplest complete graph state, so the contribution cannot be considered valid in its current form.

major comments (4)

[Section III, Step 3 and discussion after Fig. 4] The Entanglement Matrix entries are posited rather than derived. The text asserts that each CZ edge contributes log2 2 and that a secondary midpoint of degree d contributes (d/2) log2 2, but no partial trace, Schmidt decomposition, or eigenvalue calculation is performed for any graph state. Since total entanglement is then defined as the sum of diagonal plus one triangular block of this matrix, all formulas in Eqs. (2)-(10) are arithmetic consequences of these definitions rather than predictions from quantum mechanics.
[Table I, Class 4] For the 3-qubit complete graph state (the GHZ-class state), tracing out any two qubits yields the reduced density matrix diag(1/2,1/2), so the von Neumann entropy of a 1-vs-2 bipartition is 1; summing the three single-qubit cuts gives 3. Table I instead reports 6 log2 2 for the fully entangled class, overcounting by a factor of two already at N=3. This is a quantitative failure of the proposed bookkeeping, not merely a missing proof.
[Section IV, 'Bipartite Entanglement Calculation' and Table III] The even-N formulas in Eqs. (3)-(6) rely on the unproven geometric claim that, in a complete graph drawn with vertices in convex position, midpoint degrees alternate between 2 and 4 on concentric circles and the central midpoint has degree N. The pattern is inferred from Table III for a few small N and extrapolated to all even N. If this midpoint-degree pattern fails for larger N, the piecewise quadratic expressions and the distinction between C/2 integer and non-integer cases collapse; no proof of this geometric claim is supplied.
[Section IV, 'A Special case of Even graph States' and Table IV] The multiples-of-12 case is handled by introducing an ad hoc replacement of one midpoint degree (2 by 6, or 4 by 8), with the paper stating that the reason is 'still not clear.' Equations (7)-(10) then adjust the counts so that the final Emax becomes 5N^2/4. This is not a derived prediction but a fitting procedure: the replacement term is chosen to preserve the desired quadratic form, and there is no independent check against actual entanglement entropies of the corresponding graph states.

minor comments (4)

[Section III] The notation for the size of the Entanglement Matrix is inconsistent: the text says it is n×n with n the total number of midpoints, but in the 4-qubit example the matrix is 5×5 while the number of qubits is 4.
[Section IV, Eq. (2)] The text refers to 'Equation (1)' when citing the odd-graph formula, but Eq. (1) is the definition of von Neumann entropy; the intended reference is Eq. (2).
[References] References [5] and [12] cite Wikipedia and an arXiv preprint for graph isomorphism, respectively; more standard sources would be appropriate for these textbook notions.
[Throughout] There are numerous typographical issues, including 'T otal', 'EM ax', 'R´enyi', and inconsistent use of log2 versus log2 2; a careful proofreading pass is needed.

Circularity Check

2 steps flagged · score 8.0 of 10

Emax formulas reduce to counting entries of a matrix whose values are assigned by definition; even-N and mod-12 results are read off from small-N tables, not computed from partial traces.

self definitional [Section III ('Adjacency Matrix') and Section IV ('Maximum Entanglement in Odd Graph States', Eq. (2))]
"For a secondary midpoint, 'S' of degree 'd' and multiplicity m = d/2, the 'SS' entry (diagonal entry), of the entanglement matrix will be m · log2 2. ... wherever there is an edge connecting two qubits via a CZ gate, the von Neumann entanglement of that edge is log2 2 = 1. ... The total entanglement captured by primary midpoints will be: E_Max = N(N+1)/2 × log2 2 + [N(N−1)/2 − N] × log2 2 = N^2 − N (2)."

The construction first assigns every primary-midpoint matrix entry the constant log2 2 (i.e., 1) and every secondary midpoint of degree d the value (d/2) log2 2. It then defines the total entanglement as the sum of the diagonal plus one triangular half of this matrix. Equation (2) is therefore a pure count of matrix entries with pre-assigned constants: N(N+1)/2 plus the number of secondary diagonal slots. No reduced density matrix of the stated graph state is ever partially traced and no S(ρ_A) is evaluated. The numbers that go into the sum are identical to the numbers that come out, so the quadratic growth of E_max is a bookkeeping identity, not a quantum-mechanical prediction.
fitted input called prediction [Section IV, 'Bipartite Entanglement Calculation' and 'A Special case of Even graph States' (Eqs. (3)-(10); Tables III and IV)]
"Now, from Table III, we observe that, midpoints of degree 2 and 4 occur alternatively on the concentric circles beginning from the outermost primary midpoint of degree 2. ... A more detailed observation from the Table III reveals that whenever the total number of qubits is even multiple of 12 then we have a midpoint of degree 4 replaced by a midpoint of degree 8... The reason why this happens is still not clear."

The even-N piecewise formulas and the multiples-of-12 anomaly are obtained by importing an alternating degree-2/4 pattern and the degree-6/8 replacement directly from Table III/IV into the counting rule. The final 'combined result' is thus a compact parameterization of the observed small-N examples, not a derivation from the von Neumann entropy of the state. The author's admission that the replacement 'reason... is still not clear' confirms that the mod-12 term is fitted to the observation rather than forced by quantum mechanics. Any subsequent claim to 'predict' enhanced entanglement at N = 12k restates the input table.

full rationale

The central derivation chain in Sections III-V is circular in the strongest sense: the Entanglement Matrix entries are assigned numerical values by definition (log2 2 per CZ edge, (d/2)log2 2 per secondary midpoint), and 'maximum entanglement' is then defined as the sum of those entries. Equations (2)-(10) are algebraic consequences of these assignments and of the geometric counts N and C, so they cannot be read as independent predictions about graph states. The discrepancy with the actual partial-trace von Neumann entropy is already visible at N=3: the complete triangle state has single-qubit reduced density matrix diag(1/2,1/2) for each cut, giving total 3, while Table I reports 6 log2 2 for the fully entangled class; this confirms that the matrix values are imposed, not computed. The even-N and mod-12 formulas additionally rest on unproved geometric pattern extrapolations from Tables III and IV, and the paper itself flags the missing justification ('The reason why this happens is still not clear'). I do not assign circularity to the references: the citations to graph-state literature are used for background and not as load-bearing support for the central formula. The result is not externally benchmarked against standard entanglement measures, so the claimed quadratic growth and multiples-of-12 enhancement are forced by the paper's own definitions rather than by quantum mechanics. Score 8 is appropriate because the main claim reduces to the input definition, though the paper does contain independent counting work that justifies not giving a 10.

Assumptions & free parameters 2 free parameters · 6 assumptions · 2 invented entities

The paper's results rest on a set of ad hoc rules for filling the Entanglement Matrix. The per-entry constant log2 2 and the degree-to-entanglement conversion are chosen by hand, and the geometric degree patterns for even graphs are extrapolated from small examples. No standard theorem from graph state entanglement is used to derive these rules.

free parameters (2)

Per-unit entanglement contribution = log2 2 = 1 ebit
Every primary-block entry and every line through a secondary midpoint is assigned entropy log2 2 with no derivation. This constant is the elementary unit of the entire Entanglement Matrix calculation.
Multiplicity coefficient for midpoint degree = m = d/2
The diagonal contribution of a secondary midpoint of degree d is set to (d/2) log2 2 by definition, converting line counts into entanglement numbers.

assumptions (6)

ad hoc to paper All entries of the N by N primary block of the Entanglement Matrix equal log2 2.
Stated in Section IV, Case 1, that each primary midpoint pair defines a bipartition with the same eigenvalues and entropy log2 2. No proof is given and it is not connected to the graph state's actual density matrix.
ad hoc to paper Total entanglement is the sum of the diagonal and one triangular part of the Entanglement Matrix.
Defined in Section III. Additivity over matrix entries is asserted, not derived from standard entanglement measures.
domain assumption In even graph states, midpoint degrees on concentric circles alternate between 2 and 4, with a central midpoint of degree N.
Extrapolated from small examples in Table III; no general proof is provided for arbitrary even N.
ad hoc to paper For N a multiple of 12, exactly one midpoint degree changes from 2 to 6 or 4 to 8, and this preserves the total entanglement.
The author states 'The reason why this happens is still not clear', so this is an ad hoc observation used to patch the formulas.
domain assumption The maximum entanglement occurs for the complete graph on N vertices.
Plausible for monotone entanglement measures but not proved for the Entanglement Matrix.
ad hoc to paper Primary midpoints contribute to off-diagonal entries; secondary midpoints contribute only to diagonal entries.
This is a design choice in the matrix construction algorithm, not a consequence of quantum mechanics.

invented entities (2)

Entanglement Matrix
purpose: A matrix whose entries are meant to quantify entanglement between midpoints; its summed entries define total entanglement.
The matrix is introduced by the authors and is not shown to correspond to any standard entanglement measure or to yield falsifiable predictions beyond its own definitions.
Primary and secondary midpoints
purpose: Geometric objects used to populate the Entanglement Matrix; primary midpoints are assigned bipartition roles and secondary midpoints diagonal contributions.
The distinction and the assignment of entanglement values to midpoints are new constructs not derived from graph state theory.

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Cite this review

Pith. "Pith review of Entanglement Classification in the Graph States: The generalization to $n$-Qubits States using the Entanglement Matrix." pith.science (2026). https://pith.science/paper/P2SHEIBO

@misc{pith2026250711458,
  author       = {Pith},
  title        = {Pith review of: Entanglement Classification in the Graph States: The generalization to $n$-Qubits States using the Entanglement Matrix},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P2SHEIBO}},
  note         = {Machine review of arXiv:2507.11458}
}

abstract

Graph states represent a significant class of multi-partite entangled quantum states with applications in quantum error correction, quantum communication, and quantum computation. In this work, we introduce a novel formalism called the Entanglement Matrix for quantifying and classifying entanglement in n-qubit graph states. Leveraging concepts from graph theory and quantum information, we develop a systematic approach to analyze entanglement by identifying primary and secondary midpoints in graph representations, where midpoints correspond to controlled-Z gate operations between qubits. Using Von Neumann entropy as our measure, we derive precise mathematical relationships for maximum entanglement in graph states as a function of qubit number. Our analysis reveals that entanglement follows a quadratic relationship with the number of qubits, but with distinct behaviors for odd versus even qubit systems. For odd n-qubit graph states, maximum entanglement follows $E_{\max} = n^2 - n$, while even n-qubit states exhibit higher entanglement with varying formulae depending on specific configurations. Notably, systems with qubit counts that are multiples of 12 demonstrate enhanced entanglement properties. This comprehensive classification framework provides valuable insights into the structure of multi-qubit entanglement, establishing an analytical foundation for understanding entanglement distribution in complex quantum systems that may inform future quantum technologies.

Figures

Figures reproduced from arXiv: 2507.11458 by the authors.

**Figure 2.** FIG. 2. Eleven non-isomorphic classes for four qubit graph [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3. Graph States with 5 and 6 Qubits. The midpoints [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: FIG. 4. 4 Qubit Graph State [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: FIG. 5. Total midpoints and number of midpoints of degree [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6. Maximum Entanglement in case of Odd Graph States. [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 7.** Figure 7: FIG. 7. Example of a graph state with even number of Qubits [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

**Figure 8.** Figure 8: FIG. 8. Maximum Entanglement in case of Even Graph [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: FIG. 9. Maximum Entanglement captured in [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

**Figure 10.** Figure 10: FIG. 10. Maximum Entanglement as a function of N qubits. [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

**Figure 11.** Figure 11: FIG. 11. Maximum Entanglement as a function of N even [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Absolutely maximally entangled pure states of multipartite quantum systems
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Reference graph

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Isomorphic Graphs Consider two simple graphs, G1 = (V1, E1) and G2 = (V2, E2). These graphs are isomorphic if there exists a bijective function f : V1 → V2 such that two vertices vi, vj are adjacent in G1 if and only if f (vi) and f (vj) are adjacent in G2 [11]

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(either clock- wise or anticlockwise) to get an ordered structure of qubits, and then take the midpoints of all the possible edges joining two qubits

Consider a graph state of n qubits; we label the qubits in numerical order as 1, 2, 3, ... (either clock- wise or anticlockwise) to get an ordered structure of qubits, and then take the midpoints of all the possible edges joining two qubits. So, for a graph state with n qubits (n - nodes), the maximum num- ber of edges possible will be n 2 . So, there can...

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This be- gins with 1′, denoting the midpoint between qubits 1 and 2, and continues progressively as 2′, 3′,

Once we have identified the midpoints, beginning from primary midpoints, we start numbering them again, either clockwise or anticlockwise. This be- gins with 1′, denoting the midpoint between qubits 1 and 2, and continues progressively as 2′, 3′, . . . , n′, where n represents the total number of qubits. Subsequently, the secondary midpoints, connecting n...

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En- tanglement matrix is a n × n square symmetric ma- trix, where n is the total number of midpoints in a given graph

We begin creating the entanglement matrix. En- tanglement matrix is a n × n square symmetric ma- trix, where n is the total number of midpoints in a given graph. Now, we join two primary midpoints (i′ and j′), to divide the graph into two subsystems, such that each subsystem has same set of eigenval- ues pi, from which we can calculate the entropy of form...

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For example, {1’1’} will give us the entanglement of the edge joining 1 and 2

The {i’i’} entry will give us the entanglement of the corresponding edge on which midpoint i′ lies. For example, {1’1’} will give us the entanglement of the edge joining 1 and 2

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Primary Midpoints contribute to both diagonal and off-diagonal entries of the Entanglement Matrix. Whereas Secondary midpoints can only contribute to the diagonal entries. For secondary mid-midpoints possessing a degree ( d), greater than 2, characterizing individual edge contribu- tions to the overall entanglement becomes challenging. The in-distinguisha...

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So, all the entries of this submatrix of the entanglement matrix of any graph of N Qubits ( N = Odd) will be log 2 (Considering von-Neumann Entropy)

Case 1: Entanglement captured by the primary midpoints: For a graph state of N qubits, the first N × N block belongs to the primary midpoints. So, all the entries of this submatrix of the entanglement matrix of any graph of N Qubits ( N = Odd) will be log 2 (Considering von-Ne...
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Case 2: Entanglement captured by Secondary midpoints: These midpoints do not contribute to the off- diagonal elements of entanglement matrix and so, the entanglement captured by secondary midpoints will belong to diagonal entries of the entanglement matrix. We know the size of...
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We also have a central midpoint of degree N

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Case 1: When C 2 is an integer: In this case, C 2 = C 2 = C 2 : Entanglement 9 from Secondary midpoints: (N · C 2 − N ) × log2 2 + (N · C 2 ) × 2 × log2 2 = (N · C 2 − N ) + 2 · (N · C 2 ) = N 2 N 2 − 1 − N + 2 N 2 N 2 − 1 = N 2 4 − N 2 − N + 2N 2 4 − 2N 2 = 3N 2 4 − 5N 2 (3) ...
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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Isomorphic Graphs Consider two simple graphs, G1 = (V1, E1) and G2 = (V2, E2). These graphs are isomorphic if there exists a bijective function f : V1 → V2 such that two vertices vi, vj are adjacent in G1 if and only if f (vi) and f (vj) are adjacent in G2 [11]

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[2] [2]

This means that there is no bijection between their vertices that preserves the edge structure [12]

Non-Isomorphic Graphs If two graphs are not isomorphic, they are considered non-isomorphic. This means that there is no bijection between their vertices that preserves the edge structure [12]. Non-isomorphic graphs are distinct graphs that can- not be transformed into each other by a simple relabelling of vertices while preserving the edge connections. Fo...

arXiv 2025

[3] [3]

(either clock- wise or anticlockwise) to get an ordered structure of qubits, and then take the midpoints of all the possible edges joining two qubits

Consider a graph state of n qubits; we label the qubits in numerical order as 1, 2, 3, ... (either clock- wise or anticlockwise) to get an ordered structure of qubits, and then take the midpoints of all the possible edges joining two qubits. So, for a graph state with n qubits (n - nodes), the maximum num- ber of edges possible will be n 2 . So, there can...

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[4] [4]

This be- gins with 1′, denoting the midpoint between qubits 1 and 2, and continues progressively as 2′, 3′,

Once we have identified the midpoints, beginning from primary midpoints, we start numbering them again, either clockwise or anticlockwise. This be- gins with 1′, denoting the midpoint between qubits 1 and 2, and continues progressively as 2′, 3′, . . . , n′, where n represents the total number of qubits. Subsequently, the secondary midpoints, connecting n...

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[5] [5]

En- tanglement matrix is a n × n square symmetric ma- trix, where n is the total number of midpoints in a given graph

We begin creating the entanglement matrix. En- tanglement matrix is a n × n square symmetric ma- trix, where n is the total number of midpoints in a given graph. Now, we join two primary midpoints (i′ and j′), to divide the graph into two subsystems, such that each subsystem has same set of eigenval- ues pi, from which we can calculate the entropy of form...

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[6] [6]

For example, {1’1’} will give us the entanglement of the edge joining 1 and 2

The {i’i’} entry will give us the entanglement of the corresponding edge on which midpoint i′ lies. For example, {1’1’} will give us the entanglement of the edge joining 1 and 2

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[7] [7]

The total entanglement of the system is given by the sum of all the elements on the primary diagonal and either of the upper or lower triangular parts

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[8] [8]

Whereas Secondary midpoints can only contribute to the diagonal entries

Primary Midpoints contribute to both diagonal and off-diagonal entries of the Entanglement Matrix. Whereas Secondary midpoints can only contribute to the diagonal entries. For secondary mid-midpoints possessing a degree ( d), greater than 2, characterizing individual edge contribu- tions to the overall entanglement becomes challenging. The in-distinguisha...

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