Detecting genuine multipartite entanglement in multi-qubit devices with restricted measurements

Domain: chemistry
Published: 2026-02-17
Editorial outcome: major_revision

Abstract
Detecting genuine multipartite entanglement (GME) is a state-characterization task that benchmarks coherence and experimental control in quantum systems. Existing GME tests often require joint measurements on many qubits, posing challenges for systems like time-bin encoded qubits and microwave photons from superconducting circuits, where qubit connectivity is limited or measurement noise grows with the number of jointly measured qubits. Here we introduce versatile GME and k-inseparability criteria applicable to any state, which only require measuring O(n2) out of 2n (at most) m-body stabilizers of n-qubit target graph states, with m upper-bounded by twice the graph’s maximum degree. For cluster or ring-graph states, only constant-weight stabilizers are needed. Using semidefinite programming (and sometimes graph-local complementations), we can reduce the number or weight of required stabilizers. Analytical and numerical results show that our criteria are noise-robust and may infer state infidelity from certified k-inseparability in microwave photonic graph states generated under realistic conditions.

Introduction
Current developments of quantum technologies have enabled the simultaneous control of increasingly large systems and the preparation of highly complex quantum states (see, e.g., refs. 1–4). These advances demand suitable techniques to characterize and benchmark the corresponding devices and the states that they produce. However, increasing system sizes mean that full state tomography quickly becomes infeasible, shifting the focus to scalable but less all-encompassing characterization tools that still capture genuine quantum features as well as the quality of system control. Useful alternatives typically relying on few measurements present themselves in techniques such as shadow tomography5,6 and entanglement certification (see, e.g., ref. 7), which represent two ends of a spectrum: the former employs randomized/arbitrary measurements to create a classical representation of the state for broad a posteriori predictions, while the latter typically relies on targeted measurements to directly extract information on quantum correlations between subsystems. Yet, even these specific measurements are often difficult to implement in practice, prompting purpose-designed solutions for entanglement certification across a variety of platforms, including, e.g., spatial8,9 and temporal10 degrees of freedom of photons, spin ensembles11,12, and recently even setups not traditionally employed for quantum-information processing like electron-photon pairs in electron microscopy13,14. Under these circumstances, it becomes essential to extend the toolbox of entanglement-detection criteria to meet various platform-specific restrictions. Here, we provide such a practically implementable criterion for detecting genuine multipartite entanglement (GME) in systems that permit joint operations and measurements only on specific subsets of qubits.

Multipartite entanglement is a crucial resource for quantum technologies15, enabling applications from quantum communication16–19, measurement-based quantum computation (MBQC)20,21, to quantum error correction22, quantum metrology23, and other quantum algorithms24. In particular, certain multipartite tasks require GME specifically16,25,26. But what is GME? It formalizes the idea that some multipartite states are entangled across every bipartition (a grouping of subsystems into two subsets), but can still be produced as statistical mixtures of states that have only bipartite entanglement across different bipartitions but no multipartite entanglement themselves. The latter are called biseparable, whereas states that cannot be decomposed into mixtures of bipartite entangled states are GME. This concept extends to the more general k-(in)separability, where entanglement across all k-partitions is considered.

The study of GME is a thriving subject of ongoing research. Since the seminal works on this topic27–29, much progress has been made in developing GME-detection methods (see, e.g., the reviews7,30,31). The two crucial pieces of information for choosing a suitable detection strategy for a given setup are (i) the target states one expects to encounter, and (ii) which measurements can reasonably be carried out in the physical platform at hand. While the former influences how well a chosen GME criterion works in practice, the latter restricts which criteria one may evaluate in the first place. Consequently, the GME-detection toolbox is continuously being expanded to cover relevant platforms and target states.

In this work, we introduce a novel family of criteria designed to detect GME in the important class of n-qubit graph states (and states close to them), in scenarios where joint measurements can only be carried out on certain “local” subsets of connected qubits, with the locality and connectivity dictated by the associated graph structure. Such restrictions naturally arise in setups producing time-bin encoded qubits, e.g., time-multiplexed photons in the near-visible spectrum generated from spontaneous parametric down-conversion in combination with Sagnac interferometers32, or pairs of optical parametric amplifiers and Mach-Zehnder interferometers33–35. There, joint measurements and operations are usually restricted to temporally adjacent photons. Our criteria are also well-suited for platforms where single-qubit measurement efficiency is limited. In such systems, the number of measurements required to estimate a weight-m observable scales exponentially with m and becomes experimentally prohibitive if m scales with the system size. This measurement-efficiency limitation is the main bottleneck encountered in characterizing microwave photons generated from superconducting circuits36–39. A similar, though less severe, limitation also appears in optical photonic platforms, where sequential arbitrary-basis single-qubit measurements rely on electro-optical modulators for fast basis switching, whose losses reduce the overall detection efficiency40,41.

Crucially, such setups naturally generate graph states central to quantum information processing, including two-dimensional cluster states for MBQC20,21, ring-graph states for fusion-based quantum computing42, or tree-graph states for error correction43 and constructing quantum repeaters44. Therefore, certifying multipartite entanglement of these states is motivated by the need to characterize resources in quantum information processing, and given the practical restrictions of the setups, producing these states becomes an imperative that makes the techniques presented here highly relevant for device benchmarking.

Numerous methods exist for detecting GME, including stabilizer-based witnesses45–47, PPT-inspired criteria48,49, fidelity bounds50,51, as well as permutation inequalities and moments52–55. However, nearly all require measuring observables acting on up to O(n) qubits, which limits their experimental feasibility when only few-body measurements are accessible. Moreover, many detect only GME versus full separability, without resolving intermediate levels of k-inseparability (see Supplementary Note 1 for a review of pertinent previous methods). This motivates the approaches developed here, which are designed for scenarios restricted to O(1)-body measurements for important classes of graph states.

The entanglement criteria we introduce here are versatile yet simple, making them applicable across a broad range of physical platforms, as they only require measuring few stabilizers of chosen target graph states. Each stabilizer is a tensor product of Pauli operators acting on at most twice the graph’s maximum degree, and the number of stabilizers to be measured scales at most quadratically with the number of vertices. Since all stabilizer states are local-unitarily equivalent to graph states56,57, our criteria can certify multipartite entanglement—a generally NP-hard task58,59—for many relevant states in quantum information processing, particularly those close to stabilizer states. In addition, we show how our criteria can be used even in setups where only a subset of the few measurements mentioned above can be carried out by utilizing graph-local complementations and semidefinite programming (SDP).

We demonstrate the performance and versatility of our method in two ways. First, we evaluate the entanglement criteria for several pertinent graph states and non-stabilizer states that are local-unitary (LU) transformed Dicke states with added white noise, providing analytical noise thresholds for each case, with the latter examples demonstrating that our method applies beyond stabilizer states. Second, we show case the applicability of our criteria for benchmarking quantum devices by numerically simulating microwave-photonic graph states generated from superconducting circuits38 and quantitatively comparing our method to other approaches.

The remainder of this article is structured as follows. In section “Background and notation”, we provide basic definitions for graph states and multipartite entanglement. In sections “GME & k-inseparability criteria” and “SDP for incomplete measurements”, we present and discuss our new GME/k-inseparability criteria, as well as our fixed k-partition inseparability criteria, and demonstrate how SDP can be used to bound certain observables when direct measurements are not feasible. We then test our criteria’s performance in a simulated experiment with microwave-photonic qubits in the remaining Results sections, showing that the certified k-inseparability may infer state infidelity. Finally, we provide a discussion and outlook.

Results
Background and notation
Let us first state some definitions from graph theory that are relevant for this paper. Let G = (V

Methods

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript “Detecting genuine multipartite entanglement in multi-qubit devices with restricted measurements” presents a family of criteria for detecting genuine multipartite entanglement (GME) and k-inseparability in multi-qubit systems under restricted measurement conditions. The study of GME is important for quantum technologies, yet it is notoriously challenging —existing tests often demand non-local measurements, an exponential number of measurements, or even quantum state tomography. The authors propose stabilizer-based graph criteria that only require few-body measurements: the number of measurements scales at most quadratically with system size, and in many cases, involves only constant-weight stabilizers. At its core, the proposed method adopts a fidelity-based approach, linking stabilizer expectation values to fidelities relative to target graph states. The work is complemented by semidefinite programming techniques for handling incomplete data, and supported by both analytical calculations of noise thresholds and numerical simulations of microwave-photonic graph states in superconducting circuits. Overall, the approach is practical and versatile, holding potential to make a valuable contribution to benchmarking multipartite entanglement in realistic experimental platforms.\nIn general, I support its publication in Nature Communications. However, the manuscript requires substantial revisions before it can be published. My specific comments are as follows:\n1. The authors should add comparisons with existing fidelity-based methods. Fidelity-based entanglement detection is already a well-studied field, and the authors must explicitly compare their approach with established fidelity-based methods (e.g., those in Refs. [43, 44]). Such a comparison is essential to clarify the novelty and concrete advantages of their proposal. Additionally, it would be valuable to discuss how the proposed stabilizer-based method relates to fidelity-based approaches for non-stabilizer target states (e.g., symmetric states), thereby expanding the contextual relevance of the work.\n2. The manuscript should also compare the proposed method with moment-based GME detection approaches (e.g., PRL 129, 260501), highlighting differences in measurement requirements, computational complexity, and performance under realistic noise conditions.\n3. Clarification is needed for the construction of Eq. (2). The paper introduces Eq. (2) as the core object of the new criteria, yet its construction and underlying intuition are not sufficiently explained. A clearer discussion of how the stabilizer combinations are selected—and why this selection yields an effective entanglement witness—would significantly improve readability. For example, the authors should address: why choose stabilizers \\(S_i\\) and \\(S_j\\) where i and j are adjacent nodes in the graph?\n4. The relationship between graph structure and the choice of parameter \\gamma requires further elaboration. \\gamma plays a crucial role in the proposed criteria, but the physical or mathematical rationale for how its optimal value depends on the graph structure of the target state remains underdeveloped. The authors should expand on this relationship and ideally provide additional examples (e.g., for different graph topologies like linear chains, star graphs, or complete graphs) to illustrate how \\gamma is tuned.\nThe writing and presentation of the manuscript need improvement. While the work is technically rigorous, its exposition is relatively unclear, with multiple instances of notation abuse and imprecision that obscure key ideas. For example:\n• The symbol <…> is used ambiguously, sometimes denoting expectation values and other times denoting generated groups;\n• In Eq. (1), the expression fails to clearly indicate that the partition of subsystems may depend on the index i;\n• The letter X—conventionally used to denote the Pauli X operator—is also used to represent a generic operator, leading to potential confusion;\n• The statement “For any connected graph, our criteria only require measuring J out of the total of …” is vague (e.g., J is not defined or contextualized) and needs clarification.\n• The manuscript would benefit from a careful, thorough revision to enhance the precision of notation, the clarity of explanations, and overall consistency throughout.", "reviewer_2": "The article is solid science on the topic of detecting genuine multipartite entanglement and non-k-separability in graph states. The key contribution is that the authors manage to reduce the number of measurements from O(2^n) to O(n^2), in some cases even to O(n). The work is a nice strengthening of previous results, but it is also restricted to stabilizer states for which already a lot is known.\nThe key contribution is the reduction in measurements for a common class of states.\nIn my view this does not pass the threshold for Nature Communications, but could be convinced if the insight behind the results translates to other settings.\nMajor comments:\nThe article needs more insight/intuition on why these criteria and why matchings / cuts show up.\nDefining a witness as in Eq. [2] or Theorem 1 is fine, but the conceptual contribution or\ninsight is not well highlighted. In Proposition 1 & 2, what are novel insights / differences\nto previous similar bounds by the authors?\nMinor:\nThe optimization in (8a) is not very scalable and can be performed only for small system sizes.", "reviewer_3": "The authors study practical (in the sense of requiring a small number of stabilizer measurements, which only act on a small number of qubits at the same time) entanglement witnesses, which can certify non only GME, but also general k-separability. The entanglement witness is tailored to a given graph state, but can also certify the GME of interesting non-stabilizer states. The authors contributions are noteworthy from both the theoretical point of view, but they also took great care to ensure their results are practically applicable. They do so by suggesting experimental implementations, simulating them in (what I presume are) realistic experimental situations, and by considering SDP to further ease the experimental implementation.\nOverall, I think the paper is very well-written, and I would expect that this could naturally lead to experiments in the near-term. The introduction in particular was enjoyable to read, providing a clear motivation for this work, and furthermore also provided enough of a literature review to put this work into context.\nI do, however, have some comments that should be addressed. My main concern is the following. The motivation for minimizing the m-body correlators is that it is hard to perform them. However, to estimate the expectation value of an m-body Pauli string, it suffices to perform single-qubit Pauli measurements and record the parity of all measurements. Single-qubit Pauli measurements are clearly easier to implement than larger weight Pauli strings (this is essentially one of the biggest motivations for studying this problem, according to the authors). Unless I am missing something, this severely limits the motivation for minimizing the weight of Pauli measurements performed.\nThere are several reasons I could think of why such single-qubit Pauli measurements might be bad, however. Clearly for each measurement one can only estimate the expectation value of Pauli measurements that are locally commutative (i.e. subsets of stabilizers whose restrictions to individual qubits pairwise commmute), which would correspond for example to the `vertex stabilizers' that form independent sets in the graph. After such a measurement, the state would need to be prepared again (note that this also motivates the concept of local measurement settings studied previously). Here I could imagine the authors' approach could provide some gain, since the m-body stabilizers S_i can be measured in succession, without having to recreate the state. At the same time, if i and j are connected, then after measuring S_i and S_j there is no use in measuring S_i * S_j, since measuring S_i and S_j projects the state into a well-defined eigenstate of their product. So this would require repreparing the state anyway when measuring the S_i*S_j stabilizers. Furthermore, it is not clear whether performing some of these multi-body measurements in succession (without having to reprepare the state) is actually beneficial in an experiment, since this incurs extra decoherence because the state is idling and one measures larger body correlators.\nIt might be that the above concern is easily addressed, since I'm not an expert on (experimental considerations) of entanglement witnesses. If it is, it would be good to highlight in the paper what the experimental drawbacks would be for using single-qubit Pauli measurements, as opposed to measuring the full Pauli strings directly/the procedure that the authors propose.\nMinor concerns:\nIt took some time to understand what was meant in theorem 1 under eq 4. It would help to be more precise here, and it would also help (but is not necessary) to have Fig. 1 on the same page.\nIt would be good to mention the notion of locally equivalent graph states after remark 1. In particular that one optimize the graph used in the entanglement witness over all graphs in the local equivalence class. For example, all graphs in Fig. A.2 are locally equivalent (if we include qubit permutations). Furthermore, the second bound in eq (3) motivates the study of optimizing the number of edges in graph states up to local complementations, which can be done using the results in for example 2506.00292.\nI think remark 2 could be sharpened, by first saying that the criterion based on G is tailored to |G>, then saying that it achieves n+\\gamma |E| for |G><G|, and then that it can still apply to other non-stabilizer states.\nProposition 1 has the definition of the anti-commutator, while it is already used in lemma 2.\nI would change the word 'define' in lemma 2 by 'let'.\n\"As the stabilizer term |<S_b>| of b \\in V_match^k *is* already paired (...)\"\nIn A.1 the authors highlight how other witnesses do not detect between different levels of k-inseperability. I feel this could be stressed more in the main text as a novel result.\n(Remarks on code availability)\nThe code does not include the calculations for the expectation values from the density matrices, as far as I can tell."}, "author_response": "Response to Reviewer 1\nThank you for reviewing our manuscript for Nature Communications. We appreciate the time and\neffort that you put into the review process and providing constructive feedback on our manuscript. We\naddressed all comments in the revised version of our manuscript. All changes are highlighted in\npurple texts in the attached manuscript. Please find our in-line response below. Note that all page and\nsection numbers refer to the attached manuscript and supplementary information in PDF with purple\ntexts that highlight the amendments.\nWith best regards,\nNicky Kai Hong Li (on behalf of all authors)\n***********************************\nInline Response:\nReviewer #1 (Remarks to the Author):\n[...]\nGeneral Response: We sincerely thank the referee for the thoughtful and encouraging assessment of\nour work and for the constructive comments and questions. In addition to addressing the specific\npoints raised, we would like to highlight two substantial improvements made in the revised\nmanuscript.\nFirst, we have clarified and emphasized an important practical aspect of our criteria: optimization\nunder local Clifford (LC) equivalence of graph states. As explained now in the main text (after\nRemark 1), LC-equivalent graph states may differ in maximum degree or number of edges, quantities\nthat directly determine stabilizer weights and the number of measurements in our criteria. Choosing\nan LC-representative with a smaller maximum degree or number of edges can therefore significantly\nreduce the experimental effort required to evaluate our criteria. This perspective is incorporated into\nthe updated abstract and Introduction (third-to-last paragraph), formalized in a new Remark 2,\nand illustrated explicitly in Appendix A.V.\nSecond, we have substantially expanded the Discussion section to articulate the broader conceptual\nscope and impact of our results. Our criteria admit an interpretation in terms of a Pauli-local\nHamiltonian whose structure resembles families of models studied in the context of\nsymmetry-protected topological (SPT) phases [86, 87]. Constructions such as “pivoting’’ [87], in\nwhich graph-local unitaries transform simple reference models (e.g., classical Ising models) into\nstabilizer-type Hamiltonians, yield a class of models with analytically tractable ground and\nlow-energy excited states, and provide a useful conceptual parallel to the Hamiltonian form\nunderlying our criteria. While originating from different motivations, our approach suggests a\ncomplementary direction by offering a theoretically tractable method for probing entanglement\nproperties of highly excited states, a regime that is often challenging for conventional numerical\ntechniques. Moreover, the relevant observables are local, in the sense that they involve only a few\nqubits for many important interaction graphs (e.g., 1D and 2D clusters), and are therefore accessible\nin current experimental platforms.\nImportantly, this Hamiltonian perspective shows that our method yields experimentally friendly\nGME/k-inseparability criteria based on mean-energy measurements, quantities that are typically\naccessible in quantum many-body experiments. This substantially broadens the applicability of our\napproach beyond graph states and stabilizer architectures and provides a concrete pathway for\ndeploying our criteria in diverse physical settings governed by local Hamiltonians. Looking ahead, the\nstructural similarity to many-body Hamiltonians points toward a broader framework in which\nentanglement witnesses or detection criteria may be systematically derived by combining insights\nfrom many-body physics with multipartite entanglement theory.\nWe believe these additions significantly strengthen the scope and practical relevance of our results,\nand demonstrate that the underlying insights of our method extend beyond stabilizer states and to\nother physical settings.\n1. The authors should add comparisons with existing fidelity-based methods. [...]\nResponse: We thank the reviewer for this valuable suggestion and we agree that comparisons with\nestablished fidelity-based methods will help to place our contribution in an appropriate context.\nIn response, we have added a direct comparison between our criteria and the\nGME/k-inseparability witnesses of [npj Quantum Inf. 5, 83 (2019)] (now cited as Ref. [47]) in\nTables I–IV and in Sec. II E. These tables showcase the performance of several witnesses/criteria on\nsimulated noisy graph states. While the witnesses in Ref. [47] perform slightly better than our criteria,\nthey generally require measuring at least O(2^n) stabilizers with a maximum degree of up to O(n)\n(where c is the chromatic number of the graph). Such measurements are not experimentally feasible in\nthe platforms that we consider when n becomes large, due to connectivity constraints or rapidly\nincreasing measurement noise.\nLet us also note that, to the best of our knowledge, other existing stabilizer/fidelity-based GME\nwitnesses do not certify general k-inseparability (except for full separability). This is why we believe\nthat the addition of the comparison with Ref. [47] already gives a representative and comprehensive\ncomparison to the relevant fidelity-based approaches. In particular, Tables I–IV now contrast our\ncriteria with two extremal classes of GME witnesses: (1) the witness in Eq. (45) of Ref. [46], which\nfulfills the same measurement restrictions as our method and uses the minimum number of stabilizers;\nand (2) the witness based on fidelity > 0.5 in Ref. [45], which requires measuring all 2^n stabilizers.\n2. The manuscript should also compare the proposed method with moment-based GME\ndetection approaches (e.g., PRL 129, 260501), highlighting differences in measurement\nrequirements, computational complexity, and performance under realistic noise conditions.\nResponse: We also thank the reviewer for this helpful suggestion and for pointing us to this valuable\nreference. We have now mentioned this method in the Introduction (4th last paragraph) and\nextended the review section in Appendix A.I to summarize the measurement requirements and\ncomputational complexity requirements of [PRL 129, 260501] (now cited as Ref. [55]).\n3. Clarification is needed for the construction of Eq. (2). [...]\nResponse: We thank the referee for this useful feedback. We have substantially expanded the\ndiscussion surrounding Eq. (2) to make its construction and intuition clearer.\nThe key idea behind our method is that the entire set of stabilizers is not actually needed to obtain\ngood bounds for certifying entanglement. Once the union of all the edges corresponding to the\nstabilizers form a connected graph, a polynomially small subset of local stabilizers, together with the\npositivity of quantum states, is already sufficient to certify important properties of many-body\nquantum states.\nMore precisely, the choice of stabilizer subset stems from the use of the anticommutativity inequality\n(Lemma 2 and Proposition 1 in Methods) and the fact that Pauli matrices anticommute, which\ntogether lead to the analytic bounds in Eqs. (4) and (8). These properties ensure that whenever two\nqubits a and b in ρ correspond to adjacent vertices in the underlying graph and belong to different\ngroups of a given k-partition of the n qubits, every k-product state of that k-partition satisfies\n|⟨S_a⟩| + |⟨S_b⟩| + |⟨S_a S_b⟩| ≤ 1, |⟨S_a⟩| + |⟨S_a S_b⟩| ≤ 1, and |⟨S_a⟩| + |⟨S_b⟩| ≤ 1. These constraints\narise because, after factorization across the partition, the operators acting on each subsystem form\nanticommuting triples or pairs whose squared expectation values sum to at most one. Alternatively or\nadditionally, one may also derive bounds by leveraging the positivity constraint via a semidefinite\nprogramming (SDP) approach, as done in Sec. II C.\nTo illustrate this more concretely, consider the example of a 5-qubit graph in Fig. 1 of the manuscript,\nand the partition of the 5 qubits/vertices into three sets, {1,4,5}, {2}, and {3}. Since Eq. (2) contains\nsums over stabilizers for all vertices and products of stabilizers corresponding to all pairs of\nneighboring vertices, the quantity in Eq. (2) contains (amongst others) terms such as\n|⟨S_1⟩| = |⟨X_1 Z_2 Z_3 Z_5⟩|, |⟨S_2⟩| = |⟨X_2 Z_1 Z_3⟩|, and |⟨S_1 S_2⟩| = |⟨Y_1 Y_2 Z_5⟩|, where the fact that vertices\n1 and 2 are connected by an edge means that the Pauli-Y terms appear as products of X and Z\noperators for both qubits 1 and 2. For any state that is separable with respect to the chosen partition in\nFig. 1, these stabilizer expectation values factorize as |⟨S_1⟩| = |⟨X_1 Z_5⟩| |⟨Z_2 Z_3⟩|,\n|⟨S_2⟩| = |⟨Z_1⟩| |⟨X_2 Z_3⟩|, and |⟨S_1 S_2⟩| = |⟨Y_1 Z_5⟩| |⟨Y_2⟩|. Hence, the sum of these three terms can be\nbounded via the Cauchy-Schwarz inequality,\n|⟨S_1⟩ + |⟨S_2⟩| + |⟨S_1 S_2⟩| = |⟨X_1 Z_5⟩| |⟨Z_2 Z_3⟩| + |⟨Z_1⟩| |⟨X_2 Z_3⟩| + |⟨Y_1 Z_5⟩| |⟨Y_2⟩|\n≤ (|⟨X_1 Z_5⟩|^2 + |⟨Z_1⟩|^2 + |⟨Y_1 Z_5⟩|^2)^1/2 (|⟨Z_2 Z_3⟩|^2 + |⟨X_2 Z_3⟩|^2 + |⟨Y_2⟩|^2)^1/2.\nFor the sums of squared expectation values under each square root, we can use the anticommutativity\nbound (Lemma 2 and Proposition 1 in Methods) to bound them by a constant (1 in this case): for each\nside of the partition, the Pauli operators {X, Y, Z} (tensored with spectator Pauli-Z’s) form a maximally\nanticommuting triple. We can also make similar arguments for any other pair of neighboring vertices\nthat is “separated” by the selected partitioning, and then repeat this procedure for all possible\nk-partitions, thus arriving at a bound for all k-separable states.\nA key structural point is that this argument applies only when the two vertices are connected by an\nedge and separated by the partition. Thus, by including all vertex stabilizers and all pairs of stabilizers\nfor all edges, we (i) maximize the number of terms that contribute to the left-hand side of the bound,\nwhile (ii) ensuring that we still have a maximal set of anticommuting operators under each square root\nafter the factorization.\nIn contrast, states that are inseparable across this partition can exceed these bounds since the three\nsums at the beginning can reach a maximum of 3, 2 and 2, respectively (e.g., when ρ = |G⟩⟨G|).\nThese structural dependence on the graph adjacencies, arising specifically from our chosen subset of\nstabilizers, explain why matchings and cuts naturally enter the formulation of the optimal bounds and,\ncrucially, why our approach can certify not only GME but also general k-inseparability.\nTo address your comment, we have now (i) added some text to clarify and guide this intuition after\nEq. (2), (ii) inserted a further clarifying remark immediately before Theorem 1, and (iii)\nsubstantially revised Fig. 1 and its caption to better illustrate the corresponding graph-theoretic\nobjects and intuition.\n4. The relationship between graph structure and the choice of parameter \\gamma requires\nfurther elaboration. [...]\nResponse: We agree that this requires further elaboration. In general, the optimal choice of the\nparameter γ for achieving maximum violation of the inequality depends on two factors: (i) the\nmeasured values of the two summation terms in Eq. (2), and (ii) the number of edges and the\nreduction term R_k, which behave differently for different underlying graphs. Since factor (i) can vary\na lot across different experiments, the optimal γ may vary from experiment to experiment even for the\nsame underlying graph. Therefore, there is generally no analytical strategy for finding the optimal γ.\nThe most straightforward strategy is to measure W_G then perform the maximization as described in\nEq. (6) numerically, which is how we obtained the optimal γ for the simulated data in Tables I–IV and\nFig. 4. Given that there’s only one parameter to be optimized, the maximization is typically\nstraightforward. We have now added a short note under Eq. (6) to elaborate on this point.\nFor the idealized examples of graph states mixed with white noise [see the Cthulhu-graph example\n(see around Observation 1 and Appendix A.III) and Appendix A.VII], the two summation terms in\nEq. (2) have a simple relationship as described in Eq. (A.16), so the optimal γ depends entirely on\nfactor (ii) which is determined by the underlying graph’s structure. In the white-noise added scenario,\nwe observed that the optimal γ for all 1D cluster, ring, star, and tree graphs is 1 for certifying k\n-inseparability for all 2 ≤ k ≤ n, whereas for 2D cluster and complete graphs, the optimal γ can be\neither 0 or 1 depending on k and the size of the graph. In particular, the optimal γ is 0 for n-vertex\ncomplete graphs where n ≥ 4 and 3 ≤ k ≤ n. The details can be found in Appendix A.VII, where\nwe have now explicitly stated the optimal γ (denoted by γ*) for each of the above cases in the\nnew version. However, we must admit that the reason why the optimal γ can switch between 0 and 1\nalternatively for different k in white-noise added 2D cluster states is still unclear to us, and could be\nan interesting future research direction.\nTo understand conceptually the interesting situation where the optimal γ (e.g., the Cthulhu-graph\nexample) is strictly between 0 and 1, we observe that Cthulhu graphs are composed of a complete\ngraph (as its “head”) and a star graph (as its “tentacles”), of which the corresponding noisy graph\nstates have optimal γ being 0 and 1, respectively, when certifying their k-inseparability for n ≥ 4 and\n3 ≤ k ≤ n. Therefore, it is not hard to believe that the optimal γ for certifying k-inseparability of\nnoisy Cthulhu-graph states interpolate the extremal values 0 and 1. However, to show that the precise\nanalytical conditions for this to happen is more subtle, which we showed around Observation 1 and in\nAppendix A.III. To provide more intuition on how to understand this special situation, we added a\nshort note after Eq. (7) and modified Fig. 2 (and its caption).\nThe writing and presentation of the manuscript need improvement. [...]\nResponse: We agree that this notation can be confusing for the readers. Therefore, we have now\ndeleted the text “i.e., Stab(|G⟩) = ⟨{S_i| i∈ V}⟩” from the 2nd\nparagraph of Sec. IIA since it doesn’t add any extra useful information compared to the previous\nline that states that these stabilizers S_i generate the full stabilizer group, and we are not using this\nnotation anymore in the remaining article.\n• In Eq. (1), the expression fails to clearly indicate that the partition of subsystems may depend on the\nindex i;\nResponse: Thanks for pointing this out. We updated the notation in Eq. (1) and added some\nclarification right after.\n• The letter X—conventionally used to denote the Pauli X operator—is also used to represent a\ngeneric operator, leading to potential confusion;\nResponse: Thanks again. We replaced the letters X and Y with A and B to denote generic operators\nafter Eq. (2) and in Proposition 1. In Eqs. (21) and Appendices A.VI & A.X, we replaced X and Y\nwith \\tilde{X} and \\tilde{Y}.\n• The statement “For any connected graph, our criteria only require measuring J out of the total of …”\nis vague (e.g., J is not defined or contextualized) and needs clarification.\nResponse: We moved the definitions (range of allowed values) of J and M earlier (see the paragraph\nbelow Eq. (2)) to avoid confusion.\n• The manuscript would benefit from a careful, thorough revision to enhance the precision of notation,\nthe clarity of explanations, and overall consistency throughout.\nResponse: Thank you very much for going through the notations of our manuscript thoroughly. We\nhave gone through the paper, clarified any confusing notations, and modified any unclear sentences to\nimprove readability.\nResponse to Reviewer 2\n[...]\nResponse: We gratefully acknowledge the referee’s careful review and positive assessment of our\nwork. The referee raises an important question regarding the generality of our framework and the\nrelevance of our insights to other settings.\nFirst, we fully agree that broad applicability is an important consideration. We would like to clarify\nthat our framework is not restricted to stabilizer states—as demonstrated in Appendix A.VII(d), our\nmethod can certify GME/k-inseparability in non-stabilizer families such as states LU-equivalent to\nnoisy Dicke states. In the revised manuscript, we have put more emphasis on this point (see the\nsecond-to-last paragraph of the Introduction and the 2nd paragraph of the Discussion).\nSecond, we have substantially expanded the Discussion section in the revised manuscript to\nhighlight the broader impact and scope of our results. In particular, we point out an important\nconceptual connection between our work and quantum many-body physics, where entanglement has\nbecome a much-considered tool for classifying phases of matter. Our criterion admits an interpretation\nas a Hamiltonian whose Pauli-local structure resembles families of models recently studied in the\ncontext of symmetry-protected topological (SPT) phases [86,87]. This type of Hamiltonian can be\nrelated to a construction known as “pivoting” [87], in which graph-local unitaries transform simple\nreference models (e.g., classical Ising models) into stabilizer-type Hamiltonians. Such an approach\nyields a class of models with analytically tractable ground and low-energy excited states. Our work\nsuggests a potential path to go beyond this paradigm by providing theoretically tractable criteria for\nstudying entanglement properties of highly excited states—a regime that is often challenging for\nconventional numerical techniques. The observables required for our method are “local” in the sense\nthat it involves only a few qubits for many important classes of interaction graphs (e.g., 1D & 2D\nclusters), making them accessible in current experimental platforms. While our criteria originate from\ndifferent motivations, the structural similarity to many-body Hamiltonians points toward a broader\nframework in which entanglement witnesses or detection criteria may be systematically derived by\ncombining insights from many-body physics with multipartite entanglement theory.\nImportantly, this Hamiltonian perspective shows that our method yields experimentally friendly GME/\nk-inseparability criteria based on mean-energy measurements—quantities that are typically accessible\nin quantum many-body experiments. This significantly broadens the reach of our approach beyond\ngraph states and stabilizer architectures and provides a concrete pathway for applying our criteria in\ndiverse physical settings governed by local Hamiltonians.\nWe believe this generalization significantly strengthens the scope and relevance of our results and\ndemonstrates that the underlying insights of our method extend beyond stabilizer states and to other\nphysical settings.\nMajor comments:\nThe article needs more insight/intuition on why these criteria and why matchings / cuts show up.\nDefining a witness as in Eq. [2] or Theorem 1 is fine, but the conceptual contribution or insight is not\nwell highlighted. In Proposition 1 & 2, what are novel insights / differences to previous similar\nbounds by the authors?\nResponse:\nWe thank the referee for raising this important point. We agree that the intuition behind our choice of\nstabilizer subset and the appearance of matchings/cuts can be better emphasized. We have therefore\nexpanded the discussion around Eq. (2) and Theorem 1 to clarify the conceptual origin of our criteria.\n[...]\nRegarding the novelty of Propositions 1 and 2: although the anticommutativity inequality of\nLemma 2 originates from Ref. [88], its previous use in multipartite entanglement detection has been\nlimited. Earlier applications targeted only GME detection (but not for the more general k\n-inseparability) and relied only on a restricted subset of two-body Pauli observables (i.e., X^2, Y^2,\nand Z^2) as in Ref. [1]. There the witnesses are constructed from average fidelities to Bell states, and\ntherefore require only expectation values of pairs of Pauli operators. However, for increasing numbers\nof qubits, this construction led to a rapid combinatorial growth in the number of terms. While these\nwitnesses could certify GME with the data available in the experiment of Ref. [1], they were\ninsensitive to detect k-inseparability for general values of k (e.g., distinguish between 2- and\n3-separability for 4 qubits), and were ineffective for high-fidelity connected graph states. Moreover, it\nremains unclear if they could, even in principle, detect any GME states beyond 5 qubits.\nIn contrast, the present work demonstrates several genuinely new features. First, we provide a\nsystematic method for constructing witnesses that can detect more complicated entanglement\nstructures using correlations of orders higher than two (between more than two qubits). These\ncorrelators involve sets of qubits determined by the local connectivity of the underlying graph, and\nhence the measurement complexity scales with meaningful graph-theoretic structure rather than with\nthe exponentially growing pairwise combinatorics. Second, our approach leads to criteria that are not\njust systematically constructible but can also efficiently detect both GME and non-GME\nk-inseparability across a broad class of states (including non-stabilizer examples), even under realistic\nnoise. Finally, our approach constitutes the first systematic use of anticommutativity relations to\ngenerate families of multipartite entanglement criteria tailored to stabilizer and graph states, thereby\nadvancing beyond previous bounds both conceptually and operationally.\nMinor:\nThe optimization in (8a) is not very scalable and can be performed only for small system sizes.\nResponse: Thank you for this comment. In general, using the most stable and commonly used SDP\nsolver MOSEK (which uses the interior-point method) in MATLAB, we can solve the SDP problem\nthat involves up to 12 qubits efficiently on a standard laptop. This means we can readily apply our\nmethod to all graph states with max [d(i) + d(j)] ≤ 12, which include many graph states of\n(i,j)∈E\npractical importance in quantum information science, such as all 1- to 3-dimensional cluster states\n[20], all ring-graph states [42], and many tree-graph states [43, 44]. Beyond 12 qubits, while solving\nthe SDP may become intractable with conventional interior-point methods, alternative methods with\nbetter scalability [80], such as augmented Lagrangian methods, can be employed for larger problems,\nalthough the development of more stable software implementations is still required. We have added\nthis comment in Sec. II C.\nGeneral Response: In addition to addressing your specific comments, we would like to bring to your\nattention that we have added to the manuscript an important practical aspect of our criteria: their\noptimization under local Clifford (LC) equivalence of graph states. We now explain in the main text\n(after Remark 1) that LC operations generate equivalence classes of graph states whose members\nmay differ in their maximum degrees or numbers of edges. Since these graph-theoretic properties\ndirectly determine the stabilizer weights and the total number of terms appearing in our criteria,\nchoosing an LC-equivalent representative with a smaller maximum degree or fewer edges can\nsignificantly reduce the experimental efforts required to evaluate our criteria. We have updated the\nabstract and the introduction (third-to-last paragraph) accordingly, and have highlighted this\noptimization strategy in the newly added Remark 2. We further illustrate this idea with an explicit\nexample in Appendix A.V, where graph-local complementations are used to reduce both the\nmaximum stabilizer weight and the number of stabilizers that need to be measured. We also point\nreaders to recent methods for identifying minimum-edge representatives within an LC-equivalence\nclass. We believe that these additions further strengthen the practical relevance of our framework.\nResponse to Reviewer 3\n[...]\nGeneral Response: We sincerely thank the referee for the thoughtful and encouraging assessment of\nour work and for the constructive comments and questions. In addition to addressing the specific\npoints raised, we would like to highlight an improvement made in the revised manuscript.\nWe have substantially expanded the Discussion section to articulate the broader conceptual scope\nand impact of our results. Our criteria admit an interpretation in terms of a Pauli-local Hamiltonian\nwhose structure resembles families of models studied in the context of symmetry-protected\ntopological (SPT) phases [86, 87]. Constructions such as “pivoting’’ [87], in which graph-local\nunitaries transform simple reference models (e.g., classical Ising models) into stabilizer-type\nHamiltonians, yield a class of models with analytically tractable ground and low-energy excited states,\nand provide a useful conceptual parallel to the Hamiltonian form underlying our criteria. While\noriginating from different motivations, our approach suggests a complementary direction by offering a\ntheoretically tractable method for probing entanglement properties of highly excited states, a regime\nthat is often challenging for conventional numerical techniques. Moreover, the relevant observables\nare local, in the sense that they involve only a few qubits for many important interaction graphs (e.g.,\n1D and 2D clusters), and are therefore accessible in current experimental platforms.\nImportantly, this Hamiltonian perspective shows that our method yields experimentally friendly GME/\nk-inseparability criteria based on mean-energy measurements, quantities that are typically accessible\nin quantum many-body experiments. This substantially broadens the applicability of our approach\nbeyond graph states and stabilizer architectures and provides a concrete pathway for deploying our\ncriteria in diverse physical settings governed by local Hamiltonians. Looking ahead, the structural\nsimilarity to many-body Hamiltonians points toward a broader framework in which entanglement\nwitnesses or detection criteria may be systematically derived by combining insights from many-body\nphysics with multipartite entanglement theory.\nWe believe these additions significantly strengthen the scope and practical relevance of our results,\nand demonstrate that the underlying insights of our method extend beyond stabilizer states and to\nother physical settings.\nI do, however, have some comments that should be addressed. My main concern is the following. The\nmotivation for minimizing the m-body correlators is that it is hard to perform them. However, to\nestimate the expectation value of an m-body Pauli string, it suffices to perform single-qubit Pauli\nmeasurements and record the parity of all measurements. Single-qubit Pauli measurements are clearly\neasier to implement than larger weight Pauli strings (this is essentially one of the biggest motivations\nfor studying this problem, according to the authors). Unless I am missing something, this severely\nlimits the motivation for minimizing the weight of Pauli measurements performed.\n[...]\nResponse: We thank the reviewer for raising this important point. We fully agree that, in principle,\nexpectation values of m-body correlators can be estimated by performing sequential single-qubit Pauli\nmeasurements and data post-processing. In some setups, e.g., ion-trap experiments such as Ref. [1],\none can easily measure correlators of many qubits (20 in Ref. [1]), exactly in the way described by the\nreviewer. Moreover, it is true that a large body of the literature on quantum-state characterization\nfocuses on minimizing the number of required measurement settings by grouping commuting\nobservables that can be measured with a common basis setting. This is one of the motivations behind\nseveral works we cited, such as [G. Tóth and O. Gühne, Phys. Rev. Lett. 94, 060501 (2005)] and [Y.\nZhou et al., npj Quantum Inf 5, 83 (2019)].\nHowever, the experimental platforms that motivate our work, most notably microwave photonic\ngenerators (e.g., superconducting circuits) and time-bin encoded optical systems (e.g.,\ntime-multiplexed photons in the near-visible spectrum generated from spontaneous parametric\ndown-conversion in combination with Sagnac interferometers, or pairs of optical parametric\namplifiers and Mach-Zehnder interferometers), operate under very different constraints. In these\nsettings, the dominant experimental limitation is not the number of measurement settings but\nthe exponential degradation of usable statistics with the weight of the measured operator.\nOne key reason is that the detection efficiency for a single photon (especially for microwave photons),\nη, is usually much less than 1. As a result, the effective success probability of detecting all m photons\nneeded to measure a weight-m Pauli string scales as η^m. Therefore, even when single-qubit\nmeasurements are used, the number of state preparations required to estimate a weight-O(n)\nobservable to a constant precision grows exponentially in the system size n.\nTo make this quantitative, in the experiment [Nat. Commun. 16, 5505 (2025)] that inspired this\ntheoretical work, measurements are performed on microwave photons, with a detection efficiency of\nη ≈ 0.25. As a consequence, estimating weight-4 Pauli operators required around 10 billion\nrepetitions. Although higher-efficiency microwave-photon detectors exist [Phys. Rev. X 8, 021003\n(2018); Nat. Phys. 14, 546–549 (2018)], they are much harder to calibrate and are not routinely\navailable.\nOptical platforms face similar challenges. State-of-the-art optical detection efficiencies are around\n90% (see, e.g., [Phys. Rev. Lett. 129, 050502 (2022)]), but when combined with optical modules\nallowing fast basis switching between photons in adjacent time bins, the effective efficiency typically\ndrops to around 80% (see, e.g., [Phys. Rev. Lett. 115, 250401 (2015); Phys. Rev. Lett. 115, 250402\n(2015)]). Furthermore, optical systems often require non-deterministic processes, such as parametric\ndown-conversion and fusion operations for entanglement generation, further amplifying the statistical\ncost of measuring higher-weight operators. Thus, even though the correlators are accessible via\nsingle-qubit measurements, measuring high-weight operators remains experimentally\nchallenging.\nFor these reasons, even though any stabilizer can in principle be estimated using single-qubit\nmeasurements, the dominant experimental cost in these platforms is proportional to η^-m.\nTherefore, reducing the maximum stabilizer weight from, say, m = O(n) to a constant independent\nof n can reduce the required number of experimental repetitions by many orders of magnitude.\nThis is precisely the motivation of our approach: by ensuring that our GME/k-inseparability criteria\ninvolve only stabilizers whose weight is bounded by twice the maximum degree of the underlying\ngraph (constant for many important graph states such as 1D/2D clusters), we dramatically reduce the\nexperimental time needed to estimate them in the photonic platforms of interest.\nTo clarify these points, we have added a corresponding explanation to the Introduction (see the\nrevised 4th paragraph).\nMinor concerns:\nIt took some time to understand what was meant in theorem 1 under eq 4. It would help to be more\nprecise here, and it would also help (but is not necessary) to have Fig. 1 on the same page.\nResponse: We agree that the notation under Eq. (5) [Eq. (4) in the previous version] in Theorem 1\nneeds further clarification and may be made more precise. We have now repositioned Fig. 1 so that it\nappears on the same page. Furthermore, we have substantially revised Fig. 1 and its caption to\nbetter illustrate the corresponding graph-theoretic objects and intuition behind our formulation.\nTo clarify the conceptual origin of our criteria, we have added explanations around Eq. (2) and\nTheorem 1 to emphasize why matchings and cuts naturally arise in our bounds. The key idea is that\nthe entire set of stabilizers is not actually needed to obtain good bounds for certifying entanglement.\nOnce the union of all the edges corresponding to the stabilizers form a connected graph, a\npolynomially small subset of local stabilizers, together with the positivity of quantum states, is already\nsufficient to certify important properties of many-body quantum states.\nMore precisely, the main tools we use are the anticommutativity inequality (Lemma 2 and Proposition\n1 in Methods) and the anticommuting structure of Pauli operators. These two ingredients jointly yield\nthe analytic bounds in Eqs. (4) and (8). In particular, whenever two qubits a and b in ρ correspond to\nadjacent vertices in the underlying graph and belong to different groups of a given k-partition of the n\nqubits, every k-product state of that k-partition necessarily satisfies |⟨S_a⟩| + |⟨S_b⟩| + |⟨S_a S_b⟩| ≤ 1,\n|⟨S_a⟩| + |⟨S_a S_b⟩| ≤ 1 and |⟨S_a⟩| + |⟨S_b⟩| ≤ 1. These constraints arise because, after factorization\nacross the partition, the operators acting on each subsystem form anticommuting triples or pairs\nwhose squared expectation values sum to at most one. Alternatively or additionally, one may also\nderive bounds by leveraging the positivity constraint via a semidefinite programming (SDP) approach,\nas done in Sec. II C.\nTo illustrate this more concretely, consider the example of a 5-qubit graph in Fig. 1 of the manuscript,\nand the partition of the 5 qubits/vertices into three sets, {1,4,5}, {2}, and {3}. Since Eq. (2) contains\nsums over stabilizers for all vertices and products of stabilizers corresponding to all pairs of\nneighboring vertices, the quantity in Eq. (2) contains (amongst others) terms such as\n|⟨S_1⟩| = |⟨X_1 Z_2 Z_3 Z_5⟩|, |⟨S_2⟩| = |⟨X_2 Z_1 Z_3⟩|, and |⟨S_1 S_2⟩| = |⟨Y_1 Y_2 Z_5⟩|, where the fact that vertices\n1 and 2 are connected by an edge means that the Pauli-Y terms appear as products of X and Z\noperators for both qubits 1 and 2. For any state that is separable with respect to the chosen partition in\nFig. 1, these stabilizer expectation values factorize as |⟨S_1⟩| = |⟨X_1 Z_5⟩| |⟨Z_2 Z_3⟩|,\n|⟨S_2⟩| = |⟨Z_1⟩| |⟨X_2 Z_3⟩|, and |⟨S_1 S_2⟩| = |⟨Y_1 Z_5⟩| |⟨Y_2⟩|. Hence, the sum of these three terms can be\nbounded via the Cauchy-Schwarz inequality,\n|⟨S_1⟩ + |⟨S_2⟩| + |⟨S_1 S_2⟩| = |⟨X_1 Z_5⟩| |⟨Z_2 Z_3⟩| + |⟨Z_1⟩| |⟨X_2 Z_3⟩| + |⟨Y_1 Z_5⟩| |⟨Y_2⟩|\n≤ (|⟨X_1 Z_5⟩|^2 + |⟨Z_1⟩|^2 + |⟨Y_1 Z_5⟩|^2)^1/2 (|⟨Z_2 Z_3⟩|^2 + |⟨X_2 Z_3⟩|^2 + |⟨Y_2⟩|^2)^1/2.\nFor the sums of squared expectation values under each square root, we can use the anticommutativity\nbound (Lemma 2 and Proposition 1 in Methods) to bound them by a constant (1 in this case): for each\nside of the partition, the Pauli operators {X, Y, Z} (tensored with spectator Pauli-Z’s) form a maximally\nanticommuting triple. We can also make similar arguments for any other pair of neighboring vertices\nthat is “separated” by the selected partitioning, and then repeat this procedure for all possible\nk-partitions, thus arriving at a bound for all k-separable states.\nA key structural point is that this argument applies only when the two vertices are connected by an\nedge and separated by the partition. Thus, by including all vertex stabilizers and all pairs of stabilizers\nfor all edges, we (i) maximize the number of terms that contribute to the left-hand side of the bound,\nwhile (ii) ensuring that we still have a maximal set of anticommuting operators under each square root\nafter the factorization.\nIn contrast, states that are inseparable across this partition can exceed these bounds since the three\nsums at the beginning can reach a maximum of 3, 2 and 2, respectively (e.g., when ρ = |G⟩⟨G|).\nThese structural dependence on the graph adjacencies, arising specifically from our chosen subset of\nstabilizers, explain why matchings and cuts naturally enter the formulation of the optimal bounds and,\ncrucially, why our approach can certify not only GME but also general k-inseparability, which most\nconventional stabilizer-based criteria cannot.\nTo highlight these points more clearly, we have now also (i) added a concise explanation after Eq.\n(2), and (ii) inserted a clarifying remark immediately before Theorem 1.\nIt would be good to mention the notion of locally equivalent graph states after remark 1. In particular\nthat one optimize the graph used in the entanglement witness over all graphs in the local equivalence\nclass. For example, all graphs in Fig. A.2 are locally equivalent (if we include qubit permutations).\nFurthermore, the second bound in eq (3) motivates the study of optimizing the number of edges in\ngraph states up to local complementations, which can be done using the results in for example\n2506.00292.\nResponse: We thank the referee for this helpful suggestion. Following the comment, we have added a\ndedicated paragraph after Remark 1 that introduces the notion of LC-equivalent graph states and\nexplains how this freedom can be used to optimize our criteria. In particular, we clarify that one may\nchoose an LC-equivalent representative whose graph has a smaller maximum degree or fewer edges,\nthereby reducing stabilizer weights or the total number of terms that need to be measured in Eq. (2).\nThis point is now highlighted in the newly added Remark 2. We have also updated the abstract\nand the introduction (third-to-last paragraph) accordingly. In addition, we describe how the\nwitness is evaluated using the LC-conjugated stabilizers of the chosen representative state, and we\nreference the methods of Ref. [76] (arXiv:2506.00292) for identifying minimum-edge representatives.\nTo better illustrate these ideas, we have added a new Appendix A.V, where we give an explicit\nexample showing how local complementations can reduce the maximum degree and the number of\nedges of a 5-vertex graph. In that appendix, we also explain how to construct a new set of criteria that\nrequire fewer stabilizers with a smaller maximum weight.\nI think remark 2 could be sharpened, by first saying that the criterion based on G is tailored to |G>,\nthen saying that it achieves n+\\gamma |E| for |G><G|, and then that it can still apply to other\nnon-stabilizer states.\nResponse: Thank you for the suggestion. We have rewritten this remark (i.e., Remark 3 in the new\nversion) according to your feedback.\nProposition 1 has the definition of the anti-commutator, while it is already used in lemma 2.\nResponse: Thanks for spotting that. We have now moved the definition of anti-commutator from\nProposition 1 to Lemma 2.\nI would change the word 'define' in lemma 2 by 'let'.\nResponse: We agree that this sounds better and have replaced the word “define” with the word “let”.\n\"As the stabilizer term |<S_b>| of b \\in V_match^k *is* already paired (...)\"\nResponse: Thanks for spotting this grammatical mistake. We already added “is” accordingly.\nIn A.1 the authors highlight how other witnesses do not detect between different levels of\nk-inseperability. I feel this could be stressed more in the main text as a novel result.\nResponse: We agree that this is an important point and merits clearer emphasis in the main text. In\nparticular, our criteria not only detect GME but also distinguish between different levels of k\n-inseparability, unlike most existing stabilizer-based witnesses (e.g., Refs. [45, 46, 50]). To make this\nnovel aspect more visible, we have added an explicit remark highlighting this distinction at the end of\nthe paragraph that includes Eq. (3) and discusses the intuition behind our construction. In the\nfollowing paragraph, we further emphasize this point and reference Appendix A.1 for additional\ndetails.\nReviewer #3 (Remarks on code availability):\nThe code does not include the calculations for the expectation values from the density matrices, as far\nas I can tell.\nResponse: We thank the referee for the comment. Indeed, initially the code freely available on the\nreferred GitHub repository only contained the notebooks used to obtain the simulated noisy states. We\nhave now added the notebooks used for the computation of the expectation values. In the process, we\nreorganized the contents of the repository and extended the readme file to clarify the function and\nexpected usage of each of the notebooks."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The concerns and issues raised in the previous round of review have been carefully and adequately addressed by the authors in the revised manuscript. The current version demonstrates improved clarity and correctness. Therefore, I recommend this work for publication.", "reviewer_2": "The authors addressed my major and minor concerns. It reads much better now with a clear explanation of the key concept. However, due to the somewhat incremental nature (using locally commuting stabilizer operators with Cauchy Schwarz), I am not 100% sure whether the paper represents an advance of significance to specialists within the field.", "reviewer_3": "The authors have addressed all my questions, in particular my concern regarding minimizing the weight of the stabilizers is sufficiently addressed. As such, I now recommend acceptance in Nature Communications."}, "author_response": "Response to Reviewer 1\nDear Referee,\nThank you again for reviewing our manuscript for Nature Communications. We appreciate the time\nand effort that you put into the review process and providing constructive feedback on our manuscript.\nPlease find our in-line response below.\nWith best regards,\nNicky Kai Hong Li (on behalf of all authors)\n***********************************\nInline Response:\nReviewer #1 (Remarks to the Author):\nThe concerns and issues raised in the previous round of review have been carefully and adequately\naddressed by the authors in the revised manuscript. The current version demonstrates improved clarity\nand correctness. Therefore, I recommend this work for publication.\nResponse: We sincerely thank the referee again for the thorough and encouraging assessment of our\nwork, and for all the constructive comments that have improved the clarity of the manuscript.\nResponse to Reviewer 2\nDear Referee,\nThank you again for reviewing our manuscript for Nature Communications. We appreciate the time\nand effort that you put into the review process and providing constructive feedback on our manuscript.\nPlease find our in-line response below.\nWith best regards,\nNicky Kai Hong Li (on behalf of all authors)\n***********************************\nInline Response:\nReviewer #2 (Remarks to the Author):\nThe authors addressed my major and minor concerns. It reads much better now with a clear\nexplanation of the key concept.\nHowever, due to the somewhat incremental nature (using locally commuting stabilizer operators with\nCauchy Schwarz), I am not 100% sure whether the paper represents an advance of significance to\nspecialists within the field.\nResponse: We thank the referee for providing useful feedback on the manuscript and for\nacknowledging that the major and minor concerns have been addressed, leading to improved clarity\nand presentation.\nWe note the referee’s comment regarding the perceived incremental nature of the approach. As\nclarified in the manuscript, our aim is to provide a practically motivated and experimentally accessible\nframework that significantly reduces measurement overhead for multipartite entanglement detection\nunder realistic constraints. We believe this perspective, together with the generality of the framework\nand its applicability to current multi-qubit platforms, will be of interest to specialists in the field.\nResponse to Reviewer 3\nDear Referee,\nThank you again for reviewing our manuscript for Nature Communications. We appreciate the time\nand effort that you put into the review process and providing constructive feedback on our manuscript.\nPlease find our in-line response below.\nWith best regards,\nNicky Kai Hong Li (on behalf of all authors)\n***********************************\nInline Response:\nReviewer #3 (Remarks to the Author):\nThe authors have addressed all my questions, in particular my concern regarding minimizing the\nweight of the stabilizers is sufficiently addressed. As such, I now recommend acceptance in Nature\nCommunications.\nResponse: We sincerely thank the referee again for the thorough and encouraging assessment of our\nwork, and for all the constructive comments, in particular regarding minimizing stabiliser weight with\nlocal complementations, which have improved the clarity of the manuscript and the applicability of\nour results."}]

Decision letter