REVIEW 3 major objections 5 minor 55 references
Revealing Hidden Non n-Locality In n-Local Star Network
T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read Filtering first, then measuring, can reveal non n-locality in star-shaped quantum networks that ordinary measurements miss, and a non-separable filter at the central node works even when one source sends a product state.
desk verdict The separable-filter extension to star networks is legitimate and worth having, but the headline non-separable-filter advantage is computed with an inapplicable formula and inconsistent numbers, so that claim needs a direct derivation before it can be believed. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the star-network n-local inequality of Eq. (11) together with its upper bound B_n-star in Eq. (13), expressed through the two largest singular values of each source's two-qubit correlation tensor. The protocol adds a preparation phase in which every party applies a local filter before measurement; because separable filters keep the post-filter state a tensor product across sources, Theorem 1 reduces the task to computing the largest singular values of each individually filtered two-qubit state. The non-separable case replaces the central party's separable filter by a joint filter F_NS acting on qubits from different sources, which couples the formerly independent subsystems and is claimed to generate violations that separable filtering cannot.
What would settle it
Take the first numerical row of Table III, explicitly optimize the edge-party measurement directions, and evaluate the left-hand side of the n-local inequality directly on the post-filter state produced by the stated F_NS; if the maximal value does not exceed 2, then Eq. (18) was not validly applied after the non-separable filter and the claimed non-separable advantage would fail. The missing measurement directions used for the Table III values are exactly what such a check would need.
Extended reading notes
Core claim
The central claim is that the n-local inequality for a star network can be violated after suitable local filtering even when the unfiltered correlations satisfy the inequality, and that the maximal violation after separable single-qubit filters is governed by the closed expression B_n-star^(seq)=2 $\sqrt$( product-of-largest-singular-values-squared plus product-of-second-singular-values-squared ) for the filtered two-qubit states, given in Eq. (18). The paper proves that if every source distributes a two-qubit Bell-CHSH local state that remains local under any filtering, no violation is possible (Theorem 2), and that if at least one source distributes a two-qubit product state, no separable-filter protocol can violate the inequality (Theorem 3). It then presents numerical examples in which a non-separable filter at the central node produces violations in exactly the product-state setting where Theorem 3 forbids separable filters, along with instances where filtering by only the central party or only the edge parties suffices to reveal hidden non-trilocality.
Load-bearing premise
The load-bearing premise is that the filtered-network bound of Eq. (18), proved for separable filters where the post-filter state factors across sources, remains a valid upper bound after a joint non-separable filter has coupled qubits from different sources; the paper gives no proof for that case.
Editorial extensions
If this is right
- Violation of the n-local inequality in the sequential protocol certifies that at least one source is entangled, while saying nothing about how many sources are entangled.
- Filtering can enhance noise robustness: for amplitude-damped states there is a range of noise parameters in which the sequential network violates the trilocal inequality even though the usual network does not.
- Hidden non-trilocality can be generated with only one source supplying a hidden-Bell-nonlocal state while the other two sources distribute Bell-CHSH local states.
- Locally accessible hidden non-trilocality exists when only the central party filters, when only edge parties filter, or when a proper subset of parties filters.
- If the non-separable examples are correct, a joint central filter is a genuine resource in star networks because it can circumvent the product-state obstruction stated in Theorem 3.
Reading between the lines
- The non-separable advantage, if it survives direct evaluation, suggests that joint filtering at a network hub is a topology-specific resource with no analogue in the bipartite Bell scenario, and it invites a systematic search over non-separable filters for other network geometries such as linear and bilocal networks.
- A natural testable extension is to derive a genuine upper bound for the n-local expression after a joint non-separable filter, since the tensor-product factorization used to prove Eq. (18) fails there; without such a bound, the Table III violations are not yet anchored by a theorem.
- The paper's examples also indicate a possible route to device-independent entanglement detection in networks whose sources are contaminated by product-state noise, which would be practically relevant for diagnosing untrusted quantum-network nodes.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies a sequential star-shaped n-local network in which parties apply local filtering operations before performing measurements, with the goal of characterizing 'hidden non n-locality'. The authors derive an upper bound, Eq. (18), on the n-local inequality for the case where the central party's filter is a tensor product of single-qubit filters, and prove two no-go results: Theorem 2, that hidden non n-locality cannot be detected if every source state remains Bell-CHSH local under all local filtering operations, and Theorem 3, that no violation is possible in the separable-filter protocol when at least one source distributes a two-qubit product state. They then provide numerical examples of hidden non-trilocality, including cases with one or two hidden nonlocal sources, and discuss locally accessible hidden non-trilocality and enhanced noise robustness. In Section VII, they claim that non-separable multi-qubit filters at the central node reveal hidden non-trilocality even when a product state is distributed, and report violations in Table III. The central claim of the paper is that joint non-separable central filtering is a new resource in star networks that has no analogue in the bipartite Bell scenario.
Significance. If the non-separable filtering advantage were established, this would be a conceptually interesting result: it would show that the star topology gives the central party a joint-access resource beyond separable SLOCC operations, leading to network nonlocality that cannot be reproduced in a bipartite Bell setup. The separable-filter upper bound Eq. (18) and the no-go theorems Theorems 2 and 3 are valuable and appear to follow from the stated arguments. The explicit numerical examples, if correct, would also demonstrate an improved noise robustness of the sequential protocol. However, the central novelty of the paper, namely the non-separable filter advantage in Section VII, is currently unsupported because the derivations and numerical tables suffer from a missing justification and internal inconsistencies. The paper's significance is therefore conditional on a successful direct check of the non-separable filter examples.
major comments (3)
- [Section VII / Eq. (18) / Appendix A] The values B^(seq)_3-star in Table III are labeled with the symbol defined in Theorem 1, but Theorem 1 and its proof in Appendix A rely essentially on the post-filter state factoring as a tensor product, Eq. (34). This factorization holds for the separable central filter of Eq. (16), but not for the non-separable filter F_NS used in Eq. (27). After applying F_NS, the central party's qubits are entangled across the different sources, so Eq. (18) has no justification in this case. The paper provides no replacement bound and no statement of how the numbers in Table III were obtained. The authors should either provide a direct evaluation of the n-local inequality Eq. (11) for explicit measurement directions, or prove a new theorem that extends Eq. (18) to non-separable central filters.
- [Table III, first row] The reported value B_3-star = 1.31468 for the first row is inconsistent with the paper's own formula Eq. (13). For the stated states, beta_1 = 0.785 and beta_2 = -0.144 are pure states of the form Eq. (28), whose correlation tensors have singular values (1, |sin 2 beta_i|, |sin 2 beta_i|), and S_3 distributes the product state Eq. (29), whose correlation tensor has only one nonzero singular value. Eq. (13) then gives B_3-star = 2, not 1.31468. This discrepancy indicates that the table entries are not reliably generated from the formulas in the paper, and it undermines confidence in the non-separable filter claim.
- [Section VII.B / Table III] Even if the non-separable filter numbers were intended to come from a direct computation of the n-local inequality rather than from Eq. (18), the manuscript does not specify the edge measurement directions or any other measurement settings used to obtain B^(seq)_3-star = 2.00716, 2.0408, and 2.0056. Without these settings, the reported violations cannot be checked or reproduced. This is a load-bearing omission because the non-separable advantage is the main new claim of the paper.
minor comments (5)
- [Section IV.A.1 and Figure 3] The text states w11 = 0.01 and success probability approximately 37%, while the Figure 3 caption states w11 = 0.05 and approximately 32% probability of success; these specifications should be reconciled.
- [Section V.A] The sentence 'none of P2, P2, P3 perform any operation' appears to contain a typo; it should list P2, P3, and P4.
- [Appendix A, Eqs. (34)-(35)] The normalization is written ambiguously: the factor 1/Π_i C_i appears inside the tensor product in Eq. (34) and again in the definition of ρ^(f)_{1,i} in Eq. (35). This should be rewritten so that each normalized two-qubit state has unit trace.
- [Section VII] The notation for the sequential network with non-separable filters (referred to as N(seq)_n-star) is too easily confused with the separable-filter network N (seq)_n-star; distinct symbols would improve readability.
- [Section VII.B, Eq. (32)] The term '0.2089∥11⟩⟨00|' contains a typo: the first ket should be '|11⟩' rather than '∥11⟩'.
Circularity Check
No circular derivation: the central bound Eq. 18 is obtained from the external upper bound Eq. 13 (Kundu et al. [30]) applied to a factorized post-filter state, and the examples are free parameter searches rather than fitted predictions.
full rationale
The paper's main derivation chain is not circular. Theorem 1 (Eq. 18) is proved in Appendix A by noting that under the separable-filter protocol the post-filter state factorizes (Eq. 34) as a tensor product of per-source filtered states, and then applying the previously known star-network upper bound of Eq. 13, cited from Kundu et al. [30]. The hidden-nonlocality criterion (Eq. 8) is taken from Pal and Ghosh [12]. The numerical examples in Sections IV and V are existence constructions: the authors choose states and filter parameters by hand and compute B_3-star^(seq) from Eq. 18; this is not fitting a parameter to the target quantity and then renaming it a prediction. Theorem 2 is derived from the standard Bell-locality singular-value condition and an AM-GM bound; Theorem 3 follows from the product-state singular-value structure of the filtered state. The only self-citations are Ref. [39] for the notion of hidden non n-locality and Ref. [52] in Theorem 4; neither is load-bearing, since the new protocol is defined in the paper and Theorem 4 is an immediate application of the external inequality bound after local filtering preserves separability. The non-separable filter section (Section VII) may be mathematically under-supported because the factorized-state derivation of Eq. 18 does not apply to the joint filter F_NS, but that is a correctness/validity gap, not a circular reduction of the claimed prediction to its inputs. No equation is used as both premise and conclusion, and no fitted parameter is relabeled as a prediction.
Assumptions & free parameters
free parameters (5)
- Source state parameters in constructed examples (w's, p's, theta's, beta's, v's) =
Many; e.g., w11=0.01, w12=0.2456, w13=0.639, p2=0.042, p3=0.2169, theta2=theta3=0.4585
- Edge filter parameters epsilon_i (Eq. 21) =
epsilon2=0.99, epsilon3=0.902, epsilon4=0.998 in the first example; other values in Tables II and III
- Central separable filter parameters epsilon_1^(j) (Eq. 22) =
e.g., 0.98, 0.5569, 0.98; also 0.99, 0.99, 0.37
- Nonseparable 3-qubit filter parameters alpha1, alpha2 (Eq. 30) =
alpha1 around 0.26 to 0.27143, alpha2 around 0.173
- Nonseparable 2-qubit filter parameters alpha3, alpha4 and alpha5, alpha6 (Eqs. 32 and 33) =
alpha3=0.88, alpha4=0.67; alpha5=1.1, alpha6=-pi/2
assumptions (6)
- standard math Horodecki CHSH criterion: for two-qubit states, the maximum CHSH value is 2 sqrt(lambda1^2 + lambda2^2), where lambda1, lambda2 are the two largest singular values of the correlation tensor.
- standard math Pal-Ghosh criterion mu2 + mu3 > mu1 for hidden Bell-CHSH nonlocality of a two-qubit state.
- domain assumption n-local source independence, Eq. 9.
- domain assumption Star-network n-local inequality and its upper bound, Eqs. 11 and 13 from Tavakoli et al. [22] and Kundu et al. [30].
- ad hoc to paper The non-separable central filters in Eqs. 30, 32, 33 are valid quantum filters satisfying F^dagger F <= I.
- ad hoc to paper Equation 18 extends to non-separable central filters even though the post-filter state is not a product across sources.
Cite this review
Pith. "Pith review of Revealing Hidden Non n-Locality In n-Local Star Network." pith.science (2026). https://pith.science/paper/HGQBH3HG
@misc{pith2026250619026,
author = {Pith},
title = {Pith review of: Revealing Hidden Non n-Locality In n-Local Star Network},
year = {2026},
howpublished = {\url{https://pith.science/paper/HGQBH3HG}},
note = {Machine review of arXiv:2506.19026}
}
read the original abstract
Keeping pace with technological advancement, in the past decade, use of scalable networks have extended the study of quantum non-classicality beyond the regime of Bell-CHSH nonlocality. Present work provide characterization of non n-locality that can be exploited by incorporating filtering operations in star-shaped n-local networks. This in turn provide a framework of sequential n-local networks capable of generating non n-local correlations by involving some suitable form of stochastic local operations assisted with classical communications(SLOCC). It is observed that for effectiveness of such sequential networks, Bell-CHSH nonlocality(upto SLOCC operations) of every individual two-qubit state, distributed in the network, is not mandatory. However, there does not exist any separable local filter, which when applied in n-local network involving only Bell local states(upto SLOCC operations), can reveal non n-locality. Interestingly, instances revealing advantage of nonseparable mutli-qubit local filters over separable mutli-qubit local filters(by central node) are obtained. Such an advantage is attributable to the specific topology of star-shaped n-local networks and thus can never be reflected in Bell scenario.
Reference graph
Works this paper leans on
-
[1]
So, S1 distribute state that do not display any hidden nonlocal- ity(violates Eq.(8))
N (seq) n−star Involving Two Hidden Nonlocal States Let one source(S1,say) distribute Bell-CHSH local Bell- diagonal state, i.e., ρ1,1=ρ(BD) 1 satisfies Eq.( 4). So, S1 distribute state that do not display any hidden nonlocal- ity(violates Eq.(8)). Let each ofS2,S3 distribute a state from the following two-qubit entangled family of states[ 50]: ρ1,i = ρH ...
-
[2]
Measurement Phase Preparation Phase: As in any n-local star network (sec.II), let each of n sources Si distribute a two-qubit state ρ1,i Table I: Table displays the details of the classically post-processed bits ˜c(i) 1 and also the functions gi(x2, x3, ...,xn+1) for n=3, 4 in the n-local inequality(Eq.(11)). n ˜c(i) 1 gi(x2, x3, ...,xn+1) 3 ˜c(1) 1 =c11,...
-
[3]
Let S3 distribute hidden nonlocal state ρH 3 (Eq.(23))
N (seq) n−star Involving Only One Hidden Nonlocal State LetS1,S2 both distribute local Bell-Diagonal state, i.e., ρ(BD) 1 , ρ(BD) 2 both satisfy Eq.( 4). Let S3 distribute hidden nonlocal state ρH 3 (Eq.(23)). For some local ρBD 1 , ρBD 2 and hidden nonlocal ρH 3 hidden non-trilocality is observed in the network under application of suitable local filters...
-
[4]
Specifications of the state, noise and filtering parameters are as mentioned in the main text
Communication Through Amplitude Damping Channel[ 48] Let each ofS1,S2 andS3 distribute an identical copy of pure entangled state: |Ψ(θ⟩) = sin θ|01⟩ + cos θ|10⟩, θ∈ (0, π 4 ) (26) Figure 6: Plotting curvesB(seq) 3−star−2 andB3−star−2. Specifications of the state, noise and filtering parameters are as mentioned in the main text. Comparison of the two curve...
-
[5]
J. S. Bell, Physics 1, 195 (1964)
1964
-
[6]
Speakable and Unspeakable in Quantum Mechanics
J. S. Bell, “Speakable and Unspeakable in Quantum Mechanics” (Cambridge University Press, Cambridge,England, 2004), 2nd ed
work page 2004
-
[7]
P .Zoller, et.al., The European Physical Journal D 36, 203 (2005)
work page 2005
-
[8]
N. Brunner, D. Cavalcanti, S. Pironio, V . Scarani, and S. Wehner, “Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014)
work page 2014
Show all 55 references
-
[9]
S.Pironio, et.al., Nature 464, 1021-1024 (2010)
2010
-
[10]
A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V .Scarani, Physical Review Letters98, 230501 (2007)
2007
-
[11]
Hirsch: Hidden Nonlocality
F. Hirsch: Hidden Nonlocality. Mas- ter thesis, University of Geneva ( 2013), http://cms.unige.ch/sciences/physique/wpcontent/ uploads/Travail-de- Master.pdf
2013
-
[12]
Gisin, Phys
N. Gisin, Phys. Lett. A 210, 151 (1996)
1996
-
[13]
Popescu, Phys
S. Popescu, Phys. Rev. Lett. 74, 2619 (1995)
1995
-
[14]
Hirsch, M
F. Hirsch, M. T. Quintino, J. Bowles, N. Brunner, Phys. Rev. Lett. 111, 160402 (2013)
2013
-
[15]
Quantum entanglement
R. Horodecki, P . Horodecki, M. Horodecki, K. Horodecki, “Quantum entanglement”, Rev. Mod. Phys. 81,865 (2009)
2009
-
[16]
R.Pal and S.Ghosh, arxiv: 1410.7574 [quant-ph] (2014)
2014 arXiv
-
[17]
Locally inaccessible hidden quantum correlations
A.F. Ducuara, Cristian E. Susa, Paul Skrzypczyk, “ Locally inaccessible hidden quantum correlations”, Phys. Rev. A 110, 022435 (2024)
2024
-
[18]
B. Paul, K. Mukherjee and D. Sarkar Phys. Rev. A 94, 052101 (2016)
2016
-
[20]
T.Fritz, New J. Phys. 14 103001 (2012)
2012
-
[21]
Branciard,D
C. Branciard,D. Rosset, N. Gisin and S. Pironio, Phys. Rev. A 85, 032119 (2012)
2012
-
[22]
Branciard, N
C. Branciard, N. Gisin, and S. Pironio, Phys. Rev. Lett. 104,170401 (2010)
2010
-
[23]
Mukherjee, B
K. Mukherjee, B. Paul and D. Sarkar, Quantum Inf Process. 14, 2025 (2015)
2015
-
[24]
Mukherjee, B
K. Mukherjee, B. Paul and D. Sarkar, Quantum Inf Process. 15, 2895 (2016)
2016
-
[25]
M.O.Renou, et al., Phys. Rev. Lett. 123, 140401 (2019)
2019
-
[26]
Skrzypczyk, D
A.Tavakoli, P . Skrzypczyk, D. Cavalcanti, A. Acín, Phys. Rev. A 90, 062109 (2014)
2014
-
[27]
Rev.A96, 020304, (2017)
N.Gisin, et al., Phys. Rev.A96, 020304, (2017)
2017
-
[29]
F.Andreoli, et al.,, Phys. Rev. A 95, 062315 (2017)
2017
-
[30]
Andreoli, et al., New.J.Phys
F. Andreoli, et al., New.J.Phys. 19, 113020 (2017)
2017
-
[31]
Mukherjee, B
K. Mukherjee, B. Paul and D. Sarkar, Phys. Rev. A 96, 022103 (2017)
2017
-
[32]
Mukherjee, B
K. Mukherjee, B. Paul and D. Sarkar, Quantum Inf Process. 18, 212 (2019)
2019
-
[33]
Mukherjee, B
K. Mukherjee, B. Paul and A.Roy, Phys. Rev. A101, 032328 (2020)
2020
-
[34]
Kundu, M.K
A. Kundu, M.K. Molla, I. Chattopadhyay and D. Sarkar, Phys. Rev. A 102, 052222 (2020)
2020
-
[35]
A. P . Kerstjens, N. Gisin and A. Tavakoli, Phys. Rev. Lett. 128, 010403 (2022)
2022
-
[36]
A.Tavakoli, C.Branciard and N.Gisin, Phys. Rev. Lett. 126, 220401 (2021)
2021
-
[37]
I.Supic, J.D.Bancal and N.Brunner, Phys. Rev. Lett. 125, 240403 (2020)
2020
-
[38]
Aberg, R
J. Aberg, R. Nery, C. Duarte, R. Chaves, Phys. Rev. Lett. 125, 110505 (2020)
2020
-
[39]
Wolfe, A
E. Wolfe, A. P .Kerstjens, M. Grinberg, D. Rosset, A. Acin, M. Navascues, Phys. Rev. X 11, 021043 (2021)
2021
-
[40]
K.Hansenne, Z.P . Xu, T. Kraft, O. Guhne, Nature Commu- nications 13, 496 (2022)
2022
-
[41]
S.K.Liao, et al., Phys. Rev. Lett. 120, 030501 (2018)
2018
-
[42]
Mukherjee, I
K. Mukherjee, I. Chakrabarty, and G. Mylavarapu, Phys. Rev. A 107, 032404 (2023)
2023
-
[43]
Hidden Non n-locality In Linear Networks
K.Mukherjee, S.Mandal, T.Patro and N.Ganguly, " Hidden Non n-locality In Linear Networks ", Phys. Rev. A 108,032416 (2023)
2023
-
[44]
Mukherjee, Phys
K. Mukherjee, Phys. Rev. A 109, 032216 (2024)
2024
-
[45]
The quantum internet
H. J. Kimble, “ The quantum internet ”, Nature 453, 1023 (2008)
2008
-
[46]
Yu, et.al., Nature 578, 240 (2020)
Y. Yu, et.al., Nature 578, 240 (2020)
2020
-
[47]
Quantum internet: A vision for the road ahead
S. Wehner, D. Elkouss, and R. Hanson, “ Quantum internet: A vision for the road ahead”, Science 362, 6412 (2018)
2018
-
[48]
Kozlowski, S
W. Kozlowski, S. Wehner,’ Proceedings of the Sixth An- nual ACM International Conference on Nanoscale Com- puting and Communication, NANOCOM ’19 (Association for Computing Machinery, New York, NY, USA,(2019)
2019
-
[49]
C. M. Lee and M. J. Hoban, Phys. Rev. Lett. 120, 020504 (2018)
2018
-
[50]
Tavakoli, A
A. Tavakoli, A. P .Kerstjens, M.X. Luo, M.O. Renou, arXiv:2104.10700 [quant-ph] (2021)
2021 arXiv
-
[51]
Entangled Bloch spheres: Bloch matrix and two- qubit state space
O.Gamel,“Entangled Bloch spheres: Bloch matrix and two- qubit state space”, Phys. Rev. A. 93, 062320 (2016)
2016
-
[52]
Quantum information science
I. Chuang and M. Nielsen: “ Quantum information science”, Cambridge University Press (2000)
2000
-
[53]
Entanglement versus Bell violations and their behaviour under local filtering operations
F.Verstraete, M.M.Wolf, “Entanglement versus Bell violations and their behaviour under local filtering operations ”, Phys. Rev. Lett. 89, 170401 (2002)
2002
-
[54]
Wojcik, J
A. Wojcik, J. Modlawska, A. Grudka, and M. Czechlewski, Phys. Lett. A 374, 4831 (2010)
2010
-
[55]
Optimal remote entanglement distribution
W. Dai, T. Peng, M.Z. Win, “Optimal remote entanglement distribution.”, IEEE J. Sel. Areas Commun. 38(3), 540–556 (2020). https://doi.org/10.1109/JSAC.2020.2969005
2020
-
[56]
Network Nonlocality Without En- tanglement Of All Sources
K.Muherjee and B.Paul,“Network Nonlocality Without En- tanglement Of All Sources”, arXiv:2410.15131v1 (2024)
2024 arXiv
-
[57]
Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model
R.F. Werner, "Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model”, Phys. Rev. A 40 (8): 4277 (1989)
1989
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