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REVIEW 3 major objections 6 minor 38 references

Construction of Sets of Orthogonal Quantum States with Minimal Nonlocality in Bipartite and Tripartite Systems of Unequal Local Dimensions

T0 review · 3 major / 6 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper constructs orthogonal state sets with minimal nonlocality in bipartite and tripartite systems whose local dimensions are unequal, filling a gap left by equal-dimension-only constructions.

desk verdict Bipartite result is real; the tripartite proof as printed has a load-bearing gap and should not be accepted without revision. read the letter →

arxiv 2607.28998 v1 pith:23KNTQ4U submitted 2026-07-31 quant-ph

Ben-Hao Zhao , Dong-Huan Jiang , Yu-Guang Yang , Guang-Bao Xu This is my paper · ORCID
classification quant-ph PACS 03.67.-a03.65.Ud
keywords minimalnonlocalityorthogonalquantumstateslocalindistinguishabilityLOCCunequaldimensionsbipartitesystemtripartitefull-sumstate
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

The paper addresses the construction problem for minimal nonlocality in quantum systems where the local dimensions are unequal, a case previous work left open. It gives explicit state sets: S2 in C^{d1} ⊗ C^{d2} for 4 ≤ d1 < d2 containing 2d2 − 2 states, and S3 in C^{d1} ⊗ C^{d2} ⊗ C^{d3} for 5 ≤ d1 < d2 < d3 containing 4d3 + 2d1 − 11 states. For each set it proves two claims: the entire set cannot be perfectly distinguished by LOCC, and after deleting the last full-sum state the remaining states can be perfectly distinguished by LOCC. If correct, this settles the minimal-nonlocality construction problem for unequal-dimensional bipartite and tripartite systems and offers concrete state families with practical communication potential.

What carries the argument

The load-bearing object is the 'full-sum state', a product state that superposes all basis states on every party with equal amplitude. It plays two roles: together with root-of-unity superposed states it forces the diagonal entries of any orthogonality-preserving POVM matrix to be equal, finishing the proof that a first measurement by any party is trivial; and deleting exactly this full-sum state is what turns the remainder into an LOCC-distinguishable set. The proof also relies on pairwise orthogonality tables, one per party and per bipartite/tripartite setting, to drive off-diagonal matrix entries to zero, and on Vandermonde-style linear systems solved by Cramer's rule.

What would settle it

For the explicit d1 = 5, d2 = 7, d3 = 9 example, compute the 35 listed states and numerically search for a non-identity POVM matrix M_B on Bob's subsystem that keeps all post-measurement states mutually orthogonal; if such a matrix exists, Theorem 3's nonlocality claim fails. A direct check is the Table 9 row citing |φ_{4d3+3d2−14}⟩, an index that does not exist in a 35-state set, so that orthogonality constraint cannot hold as written.

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Extended reading notes

Core claim

The central claim is Theorem 2 and Theorem 3: for 4 ≤ d1 < d2 the set S2 of 2d2 − 2 orthogonal quantum states in C^{d1} ⊗ C^{d2}, and for 5 ≤ d1 < d2 < d3 the set S3 of 4d3 + 2d1 − 11 orthogonal quantum states in C^{d1} ⊗ C^{d2} ⊗ C^{d3}, each possess minimal nonlocality. Local indistinguishability is argued by showing that any POVM element on an individual party that preserves the mutual orthogonality of the post-measurement states must be proportional to the identity: pairwise orthogonality constraints force all off-diagonal matrix elements to zero, and orthogonality with a uniform full-sum state forces all diagonal elements equal, with the equalities completed by root-of-unity systems of

Load-bearing premise

The tripartite local-indistinguishability proof assumes that the listed pairwise orthogonality constraints on Bob's and Charlie's POVM matrices are exhaustive enough to force every off-diagonal entry to zero, and at least one listed constraint in Table 9 refers to a state index that is not defined in S3.

Editorial extensions

If this is right

  • If Theorems 2 and 3 hold, unequal-dimensional quantum systems for the first time have state sets whose nonlocality is exactly one-state-deep: deleting a single full-sum state converts a globally indistinguishable set into an LOCC-distinguishable one.
  • The explicit state families, including the C4 ⊗ C7 and C5 ⊗ C7 ⊗ C9 examples, give ready-made resources for quantum communication protocols that need to block an adversary while allowing a legitimate receiver to decode after applying one local deletion.
  • The construction method is not tied to the specific dimensions 4 and 7, so the same pattern may extend to larger unequal local dimensions and to multipartite systems with more than three parties.
  • The proof that the deleted remainder is LOCC-distinguishable supplies an explicit measurement sequence, which is directly implementable in distributed settings where parties can broadcast classical measurement outcomes.
  • The tripartite construction reduces, at several points, to the bipartite construction on a subsystem, suggesting that the bipartite theorem is a reusable building block for higher-partite minimal-nonlocality results.

Reading between the lines

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

  • The paper's meaning of 'minimal nonlocality' is the one-state-deletion property of Definition 3, not minimal cardinality among all locally indistinguishable sets; whether 2d2 − 2 and 4d3 + 2d1 − 11 are also the smallest possible cardinalities for unequal dimensions is not established and remains open.
  • The tripartite proof relies on asserted reductions to the bipartite theorem after deleting the full-sum state; if those reductions are made fully rigorous they would yield a recursive framework for constructing minimally nonlocal sets in arbitrary multipartite unequal-dimensional systems.
  • Because the constructed states are entangled rather than merely product states, the result indicates that minimal nonlocality is not an exclusive feature of orthogonal product states, widening the known landscape of such sets.
  • A concrete testable extension is to run numerical LOCC or POVM searches on the explicit 12-state and 35-state sets to independently confirm that no non-identity orthogonality-preserving POVM exists, checking the pairwise orthogonality tables before relying on them.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper proposes constructions of orthogonal quantum state sets with minimal nonlocality in unequal-dimensional systems. Theorem 1 gives a 12-state set in C^4⊗C^7; Theorem 2 gives a (2d2−2)-state set in C^{d1}⊗C^{d2} for 4≤d1<d2; Theorem 3 gives a (4d3+2d1−11)-state set in C^{d1}⊗C^{d2}⊗C^{d3} for 5≤d1<d2<d3. For each set, the authors argue that any LOCC protocol that preserves orthogonality must begin with a trivial measurement (local indistinguishability), and that deleting one full-sum state leaves a set perfectly distinguishable by LOCC (minimality). The bipartite proof is largely explicit and self-contained. The tripartite proof is not: the Bob POVM analysis has undefined state indices and missing constraints, and the post-removal distinguishability proof relies on asserted reductions to Theorem 2 without a written mapping.

Significance. If the results are correct, this would be the first construction of minimal-nonlocality sets for unequal local dimensions, extending the equal-dimensional framework of Zhu et al. The bipartite part is constructive, systematic, and appears sound; the C4⊗C7 example illustrates the method concretely. However, the tripartite theorem — a central advertised contribution — is not established as printed. The gaps are load-bearing because both halves of minimal nonlocality (indistinguishability of the full set and LOCC-distinguishability after one deletion) are required. The paper is worth publishing after substantial revision, but the current version does not support the abstract's claim that the tripartite construction problem is settled.

major comments (3)
  1. [§IV, Theorem 3, Tables 9–10] The Bob POVM proof is incomplete. Table 9 cites a state |φ_{4d3+3d2−14}⟩ that is not in S3; the intended index appears to be |φ_{3d3+3d2−14}⟩, the i=d2−d1 member of |φ_{3d3+2d2+d1−14+i}⟩. More importantly, the listed constraints do not force M_B†M_B to be proportional to identity. The only states with Bob support on |j⟩_B for j∈{d1,...,d2−2} are |φ_{3d3+2d2+d1−14+i}⟩ with j=d1−1+i, and each such state has Bob support on a single basis vector. Tables 9–10 contain no row that couples this subspace to the low-index Bob subspace through a common Alice/Charlie factor, and no row that equates the diagonal entries b_{jj} for j∈{d1,...,d2−2} to b_{00} or b_{11}. Thus a positive diagonal POVM element with arbitrary weights on span{|d1⟩_B,...,|d2−2⟩_B} and proportional to identity on the complementary subspace preserves the orthogonality of all pairs in S3. The conclusion that Bob's first measurem
  2. [§IV, diagonal part of M_B†M_B] Even if all off-diagonal entries were shown to vanish, the diagonal entries for Bob's high-interior subspace are undetermined. Table 10 equates b_{00} with b_{11},...,b_{(d2−2)(d2−2)} and b_{11} with b_{(d2−1)(d2−1)}, but it gives no relation between b_{00} and b_{jj} for j=d1,...,d2−2. For Charlie, Eq. (3) plus Table 12 connects the high-index block to c_{11}; no analogous Fourier system exists for Bob because the states |φ_{2d3+d2+d1−8+i}⟩, which generate Eq. (3), have Bob support only on |d2−4⟩. A diagonal Bob matrix with a different weight on the subspace spanned by |d1⟩_B,...,|d2−2⟩_B preserves S3's orthogonality under the stated conditions. This is a second, independent reason that the local-indistinguishability half of Theorem 3 fails as written.
  3. [§IV, LOCC-distinguishability of S3−{|φ_{4d3+2d1−11}⟩}] The proof asserts two reductions to Theorem 2 — after Alice's A1 outcome followed by Bob's B0, and after Bob's B2 — with the phrases 'equivalent to the constructions for Alice and Charlie subsystems in Theorem 2' and 'conform to the constructions in Theorem 2'. No state-by-state mapping is provided. The collapsed sets listed do not match the S2 template in an obvious way: the Alice superpositions (e.g., |1−2⟩_A), the Charlie indices (e.g., |d3−2−i⟩_C), and the ordering of the states differ from the S2 definitions. A reader cannot verify that these sets are LOCC-distinguishable without an explicit identification or a separate proof of local equivalence. Since minimal nonlocality requires this second half, Theorem 3 is incomplete.
minor comments (6)
  1. [Throughout] Typos and language issues: 'quantun systems' in the Conclusion; 'ststes' at the start of Section IV; 'Noniagonal' in Table 4's header; 'of of' in Section III; 'by via LOCC' in the Conclusion. These should be corrected.
  2. [§IV, Table 9] The state |φ_{4d3+3d2−14}⟩ is not defined anywhere in S3. The intended state is almost certainly |φ_{3d3+3d2−14}⟩ = |φ_{3d3+2d2+d1−14+i}⟩ with i=d2−d1. Please fix the index.
  3. [§IV, Table 10] The row '|φ_{2d−3+d2+d1−8}⟩' contains a typo in the subscript: it should be |φ_{2d3+d2+d1−8}⟩.
  4. [§III, Theorem 1, Table 3] For the pair (|φ3⟩,|φ9⟩), the table records b30=b06=0 and b03=b60=0. Direct expansion of ⟨φ3|I⊗M_B†M_B|φ9⟩ and its conjugate gives b30=b06 and b03=b60, not automatic vanishing of each entry. Either the table should state the equalities, or additional pairs that force each entry to zero should be cited.
  5. [§II, Lemma 1] 'Kramer's rule' should be 'Cramer's rule'. Also, the lemma states a unique solution when the determinant is nonzero; this is standard, but the wording could be tightened.
  6. [§IV, C5⊗C7⊗C9 example] In the example, 'ω=e^{π√−1}' is correct because d3−d2=2, but writing ω=−1 would be clearer and avoid confusion with the general definition.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the constructions and proofs are self-contained; identified issues are rigor gaps, not circular dependencies.

full rationale

The paper's derivation chain is self-contained against standard definitions and prior equal-dimensional benchmarks. The claimed minimal-nonlocality results for Theorems 1-3 are established by direct POVM-constraint computations on the constructed state sets and by explicit LOCC protocols after removal of one state. No parameter is fitted to the target conclusion, no 'prediction' is obtained by renaming an input quantity, and no theorem's conclusion is assumed as an input. Self-citations such as Refs. [33] and [34] provide the conceptual definition of minimal nonlocality and prior equal-dimensional constructions, but the central local-indistinguishability arguments do not rest on those citations: Theorem 3's use of Theorem 2 is an internal reduction within the same paper, not an appeal to external self-citation. The apparent undefined index |φ_{4d3+3d2−14}⟩ in Table 9, the possibly non-exhaustive orthogonality tables, and the abbreviated 'reduction to Theorem 2' in the post-removal distinguishability proof are correctness/completeness concerns, not circularity: they do not make the claimed result equivalent to its own premises by construction. Therefore the honest finding is no significant circularity, score 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim needs no fitted parameters and no new physical entities. It rests on standard LOCC orthogonality-preservation arguments, root-of-unity algebra, and prior definitions of minimal nonlocality.

assumptions (3)
  • domain assumption A successful LOCC protocol's first measurement can be assumed orthogonality-preserving
    Used throughout Sections III and IV to reduce LOCC indistinguishability to showing that every orthogonality-preserving POVM element is proportional to the identity. This is standard in the local-distinguishability literature but not proven in the paper.
  • standard math Root-of-unity Vandermonde systems imply the required diagonal equalities (Eqs. (1), (2), (3))
    Invoked as Lemma 1 (Cramer's rule). The systems actually have a one-parameter solution space, so the word 'unique' is loose, but the conclusion that all relevant diagonal entries are equal up to a common scale is correct.
  • domain assumption Definitions of minimal nonlocality and trivial/nontrivial measurement are imported from prior work
    Definition 3 (minimal nonlocality) and Definition 2 (trivial measurement) are taken from Zhu et al. [33] and Xu et al. [36]; the paper's claims are evaluated against these definitions.

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Pith. "Pith review of Construction of Sets of Orthogonal Quantum States with Minimal Nonlocality in Bipartite and Tripartite Systems of Unequal Local Dimensions." pith.science (2026). https://pith.science/paper/23KNTQ4U

@misc{pith2026260728998,
  author       = {Pith},
  title        = {Pith review of: Construction of Sets of Orthogonal Quantum States with Minimal Nonlocality in Bipartite and Tripartite Systems of Unequal Local Dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/23KNTQ4U}},
  note         = {Machine review of arXiv:2607.28998}
}
abstract

The research on minimal nonlocality aims to determine the minimal cardinality of nonlocal sets of quantum states. However, the construction of a nonlocal set of states in bipartite or tripartite quantum systems with unequal local dimensions remains unsolved. In this paper, we first give a method to construct a set of orthogonal quantum states with minimal nonlocality in $\mathbb{C}^{4} \otimes \mathbb{C}^{7}$ quantum system. Then we give a general method to construct a set of orthogonal quantum states with minimal nonlocality in a bipartite quantum system with unequal local dimensions. Furthermore, we generalize the construction method to tripartite quantum system with unequal local dimensions, and construct a set of orthogonal quantum states with minimal nonlocality in $\mathbb C^{d_1} \otimes \mathbb C^{d_2}\otimes \mathbb C^{d_3}$ quantum system for $5\le d_1 < d_2 < d_3$. Our work settles the construction problem of a set of orthogonal states with minimal nonlocality in both bipartite and tripartite systems with unequal local dimensions.

Figures

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