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REVIEW 3 major objections 6 minor 44 references

Unextendible and strongly uncompletable product bases

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that a 'quasi U-tile' tile structure is a sufficient condition for an orthogonal product set to be a strongly uncompletable product basis, and uses it to construct a UPB of size $d^3-3d^2+1$ in…

desk verdict A useful construction paper that likely answers the one/two-bipartition SUCPB question and shrinks the known all-bipartition example, but the proof of Lemma 1 skips the one step that makes the whole thing work. read the letter →

arxiv 2411.18036 v1 pith:IA33VRXP submitted 2024-11-27 quant-ph

Xiao-Fan Zhen , Hui-Juan Zuo , Fei Shi , Shao-Ming Fei This is my paper · ORCID
classification quant-ph
keywords unextendibleproductbasisstronglyuncompletablequasiU-tilestructuretripartitequantumsystemsboundentanglementtile
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 gives a geometric sufficient condition for a set of orthogonal product states to be a strongly uncompletable product basis (SUCPB): the underlying tile structure must be a quasi U-tile structure. With this condition the authors settle the trichotomy question left open since 2003: in tripartite systems a UPB may be SUCPB in two bipartitions, in at most one, or in every bipartition. They build examples of each, and the headline construction is a UPB in $\mathbb{C}^d\otimes\mathbb{C}^d\otimes\mathbb{C}^d$ with $d^3-3d^2+1$ states that is SUCPB in every bipartition, a smaller cardinality than the previous known example. The proof machinery is Lemma 1, which shows that in a quasi U-tile structure any product state orthogonal to the stopper state is forced to lie in one of $m$ mutually orthogonal blocks, so the product states left in the complement cannot span it.

What carries the argument

The load-bearing object is the quasi U-tile structure: a tile structure whose tiles can be grouped into at least five new tiles, each new tile formed by old tiles sharing a row or column index, each new tile non-extendable except to the whole structure, and the new-tile structure itself a U-tile structure. Lemma 1 attaches to each tile a family of Fourier-type product states and adds the uniform stopper state $|S\rangle$; any product state in the complement corresponds to a rank-one matrix with zero sum of entries. The quasi U-tile conditions are used to force that matrix's support into a single new tile, and because the new tiles are mutually orthogonal, all product states in the complement lie in a space of dimension at most $n-m\le n-5$, strictly below the complement dimension $n-1$. This dimension shortfall is exactly the defining property of an SUCPB.

What would settle it

Find a quasi U-tile structure and a solution of the zero-sum, rank-one conditions on its matrix $M$ whose nonzero entries touch two different new tiles. The simplest place to look is the paper's $3\times 5$ example with six tiles, where solving the equations for the six coefficients $a_1,\dots,a_6$ would either confirm or refute the 'only $m$ cases' assertion; a counterexample would directly falsify Lemma 1 and the SUCPB claims built on it.

Watch

Extended reading notes

Core claim

The central claim is that a $d_1\times d_2$ tile structure is a quasi U-tile structure—its $n$ tiles can be partitioned into $m\ge 5$ new tiles, each new tile a union of old tiles sharing a common row or column index, none extendable except to the whole structure, and the new tiles themselves forming a U-tile structure—then the associated orthogonal product set is an SUCPB of size $d_1d_2-n+1$. In the tripartite setting the paper claims that by viewing each of the three bipartitions separately, the quasi U-tile criterion yields UPBs that are SUCPBs in two bipartitions, in at most one, and in every bipartition, completing the classification. The main object is the UPB $U^4$ in $\mathbb{C}^d\otimes\mathbb{C}^d\otimes\mathbb{C}^d$ of size $d^3-3d^2+1$, built from vertical, horizontal, and crossed tile states plus the uniform stopper state, which is SUCPB in every bipartition for $d\ge 6$.

Load-bearing premise

The load-bearing premise is that a product state orthogonal to the constructed basis can only have nonzero coefficients inside one of the $m$ new tiles; if this classification is false, the dimension count that makes the basis strongly uncompletable collapses.

Editorial extensions

If this is right

  • For odd $d\ge 3$, the set $U^1$ in $\mathbb{C}^d\otimes\mathbb{C}^d\otimes\mathbb{C}^2$ is a UPB of size $2d^2-4d+4$ that is SUCPB in the two bipartitions pairing the qubit with one qudit; for even $d\ge 4$, $U^{1\prime}$ has size $2d^2-4d+8$ and the same property.
  • In $\mathbb{C}^d\otimes\mathbb{C}^2\otimes\mathbb{C}^2$, the sets $U^2$ (size $4d-4$) and $U^3$ (size $4d-7$) show that UPBs that are SUCPB in no bipartition and in exactly one bipartition both exist, so all three logical cases occur.
  • The main construction $U^4$ is a UPB of size $d^3-3d^2+1$ in $\mathbb{C}^d\otimes\mathbb{C}^d\otimes\mathbb{C}^d$ for $d\ge 6$ and is SUCPB in every one of the three bipartitions; for $d=6$ this yields 109 states, fewer than the previous 200-state example.
  • Any orthogonal product set whose tile structure is quasi U-tile is automatically SUCPB by Lemma 1, giving a reusable sufficient condition that avoids checking complement completions case by case.

Reading between the lines

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

  • The quasi U-tile criterion is formulated for rectangular tiles, but the same rank-one/zero-sum mechanism should extend to cube-like tiles in N-partite systems with $N\ge 4$, producing SUCPBs in every bipartition by applying the argument cut by cut.
  • The new construction requires $d\ge 6$, so the minimal size of a UPB that is SUCPB in every bipartition in dimensions $d=3,4,5$ remains open; the bound $d^3-3d^2+1$ suggests the true scaling is $d^3-O(d^2)$, and combinatorial searches over tile partitions could test smaller cases.
  • The quasi U-tile recipe decouples the SUCPB property from the explicit vector details: any tile partition satisfying the three geometric conditions yields the same dimension shortfall, so one can search for new SUCPBs by purely combinatorial data on tiles.
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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 introduces a geometric sufficient condition, called a quasi U-tile structure, for constructing strongly uncompletable product bases (SUCPBs). Lemma 1 claims that any bipartite tile structure satisfying this condition yields an SUCPB of size d1d2 - n + 1. The authors then apply this lemma to tripartite systems: they construct UPBs in C^d ⊗ C^d ⊗ C^2 that are SUCPBs in two bipartitions, UPBs in C^d ⊗ C^2 ⊗ C^2 that are SUCPBs in at most one bipartition, and a UPB in C^d ⊗ C^d ⊗ C^d of size d^3 - 3d^2 + 1 that is an SUCPB in every bipartition. The paper's main advertised achievement is this smaller-cardinality all-bipartition SUCPB, together with a systematic survey of the possible numbers of bipartitions in which a UPB can be an SUCPB.

Significance. If Lemma 1 is correct, the quasi U-tile criterion is a useful and genuinely new tool for constructing SUCPBs, and the reported examples answer the open question about existence of UPBs that are SUCPBs in only one or two bipartitions. The explicit constructions are nontrivial and the cardinality improvement from the previously known all-bipartition SUCPB to d^3 - 3d^2 + 1 is a concrete contribution. The paper does not fit parameters and does not assume its main conclusion as an input; it builds on the established U-tile characterization of UPBs from Ref. 28 and the SUCPB criterion from Ref. 29, which is a reasonable foundation if those results are correctly invoked. However, the central lemma's decisive step is presently asserted rather than proved, and several later verifications are delegated to visual inspection of figures, so the significance of the paper depends on closing these gaps.

major comments (3)
  1. [Lemma 1, proof, Eq. (5)] The load-bearing step of Lemma 1 is the sentence 'From the conditions (i)-(iii), if Sum(M)=0 and rank(M)=1, there are only m cases for the matrix M.' This classification is not proved, and it is not a direct consequence of the stated conditions. A rank-one matrix has support equal to a product of a row set and a column set; such a support can intersect several groups l_j while containing none of them as a full rectangle. Condition (i) only forbids extending an entire l_j to a larger tile, and condition (iii) only constrains the coarse tile structure T_U. Neither condition, as written, rules out a support that uses only some tiles from each of several l_j and is closed under the row-column Cartesian product. Without Eq. (5), the dimension count Dim(O_1+...+O_m)=n-m is unsupported, and the conclusion Dim(H⊥_US) ≤ n-5, hence the SUCPB property, does not follow. The same gap is inherited by Theorems 1, 3, 4 and Proposition 3, which all invoke Lemma 1. Please provide a complete proof of the support classification, or add and justify an explicit extra condition on the quasi U-tile structure that makes it true.
  2. [Section V, proofs of Proposition 3 and Theorem 4] The proofs that the new tile structures ∪ l_j are U-tile structures in each bipartition are asserted rather than demonstrated. For Proposition 3 this is a visual check in Fig. 11, and for Theorem 4 it is stated as 'they form a U-tile structure' without a general combinatorial argument. To apply Lemma 1, condition (iii) must be verified for all d, which requires showing that no special rectangle in the new tile structure can be decomposed into two smaller special rectangles or tiles. The current text does not supply such a verification, and the analogy with the d=6 case is not a proof for general d. This point is load-bearing for the claim that the constructed sets are SUCPBs in every bipartition.
  3. [Theorem 2, proof and general construction U2] The proof of Theorem 2 states that the four states |ψ1⟩,...,|ψ4⟩ are orthogonal product states in H⊥_{U2_A|BC}, but it does not verify their orthogonality or their containment in the complementary subspace for general d. Since the conclusion that U2 is not an SUCPB in any bipartition rests on these four states spanning a four-dimensional complement, this verification should be given explicitly. The same applies to the analogous four-state argument in Example 2, where the states are listed without a check that they are mutually orthogonal and orthogonal to all members of the UPB.
minor comments (6)
  1. [Abstract and Introduction] There are several typos: 'sized3 −3d2 +1' should read 'size d^3 − 3d^2 + 1', and 'bipartions' should be 'bipartitions'.
  2. [Section V, after Proposition 3] The sentence 'In Ref. ??, Shi et al. proposed a UPB of size 200...' contains a missing reference placeholder. This must be filled in before publication.
  3. [Eq. (14), general construction for Theorem 3] The displayed formula for A5 has a typographical error: it reads '|β_j⟩_A |η_s⟩_V |1⟩_C}' with a subscript V and an extra brace. It should presumably be '|β_j⟩_A |η_s⟩_B |1⟩_C'.
  4. [Section II, definition of SUCPB] The definition of SUCPB is awkward: 'for all Hext' is unclear. The intended meaning appears to be that the incompletability holds for every bipartition of the multipartite system; the text should say this explicitly.
  5. [Theorem 4, proof] In the proof of Theorem 4, the index set for k and i is written as Z6 even though the statement is for general d ≥ 6. This should be Z_d.
  6. [Example 2 and Fig. 7/Fig. 8] The text says the 4 × 4 tile structure T^{V2}_{A|BC} is shown in Fig. 7, but the figure labeled Fig. 8 is the one displaying a 4 × 4 tile structure; the references to the figures are inconsistent.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the quasi U-tile construction is new and its conclusions are not assumed as inputs; the self-citations to Refs. 28/29 are background tools, not a circular chain.

full rationale

The central derivation proceeds by defining a quasi U-tile structure in Sec. II, producing explicit orthogonal product states from each tile, and then proving in Lemma 1 that the stopper-state set is an SUCPB. Nothing in the definition of the quasi U-tile structure or in the construction of U1-U4 is fitted to the claimed conclusion: the states are explicit, the bipartition cases are checked by direct tile partitions, and the cardinalities d^3-3d^2+1, 2d^2-4d+4, etc. are not fitted parameters. The only step that is not fully argued is the sentence before Eq. (5): 'From the conditions (i)-(iii), if Sum(M)=0 and rank(M)=1, there are only m cases for the matrix M.' This is an unproved support-classification claim (a rank-one zero-sum matrix may in principle use proper subrectangles of several l_j), and if false the dimension count n-m<n-1 fails. But this is a correctness gap in the proof, not circularity: the conclusion is not contained in the premise by definition, and the proof does not merely restate the target. The paper also relies on Refs. 28 and 29, which are by overlapping authors (F. Shi, S.-M. Fei), for the tile-structure/UPB equivalence and the SUCPB criterion; these are prior stated theorems used as background, not a uniqueness result imported to forbid alternatives, and the present constructions have independent content. The unresolved 'In Ref. ??' placeholder in the comparison with the size-200 UPB is a missing-reference issue, not circularity. No parameter fitting, renaming, or ansatz-by-citation is present, so the circularity score is low; 2 reflects the minor self-citation burden rather than a circular derivation.

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

No physical entities are postulated, and no numerical parameters are fitted; the quasi U-tile structure is a mathematical definition rather than a physical object. The central proof burden rests on the unproved rank-one support assertion in Lemma 1.

assumptions (3)
  • domain assumption A tile structure with n tiles corresponds to a UPB of size mn - n + 1 if and only if the tile structure is a U-tile structure.
    Cited theorem from Ref 28, used in Lemma 1, Proposition 2, and Theorem 3 to convert tile structures into UPBs and SUCPBs. One author of the present paper is an author of Ref 28, but the theorem is treated as an external benchmark.
  • domain assumption Any orthogonal product set in C2 ⊗ Cn can be extended to an orthogonal product basis.
    Used in Sections III and IV to conclude that the C|AB or B|CA bipartition is completable whenever one subsystem is a qubit, referencing Refs 27 and 30.
  • ad hoc to paper If Sum(M) = 0 and rank(M) = 1, then a matrix with the block structure of a quasi U-tile has nonzero coefficients only inside one of the new tiles l_j.
    This is the central assertion of the proof of Lemma 1; it is stated without a detailed derivation and is load-bearing for all SUCPB claims in the paper.

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Cite this review

Pith. "Pith review of Unextendible and strongly uncompletable product bases." pith.science (2026). https://pith.science/paper/IA33VRXP

@misc{pith2026241118036,
  author       = {Pith},
  title        = {Pith review of: Unextendible and strongly uncompletable product bases},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IA33VRXP}},
  note         = {Machine review of arXiv:2411.18036}
}
abstract

In 2003, DiVincenzo {\it et al}. put forward the question that whether there exists an unextendible product basis (UPB) which is an uncompletable product basis (UCPB) in every bipartition [\href{https://link.springer.com/article/10.1007/s00220-003-0877-6}{DiVincenzo {\it et al}. Commun. Math. Phys. \textbf{238}, 379-410(2003)}]. Recently, Shi {\it et al}. presented a UPB in tripartite systems that is also a strongly uncompletable product basis (SUCPB) in every bipartition [\href{https://iopscience.iop.org/article/10.1088/1367-2630/ac9e14}{Shi {\it et al}. New J. Phys. \textbf{24}, 113-025 (2022)}]. However, whether there exist UPBs that are SUCPBs in only one or two bipartitions remains unknown. We provide a sufficient condition for the existence of SUCPBs based on a quasi U-tile structure. We analyze all possible cases about the relationship between UPBs and SUCPBs in tripartite systems. In particular, we construct a UPB with smaller size $d^3-3d^2+1$ in $\mathbb{C}^{d}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}$, which is an SUCPB in every bipartition and has a smaller cardinality than the existing one.

Figures

Figures reproduced from arXiv: 2411.18036 by the authors.

Figure 1
Figure 1. FIG. 1. The Venn diagram of the inclusive relationship among the three concepts: UPBs, SUCPBs and UCPBs. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The quasi U-tile structure [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (5 more)
Figure 5
Figure 5. Figure 5: FIG. 5. For odd [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The quasi U-tile structures of [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The 4 [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The 4 [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]
Figure 11
Figure 11. Figure 11: FIG. 11. The quasi U-tile structures of [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]

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Reference graph

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    author author L. Chen \ and\ author D. Z. \ Dokovic ,\ title title Multiqubit upb: The method of formally orthogonal matrices , \ @noop journal journal J. Phys. A: Math. Theor. \ volume 51 ,\ pages 265302 ( year 2018 ) NoStop

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    author author L. Chen \ and\ author D. Z. \ Dokovic ,\ title title The unextendible product bases of four qubits: Hasse diagrams , \ @noop journal journal Quantum Inf. Process. \ volume 18 ,\ pages 143 ( year 2019 ) NoStop

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    author author K. Wang \ and\ author L. Chen ,\ title title The construction of 7-qubit unextendible product bases of size ten , \ @noop journal journal Quantum Inf. Process. \ volume 19 ,\ pages 185 ( year 2020 ) NoStop

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    author author F. Shi , author M. S. \ Li , author L. Chen , \ and\ author X. D. \ Zhang ,\ title title Strong quantum nonlocality for unextendible product bases in heterogeneous systems , \ @noop journal journal J. Phys. A: Math. Theor \ volume 55 ,\ pages 015305 ( year 2021 ) NoStop

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    2022

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Reviewed August 12, 2026 · model on record in the stance chip above.