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Genuinely Unextendible Product Bases from Maximum Distance Separable Codes

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Pith's one-line read Every number of parties $N\ge 3$ has a genuinely unextendible product basis, built from any linear MDS code over a prime field.

desk verdict Resolves the GUPB existence problem with a clean MDS-code construction; the prime-field restriction is real but explicit and doesn't stop the construction from working for all N. read the letter →

arxiv 2608.09504 v1 pith:RXRLACM6 submitted 2026-08-10 quant-ph

Mao-Sheng Li This is my paper · ORCID
classification quant-ph MSC 81P4081P4594B05 PACS 03.67.Mn03.67.-a
keywords genuinelyunextendibleproductbasesMDScodesmaximumdistanceseparableentangledsubspaceboundentanglementpartialtranspositionGMEwitnesseslocaldiscrimination
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

A genuinely unextendible product basis (GUPB) is an incomplete orthogonal family of fully product states whose orthogonal complement contains no product vector across any bipartition. This paper claims to settle the open existence problem by constructing a GUPB for every $N\ge 3$ and every prime $p$ from any linear $[N^2,N,N^2-N+1]_p$ maximum distance separable (MDS) code. The construction tiles the computational basis using the code's generator matrix, places Fourier modes on each tile, deletes one mode per tile, and adds a global stopper state. If the proof is correct, the orthogonal complement of each such family is a genuinely entangled subspace whose normalized projector is invariant under partial transposition across every bipartition, giving explicit multipartite bound entangled states that no fully decomposable witness can detect.

What carries the argument

The central object is the generator matrix $G$ of an MDS code, whose defining property is that every $N$ of its $N^2$ columns are linearly independent. That minor condition is used twice: transversal minors assign each computational cell to exactly one product tile, and replacement minors control the linear transformations between neighboring fixed-address slices. Together they yield the rectangle rigidity lemma, which forces any Cartesian union of more than one tile to fill the whole grid across every bipartition. Fourier-mode deletion and the stopper state then rule out the only rectangular configurations the rigidity lemma permits.

What would settle it

Run an exhaustive search for small parameters, say $N=3$ and $p=3$: enumerate all subsets $\Lambda\subseteq\mathbb{F}_3^3$ with $|\Lambda|\ge 2$ and test whether the union of the tiles $T_t$ for $t\in\Lambda$ is a Cartesian rectangle across any nontrivial bipartition; finding one would disprove Lemma 1. Alternatively, search the subspace $\{\sum_t a_t|\psi_t\rangle : \sum_t a_t=0\}$ for a nonzero vector that factors across some bipartition; finding one would refute Theorem 1.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1: for every $N\ge 3$, every prime $p$, and every linear $[N^2,N,N^2-N+1]_p$ MDS code $C$, the orthogonal family $U_C$ constructed by deleting one Fourier mode per tile and adjoining a global stopper is a genuinely unextendible product basis in $(\mathbb{C}^{Np})^{\otimes N}$. In plain terms, the complement of $U_C$ contains no nonzero vector that factors across any nontrivial bipartition of the $N$ parties. The heart of the proof is Lemma 1, MDS rectangle rigidity: if a union of at least two tiles is a Cartesian rectangle across some bipartition, then the union must be the entire computational grid. Since every vector in the complement is a superposition of one deleted mode per tile with coefficients summing to zero, a product vector in the complement would force such a rectangle; rigidity forces every tile to appear, and then the coefficient equations make the vector proportional to the stopper state, which is impossible because the stopper belongs to the original family.

Load-bearing premise

The load-bearing premise is that the code's symbol field is a prime field $\mathbb{F}_p$: the rigidity lemma fills missing positions by repeatedly adding a fixed nonzero difference, and this fills every slot only when the additive group is cyclic of prime order, as it is for $\mathbb{F}_p$ but not for fields such as $\mathbb{F}_4$.

Editorial extensions

If this is right

  • For every prime $p\ge N^2$, generalized Reed\textendash Solomon codes give an explicit GUPB of size $p^N(N^N-1)+1$ in $(\mathbb{C}^{Np})^{\otimes N}$.
  • The normalized projector onto the complement of each GUPB is PPT with respect to every bipartition while supported on a genuinely entangled subspace, yielding an explicit family of multipartite bound entangled states.
  • A canonical witness $W_G=P_U-\varepsilon_G I$ detects these states and is nondecomposable with respect to every bipartition, while no fully decomposable GME witness detects any member of the family.
  • Every finite tensor power $U_C^{\otimes \ell}$ remains a GUPB, and a PPT-invariant measurement distinguishes $\rho_U^{\otimes \ell}$ from $\rho_G^{\otimes \ell}$ perfectly even though every bipartite-separable measurement suffers a constant-factor gap.

Reading between the lines

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

  • Editorial inference: the rigidity lemma is proved only for prime fields, but the underlying MDS minor structure may survive over prime powers with a different filling argument; testing whether two tiles in $\mathbb{F}_4$ can form a Cartesian rectangle would show whether the prime-field restriction is an artifact of the proof.
  • Editorial inference: because the construction assigns one tile per MDS codeword, the size and detection constant of the GUPB are tied to code parameters such as distance and dual distance, suggesting code-dependent entanglement measures independent of this paper.
  • Editorial inference: the asymptotic discrimination exponent $\xi_{X|Y}$ defined from $\kappa_{\ell,X|Y}$ is a new finite-copy quantity attached to the MDS code; its value for Reed\textendash Solomon codes is left open and could be computed explicitly.
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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

0 major / 5 minor

Summary. The paper constructs genuinely unextendible product bases (GUPBs) for any number N>=3 of parties in (C^{Np})^{⊗N}, using linear MDS codes over prime fields. Starting from an [N^2,N,N^2-N+1]_p MDS code, the author partitions the computational grid into product tiles, places a local Fourier product basis on each tile, deletes the uniform Fourier mode from each tile, and adds a global stopper state. Theorem 1 asserts that the resulting orthogonal family is a GUPB; the proof combines an MDS rectangle-rigidity lemma with a projected-tile connectivity lemma, both proved in the Supplemental Material. The paper then shows that the complementary subspace is a genuinely entangled subspace whose normalized projector is invariant under partial transposition across every bipartition, yielding full-rank PPT genuine multipartite bound entangled states. It also constructs a GME witness that is nondecomposable with respect to every bipartition, proves a one-copy discrimination tradeoff, and establishes finite-copy indistinguishability under measurements separable across any bipartition, with a sharp ratio constant.

Significance. If correct, this settles the previously open existence problem for finite-dimensional orthogonal GUPBs and gives an explicit infinite family via generalized Reed–Solomon codes for every N>=3 (using Bertrand's postulate to choose a prime p with N^2 <= p < 2N^2). The proof is transparent and self-contained: the main theorem reduces to two auxiliary lemmas that are proved in detail, the construction has no fitted or free parameters, and the resulting states and witnesses are given by explicit formulas. The connection between classical MDS codes and genuinely multipartite bound entanglement is a new and potentially influential idea. The paper also provides falsifiable quantitative statements, such as the sharp one-copy and finite-copy discrimination ratios, which are strong positive features.

minor comments (5)
  1. [Main text, Section 'Genuine bound entanglement and its witness'] The phrase 'We have showed' should be 'We have shown', and the heading of the following section contains the typo 'finte copy local discrimination' instead of 'finite copy local discrimination'.
  2. [Main text, Eq. (50)] The displayed formula for κ_{ℓ,X|Y} is typeset in a confusing way: the minimization should be over normalized vectors A,B with p_G^{(ℓ)}(A,B)>0 of the ratio p_U^{(ℓ)}(A,B)/p_G^{(ℓ)}(A,B), multiplied by (R/K)^ℓ. The current rendering makes the fraction and the positivity condition ambiguous and should be corrected for readability.
  3. [Main text, before Eq. (37) and in the measurement section] Two cross-references appear as 'As shown in Supplemental Sec. ,' with an empty section number; these should be filled in or replaced by a reference to the specific proposition in the Supplemental Material.
  4. [Theorem 1] The universal quantification over all primes p is vacuous for primes for which no [N^2,N,N^2-N+1]_p MDS code exists (in particular, the explicit Reed–Solomon construction requires p >= N^2). The theorem is mathematically correct as an implication, but the wording 'for every prime p for which such a code exists' would more accurately reflect the content.
  5. [Theorem 2] The family ρ_λ is defined for 0<λ≤1 and the GME interval is stated as 0<λ<κ_G; it may be worth noting explicitly that the endpoint λ=0, namely ρ_G itself, is already GME by Theorem 1, even though the family as defined starts at λ>0.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the GUPB construction is derived from the assumed MDS code property and explicit formulas, with no fitted parameter or load-bearing self-citation.

full rationale

The paper's central claim is Theorem 1: for every prime p and every linear [N^2,N,N^2-N+1]_p MDS code C, the explicitly defined family U_C in Eq. (13) is a genuinely unextendible product basis. The proof is self-contained relative to the stated assumption that C is MDS. The key rigidity input, Lemma 1, is proved in the Supplemental Material from the nonvanishing-minor property of the MDS generator matrix (Lemmas 2 and 3), the prime-field translation lemma (Lemma 4), and the Cartesian-slice filling argument (Lemma 5). None of these steps presupposes that U_C is a GUPB; they show it from the code's defining minor condition. The stopper state and Fourier-mode deletion are used only after the rigidity lemma to exclude the two surviving rectangular cases, and they are not adjusted to force the conclusion. The witness quantity ε_G in Eq. (39) is a genuine minimum over the compact biseparable set, and its positivity follows from Theorem 1 rather than being fitted to make the witness work. Self-citations in the reference list (Refs. 35, 37, 40) concern background results on related multipartite product-state phenomena and are not load-bearing for the construction or its proof. The prime-field restriction is an explicit hypothesis of Theorem 1, and the Supplemental Material identifies the additive-mixing reason it is needed; this is a stated limitation, not a circular import. The paper does not rename a known result, does not invoke an author-supplied uniqueness theorem, and does not smuggle an ansatz via citation. The derivation chain is independent of its conclusion.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard MDS code properties (existence via Reed-Solomon, full minor independence) and standard quantum information facts. No free parameters are fitted to data. The only construction element that could be called invented is the stopper state, which is an explicit vector in the construction, not a postulate about nature.

assumptions (5)
  • standard math Existence of [N^2, N, N^2-N+1]_p MDS codes over F_p for prime p ≥ N^2
    Theorem 1 requires such a code; the explicit family is given by generalized Reed-Solomon codes (Eq. 32).
  • standard math Every N columns of the MDS generator matrix are linearly independent
    This is the defining property of MDS codes, used in the tile partition (Eq. 8) and in Lemmas 3, 5, 6.
  • standard math For a prime field F_p, a nonempty subset E with E+δ=E for δ≠0 must equal F_p
    Lemma 4, used in Lemma 5 to fill the Cartesian slice; this is the reason the construction is restricted to prime fields.
  • domain assumption A state that is PPT with respect to every bipartition is nondistillable (bound entangled)
    Standard quantum information fact cited as Ref. [4]; used to conclude ρ_G and ρ_λ are bound entangled.
  • domain assumption Fully decomposable GME witnesses are nonnegative on states that are PPT with respect to every bipartition
    Standard result cited as Ref. [44]; used in Theorem 2.

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Pith. "Pith review of Genuinely Unextendible Product Bases from Maximum Distance Separable Codes." pith.science (2026). https://pith.science/paper/RXRLACM6

@misc{pith2026260809504,
  author       = {Pith},
  title        = {Pith review of: Genuinely Unextendible Product Bases from Maximum Distance Separable Codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RXRLACM6}},
  note         = {Machine review of arXiv:2608.09504}
}
abstract

The existence of genuinely unextendible product bases (GUPBs), incomplete orthogonal sets of fully product states whose orthogonal complements contain no product vector across any bipartition, has remained an open problem. Here we construct GUPBs for any number $N\geq3$ of parties using classical maximum distance separable (MDS) codes. The MDS property imposes a rigidity on the induced product tiling across every bipartition; combined with Fourier mode deletion and a stopper state, this rigidity enforces genuine unextendibility. Consequently, the orthogonal complement of each GUPB is a genuinely entangled subspace whose normalized projector is invariant under partial transposition across every bipartition, yielding an explicit family of multipartite bound entangled states. We further construct GME witnesses that detect these states even though no fully decomposable witness can do so. Moreover, the resulting indistinguishability persists under arbitrary finite tensor powers and measurements separable across any bipartition. These results establish a direct connection between error-correcting codes and multipartite entanglement and provide an algebraic route to certifying genuinely multipartite bound entanglement.

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Works this paper leans on

53 extracted references · 53 canonical work pages

  1. [1]

    C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Physical Review Letters82, 5385 (1999)

    work page 1999

  2. [2]

    D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Communications in Mathematical Physics238, 379 (2003)

    work page 2003

  3. [3]

    A. O. Pittenger, Linear Algebra and its Applications359, 235 (2003)

    work page 2003

  4. [4]

    Horodecki, P

    M. Horodecki, P. Horodecki, and R. Horodecki, Physical Review Letters80, 5239 (1998)

    work page 1998

  5. [5]

    Bej and S

    P. Bej and S. Halder, Physics Letters A386, 126992 (2021)

    work page 2021

  6. [6]

    C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin, and W. K. Wootters, Physical Review A59, 1070 (1999)

    work page 1999

  7. [7]

    De Rinaldis, Physical Review A70, 022309 (2004)

    S. De Rinaldis, Physical Review A70, 022309 (2004)

    work page 2004

  8. [8]

    Hayden, D

    P. Hayden, D. Leung, and G. Smith, Physical Review A71, 062339 (2005)

    work page 2005

Show all 53 references
  1. [9]

    S. M. Cohen, Physical Review A77, 012304 (2008)

    2008

  2. [10]

    Augusiak, J

    R. Augusiak, J. Stasi´nska, C. Hadley, J. K. Korbicz, M. Lewen- stein, and A. Acín, Physical Review Letters107, 070401 (2011)

    2011

  3. [11]

    Augusiak, T

    R. Augusiak, T. Fritz, M. Kotowski, M. Kotowski, M. Pawłowski, M. Lewenstein, and A. Acín, Physical Review A85, 042113 (2012)

    2012

  4. [12]

    Alon and L

    N. Alon and L. Lovász, Journal of Combinatorial Theory, Series A95, 169 (2001)

    2001

  5. [13]

    Feng, Discrete Applied Mathematics154, 942 (2006)

    K. Feng, Discrete Applied Mathematics154, 942 (2006)

    2006

  6. [14]

    Johnston, Journal of Physics A: Mathematical and Theoretical 47, 424034 (2014)

    N. Johnston, Journal of Physics A: Mathematical and Theoretical 47, 424034 (2014)

    2014

  7. [15]

    Chen and N

    J. Chen and N. Johnston, Communications in Mathematical Physics333, 351 (2015)

    2015

  8. [16]

    Chen and D

    L. Chen and D. Z. ˇDokovi´c, Journal of Physics A: Mathematical and Theoretical51, 265302 (2018)

    2018

  9. [17]

    Y .-H. Yang, F. Gao, G.-B. Xu, H.-J. Zuo, Z.-C. Zhang, and Q.-Y . Wen, Scientific Reports5, 11963 (2015)

    2015

  10. [18]

    F. Shi, X. Zhang, and L. Chen, Physical Review A101, 062329 (2020)

    2020

  11. [19]

    Wang and L

    K. Wang and L. Chen, Quantum Information Processing19, 185 (2020)

    2020

  12. [20]

    S. You, C. Wang, F. Shi, S. Hu, and X. Zhang, Journal of Physics A: Mathematical and Theoretical56, 015303 (2023)

    2023

  13. [21]

    F. Shi, G. Bai, X. Zhang, Q. Zhao, and G. Chiribella, Physical Review Research5, 033144 (2023)

    2023

  14. [22]

    K. R. Parthasarathy, Proceedings of the Indian Academy of Sci- ences: Mathematical Sciences114, 365 (2004)

    2004

  15. [23]

    B. V . R. Bhat, International Journal of Quantum Information4, 325 (2006)

    2006

  16. [24]

    Demianowicz and R

    M. Demianowicz and R. Augusiak, Physical Review A98, 012313 (2018)

    2018

  17. [25]

    Agrawal, S

    S. Agrawal, S. Halder, and M. Banik, Physical Review A99, 032335 (2019)

    2019

  18. [26]

    Demianowicz and R

    M. Demianowicz and R. Augusiak, Physical Review A100, 062318 (2019)

    2019

  19. [27]

    Demianowicz and R

    M. Demianowicz and R. Augusiak, Quantum Information Pro- cessing19, 199 (2020)

    2020

  20. [28]

    Demianowicz, G

    M. Demianowicz, G. Rajchel-Mieldzio´c, and R. Augusiak, New Journal of Physics23, 103016 (2021)

    2021

  21. [29]

    K. V . Antipin, Journal of Physics A: Mathematical and Theoreti- cal54, 505303 (2021)

    2021

  22. [30]

    Demianowicz, Quantum6, 854 (2022)

    M. Demianowicz, Quantum6, 854 (2022)

    2022

  23. [31]

    Johnston, B

    N. Johnston, B. Lovitz, and A. Vijayaraghavan, Physical Review A106, 062443 (2022)

    2022

  24. [32]

    Niset and N

    J. Niset and N. J. Cerf, Physical Review A74, 052103 (2006)

    2006

  25. [33]

    Halder, M

    S. Halder, M. Banik, S. Agrawal, and S. Bandyopadhyay, Physi- cal Review Letters122, 040403 (2019)

    2019

  26. [34]

    P. Yuan, G. Tian, and X. Sun, Physical Review A102, 042228 (2020)

    2020

  27. [35]

    Shi, M.-S

    F. Shi, M.-S. Li, L. Chen, and X. Zhang, Journal of Physics A: Mathematical and Theoretical55, 015305 (2022)

    2022

  28. [36]

    F. Shi, Z. Ye, L. Chen, and X. Zhang, Physical Review A105, 022209 (2022)

    2022

  29. [37]

    Shi, M.-S

    F. Shi, M.-S. Li, M. Hu, L. Chen, M.-H. Yung, Y .-L. Wang, and X. Zhang, Quantum6, 619 (2022)

    2022

  30. [38]

    H. Zhou, T. Gao, and F. Yan, Physical Review A106, 052209 (2022)

    2022

  31. [39]

    Y . He, F. Shi, and X. Zhang, Quantum8, 1349 (2024)

    2024

  32. [40]

    Shi, M.-S

    F. Shi, M.-S. Li, X. Zhang, and Q. Zhao, New Journal of Physics 24, 113025 (2022)

    2022

  33. [41]

    Zhen, H.-J

    X.-F. Zhen, H.-J. Zuo, F. Shi, and S.-M. Fei, Journal of Mathe- matical Physics65, 112201 (2024)

    2024

  34. [42]

    Demianowicz, Physical Review A106, 012442 (2022)

    M. Demianowicz, Physical Review A106, 012442 (2022)

    2022

  35. [43]

    Demianowicz, Quantum Information Processing25, 67 (2026)

    M. Demianowicz, Quantum Information Processing25, 67 (2026)

    2026

  36. [44]

    Jungnitsch, T

    B. Jungnitsch, T. Moroder, and O. Gühne, Physical Review Letters106, 190502 (2011)

    2011

  37. [45]

    Huber, F

    M. Huber, F. Mintert, A. Gabriel, and B. C. Hiesmayr, Physical Review Letters104, 210501 (2010)

    2010

  38. [46]

    Huber and R

    M. Huber and R. Sengupta, Physical Review Letters113, 100501 (2014)

    2014

  39. [47]

    R. C. Singleton, IEEE Transactions on Information Theory10, 6 116 (1964)

    1964

  40. [48]

    I. S. Reed and G. Solomon, Journal of the Society for Industrial and Applied Mathematics8, 300 (1960)

    1960

  41. [49]

    A. J. Scott, Physical Review A69, 052330 (2004)

    2004

  42. [50]

    Gour and N

    G. Gour and N. R. Wallach, Physical Review A76, 042309 (2007)

    2007

  43. [51]

    Goyeneche and K

    D. Goyeneche and K. ˙Zyczkowski, Physical Review A90, 022316 (2014)

    2014

  44. [52]

    Goyeneche, D

    D. Goyeneche, D. Alsina, J. I. Latorre, A. Riera, and K. ˙Zy- czkowski, Physical Review A92, 032316 (2015)

    2015

  45. [53]

    Genuinely Unextendible Product Bases from Maximum Distance Separable Codes

    F. Huber and M. Grassl, Quantum4, 284 (2020). 7 Supplemental Material for “Genuinely Unextendible Product Bases from Maximum Distance Separable Codes” Mao-Sheng Li Common notation.—The notation is identical to that of the Letter. In particular,Gis the generator matrix,g P,j is...

    2020

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