bibkey: santalla2026hyperlink
authors: Silvia N. Santalla, Sudipto Singha Roy, Germán Sierra, Javier Rodríguez-Laguna
year: 2026
title: "The hyperlink representation of entanglement and the inclusion-exclusion principle"
doi: 10.1103/gmjw-rb13
url: https://arxiv.org/abs/2601.17926v2
claim: "The entanglement hyperlinks J_I = Σ_{A⊆I} (−1)^{|I|−|A|} S_A of a pure state reconstruct its entanglement entropies through all ranks (proved). Appendix B conjectures the even-legged reconstruction S_A = Σ_{|I| even} Λ_{|I|,|I∩A|} J_I with leg-factors independent of N, lists Λ_{2l,p} for 2l ≤ 8 in (B5), and reports checks up to N = 9."
strata_touched:
  - D5/S3/Quantum/Information/EvenLeggedHyperlinkReconstruction
license: citation-only
triage: anchor

Santalla–Singha Roy–Sierra–Rodríguez-Laguna, the hyperlink representation of entanglement

S. N. Santalla, S. Singha Roy, G. Sierra and J. Rodríguez-Laguna, The hyperlink representation of entanglement and the inclusion-exclusion principle, arXiv:2601.17926 [quant-ph] (v2 27 July 2026); Phys. Rev. A 114, 022431 (17 August 2026), DOI 10.1103/gmjw-rb13.

Verified locator

DOI: 10.1103/gmjw-rb13 (Phys. Rev. A 114, 022431, from Crossref metadata; the journal full text was not read). Primary version read: https://arxiv.org/abs/2601.17926v2 (the latest arXiv version). Source: https://arxiv.org/pdf/2601.17926v2 (text read from the PDF):

• §2, Eqs. (5) and (6), the entanglement hyperlinks;
• Appendix B, “Even-legged edge reconstruction”, (B1)–(B5) and the closing paragraph.

Source statements

Eq. (5): “Let us define the entanglement hyperlinks (EHL) of a pure state in analogy to the inclusion-exclusion principle, J_I ≡ Σ_{A⊆I} (−1)^{|I|−|A|} S_A.”

Appendix B: “Let us put forward our proposal for the reconstruction formula, S_A = Σ_{|I| even} Λ_{|I|,|I∩A|} J_I, (B2) i.e. we sum over all EHL, each of them multiplied by a leg-factor Λ_{2l,p}, which depends on the rank of the EHL, |I| = 2l and the number of legs that the EHL has in block A, p = |I∩A|. If p = 0 or p = 2l then Λ_{2l,p} = 0 … Of course, expressions (B2) and (B3) remain a conjecture at this stage. It is still to be proved that the leg-factors Λ_{2l,p} exist and do not depend on N. If they exist, then complementarity symmetry imposes that Λ_{2l,p} = Λ_{2l,2l−p}.”

(B5): “Λ_{2,1} = −1/2, Λ_{4,1} = 1/4, Λ_{4,2} = 1/2, Λ_{6,1} = −1/2, Λ_{6,2} = −1, Λ_{6,3} = −5/4, Λ_{8,1} = 17/8, Λ_{8,2} = 17/4, Λ_{8,3} = 47/8, Λ_{8,4} = 13/2.”

Appendix B: “We should stress that the validity of Eq. (B2), with leg-factors that do not depend on N, has only been checked up to N = 9, and remains conjectural for larger sizes.” … “Since all three reconstruction operations are again linear, we may claim that Eq. (5) has several different inverses. This apparent contradiction is due to the fact that we are considering vectors S with a special symmetry, i.e.: those corresponding to pure states, in which S_A = S_Ā.”