Pith. sign in

REVIEW 3 major objections 5 minor 34 references

This paper proves that the entanglement of formation of every two-mode Gaussian state equals its Gaussian restriction, resolving Open Quantum Problem 29.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

For every two-mode Gaussian state, the unrestricted entanglement of formation equals its Gaussian restriction, resolving the two-mode instance of Open Quantum Problem 29.

T0 review reviewed 2026-08-04 challenge →

load-bearing objection A serious and original proof attempt at the two-mode Gaussian EOF problem, with the main caveat being an unstated and unverified external log-Sobolev-type theorem that the proof leans on. the 3 major comments →

arxiv 2608.01909 v1 pith:EUP7ZMBY submitted 2026-08-03 quant-ph cond-mat.stat-mechmath-phmath.MP

Optimality of Gaussian Entanglement of Formation

Gerardo Adesso This is my paper
classification quant-ph cond-mat.stat-mechmath-phmath.MP MSC 81P4281P4594A17
keywords entanglement of formationGaussian statesconvex roofcontinuous variablesEPR observablelogarithmic Sobolev inequalityopen quantum problem 29bisymmetric multimode states
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 proves a 2005 open problem in continuous-variable quantum information: for every two-mode Gaussian state, the unrestricted entanglement of formation is exactly equal to the Gaussian-restricted value. A non-Gaussian pure-state decomposition never beats the best Gaussian one. The proof rests on a sharp affine inequality linking pure-state entanglement to a generalized EPR variance, valid for all pure two-mode states. Averaging this inequality over arbitrary ensembles forces every decomposition to lie above the Gaussian candidate, while a Gaussian mixture achieves it. If correct, existing Gaussian formulas become exact for the unrestricted measure, and the same inequality yields a measurable lower bound for non-Gaussian states.

Core claim

The central discovery is the equality E_F(ρ_V) = E_F^G(V) for every two-mode Gaussian state. The load-bearing result is Theorem 1: for every normalized pure two-mode state |ψ> of finite mean energy and every 0 < q < t ≤ 1, E(ψ) + λ_{q,t}(⟨D_t⟩_ψ − d_t(q)) ≥ E(q), where D_t is a gain-asymmetric generalized EPR observable. Because the inequality is affine in the measured expectation value, it survives averaging over any pure-state ensemble; applying it to the canonical covariance form V_q + N_{t,u,v} shows both that a Gaussian decomposition reaches E(q) and that no other ensemble can go below. This settles the full two-mode instance of the open problem, and the same affine witness provides a r

What carries the argument

The generalized EPR observable D_t = (t a − b†)†(t a − b†) for gain 0 < t ≤ 1, with expectation encoding the weighted quadrature variances (t x_A − x_B)^2 and (t p_A + p_B)^2. The proof of Theorem 1 combines three ingredients: a singular-value defect inequality showing the EPR functional dominates the weighted positive drops of the Schmidt sequence; a geometric isotonic compression (pool-adjacent-violators with weights t^{2n}) that restores the ratio condition c_{n+1} ≤ t c_n while not increasing entropy or Dirichlet energy; and a thermal logarithmic-Sobolev inequality for phase-covariant Gaussian channels, which shows the scalar trade-off is minimized by the geometric two-mode-squeezed-vacu

Load-bearing premise

The proof assumes the meta logarithmic-Sobolev inequality of Ref. [22] applies to the chosen operator and parameter range, so that the infimum over single-mode states is attained by thermal states; if that theorem fails in this regime, the affine bound of Theorem 1 does not follow.

What would settle it

A specific way to refute the central claim would be to find a normalized pure two-mode state with finite mean energy and some 0 < q < t ≤ 1 for which Eq. (7) is violated, or to find a single-mode state with finite mean photon number where the infimum in Eq. (S51) over thermal states is strictly larger than the infimum over all states, contradicting the imported log-Sobolev theorem. A numerical optimization of H(d^2) + λ F_t(d) over randomly generated normalized nonnegative sequences d_n could serve as a practical search.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • For any two-mode Gaussian state, the unrestricted entanglement of formation equals the previously computed Gaussian restriction, so existing analytical formulas and one-parameter numerical minimizations now give the exact value.
  • The same affine witness gives a rigorous lower bound on the entanglement of formation of arbitrary non-Gaussian states from two phase-resolved quadrature variances, with no need for full state tomography.
  • The result extends to bisymmetric (m+n)-mode Gaussian states, reducing their entanglement of formation to that of a correlated two-mode core.
  • For a validated Gaussian source, a measured 4×4 covariance matrix determines the exact entanglement of formation, while the same records give a conservative lower bound if Gaussianity is not assumed.
  • The proof replaces an infinite-dimensional convex-roof optimization over arbitrary Schmidt decompositions by a one-parameter optimization over a squeezing parameter, pointing toward other bosonic convex-roof problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The singular-value defect inequality and geometric isotonic compression may transfer to other bosonic convex-roof problems governed by weighted ladder-operator Dirichlet forms, such as Renyi entanglement measures or other EPR-variance-based witnesses, though the paper only suggests this possibility.
  • The device-dependent witness could be turned into a practical finite-sample certification protocol; the paper gives the statistical confidence treatment but not a full protocol.
  • Because the proof imports the thermal logarithmic-Sobolev theorem without restating its hypotheses, a direct verification of that theorem in the required parameter regime is the natural next check before relying on the affine bound.
  • The equality for Gaussian states strengthens the interpretation of covariance-matrix-based entanglement quantifiers in hybrid optical-microwave experiments where phase-resolved covariance reconstruction is already routine.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper claims to prove that the entanglement of formation of every two-mode Gaussian state equals the Gaussian-restricted entanglement of formation, thereby resolving the two-mode case of Open Quantum Problem 29. The main engine is Theorem 1, an affine entanglement--EPR inequality, Eq. (7), asserted for every pure two-mode state with finite mean energy. The proof proceeds through a singular-value defect inequality (Theorem S1), a geometric isotonic compression step (Theorem S2), and a scalar 'thermal log-Sobolev closure' (Eq. S51). The authors then use a canonical covariance form (Lemma S3) to construct an admissible Gaussian decomposition and match the lower bound from Theorem 1. The paper also extends the result to bisymmetric multimode Gaussian states and derives an experimentally accessible lower bound on the entanglement of formation of arbitrary two-mode non-Gaussian states, with a numerical example.

Significance. If correct, the result settles a twenty-year-old open problem and converts all established Gaussian covariance-matrix algorithms for E_F^G into exact evaluations of the unrestricted E_F. The paper is also commendable for providing a structured supplementary proof, for explicitly identifying the single missing conjecture of Ref. [18], and for shipping a reproducible numerical example. The claimed non-Gaussian lower bound, Eq. (9), is a potentially useful experimental witness. These are substantial contributions. However, the central proof relies on a black-box external theorem whose exact hypotheses are not stated, and on an infinite-dimensional limiting argument that is compressed; these points need to be resolved before the claim can be accepted.

major comments (3)
  1. [Supplemental Material, 'Thermal log-Sobolev closure', Eqs. (S46)--(S51)] Equation (S51) is the scalar inequality that closes the proof of Theorem 1, but it is derived entirely from an unstated external result of Ref. [22]. As written, the manuscript asserts that for arbitrary ν0,ν1,ω≥0 the functional Υ in Eq. (S46) has its infimum attained by thermal states. This is a very strong claim; Υ is not convex in ρ, so thermal extremality is not immediate. The choice ν0=λt, ν1=0, ω=λ(1−t)^2 must be shown to lie within the hypotheses of the cited theorem, and the theorem must be stated precisely. If the actual theorem has extra conditions—e.g., a restricted parameter range, Gaussian inputs, or an additional positive remainder term—then Eq. (S51) is unsupported and the affine bound (7), and hence Eq. (19), does not follow. This is the load-bearing step that controls arbitrary non-Gaussian decompositions; it must be made self-contained or rigorously referenced.
  2. [Supplemental Material, 'Geometric isotonic compression' and 'Domain remarks'] The passage from finite-rank truncations to the infinite-dimensional statement needed for every finite-mean-energy pure state is compressed. The text says that 'applying the finite-rank inequalities and then taking M→∞ proves the infinite-dimensional statements by lower semicontinuity,' but von Neumann entropy is not lower semicontinuous without appropriate energy constraints. The authors cite Winter's energy-constrained continuity, yet they do not spell out the uniform energy bounds that justify convergence of H(c^2) and of the graph norms of a† for the compressed sequences. Since Theorem 1 is claimed for all finite-mean-energy pure states, this limiting step is load-bearing. I request a detailed proof that the truncations satisfy the hypotheses of the cited continuity bound and that the inequalities (S31)--(S33) pass to the limit.
  3. [Supplemental Material, 'Canonical covariance geometry'] Lemma S3 is central to the upper bound in the Gaussian case: it guarantees that every entangled two-mode covariance matrix can be brought to the form V_q+N_{t,u,v} with 0<q≤t≤1 and N≥0. The Supplement only summarizes the argument from Refs. [14,18] and does not give a complete proof. This is a known result, but because it is used to assert both the existence of the Gaussian decomposition and the identity Tr(ρ_{V'}D_t)=d_t(q), the manuscript should state the precise theorem, including any assumptions on the squeezing branch and the ordering q≤t, or provide a full proof in the Supplement.
minor comments (5)
  1. [Letter, Eq. (21)] The definition of h(ν) is easy to misread: the parentheses are ambiguous between ν+(1/2) and (ν+1)/2. Please write ν + 1/2 and ν − 1/2 with explicit parentheses or a separate displayed definition.
  2. [Letter, Eq. (13)] The notation F_t(d) is not defined in the main text; the reader must consult Eq. (S43). Please define it when Eq. (13) is introduced, and clarify that no positive part appears in this scalar functional.
  3. [Supplemental Material, Eq. (S27)] The inequalities P≥t c_i D and Q≤c_j D are correct but are stated without explanation. A brief parenthetical reminding the reader that c_k≥c_i for k≤i−1 and c_{k+1}≤c_j for k≥j would improve readability.
  4. [Supplemental Material, 'Geometric isotonic compression'] The proof that pooling does not increase Eq. (S38) considers only the two pooled blocks and their external neighbors. It would help to state explicitly why this local analysis is sufficient for the full sum, e.g., because all other terms are unchanged.
  5. [Conclusion] The paper's acknowledgment of the use of an AI assistant is transparent and appropriate. No further action is needed, but the reproducible script mentioned in the Supplement should be made available in a stable archival repository.

Circularity Check

0 steps flagged

No significant circularity: the Gaussian optimality result is derived from an independent log-Sobolev theorem, a proved singular-value inequality, and a cited covariance lemma that explicitly does not assume optimality.

full rationale

The central claim, Eq. (19), is obtained by a genuine sandwich: the canonical covariance construction (Lemma S3, taken from Refs. [14,18], not from the present author) provides an admissible Gaussian decomposition giving the upper bound EF(ρV') ≤ E(q) (Eqs. (17)/(S71)), while Theorem 1 (Eqs. (7)/(S60)) gives the matching lower bound for arbitrary pure-state ensembles (Eqs. (18)/(S72)). Theorem 1 is not assumed; it is proved from Theorem S1 (singular-value defect inequality), Theorem S2 (geometric isotonic compression), and Eq. (S51), which is imported from the independent meta logarithmic-Sobolev theorem of Beigi and Rahimi-Keshari [22]. The paper never fits a parameter to the quantity it later predicts: λq,t is chosen after the fact to make the scalar thermal objective stationary at q (Eqs. (S52)-(S58)), and the covariance lemma explicitly states that no optimality property of q is assumed. The cited covariance lemma and the [22] theorem are the main load-bearing external inputs, but they are not the target statement and are not authored by Adesso, so their use is not circular. The self-citations present in the manuscript ([1], [24], [25], [30], [32]) are used for context, for prior Gaussian covariance formulas, and for the bisymmetric structural reduction; none smuggles in the two-mode Gaussian convex-roof equality itself. The bisymmetric extension does rely on the structural theorem of Ref. [25] (co-authored by Adesso), but that is an independent, earlier mathematical characterization of bisymmetric Gaussian states, not an assumption of the present result. Any concern that Ref. [22]'s theorem may have unstated hypotheses restricting its applicability is a correctness or verification issue, not a circularity: the manuscript's derivation chain does not reduce to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

No free parameters are fitted: all inequalities are proven for a full parameter range; q,t,u,v are state-derived canonical parameters and λ is a derived Lagrange multiplier. The three listed axioms are external theorems or lemmas. No new physical entities are introduced.

axioms (3)
  • domain assumption Beigi-Rahimi-Keshari meta logarithmic-Sobolev theorem for phase-covariant Gaussian channels: the infimum of Υ in Eq. (S46) is attained by thermal states.
    Used in the Thermal Log-Sobolev Closure to prove Eq. (S51)/(S58), the scalar supporting-line inequality that closes Theorem 1. The manuscript does not state the theorem's hypotheses or proof.
  • domain assumption Canonical formation form: every entangled two-mode Gaussian covariance matrix is locally symplectically equivalent to V_q + N_{t,u,v} with 0<q≤t≤1 (Lemma S3, from Refs. [14,18]).
    Basis of the Gaussian optimality proof in Eq. (14)/(S66). It is an external covariance geometry lemma, independent of the non-Gaussian extremality being proven.
  • standard math Energy-constrained continuity of von Neumann entropy for finite-mean-energy sequences (Ref. [23]).
    Justifies the infinite-rank limiting argument after Theorem S2, including convergence of truncated Schmidt entropies and graph norms.

reviewed 2026-08-04 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Optimality of Gaussian Entanglement of Formation." pith.science (2026). https://pith.science/paper/EUP7ZMBY

@misc{pith2026260801909,
  author       = {Pith},
  title        = {Pith review of: Optimality of Gaussian Entanglement of Formation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EUP7ZMBY}},
  note         = {Machine review of arXiv:2608.01909}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

We prove that the entanglement of formation of every two-mode Gaussian state coincides with its Gaussian restriction, solving a longstanding open problem in continuous variable quantum information theory. The key result is a sharp affine relation between entanglement and a generalized Einstein-Podolsky-Rosen observable, valid for arbitrary pure two-mode states, including non-Gaussian ones. Our result extends to bisymmetric multimode Gaussian states, and also provides a measurable lower bound on the entanglement of formation of arbitrary non-Gaussian states.

Figures

Figures reproduced from arXiv: 2608.01909 by Gerardo Adesso.

Figure 1
Figure 1. Figure 1: FIG. 1. Exact entanglement of formation and variance-based lower [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

34 extracted references · 1 canonical work pages · 1 internal anchor

  1. [1]

    Turner, M

    L. Turner, M. Gut ¸˘a, and G. Adesso, All non-gaussian states are advantageous for channel discrimination: Robustness of non- convex continuous variable quantum resources, New J. Phys.27, 094507 (2025)

    2025

  2. [2]

    For 0<t≤ 1, define the generalized EPR observable bDt =(ta−b †)†(ta−b †) = 1 2 h (tx A−x B)2 +(tp A +p B)2 +1−t 2i .(3) The parameter t is a relative gain; t =1 gives the familiar balanced EPR combination, while t< 1 accommodates asym- metric states and channels. For the two-mode squeezed vacuum (TMSV) state |Ψq⟩= q 1−q 2 ∞X n=0 qn|n,n⟩,0≤q<1,(4) its enta...

    Pith/arXiv arXiv 2026

  3. [3]

    S. L. Braunstein and P. van Loock, Quantum information with continuous variables, Rev. Mod. Phys.77, 513 (2005)

    2005

  4. [4]

    Weedbrook, S

    C. Weedbrook, S. Pirandola, R. Garc´ıa-Patr´on, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum informa- tion, Rev. Mod. Phys.84, 621 (2012)

    2012

  5. [5]

    Adesso, S

    G. Adesso, S. Ragy, and A. R. Lee, Continuous variable quantum information: Gaussian states and beyond, Open Syst. Inf. Dyn. 21, 1440001 (2014)

    2014

  6. [6]

    Furusawa, J

    A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, Unconditional quantum teleportation, Science282, 706 (1998)

    1998

  7. [7]

    Grosshans, G

    F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, Quantum key distribution using gaussian- modulated coherent states, Nature421, 238 (2003)

    2003

  8. [8]

    X. Guo, C. R. Breum, J. Borregaard, S. Izumi, M. V . Larsen, T. Gehring, M. Christandl, J. S. Neergaard-Nielsen, and U. L. Andersen, Distributed quantum sensing in a continuous-variable entangled network, Nat. Phys.16, 281 (2020)

    2020

  9. [9]

    X. Jia, C. Zhai, X. Zhu,et al., Continuous-variable multipar- tite entanglement in an integrated microcomb, Nature639, 329 (2025)

    2025

  10. [10]

    Y . Lou, Y . Lv, J. Wang, S. Liu, and J. Jing, Deterministic all- optical continuous-variable quantum telecloning, Phys. Rev. Lett. 132, 160803 (2024)

    2024

  11. [11]

    R. Sahu, L. Qiu, W. Hease, G. Arnold, Y . Minoguchi, P. Rabl, and J. M. Fink, Entangling microwaves with light, Science380, 718 (2023)

    2023

  12. [12]

    Meesala, D

    S. Meesala, D. P. Lake, S. Wood,et al., Quantum entanglement between optical and microwave photonic qubits, Phys. Rev. X 14, 031055 (2024)

    2024

  13. [13]

    H. Zhao, W. D. Chen, A. Kejriwal, and M. Mirhosseini, Quantum-enabled microwave-to-optical transduction via silicon nanomechanics, Nat. Nanotechnol.20, 602 (2025)

    2025

  14. [14]

    C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Woot- ters, Mixed-state entanglement and quantum error correction, Phys. Rev. A54, 3824 (1996)

    1996

  15. [15]

    M. M. Wolf, G. Giedke, O. Kr¨uger, R. F. Werner, and J. I. Cirac, Gaussian entanglement of formation, Phys. Rev. A69, 052320 (2004)

    2004

  16. [16]

    Kr ¨uger and R

    O. Kr ¨uger and R. F. Werner, Some open problems in quantum information theory, arXiv:quant-ph/0504166 10.48550/arXiv.quant-ph/0504166 (2005)

    Pith/arXiv arXiv doi:10.48550/arxiv.quant-ph/0504166 2005

  17. [17]

    Giedke, M

    G. Giedke, M. M. Wolf, O. Kr¨uger, R. F. Werner, and J. I. Cirac, Entanglement of formation for symmetric gaussian states, Phys. Rev. Lett.91, 107901 (2003)

    2003

  18. [18]

    Marian and T

    P. Marian and T. A. Marian, Entanglement of formation for an arbitrary two-mode gaussian state, Phys. Rev. Lett.101, 220403 (2008)

    2008

  19. [19]

    J. S. Ivan and R. Simon, Entanglement of formation for gaussian states, arXiv:0808.1658 10.48550/arXiv.0808.1658 (2008)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.0808.1658 2008

  20. [20]

    Akbari-Kourbolagh and H

    Y . Akbari-Kourbolagh and H. Alijanzadeh-Boura, On the entan- glement of formation of two-mode gaussian states: a compact form, Quantum Inf. Process.14, 4179 (2015)

    2015

  21. [21]

    Tserkis and T

    S. Tserkis and T. C. Ralph, Quantifying entanglement in two- mode gaussian states, Phys. Rev. A96, 062338 (2017)

    2017

  22. [22]

    Tserkis, S

    S. Tserkis, S. Onoe, and T. C. Ralph, Quantifying entanglement of formation for two-mode gaussian states: Analytical expres- sions for upper and lower bounds and numerical estimation of its exact value, Phys. Rev. A99, 052337 (2019)

    2019

  23. [23]

    Beigi and S

    S. Beigi and S. Rahimi-Keshari, A meta logarithmic-sobolev inequality for phase-covariant gaussian channels, Ann. Henri Poincar´e26, 2737 (2025)

    2025

  24. [24]

    Winter, Tight uniform continuity bounds for quantum en- tropies: conditional entropy, relative entropy distance and energy constraints, Commun

    A. Winter, Tight uniform continuity bounds for quantum en- tropies: conditional entropy, relative entropy distance and energy constraints, Commun. Math. Phys.347, 291 (2016)

    2016

  25. [25]

    Adesso and F

    G. Adesso and F. Illuminati, Gaussian measures of entanglement versus negativities: Ordering of two-mode gaussian states, Phys. Rev. A72, 032334 (2005)

    2005

  26. [26]

    Serafini, G

    A. Serafini, G. Adesso, and F. Illuminati, Unitarily localizable entanglement of gaussian states, Phys. Rev. A71, 032349 (2005)

    2005

  27. [27]

    W. K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett.80, 2245 (1998)

    1998

  28. [28]

    Friis, G

    N. Friis, G. Vitagliano, M. Malik, and M. Huber, Entanglement certification—from theory to experiment, Nat. Rev. Phys.1, 72 (2019)

    2019

  29. [29]

    X. Wang, J. Fu, and J. Jing, Experimental measurement-device- independent verification of continuous-variable entanglement, Phys. Rev. Lett.135, 150201 (2025)

    2025

  30. [30]

    Fiur´aˇsek and N

    J. Fiur´aˇsek and N. J. Cerf, How to measure squeezing and entan- glement of gaussian states without homodyning, Phys. Rev. Lett. 93, 063601 (2004)

    2004

  31. [31]

    Laurat, G

    J. Laurat, G. Keller, J. A. Oliveira-Huguenin, C. Fabre, T. Coudreau, A. Serafini, G. Adesso, and F. Illuminati, En- tanglement of two-mode gaussian states: characterization and experimental production and manipulation, J. Opt. B: Quant. Semiclass. Opt.7, S577 (2005)

    2005

  32. [32]

    E. P. Menzel, R. Di Candia, F. Deppe, P. Eder, L. Zhong, M. Ih- mig, M. Haeberlein, A. Baust, E. Hoffmann, D. Ballester, K. In- omata, T. Yamamoto, Y . Nakamura, E. Solano, A. Marx, and R. Gross, Path entanglement of continuous-variable quantum microwaves, Phys. Rev. Lett.109, 250502 (2012)

    2012

  33. [33]

    Optimality of Gaussian Entanglement of Formation

    G. Adesso, Towards the ultimate brain: Exploring scientific discovery with ChatGPT AI, AI Magazine44, 328 (2023). 5 Supplemental Material for “Optimality of Gaussian Entanglement of Formation” CONVENTIONS AND COEFFICIENT OPERA TORS We use quadratures satisfying [x j,p k]= iδ jk and annihila- tion operators a =( xA + ip A)/ √ 2 and b =( xB + ip B)/ √

    2023

  34. [34]

    A normalized pure two-mode state is represented by a Hilbert– Schmidt coefficient operator |ψC⟩= X m,n≥0 Cmn|m,n⟩,∥C∥ 2 =1.(S1) If c0≥c 1≥···≥ 0 are the singular values of C, then pn = c2 n are the Schmidt probabilities and E(ψC)=H(p)=− X n pn lnp n.(S2) The vectorization identities are (A⊗I)|C⟩⟩=|AC⟩⟩,(I⊗B)|C⟩⟩=|CB T⟩⟩.(S3) The generalized EPR observable...

This paper was first reviewed by deepseek-v4-flash on August 4, 2026.