slug: kumari-2017-cglmp-monogamy
bibkey: kumari2017sufficient
doi: 10.1103/PhysRevA.96.012128
url: https://arxiv.org/abs/1704.06516v2
triage: theorem
motivation_gids:
  - D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.result

Monogamy of the qutrit CGLMP inequality

Problem

M. Kumari, S. Ghose and R. B. Mann, Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities, Phys. Rev. A 96, 012128 (2017), arXiv:1704.06516v2, Eq. (35):

The CGLMP inequality for 2-qutrit states is monogamous, that is, if ${\rho }_{ABC}$ is any 3-qutrit state such that ${\rho }_{AB}$, ${\rho }_{BC}$ and ${\rho }_{AC}$ are its three 2-qutrit RDMs, at most one of these violates the CGLMP inequality.

Here ${\mathcal{B}}_{CGLMP}(\rho )$ is the maximum of the CGLMP expression ${\mathcal{I}}_3$ (Eq. (CGLMP1)) over the angles of $A_k=U_{\text{FT}}U({\vec{\varphi }}_k)$ and $B_l=U_{\text{FT}}^{\ast }U({\vec{\phi }}_l)$, with $U(\vec{\varphi })=\mathrm{diag}(e^{-i\varphi (j)})$ and a computational-basis measurement after each operator (Eq. (CGLMP2)). A state violates the inequality when ${\mathcal{B}}_{CGLMP}>2$.

Motivation

With the paper’s Theorem 3, Eq. (35) was offered as a test for the absence of symmetric extensions of two-qutrit states. The frozen declaration D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.result shows that Eq. (35) is false.

Gap

Issue #12528 classifies the conjecture as Tier 1 and records the checks made before any Lean:

• arXiv:1704.06516 v2 is the latest version; the conjecture rests on numerical studies;
• the forward citations listed by Semantic Scholar (arXiv:2407.12933, arXiv:1904.02692) do not treat CGLMP monogamy; Lee–Son, Entropy 22, 1282 (2020), arXiv:2010.09999, l. 581–582, still offer CGLMP monogamy as a possible explanation of non-violation;
• Augusiak et al., PRA 90, 052323 (2014), arXiv:1307.6390, Theorem 1, is a no-signalling relation with settings shared inside one tripartite distribution, so it does not decide Eq. (35), in which each reduced state is optimized separately;
• Cieśliński et al., arXiv:2312.04373, and Panda–Datta–Agrawal, arXiv:2601.02925, concern qubit and Dicke-state scenarios.

These are orchestrator-checked and seat-reported readings, not-found-in-searched-scope; they do not establish exhaustive worldwide novelty.

Route

1. $v=|002\rangle+|011\rangle+2|020\rangle+|100\rangle+2|112\rangle-|121\rangle +2|210\rangle+2|222\rangle$,$|v|^2=20$,$\rho=vv^\dagger/20$.
2. For a reduced state of a pure state, each joint probability is $\frac{1}{20}{\sum }_c|((A\otimes B)v_{\cdot \cdot c}{)}_{jk}{|}^2$. With angles in units of $\pi /6$, every amplitude is a sum of twelfth roots of unity divided by $3$.
3. On ${\rho }_{AB}$ take ${\vec{\varphi }}_1=(0,2,7)$, ${\vec{\varphi }}_2=(0,2,1)$, ${\vec{\phi }}_1=(0,8,4)$, ${\vec{\phi }}_2=(0,10,2)$; on ${\rho }_{AC}$ take ${\vec{\varphi }}_1=(0,6,9)$, ${\vec{\varphi }}_2=(0,6,3)$, ${\vec{\phi }}_1=(0,10,2)$, ${\vec{\phi }}_2=(0,0,6)$. In both cases the eight event probabilities, in the order of Eq. (CGLMP1), are $\frac{23}{60}+\frac{\sqrt{3}}{5}$, $\frac{23}{60}+\frac{\sqrt{3}}{5}$, $\frac{1}{2}$, $\frac{1}{2}$, $\frac{23}{60}-\frac{\sqrt{3}}{5}$, $\frac{23}{60}-\frac{\sqrt{3}}{5}$, $\frac{1}{4}-\frac{\sqrt{3}}{15}$, $\frac{1}{4}-\frac{\sqrt{3}}{15}$, so ${\mathcal{I}}_3=\frac{1}{2}+\frac{14\sqrt{3}}{15}\approx 2.1166$.

Falsifier

The kernel-checked result is the negation of the following statement. For every positive semidefinite $27\times 27$ complex matrix $\rho$ of trace one, indexed by $(a,b,c)$, and for all reals $b_{AB},b_{BC},b_{AC}$ that are the maxima of ${\mathcal{I}}_3$ over the twelve angles on $\rho_{AB},\rho_{BC}, \rho_{AC}$(‘IsCglmpValue‘,thatis$\mathcal B_{CGLMP}$ of Eq. (CGLMP2)), $b_{AB}>2$ implies $b_{BC}\le 2$ and $b_{AC}\le 2$. This is the displayed implication of Eq. (35), which the conjecture contains.

The definitions are as follows:

• the reduced states are the frozen partialTraceRight and partialTraceLeft of PartialTraceMutualInformation, with the first-named party of each pair measured by $A_k$;
• $U_{\text{FT}}$ has entries ${\omega }^{jk}/\sqrt{3}$, with the frozen omega $=e^{2\pi i/3}$ of FiniteLocalLatitudeGeometry;
• outcome relations such as $B_1=A_2+1$ are read in Fin 3;
• cglmpAt is ${\mathcal{I}}_3$ at a given choice of the twelve angles, and IsCglmpValue ρ b says that $b$ is its greatest value over all angles.

Under the opposite sign convention for the Fourier transform, the probabilities equal those at the negated angles, because $v$ is real. The refutation is therefore convention independent. Since the paper’s family is a subset of all local projective measurements, the same values also refute the conjecture read with unrestricted measurements.

Evidence

Exact recomputation (SymPy, issue #12528): the eight event probabilities and ${\mathcal{I}}_3$ for both reduced states. The literature seat recomputed them independently. BFGS optimization over the paper’s family gives $\max {\mathcal{I}}_3({\rho }_{AB})=\max {\mathcal{I}}_3({\rho }_{AC})\approx 2.1278$ and $\max {\mathcal{I}}_3({\rho }_{BC})\approx 0$.

The canonical source is D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.lean. It reuses the frozen QutritThresholdSharing.normalization ($(1/\sqrt{3}{)}^2=1/3$), made public for this purpose. Its public declarations are dft, phase, jointProb, eventProb, cglmpI3, cglmpAt, IsCglmpValue, rhoAB, rhoBC, rhoAC, claim, stateVec, rho, ang and result. The frozen module state has statement identity sha256:d702782b03881bd0fe04184abc5c12dd2f56c345b4e8b79d37b27c6366d172fd. The result declaration has statement identity sha256:9db90f908beb62a379f1fc501b8cd0b9b122407fc08b9390544e7b57547d5a96. The Freeze event is sha256:712f68a6d1b05afac438770e5b0bfab63c5d138e8cc655c55cfa93d9aeeed287; its project-level frozen prerequisites are the Freeze events of the three reused modules. The proof uses only the standard axioms propext, Classical.choice and Quot.sound; no sorry, native_decide, or new axiom.

Triage

Tier 1 conjecture of a 2017 journal article; resolution Refuted by D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.result. proof_shape: bind-only (the settlement of the named conjecture is the new content; no escape witness); admission_basis: open-problem-resolution (issue #12528). Utility kind certified-instance, basis refutes the module’s claim.

What the settlement shows

• Proved by result: some three-qutrit state has ${\mathcal{B}}_{CGLMP}({\rho }_{AB})>2$ and ${\mathcal{B}}_{CGLMP}({\rho }_{AC})>2$, so the CGLMP inequality is not monogamous in the paper’s sense.
• Proved inside the proof of result: for every two-qutrit matrix, ${\mathcal{I}}_3$ attains its maximum over the angles: it is continuous, it is unchanged when an angle moves by $2\pi$, and the box $[0,2\pi {]}^{24}$ is compact. For each reduced state of $v{v}^{†}/20$, every joint probability is one twentieth of a sum of squared amplitudes. The two values ${\mathcal{I}}_3=\frac{1}{2}+\frac{14\sqrt{3}}{15}$ hold at the stated angles, and $\rho$ is positive semidefinite with trace one.
• Mechanism (orchestrator argument, not stated in Lean):
  • Every basis state $|abc⟩$ in the support of $v$ has $a-b-c\equiv 1(\mathrm{mod}3)$. For fixed $c$, the slice $v_{\cdot \cdot c}$ is supported on $b=a-c-1$, so for the events $B=A+s$ the squared amplitude does not depend on the outcome $j$ of $A$, and each event probability is a sum of three squared moduli of phase-twisted slices.
  • The two Bell tests use different settings on $A$. A no-signalling monogamy relation with shared settings (for example Augusiak et al., Theorem 1) therefore does not apply.
• Effect on the paper’s other conclusions (checked by reading the v2 source):
  • The application to symmetric extensions (l. 407; abstract, l. 82; Discussion, l. 413) no longer follows from Eq. (35). It still holds: by Terhal–Doherty–Schwab, PRL 90, 157903 (2003), a state with a symmetric extension to two copies of $B$ admits a local model for every Bell test with two settings on $B$, so a CGLMP violation excludes such an extension. The paper cites that work (l. 96).
  • The statements that the numerical evidence supports monogamy of the 2-qutrit CGLMP inequality (abstract, l. 82; introduction, l. 99; Discussion, l. 413) are contradicted.
  • The qubit results (Theorems 1 and 2) and the plotted families of Figs. 2–3, where only one reduced state violates, are unaffected.
• Prior related examples (orchestrator-checked):
  • Brunner–Vértesi, PRA 86, 042113 (2012), arXiv:1207.3986, l. 338–393, give three-qutrit states whose three marginals all violate CHSH with binary measurements on a qubit subspace.
  • Their state gives ${\mathcal{I}}_3=0$ for every angle in the paper’s family (seat’s computation). Optimizing ${\mathcal{I}}_3$ over arbitrary $3\times 3$ unitaries reaches only $2.0$ on each marginal (orchestrator’s BFGS, 25 restarts, $a=0.6469$; positive control $2.12779$ on this module’s ${\rho }_{AB}$).
• Open here: whether all three reduced states of one three-qutrit state can violate the CGLMP inequality, and the largest pair $({\mathcal{B}}_{CGLMP}({\rho }_{AB}),{\mathcal{B}}_{CGLMP}({\rho }_{AC}))$.

ASSUMED-UNVERIFIED

The literature checks do not establish worldwide priority, and a complete forward-citation graph was not obtained. The Lean kernel verifies the encoded statement and its axiom closure. Its correspondence to the paper, including the Fourier-transform convention and the reading of ${\mathcal{B}}_{CGLMP}$ as the maximum over the angles, is checked by reading the source and the definitions.