bibkey: kumari2017sufficient authors: Meenu Kumari, Shohini Ghose, Robert B. Mann year: 2017 title: “Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities” doi: 10.1103/PhysRevA.96.012128 url: https://arxiv.org/abs/1704.06516v2 claim: “For a two-qutrit state, the CGLMP expression I_3 = P(A_1=B_1) + P(B_1=A_2+1) + P(A_2=B_2) + P(B_2=A_1) - P(A_1=B_1-1) - P(B_1=A_2) - P(A_2=B_2-1) - P(B_2=A_1-1) is evaluated with A_k = U_FT U(phi_k) and B_l = U_FT^* U(phi’_l), U(phi) = diag(exp(-i phi(j))), followed by a computational-basis measurement, and B_CGLMP(rho) is its maximum over the twelve angles (Eqs. (CGLMP1), (CGLMP2)). Based on numerical studies the paper conjectures (Eq. (35)) that the CGLMP inequality is monogamous: for every three-qutrit state at most one of rho_AB, rho_BC, rho_AC has B_CGLMP > 2.” strata_touched:

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Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities

Meenu Kumari, Shohini Ghose and Robert B. Mann, Phys. Rev. A 96, 012128 (2017); arXiv:1704.06516v2. Quotations are from the arXiv v2 source.

The CGLMP expression (Eq. (CGLMP1)):

${\mathcal{I}}_3(\rho )=P(A_1=B_1)+P(B_1=A_2+1)+P(A_2=B_2)+P(B_2=A_1)-P(A_1=B_1-1)-P(B_1=A_2)-P(A_2=B_2-1)-P(B_2=A_1-1)$

The measurement family:

Let $U(\overrightarrow{{\varphi }_k})$ and $U(\overrightarrow{{\phi }_l})$ be $3\times 3$ unitary operators whose diagonal elements are $\exp (-\mathrm{i}{\varphi }_k(j))$ and $\exp (-\mathrm{i}{\phi }_l(j))$, and off-diagonal elements are zero. Let $U_{\text{FT}}$ and $U_{\text{FT}}^{\ast }$ be the respective 3-dimensional discrete Fourier transform and inverse.

with $A_k=U_{\text{FT}}U({\vec{\varphi }}_k)$ and $B_l=U_{\text{FT}}^{\ast }U({\vec{\phi }}_l)$ “followed by a measurement in the $\{|0⟩,|1⟩,|2⟩\}$ basis”, the probabilities $P(A_m=j,B_n=k)=\mathrm{tr}({\Pi }_j\otimes {\Pi }_k{A}_m\otimes B_n\rho A_m^{†}\otimes B_n^{†})$, and ${\mathcal{B}}_{CGLMP}(\rho )={\max }_{{\vec{\varphi }}_k,{\vec{\phi }}_l}{\mathcal{I}}_3$ (Eq. (CGLMP2)).

The conjecture (Eq. (35)):

Based on our numerical studies, we make the following conjecture. 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.

followed by the display ${\mathcal{B}}_{\text{CGLMP}}({\rho }_{AB})>2\Rightarrow {\mathcal{B}}_{\text{CGLMP}}({\rho }_{BC})\le 2$ and ${\mathcal{B}}_{\text{CGLMP}}({\rho }_{AC})\le 2$, “with the same result holding for any permutation of $(A,B,C)$”. With Theorem 3 (a monogamous two-qudit Bell inequality certifies that a violating state has no symmetric extension) the paper concludes that a CGLMP violation would exclude a three-qutrit symmetric extension.

The encoding takes outcomes in Fin 3 with modular addition; $U_{\text{FT}}$ has entries ${\omega }^{jk}/\sqrt{3}$ with the existing omega $=e^{2\pi i/3}$, and $U_{\text{FT}}^{\ast }$ is its conjugate transpose; ${\mathcal{B}}_{CGLMP}$ is the supremum over all angle vectors; the reduced states are the existing partial traces, with the first-named party of each pair measured by $A_k$.

Verified locator

• DOI: https://doi.org/10.1103/PhysRevA.96.012128 (Phys. Rev. A 96, 012128 (2017)).
• URL: https://arxiv.org/abs/1704.06516v2 (v2, 2017-08-07, the latest version; source Symmetric_extension_and_Bell_inequality_paper_revision_for_arxiv.tex, md5 1fd49d0919d1b80fc742512d732f3b53): Eq. (CGLMP1) (l. 128–132), the measurement family and Eq. (Probabilities) (l. 133–146), Eq. (CGLMP2) (l. 148–152), Theorem 3 (l. 338) and the conjecture (l. 400–406).