bibkey: cereceda2017cabello authors: José L. Cereceda year: 2017 title: “Cabello’s nonlocality for generalized three-qubit GHZ states” doi: 10.1007/s40509-016-0093-7 url: https://arxiv.org/abs/1609.04763v2 claim: “For three qubits with local ±1-valued projective observables U_k, D_k (eigenvectors u_k^±, d_k^±), Cabello’s argument uses P = P(U_1,U_2,U_3|+++), the vanishing of P(D_1,U_2,U_3|+++), P(U_1,D_2,U_3|+++), P(U_1,U_2,D_3|+++), and Q = P(D_1,D_2,D_3|—); it contradicts local realism when Q < P. For the generalized GHZ states t|v_1v_2v_3> + |w_1w_2w_3> the maximum of C = P - Q under the three constraints is 9/64, at t = 1. The paper conjectures that for Q > 0 this maximum, 9/64, is the maximum over all possible states and choices of observables.” strata_touched:

• D5/S3/Quantum/Entanglement/ThreeQubitCabelloRefutation license: citation-only triage: anchor

Cabello’s nonlocality for generalized three-qubit GHZ states

José L. Cereceda, arXiv:1609.04763v2 [quant-ph] (2016); Quantum Studies: Mathematics and Foundations 4(3) (2017) 205–215. Quotations are from the arXiv source.

The measurements (Section 2):

Each of the qubits then enters its own measuring station where, for each run of the experiment, one of two alternative measurements is performed: $U_1$ or $D_1$ for qubit 1, $U_2$ or $D_2$ for qubit 2, and $U_3$ or $D_3$ for qubit 3. Each measurement gives the possible outcomes $+1$ or $-1$. The observables $U_k$ and $D_k$ have associated operators ${\hat{U}}_k=|u_k^{+}⟩⟨u_k^{+}|-|u_k^{-}⟩⟨u_k^{-}|$ and ${\hat{D}}_k=|d_k^{+}⟩⟨d_k^{+}|-|d_k^{-}⟩⟨d_k^{-}|$

The conditions of Cabello’s argument are $P(U_1,U_2,U_3|+++)=P$, $P(D_1,U_2,U_3|+++)=P(U_1,D_2,U_3|+++)=P(U_1,U_2,D_3|+++)=0$ and $P(D_1,D_2,D_3|---)=Q$, where

$P(U_1,U_2,U_3|+++)$ denotes the joint probability that a measurement of $U_1$, $U_2$, and $U_3$ on qubits 1, 2, and 3, respectively, gives the outcome $+1$ for each of them.

For the generalized GHZ states the paper finds the maximum $C=P-Q=9/64$ at $t=1$, $x=y=z=1$, $\gamma =-\arccos (7/8)$, and then states:

Motivated by this fact, we conjecture that, for the case in which $Q>0$, the maximum quantum violation of inequality (ineq2) (subject to the fulfillment of conditions (hc2)-(hc4)) is achieved, over all possible states and choice of observables, for the GHZ state $|\psi ⟩=\frac{1}{\sqrt{2}}(|v_1,v_2,v_3⟩+|w_1,w_2,w_3⟩)$, with the maximum violation being equal to $C_{\text{max}}=\frac{9}{64}$.

Here inequality (ineq2) is $C=P-Q\le 0$. The conclusions restate it:

We conjectured that the value $\frac{9}{64}$ is indeed optimal over all entangled states of three qubits subjected to arbitrary local projective measurements.

Verified locator

• DOI: https://doi.org/10.1007/s40509-016-0093-7 (the Crossref record gives the title, the author, volume 4, issue 3 and pages 205–215; the journal text was not read).
• URL: https://arxiv.org/abs/1609.04763v2 (source cabello2.tex): the measurements and the conditions (Section 2, l. 150–170), the conjecture (Section 2, l. 281–286) and the conclusions (Section 4, l. 463–465).