bibkey: vandam2005statisticalstrength authors: W. van Dam; R. D. Gill; P. D. Grünwald year: 2005 title: “The Statistical Strength of Nonlocality Proofs” doi: 10.1109/TIT.2005.851738 url: https://arxiv.org/abs/quant-ph/0307125v2 claim: “There is an experiment on pairs of Bell singlets, of the 2×4×4 type, more than twice as strong as CHSH, and involving joint measurements on the pairs.” strata_touched:

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Statistical strength of nonlocality proofs

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DOI: https://doi.org/10.1109/TIT.2005.851738

Source: https://arxiv.org/abs/quant-ph/0307125v2

IEEE Transactions on Information Theory 51(8), 2812–2835. The arXiv v2 PDF gives the KL conventions in Section IV.A, p. 6; local response mixtures in Section II.D, p. 5; the three strengths in Section IV.B, p. 8, Definitions 1–3 and equations (8)–(10); the strengths table in Section V, p. 8; and Conjecture 5 in Section VI.D, p. 11. The CHSH quantum and local tables appear in Appendix III.C, pp. 18–20.

Section VI.D, Conjecture 5, p. 11:

There is an experiment on pairs of Bell singlets, of the $2\times 4\times 4$ type, more than twice as strong as CHSH, and involving joint measurements on the pairs.

The type counts two parties, four settings per party, and four outcomes per setting. Section IV.B, Definition 1, p. 8:

When each measurement setting is sampled with equal probability, the resulting strength $S_Q^{\mathrm{UNI}}$ is defined by $$S_Q^{\mathrm{UNI}}:=D(Q_{{\sigma }^{\circ }}\Vert {\mathcal{P}}_{{\sigma }^{\circ }})=\underset{\pi \in \Pi }{\inf }D(Q_{{\sigma }^{\circ }}\Vert P_{{\sigma }^{\circ },\pi }),$$ where ${\sigma }^{\circ }$ denotes the uniform distribution over the settings.

Definition 2, p. 8:

When the experimenter QM is allowed to choose any distribution on measurement settings, as long as the distribution for each party is uncorrelated with the distributions of the other parties, the resulting strength $S_Q^{\mathrm{UC}}$ is defined by $$S_Q^{\mathrm{UC}}:=\underset{\sigma \in {\Sigma }^{\mathrm{UC}}}{\sup }D(Q_{\sigma }\Vert {\mathcal{P}}_{\sigma })=\underset{\sigma \in {\Sigma }^{\mathrm{UC}}}{\sup }\underset{\pi \in \Pi }{\inf }D(Q_{\sigma }\Vert P_{\sigma ,\pi }),$$ where $\sigma \in {\Sigma }^{\mathrm{UC}}$ denotes the use of uncorrelated settings.

Definition 3, p. 8:

When the experimeniter QM is allowed to choose any distribution on measurement settings (including correlated distributions), the resulting strength $S_Q^{\mathrm{COR}}$ is defined by $$S_Q^{\mathrm{COR}}:=\underset{\sigma \in \Sigma }{\sup }D(Q_{\sigma }\Vert {\mathcal{P}}_{\sigma })=\underset{\sigma \in \Sigma }{\sup }\underset{\pi \in \Pi }{\inf }D(Q_{\sigma }\Vert P_{\sigma ,\pi }),$$ where $\sigma \in \Sigma$ denoted the use of correlated settings.

The spelling “experimeniter” and “denoted” are those of the source. The notation above expands the source small-capital strength macros and its probability simplexes. KL is in bits; zero quantum mass contributes zero, and positive quantum mass above zero local mass contributes infinity. The results table lists CHSH strength 0.0462738469 in all three columns.

Two-singlet encoding

The local Hilbert indices are Fin 2 × Fin 2. The global indices group Alice’s two qubits and Bob’s two qubits. The shared amplitudes multiply the two literal singlet amplitudes (01 − 10)/sqrt 2, one on each corresponding Alice–Bob qubit pair. Each party has four nonzero, four-outcome projective measurements. A joint measurement contains a projector that cannot be expressed as the Kronecker product of two single-qubit operators.

The magic-square construction uses the commuting rows and columns of [[ZI, IZ, ZZ], [IX, XI, XX], [ZX, XZ, YY]]. The fourth setting measures YI and IY. Bob’s projectors are transposed and conjugated by [[0, −1], [1, 0]] on both qubits. The same local conjugation realizes the source’s CHSH table on a literal singlet; its displayed source realization uses the Bell state Phi-plus.

Magic-square credit

The parity construction originates in N. D. Mermin, “Simple unified form for the major no-hidden-variables theorems”, Physical Review Letters 65, 3373–3376 (1990), DOI 10.1103/PhysRevLett.65.3373, and A. Peres, “Incompatible results of quantum measurements”, Physics Letters A 151, 107–108 (1990), DOI 10.1016/0375-9601(90)90172-K. Its two-party pseudo-telepathy use is due to G. Brassard, A. Broadbent and A. Tapp, “Quantum Pseudo-Telepathy”, Foundations of Physics 35, 1877–1907 (2005), https://arxiv.org/abs/quant-ph/0407221. The statistical-strength comparison combines this established construction with finite log-sum and a uniform CHSH reference theory.