bibkey: bhattacharya2015hardy authors: Some Sankar Bhattacharya; Arup Roy; Amit Mukherjee; Ramij Rahaman year: 2015 title: “Witnessing Genuine Mutipartite Non-locality” doi: 10.48550/arXiv.1507.07327 url: https://arxiv.org/abs/1507.07327v1 claim: “For N parties with inputs u, v and outcomes 1, …, d, the relaxed Hardy conditions (relH) are P(1…1|u…u) = q > 0, P(1…1 not-d_r 1…1|v at r) = 0 for all r, and P(1…1 d_i 1…1 d_j 1…1|v at i and j) = 0 for all i != j with j fixed. Under no-signalling the paper observes the optimal q to be 1/3 for n = 3, 4 and d = 2, …, 5, and conjectures that the optimal q is 1/3 for any dimension and any number of parties.” strata_touched:

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

Witnessing Genuine Mutipartite Non-locality

Some Sankar Bhattacharya, Arup Roy, Amit Mukherjee, Ramij Rahaman, arXiv:1507.07327v1 [quant-ph] (2015). The title is spelled as printed. Quotations are from the arXiv source.

The outcomes (Section 3, “Modified Hardy type argument for arbitrary dimension”):

The possible outcomes $x_i$ of each such local (i-th party) measurement can be $1,\dots d_i$.

The relaxed Hardy conditions (eq. relH) are $P(1\dots 1|{\hat{u}}_1\dots {\hat{u}}_N)=q>0$, $P(1\dots 1\neg d_{r}1\dots 1|{\hat{u}}_1\dots {\hat{u}}_{r-1}{\hat{v}}_r{\hat{u}}_{r+1}\dots {\hat{u}}_N)=0$ for all $r$, and $P(1\dots 1d_{i}1\dots 1d_{j}1\dots 1|{\hat{u}}_1\dots {\hat{v}}_i\dots {\hat{v}}_j\dots {\hat{u}}_N)=0$ for all $i\ne j$, where

Here $\neg d_r$ denotes other than $d_r$ it can be any positive integer less or equal to $d_r-1$ and $j\in \{1,\dots .,N\}$ is kept fixed.

The no-signalling condition requires that summing over the outcome of party 1 gives a marginal independent of party 1’s input, and

This condition also holds for the cyclic permutation of the parties.

The observation and the conjecture (Section 3, “Multi-partite Hardy Paradox in GNST”):

The optimal value of the probability of success, $q$ in(\ref{relH}) is $\frac{1}{3}$ for $n=3,4$ and $d=2,3,4,5$ in GNST.

The optimal value of the probability of success, $q$ in(\ref{relH}) is $\frac{1}{3}$ for any arbitrary dimension for any number of parties in GNST.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.1507.07327 (arXiv-issued; v1 is the only version and the record lists no journal reference).
• URL: https://arxiv.org/abs/1507.07327v1 (source Gen_Hardy_2015_07_27.tex with CRLF line endings, md5 of the raw decompressed bytes 84c375d26a8c9c405f9b62d1b6668395; the gzip file from https://arxiv.org/src/1507.07327v1 has md5 1fc0975c85ac62196a2c7e24e8e00bd6): the outcomes (l. 170), the relaxed Hardy conditions (l. 173–182), the normalization and no-signalling conditions (l. 219–221, 230–235), the observation (l. 238–240) and the conjecture (l. 243–245).