Constant-sized correlations are sufficient to self-test maximally entangled states with unbounded dimension

Honghao Fu

Joint Center for Quantum Information and Computer Science, Institute for Advanced Computer Studies and Department of Computer Science, University of Maryland, College Park, MD 20742, USA

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Let $p$ be an odd prime and let $r$ be the smallest generator of the multiplicative group $\mathbb{Z}_p^\ast$. We show that there exists a correlation of size $\Theta(r^2)$ that self-tests a maximally entangled state of local dimension $p-1$. The construction of the correlation uses the embedding procedure proposed by Slofstra ($\textit{Forum of Mathematics, Pi.}$ ($2019$)). Since there are infinitely many prime numbers whose smallest multiplicative generator is in the set $\{2,3,5\}$ (D.R. Heath-Brown $\textit{The Quarterly Journal of Mathematics}$ ($1986$) and M. Murty $\textit{The Mathematical Intelligencer}$ ($1988$)), our result implies that constant-sized correlations are sufficient for self-testing of maximally entangled states with unbounded local dimension.

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[2] Thomas Vidick, "Almost synchronous quantum correlations", Journal of Mathematical Physics 63 2, 022201 (2022).

[3] Harshank Shrotriya, Kishor Bharti, and Leong-Chuan Kwek, "Robust semi-device-independent certification of all pure bipartite maximally entangled states via quantum steering", Physical Review Research 3 3, 033093 (2021).

[4] Laura Mančinska, Jitendra Prakash, and Christopher Schafhauser, "Constant-sized robust self-tests for states and measurements of unbounded dimension", arXiv:2103.01729.

[5] Zhengfeng Ji, Debbie Leung, and Thomas Vidick, "A three-player coherent state embezzlement game", arXiv:1802.04926.

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