Robust self-testing of nonmaximal entanglement from a reduced inner product game

Abstract: We reduce Lalonde's pseudo-telepathy inner product game from six to five dimensions, keeping its four Alice questions and three Bob questions and reducing each answer alphabet to five. We prove that the reduced game self-tests its nonmaximally entangled state and all 35 measurement effects. The self-test is also robust: for a strategy winning with probability at least $1-\varepsilon$ we obtain $$d_{\mathrm{ext}}\le\min{1,2<sup>{21}\sqrt{\varepsilon}}$$, where $d_{\mathrm{ext}}=\sqrt{1-Ξ}$ and $Ξ$ is the optimal squared fidelity with the reference state under local extraction channels. A compactness argument then gives simultaneous qualitative robustness for all measurement actions on the state under common local isometries. As a consequence of the self-test, no maximally entangled state of any finite dimension can win the game perfectly.