Test for a large amount of entanglement, using few measurements

Rui Chao1, Ben W. Reichardt1, Chris Sutherland1, and Thomas Vidick2

1University of Southern California
2California Institute of Technology

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Bell-inequality violations establish that two systems share some quantum entanglement. We give a simple test to certify that two systems share an asymptotically large amount of entanglement, $n$ EPR states. The test is efficient: unlike earlier tests that play many games, in sequence or in parallel, our test requires only one or two CHSH games. One system is directed to play a CHSH game on a random specified qubit $i$, and the other is told to play games on qubits $\{i,j\}$, without knowing which index is $i$.
The test is robust: a success probability within $\delta$ of optimal guarantees distance $O(n^{5/2} \sqrt{\delta})$ from $n$ EPR states. However, the test does not tolerate constant $\delta$; it breaks down for $\delta = \tilde\Omega (1/\sqrt{n})$. We give an adversarial strategy that succeeds within delta of the optimum probability using only $\tilde O(\delta^{-2})$ EPR states.

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[2] Spencer Breiner, Amir Kalev, and Carl A. Miller, "Parallel Self-Testing of the GHZ State with a Proof by Diagrams", Electronic Proceedings in Theoretical Computer Science 287, 43 (2019).

[3] Chen-Hung Wu, Cheng-Yun Hsieh, Jiun-Yun Li, and James Chien-Mo Li, 2020 IEEE International Test Conference (ITC) 1 (2020) ISBN:978-1-7281-9113-3.

[4] Alex B. Grilo, William Slofstra, and Henry Yuen, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS) 611 (2019) ISBN:978-1-7281-4952-3.

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[14] Rotem Arnon-Friedman and Henry Yuen, "Noise-tolerant testing of high entanglement of formation", arXiv:1712.09368.

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