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

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.

► BibTeX data

► References

[1] Rotem Arnon-Friedman and Jean-Daniel Bancal. Device-independent certification of one-shot distillable entanglement. 2017, arXiv:1712.09369 [quant-ph].

[2] Rotem Arnon-Friedman and Henry Yuen. Noise-tolerant testing of high entanglement of formation. In Proc. 45th ICALP, pages 11:1-11:12, 2018, arXiv:1712.09368 [quant-ph].

[3] Manuel Blum, Michael Luby, and Ronitt Rubinfeld. Self-testing/​correcting with applications to numerical problems. J. Comput. Syst., 47(3):549-595, 1993.

[4] Cédric Bamps and Stefano Pironio. Sum-of-squares decompositions for a family of Clauser-Horne-Shimony-Holt-like inequalities and their application to self-testing. Phys. Rev. A, 91(5):052111, 2015, arXiv:1504.06960 [quant-ph].

[5] John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett., 23:880-884, 1969.

[6] Matthew Coudron and Anand Natarajan. The parallel-repeated magic square game is rigid. 2016, arXiv:1609.06306 [quant-ph].

[7] Andrea W. Coladangelo. Parallel self-testing of (tilted) EPR pairs via copies of (tilted) CHSH and the magic square game. Quantum Inf. Comput., 17(9&10):831-865, 2017, arXiv:1609.03687 [quant-ph].

[8] Rui Chao, Ben W. Reichardt, Chris Sutherland, and Thomas Vidick. Overlapping qubits. In Proc. 8th Innovations in Theoretical Computer Science Conference (ITCS), volume 67, pages 48:1-48:21, 2017, arXiv:1701.01062 [quant-ph].

[9] Andrea Coladangelo and Jalex Stark. Robust self-testing for linear constraint system games. 2017, arXiv:1709.09267 [quant-ph].

[10] Marissa Giustina et al. Significant-loophole-free test of Bell's theorem with entangled photons. Phys. Rev. Lett., 115:250401, 2015, arXiv:1511.03190 [quant-ph].

[11] B. Hensen et al. Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres. Nature, 526:682-686, 2015, arXiv:1508.05949 [quant-ph].

[12] B. Hensen et al. Loophole-free Bell test using electron spins in diamond: second experiment and additional analysis. Scientific Reports, 6:30289, 2016, arXiv:1603.05705 [quant-ph].

[13] Tsuyoshi Ito, Hirotada Kobayashi, and Keiji Matsumoto. Oracularization and two-prover one-round interactive proofs against nonlocal strategies. In Proc. 24th IEEE Conf. on Computational Complexity (CCC), pages 217-228. IEEE Computer Society, 2009, arXiv:0810.0693 [quant-ph].

[14] Julia Kempe, Hirotada Kobayashi, Keiji Matsumoto, Ben Toner, and Thomas Vidick. Entangled games are hard to approximate. J. ACM, 40(3):848-877, 2011, arXiv:0704.2903 [quant-ph]. Earlier version in FOCS'08.

[15] Matthew McKague. Self-testing in parallel. New J. Phys., 18:045013, 2016, arXiv:1511.04194 [quant-ph].

[16] N. David Mermin. Simple unified form for the major no-hidden-variables theorems. Phys. Rev. Lett., 65:3373, 1990.

[17] Frédéric Magniez, Dominic Mayers, Michele Mosca, and Harold Ollivier. Self-testing of quantum circuits. In Proc. 33rd ICALP, pages 72-83, 2006, arXiv:quant-ph/​0512111.

[18] Matthew McKague, Tzyh Haur Yang, and Valerio Scarani. Robust self-testing of the singlet. J. Phys. A: Math. Theor., 45:455304, 2012, arXiv:1203.2976 [quant-ph].

[19] Anand Natarajan and Thomas Vidick. A quantum linearity test for robustly verifying entanglement. In Proc. 49th ACM STOC, pages 1003-1015, 2017, arXiv:1610.03574 [quant-ph].

[20] Anand Natarajan and Thomas Vidick. Low-degree testing for quantum states, and a quantum entangled games PCP for QMA. 2018, arXiv:1801.03821 [quant-ph].

[21] Dimiter Ostrev and Thomas Vidick. Entanglement of approximate quantum strategies in XOR games. 2016, arXiv:1609.01652 [quant-ph].

[22] Asher Peres. Incompatible results of quantum measurements. Phys. Lett. A, 151(3-4):107-108, 1990.

[23] Ben W. Reichardt, Falk Unger, and Umesh Vazirani. A classical leash for a quantum system: Command of quantum systems via rigidity of CHSH games. 2012, arXiv:1209.0448 [quant-ph].

[24] Ben W. Reichardt, Falk Unger, and Umesh Vazirani. Classical command of quantum systems. Nature, 496:456-460, 2013.

[25] Lynden K. Shalm et al. Strong loophole-free test of local realism. Phys. Rev. Lett., 115:250402, 2015, arXiv:1511.03189 [quant-ph].

[26] Xingyao Wu, Jean-Daniel Bancal, Matthew McKague, and Valerio Scarani. Device-independent parallel self-testing of two singlets. Phys. Rev. A, 93:062121, 2016, arXiv:1512.02074 [quant-ph].

[27] Tzyh Haur Yang and Miguel Navascués. Robust self-testing of unknown quantum systems into any entangled two-qubit states. Phys. Rev. A, 87(5):050102, 2013, arXiv:1210.4409 [quant-ph].

Cited by

[1] Rotem Arnon-Friedman and Henry Yuen, "Noise-tolerant testing of high entanglement of formation", arXiv:1712.09368 (2017).

[2] Rotem Arnon-Friedman and Jean-Daniel Bancal, "Device-independent Certification of One-shot Distillable Entanglement", arXiv:1712.09369 (2017).

[3] Spencer Breiner, Amir Kalev, and Carl A. Miller, "Parallel Self-Testing of the GHZ State with a Proof by Diagrams", arXiv:1806.04744 (2018).

[4] Zhengfeng Ji, Debbie Leung, and Thomas Vidick, "A three-player coherent state embezzlement game", arXiv:1802.04926 (2018).

[5] Rui Chao, Ben W. Reichardt, Chris Sutherland, and Thomas Vidick, "Overlapping qubits", arXiv:1701.01062 (2017).

[6] Jedrzej Kaniewski, "Self-testing of binary observables based on commutation", Physical Review A 95 6, 062323 (2017).

[7] Joseph Fitzsimons, Zhengfeng Ji, Thomas Vidick, and Henry Yuen, "Quantum proof systems for iterated exponential time, and beyond", arXiv:1805.12166 (2018).

[8] Andrea Coladangelo, "Generalization of the Clauser-Horne-Shimony-Holt inequality self-testing maximally entangled states of any local dimension", Physical Review A 98 5, 052115 (2018).

The above citations are from Crossref's cited-by service (last updated 2019-01-23 05:17:12) and SAO/NASA ADS (last updated 2019-01-23 05:17:13). The list may be incomplete as not all publishers provide suitable and complete citation data.