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

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

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].
arXiv:1712.09369

[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].
https:/​/​doi.org/​10.4230/​LIPIcs.ICALP.2018.11
arXiv:1712.09368

[3] Manuel Blum, Michael Luby, and Ronitt Rubinfeld. Self-testing/​correcting with applications to numerical problems. J. Comput. Syst., 47(3):549–595, 1993.
https:/​/​doi.org/​10.1016/​0022-0000(93)90044-W

[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].
https:/​/​doi.org/​10.1103/​physreva.91.052111
arXiv:1504.06960

[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.
https:/​/​doi.org/​10.1103/​PhysRevLett.23.880

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

[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].
arXiv:1609.03687

[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].
https:/​/​doi.org/​10.4230/​LIPIcs.ITCS.2017.48
arXiv:1701.01062

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

[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].
https:/​/​doi.org/​10.1103/​PhysRevLett.115.250401
arXiv:1511.03190

[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].
https:/​/​doi.org/​10.1038/​nature15759
arXiv:1508.05949

[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].
https:/​/​doi.org/​10.1038/​srep30289
arXiv:1603.05705

[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].
https:/​/​doi.org/​10.1109/​CCC.2009.22
arXiv:0810.0693

[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.
https:/​/​doi.org/​10.1137/​090751293
arXiv:0704.2903

[15] Matthew McKague. Self-testing in parallel. New J. Phys., 18:045013, 2016, arXiv:1511.04194 [quant-ph].
https:/​/​doi.org/​10.1088/​1367-2630/​18/​4/​045013
arXiv:1511.04194

[16] N. David Mermin. Simple unified form for the major no-hidden-variables theorems. Phys. Rev. Lett., 65:3373, 1990.
https:/​/​doi.org/​10.1103/​PhysRevLett.65.3373

[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.
https:/​/​doi.org/​10.1007/​11786986_8
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].
https:/​/​doi.org/​10.1088/​1751-8113/​45/​45/​455304
arXiv:1203.2976

[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].
https:/​/​doi.org/​10.1145/​3055399.3055468
arXiv:1610.03574

[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].
arXiv:1801.03821

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

[22] Asher Peres. Incompatible results of quantum measurements. Phys. Lett. A, 151(3-4):107–108, 1990.
https:/​/​doi.org/​10.1016/​0375-9601(90)90172-K

[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].
arXiv:1209.0448

[24] Ben W. Reichardt, Falk Unger, and Umesh Vazirani. Classical command of quantum systems. Nature, 496:456–460, 2013.
https:/​/​doi.org/​10.1038/​nature12035

[25] Lynden K. Shalm et al. Strong loophole-free test of local realism. Phys. Rev. Lett., 115:250402, 2015, arXiv:1511.03189 [quant-ph].
https:/​/​doi.org/​10.1103/​PhysRevLett.115.250402
arXiv:1511.03189

[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].
https:/​/​doi.org/​10.1103/​PhysRevA.93.062121
arXiv:1512.02074

[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].
https:/​/​doi.org/​10.1103/​PhysRevA.87.050102
arXiv:1210.4409

Cited by

[1] William Slofstra and Thomas Vidick, "Entanglement in Non-local Games and the Hyperlinear Profile of Groups", Annales Henri Poincaré 19 10, 2979 (2018).

[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] 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.

[4] Rotem Arnon-Friedman and Jean-Daniel Bancal, "Device-independent certification of one-shot distillable entanglement", New Journal of Physics 21 3, 033010 (2019).

[5] 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).

[6] Ivan Šupić and Joseph Bowles, "Self-testing of quantum systems: a review", Quantum 4, 337 (2020).

[7] Andrea Coladangelo, Koon Tong Goh, and Valerio Scarani, "All pure bipartite entangled states can be self-tested", Nature Communications 8, 15485 (2017).

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

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

[10] Amir Kalev and Carl A. Miller, "Rigidity of the magic pentagram game", Quantum Science and Technology 3 1, 015002 (2018).

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

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

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

[14] Rui Chao and Ben W. Reichardt, "Quantum dimension test using the uncertainty principle", arXiv:2002.12432.

The above citations are from Crossref's cited-by service (last updated successfully 2020-10-23 05:45:15) and SAO/NASA ADS (last updated successfully 2020-10-23 05:45:16). The list may be incomplete as not all publishers provide suitable and complete citation data.