No Access Submitted: 22 September 2017 Accepted: 02 January 2018 Published Online: 06 February 2018
J. Math. Phys. 59, 022103 (2018);
more...View Affiliations
View Contributors
  • Carl A. Miller
  • Roger Colbeck
  • Yaoyun Shi
If a measurement is made on one half of a bipartite system, then, conditioned on the outcome, the other half has a new reduced state. If these reduced states defy classical explanation—that is, if shared randomness cannot produce these reduced states for all possible measurements—the bipartite state is said to be steerable. Determining which states are steerable is a challenging problem even for low dimensions. In the case of two-qubit systems, a criterion is known for T-states (that is, those with maximally mixed marginals) under projective measurements. In the current work, we introduce the concept of keyring models—a special class of local hidden state models. When the measurements made correspond to real projectors, these allow us to study steerability beyond T-states. Using keyring models, we completely solve the steering problem for real projective measurements when the state arises from mixing a pure two-qubit state with uniform noise. We also give a partial solution in the case when the uniform noise is replaced by independent depolarizing channels.
We are grateful to Kim Winick for numerous helpful discussions, to Emanuel Knill, Sania Jevtic, Stephen Jordan, and Chau Nguyen for useful feedback on an earlier version of the manuscript, and to Nicholas Brunner, Daniel Cavalcanti, and Ivan Supic for pointers to the literature. R.C. is supported by the EPSRC’s Quantum Communications Hub (Grant No. EP/M013472/1) and by an EPSRC First (Grant No. EP/P016588/1). C.A.M. and Y.S. were supported in part by US NSF Grant Nos. 1500095, 1526928, and 1717523. Y.S. was also supported in part by University of Michigan.
  1. 1. J. S. Bell, “On the Einstein-Podolsky-Rosen paradox,” in Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, 1987), Chap. 2. Google Scholar
  2. 2. D. Mayers and A. Yao, “Quantum cryptography with imperfect apparatus,” in Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280) (IEEE Computer Society, 1998), pp. 503–509. Google ScholarCrossref
  3. 3. J. Barrett, L. Hardy, and A. Kent, “No signalling and quantum key distribution,” Phys. Rev. Lett. 95, 010503 (2005)., Google ScholarCrossref
  4. 4. H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007)., Google ScholarCrossref, ISI
  5. 5. C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, “One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering,” Phys. Rev. A 85, 010301 (2012)., Google ScholarCrossref
  6. 6. M. Piani and J. Watrous, “Necessary and sufficient quantum information characterization of Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 114, 060404 (2015)., Google ScholarCrossref
  7. 7. D. Cavalcanti, L. Guerini, R. Rabelo, and P. Skrzypczyk, “General method for constructing local hidden variable models for entangled quantum states,” Phys. Rev. Lett. 117, 190401 (2016)., Google ScholarCrossref
  8. 8. F. Hirsch, M. T. Quintino, T. Vértesi, M. F. Pusey, and N. Brunner, “Algorithmic construction of local hidden variable models for entangled quantum states,” Phys. Rev. Lett. 117, 190402 (2016)., Google ScholarCrossref
  9. 9. R. F. Werner, “Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model,” Phys. Rev. A 40, 4277–4281 (1989)., Google ScholarCrossref, ISI
  10. 10. S. Jevtic, M. J. W. Hall, M. R. Anderson, M. Zwierz, and H. M. Wiseman, “Einstein–Podolsky–Rosen steering and the steering ellipsoid,” J. Opt. Soc. Am. B 32, A40–A49 (2015)., Google ScholarCrossref
  11. 11. H. C. Nguyen and T. Vu, “Nonseparability and steerability of two-qubit states from the geometry of steering outcomes,” Phys. Rev. A 94, 012114 (2016)., Google ScholarCrossref
  12. 12. H. C. Nguyen and T. Vu, “Necessary and sufficient condition for steerability of two-qubit states by the geometry of steering outcomes,” Europhys. Lett. 115, 10003 (2016)., Google ScholarCrossref
  13. 13. S. J. Jones, H. M. Wiseman, and A. C. Doherty, “Entanglement, Einstein-Podolsky-Rosen correlations, Bell nonlocality, and steering,” Phys. Rev. A 76, 052116 (2007)., Google ScholarCrossref, ISI
  14. 14. J. Bowles, F. Hirsch, M. T. Quintino, and N. Brunner, “Sufficient criterion for guaranteeing that a two-qubit state is unsteerable,” Phys. Rev. A 93, 022121 (2016)., Google ScholarCrossref
  15. 15. S. J. Jones and H. M. Wiseman, “Nonlocality of a single photon: Paths to an Einstein-Podolsky-Rosen-steering experiment,” Phys. Rev. A 84, 012110 (2011)., Google ScholarCrossref
  16. 16. R. Uola, K. Luoma, T. Moroder, and T. Heinosaari, “Adaptive strategy for joint measurements,” Phys. Rev. A 94, 022109 (2016)., Google ScholarCrossref
  17. 17. S. Jevtic, M. Pusey, D. Jennings, and T. Rudolph, “Quantum steering ellipsoids,” Phys. Rev. Lett. 13, 020402 (2014)., Google ScholarCrossref
  18. 18. G. F. Simmons, Introduction to Topology and Modern Analysis, reprint ed. (Krieger Publishing Company, 1983). Google Scholar
  19. 19. M. T. Quintino et al., “Inequivalence of entanglement, steering, and Bell nonlocality for general measurements,” Phys. Rev. A 92, 032107 (2015)., Google ScholarCrossref
  20. 20. J. Bavaresco et al., “Most incompatible measurements for robust steering tests,” Phys. Rev. A 96, 022110 (2017)., Google Scholar
  21. 21. A. Acín, N. Gisin, and B. Toner, “Grothendieck’s constant and local models for noisy entangled quantum states,” Phys. Rev. A 73, 062105 (2006)., Google ScholarCrossref
  22. 22. A. Grothendieck, “Résumé de la théorie métrique des produits tensoriels topologiques,” Bol. Soc. Mat. São Paulo 8, 1–79 (1953). Google Scholar
  23. 23. S. Brierley, M. Navascués, and T. Vértesi, “Convex separation from convex optimization for large-scale problems,” e-print arXiv:1609.05011 (2016). Google Scholar
  24. 24. F. Hirsch, M. T. Quintino, T. Vértesi, M. Navascués, and N. Brunner, “Better local hidden variable models for two-qubit Werner states and an upper bound on the Grothendieck constant KG(3),” Quantum 1, 3 (2017)., Google ScholarCrossref
  25. 25. J. Barrett, “Nonsequential positive-operator-valued measurements on entangled mixed states do not always violate a Bell inequality,” Phys. Rev. A 65, 042302 (2002)., Google ScholarCrossref
  1. © 2018 Author(s). Published by AIP Publishing.