Abstract

The question of which two-qubit states are steerable [i.e., permit a demonstration of Einstein–Podolsky–Rosen (EPR) steering] remains open. Here, a strong necessary condition is obtained for the steerability of two-qubit states having maximally mixed reduced states, via the construction of local hidden state models. It is conjectured that this condition is in fact sufficient. Two provably sufficient conditions are also obtained, via asymmetric EPR-steering inequalities. Our work uses ideas from the quantum steering ellipsoid formalism, and explicitly evaluates the integral of n/(nAn)2 over arbitrary unit hemispheres for any positive matrix A.

© 2015 Optical Society of America

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References

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  1. J. S. Bell, “On the Einstein–Podolsky–Rosen paradox,” Physics 1, 195–200 (1964). Reprinted in Ref. [24].
  2. R. F. Werner and M. M. Wolf, “Bell inequalities and entanglement,” Quantum Inform. Comput. 1, 1–25 (2001).
  3. J.-Å. Larsson, “Loopholes in Bell inequality tests of local realism,” J. Phys. A 47, 424003 (2014).
    [Crossref]
  4. J. S. Bell, “The theory of local beables,” Epistemological Lett. 9, 11–24 (1976). Reprinted in Ref. [24].
  5. H. M. Wiseman, “The two Bell’s theorems of John Bell,” J. Phys. A 47, 424001 (2014).
    [Crossref]
  6. B. M. Terhal, “Bell inequalities and the separability criterion,” Phys. Lett. A 271, 319–326 (2000).
    [Crossref]
  7. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
  8. E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935).
    [Crossref]
  9. 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).
    [Crossref]
  10. 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).
    [Crossref]
  11. E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
    [Crossref]
  12. F. Verstraete, “A study of entanglement in quantum information theory,” Ph.D. thesis (Katholieke Universiteit Leuven, 2002).
  13. M. Shi, F. Jiang, C. Sun, and J. Du, “Geometric picture of quantum discord for two-qubit quantum states,” New. J. Phys. 13, 073016 (2011).
    [Crossref]
  14. S. Jevtic, M. Pusey, D. Jennings, and T. Rudolph, “Quantum steering ellipsoids,” Phys. Rev. Lett. 113, 020402 (2014).
    [Crossref]
  15. R. Horodecki and M. Horodecki, “Information-theoretic aspects of inseparability of mixed states,” Phys. Rev. A 54, 1838–1843 (1996).
    [Crossref]
  16. 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).
    [Crossref]
  17. A. Acín, N. Gisin, and L. Masanes, “From Bell’s theorem to secure quantum key distribution,” Phys. Rev. Lett. 97, 120405 (2006).
    [Crossref]
  18. J. Barrett, L. Hardy, and A. Kent, “No signalling and quantum key distribution,” Phys. Rev. Lett. 95, 010503 (2005).
    [Crossref]
  19. A. J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein–Podolsky–Rosen steering allowing a demonstration over 1  km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).
  20. 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).
    [Crossref]
  21. For ellipsoids of separable states, there is a further ambiguity in the “chirality” of Alice’s local basis; that is, we may determine ρ up to a local unitary and a partial transpose on Alice’s system [14].
  22. J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014).
    [Crossref]
  23. R. F. Werner, “Steering, or maybe why Einstein did not go all the way to Bell’s argument,” J. Phys. A 47, 424008 (2014).
    [Crossref]
  24. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics,2nd ed. (Cambridge University, 2004).

2014 (5)

J.-Å. Larsson, “Loopholes in Bell inequality tests of local realism,” J. Phys. A 47, 424003 (2014).
[Crossref]

H. M. Wiseman, “The two Bell’s theorems of John Bell,” J. Phys. A 47, 424001 (2014).
[Crossref]

S. Jevtic, M. Pusey, D. Jennings, and T. Rudolph, “Quantum steering ellipsoids,” Phys. Rev. Lett. 113, 020402 (2014).
[Crossref]

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

R. F. Werner, “Steering, or maybe why Einstein did not go all the way to Bell’s argument,” J. Phys. A 47, 424008 (2014).
[Crossref]

2012 (2)

A. J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein–Podolsky–Rosen steering allowing a demonstration over 1  km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).

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).
[Crossref]

2011 (2)

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).
[Crossref]

M. Shi, F. Jiang, C. Sun, and J. Du, “Geometric picture of quantum discord for two-qubit quantum states,” New. J. Phys. 13, 073016 (2011).
[Crossref]

2009 (1)

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[Crossref]

2007 (2)

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).
[Crossref]

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).
[Crossref]

2006 (1)

A. Acín, N. Gisin, and L. Masanes, “From Bell’s theorem to secure quantum key distribution,” Phys. Rev. Lett. 97, 120405 (2006).
[Crossref]

2005 (1)

J. Barrett, L. Hardy, and A. Kent, “No signalling and quantum key distribution,” Phys. Rev. Lett. 95, 010503 (2005).
[Crossref]

2001 (1)

R. F. Werner and M. M. Wolf, “Bell inequalities and entanglement,” Quantum Inform. Comput. 1, 1–25 (2001).

2000 (1)

B. M. Terhal, “Bell inequalities and the separability criterion,” Phys. Lett. A 271, 319–326 (2000).
[Crossref]

1996 (1)

R. Horodecki and M. Horodecki, “Information-theoretic aspects of inseparability of mixed states,” Phys. Rev. A 54, 1838–1843 (1996).
[Crossref]

1976 (1)

J. S. Bell, “The theory of local beables,” Epistemological Lett. 9, 11–24 (1976). Reprinted in Ref. [24].

1964 (1)

J. S. Bell, “On the Einstein–Podolsky–Rosen paradox,” Physics 1, 195–200 (1964). Reprinted in Ref. [24].

1935 (2)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).

E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935).
[Crossref]

Acín, A.

A. Acín, N. Gisin, and L. Masanes, “From Bell’s theorem to secure quantum key distribution,” Phys. Rev. Lett. 97, 120405 (2006).
[Crossref]

Barrett, J.

J. Barrett, L. Hardy, and A. Kent, “No signalling and quantum key distribution,” Phys. Rev. Lett. 95, 010503 (2005).
[Crossref]

Bell, J. S.

J. S. Bell, “The theory of local beables,” Epistemological Lett. 9, 11–24 (1976). Reprinted in Ref. [24].

J. S. Bell, “On the Einstein–Podolsky–Rosen paradox,” Physics 1, 195–200 (1964). Reprinted in Ref. [24].

J. S. Bell, Speakable and Unspeakable in Quantum Mechanics,2nd ed. (Cambridge University, 2004).

Bennet, A. J.

A. J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein–Podolsky–Rosen steering allowing a demonstration over 1  km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).

Bowles, J.

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

Branciard, C.

A. J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein–Podolsky–Rosen steering allowing a demonstration over 1  km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).

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).
[Crossref]

Brunner, N.

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

Cavalcanti, E. G.

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).
[Crossref]

A. J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein–Podolsky–Rosen steering allowing a demonstration over 1  km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[Crossref]

Doherty, A. C.

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).
[Crossref]

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).
[Crossref]

Du, J.

M. Shi, F. Jiang, C. Sun, and J. Du, “Geometric picture of quantum discord for two-qubit quantum states,” New. J. Phys. 13, 073016 (2011).
[Crossref]

Einstein, A.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).

Evans, D. A.

A. J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein–Podolsky–Rosen steering allowing a demonstration over 1  km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).

Gisin, N.

A. Acín, N. Gisin, and L. Masanes, “From Bell’s theorem to secure quantum key distribution,” Phys. Rev. Lett. 97, 120405 (2006).
[Crossref]

Hardy, L.

J. Barrett, L. Hardy, and A. Kent, “No signalling and quantum key distribution,” Phys. Rev. Lett. 95, 010503 (2005).
[Crossref]

Horodecki, M.

R. Horodecki and M. Horodecki, “Information-theoretic aspects of inseparability of mixed states,” Phys. Rev. A 54, 1838–1843 (1996).
[Crossref]

Horodecki, R.

R. Horodecki and M. Horodecki, “Information-theoretic aspects of inseparability of mixed states,” Phys. Rev. A 54, 1838–1843 (1996).
[Crossref]

Jennings, D.

S. Jevtic, M. Pusey, D. Jennings, and T. Rudolph, “Quantum steering ellipsoids,” Phys. Rev. Lett. 113, 020402 (2014).
[Crossref]

Jevtic, S.

S. Jevtic, M. Pusey, D. Jennings, and T. Rudolph, “Quantum steering ellipsoids,” Phys. Rev. Lett. 113, 020402 (2014).
[Crossref]

Jiang, F.

M. Shi, F. Jiang, C. Sun, and J. Du, “Geometric picture of quantum discord for two-qubit quantum states,” New. J. Phys. 13, 073016 (2011).
[Crossref]

Jones, S. J.

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).
[Crossref]

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[Crossref]

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).
[Crossref]

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).
[Crossref]

Kent, A.

J. Barrett, L. Hardy, and A. Kent, “No signalling and quantum key distribution,” Phys. Rev. Lett. 95, 010503 (2005).
[Crossref]

Larsson, J.-Å.

J.-Å. Larsson, “Loopholes in Bell inequality tests of local realism,” J. Phys. A 47, 424003 (2014).
[Crossref]

Masanes, L.

A. Acín, N. Gisin, and L. Masanes, “From Bell’s theorem to secure quantum key distribution,” Phys. Rev. Lett. 97, 120405 (2006).
[Crossref]

Podolsky, B.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).

Pryde, G. J.

A. J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein–Podolsky–Rosen steering allowing a demonstration over 1  km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).

Pusey, M.

S. Jevtic, M. Pusey, D. Jennings, and T. Rudolph, “Quantum steering ellipsoids,” Phys. Rev. Lett. 113, 020402 (2014).
[Crossref]

Quintino, M. T.

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

Reid, M. D.

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[Crossref]

Rosen, N.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).

Rudolph, T.

S. Jevtic, M. Pusey, D. Jennings, and T. Rudolph, “Quantum steering ellipsoids,” Phys. Rev. Lett. 113, 020402 (2014).
[Crossref]

Saunders, D. J.

A. J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein–Podolsky–Rosen steering allowing a demonstration over 1  km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).

Scarani, V.

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).
[Crossref]

Schrödinger, E.

E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935).
[Crossref]

Shi, M.

M. Shi, F. Jiang, C. Sun, and J. Du, “Geometric picture of quantum discord for two-qubit quantum states,” New. J. Phys. 13, 073016 (2011).
[Crossref]

Sun, C.

M. Shi, F. Jiang, C. Sun, and J. Du, “Geometric picture of quantum discord for two-qubit quantum states,” New. J. Phys. 13, 073016 (2011).
[Crossref]

Terhal, B. M.

B. M. Terhal, “Bell inequalities and the separability criterion,” Phys. Lett. A 271, 319–326 (2000).
[Crossref]

Verstraete, F.

F. Verstraete, “A study of entanglement in quantum information theory,” Ph.D. thesis (Katholieke Universiteit Leuven, 2002).

Vértesi, T.

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

Walborn, S. P.

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).
[Crossref]

Werner, R. F.

R. F. Werner, “Steering, or maybe why Einstein did not go all the way to Bell’s argument,” J. Phys. A 47, 424008 (2014).
[Crossref]

R. F. Werner and M. M. Wolf, “Bell inequalities and entanglement,” Quantum Inform. Comput. 1, 1–25 (2001).

Wiseman, H. M.

H. M. Wiseman, “The two Bell’s theorems of John Bell,” J. Phys. A 47, 424001 (2014).
[Crossref]

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).
[Crossref]

A. J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein–Podolsky–Rosen steering allowing a demonstration over 1  km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).

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).
[Crossref]

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[Crossref]

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).
[Crossref]

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).
[Crossref]

Wolf, M. M.

R. F. Werner and M. M. Wolf, “Bell inequalities and entanglement,” Quantum Inform. Comput. 1, 1–25 (2001).

Epistemological Lett. (1)

J. S. Bell, “The theory of local beables,” Epistemological Lett. 9, 11–24 (1976). Reprinted in Ref. [24].

J. Phys. A (3)

H. M. Wiseman, “The two Bell’s theorems of John Bell,” J. Phys. A 47, 424001 (2014).
[Crossref]

J.-Å. Larsson, “Loopholes in Bell inequality tests of local realism,” J. Phys. A 47, 424003 (2014).
[Crossref]

R. F. Werner, “Steering, or maybe why Einstein did not go all the way to Bell’s argument,” J. Phys. A 47, 424008 (2014).
[Crossref]

New. J. Phys. (1)

M. Shi, F. Jiang, C. Sun, and J. Du, “Geometric picture of quantum discord for two-qubit quantum states,” New. J. Phys. 13, 073016 (2011).
[Crossref]

Phys. Lett. A (1)

B. M. Terhal, “Bell inequalities and the separability criterion,” Phys. Lett. A 271, 319–326 (2000).
[Crossref]

Phys. Rev. (1)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).

Phys. Rev. A (5)

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).
[Crossref]

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[Crossref]

R. Horodecki and M. Horodecki, “Information-theoretic aspects of inseparability of mixed states,” Phys. Rev. A 54, 1838–1843 (1996).
[Crossref]

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).
[Crossref]

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).
[Crossref]

Phys. Rev. Lett. (5)

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

A. Acín, N. Gisin, and L. Masanes, “From Bell’s theorem to secure quantum key distribution,” Phys. Rev. Lett. 97, 120405 (2006).
[Crossref]

J. Barrett, L. Hardy, and A. Kent, “No signalling and quantum key distribution,” Phys. Rev. Lett. 95, 010503 (2005).
[Crossref]

S. Jevtic, M. Pusey, D. Jennings, and T. Rudolph, “Quantum steering ellipsoids,” Phys. Rev. Lett. 113, 020402 (2014).
[Crossref]

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).
[Crossref]

Phys. Rev. X (1)

A. J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein–Podolsky–Rosen steering allowing a demonstration over 1  km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).

Physics (1)

J. S. Bell, “On the Einstein–Podolsky–Rosen paradox,” Physics 1, 195–200 (1964). Reprinted in Ref. [24].

Proc. Cambridge Philos. Soc. (1)

E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935).
[Crossref]

Quantum Inform. Comput. (1)

R. F. Werner and M. M. Wolf, “Bell inequalities and entanglement,” Quantum Inform. Comput. 1, 1–25 (2001).

Other (3)

F. Verstraete, “A study of entanglement in quantum information theory,” Ph.D. thesis (Katholieke Universiteit Leuven, 2002).

For ellipsoids of separable states, there is a further ambiguity in the “chirality” of Alice’s local basis; that is, we may determine ρ up to a local unitary and a partial transpose on Alice’s system [14].

J. S. Bell, Speakable and Unspeakable in Quantum Mechanics,2nd ed. (Cambridge University, 2004).

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Figures (1)

Fig. 1.
Fig. 1.

Correlation bounds for T-states, with s i = | t i | . (a) The red plane separates separable (left) and entangled (right) T-states. The sandwiched blue surface corresponds to the necessary condition for EPR steerability generated by our deterministic LHS model in Section 4.B: all T-states to the left of this surface are not EPR-steerable. We conjecture that this condition is also sufficient, i.e., that all states to the right of the blue surface are EPR-steerable. For comparison, the green plane corresponds to the sufficient condition for EPR steerability in Eq. (20) of Section 5.A: all T-states to the right of this surface are EPR-steerable. Only a portion of the surfaces are shown, as they are symmetric under permutations of s 1 , s 2 , s 3 . (b) Cross section through the top figure at s 1 = s 2 , where the necessary condition can be determined analytically (see Section 4.D). The additional black dashed curve corresponds to the nonlinear sufficient condition for EPR steerability in Eq. (22).

Equations (63)

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p ( a , b | A , B ) = λ P ( λ ) p ( a | A , λ ) p ( b | B , λ ) ,
p ( a , b | A , B ) = λ P ( λ ) p ( a | A , λ ) p Q ( b | B , λ ) ,
p Q ( b | B , λ ) = tr [ ρ B ( λ ) F b B ] .
p E ρ B E := tr A [ ρ E 1 ] = λ P ( λ ) p ( 1 | E , λ ) ρ B ( λ ) ,
p E = tr [ ρ E I ] = λ P ( λ ) p ( 1 | E , λ ) .
ρ = 1 4 ( 1 1 + a · σ 1 + 1 b · σ + j , k T j k σ j σ k ) .
a j = tr [ ρ σ j 1 ] , b j = tr [ ρ 1 σ j ] , T j k = tr [ ρ σ j σ k ] .
ρ B E = tr A [ ρ E 1 ] tr [ ρ E 1 ] .
b ( e ) = 1 2 p e ( b + T e ) ,
p e tr [ ρ ( E 1 ) ] = 1 2 ( 1 + a · e ) ,
c = b T a 1 a 2 ,
Q = 1 1 a 2 ( T b a ) ( 1 + a a 1 a 2 ) ( T a b ) .
ρ = 1 4 ( 1 1 + j t j σ j σ j )
f ( θ , ϕ ) 2 sin 2 θ cos 2 ϕ s 1 2 + sin 2 θ sin 2 ϕ s 2 2 + cos 2 θ s 3 2 .
p e = 1 / 2 , b ( e ) = T e = T e .
λ P ( λ ) p ( 1 | e , λ ) = 1 2 , λ P ( λ ) p ( 1 | e , λ ) n ( λ ) = 1 2 T e ,
P ( n ) p ( 1 | e , n ) d 2 n = 1 2 ,
P ( n ) p ( 1 | e , n ) n d 2 n = 1 2 T e ,
P ( n ) = N T [ f ( θ , ϕ ) ] m N T [ n T 2 n ] m / 2
N T R [ e ] [ n T 2 n ] m / 2 n d 2 n = 1 2 T e ,
m = 4 , R [ e ] = { n : n T 1 e 0 } ,
2 π N T | det T | = 1 .
n T 2 n d 2 n = 2 π .
s 3 = { [ 1 + arctan ( u 2 1 ) u 2 u 2 1 ] 1 u < 1 , [ 1 1 u 2 2 ( u 2 1 ) ln | 1 1 u 2 | 1 + 1 u 2 ] 1 u > 1 ,
C j k σ j σ k σ j 1 1 σ k = T j k a j b k ,
c 1 + c 2 + c 3 3 2 1 b 2 .
s 1 + s 2 + s 3 > 3 2
1 π π / 2 π / 2 A ϕ σ ϕ d ϕ 2 π [ p + 1 1 σ 3 + 2 + p 1 1 σ 3 2 ] ,
| t 1 | + | t 2 | 2 π [ ( 1 + a 3 ) 2 ( t 3 + b 3 ) 2 + ( 1 a 3 ) 2 ( t 3 b 3 ) 2 ] ,
f ( s 1 , s 2 , s 3 ) s 1 + s 2 4 π 1 s 3 2 0 .
max { f ( s 1 , s 2 , s 3 ) , f ( s 2 , s 3 , s 1 ) , f ( s 3 , s 1 , s 2 ) } > 0
n · v 0 n d 2 n ( n T 2 n ) 2 = π | det T | T 2 v | T v | .
d 2 n P ( n ) n · v 0 d 2 v v · n = π ,
Q = diag ( a , b , c ) = ( t 1 2 , t 2 2 , t 3 2 ) ,
q ( v ) n · v 0 n d 2 n ( n Q n ) 2 .
n = ( sin θ cos ϕ , sin θ sin ϕ , cos θ ) ,
v = ( sin α cos β , sin α sin β , cos α ) ,
R = ( cos β sin β 0 sin β cos β 0 0 0 1 ) ( cos α 0 sin α 0 1 0 sin α 0 cos α ) = ( cos α cos β sin β sin α cos β cos α sin β cos β sin α sin β sin α 0 cos α ) ,
( x y z ) = R ( cos γ sin γ 0 ) = ( cos α cos β cos γ sin β sin γ cos α sin β cos γ + cos β sin γ sin α cos γ ) .
cos γ = ± cos ( ϕ β ) [ cos 2 ( ϕ β ) + cos 2 α sin 2 ( ϕ β ) ] 1 / 2 .
cos χ = sin α cos ( ϕ β ) [ cos 2 ( ϕ β ) + cos 2 α sin 2 ( ϕ β ) ] 1 / 2 = sin α cos ( ϕ β ) [ cos 2 α + sin 2 α cos 2 ( ϕ β ) ] 1 / 2 .
sin χ = cos α [ cos 2 α + sin 2 α cos 2 ( ϕ β ) ] 1 / 2 ,
0 2 π 0 χ ( ϕ ) ( sin θ cos ϕ , sin θ sin ϕ , cos θ ) T sin θ d θ d ϕ ( a sin 2 θ cos 2 ϕ + b sin 2 θ sin 2 ϕ + c cos 2 θ ) 2 .
0 χ ( ϕ ) sin θ cos θ d θ ( a sin 2 θ cos 2 ϕ + b sin 2 θ sin 2 ϕ + c cos 2 θ ) 2 ,
1 2 c sin 2 χ a sin 2 χ cos 2 ϕ + b sin 2 χ sin 2 ϕ + c cos 2 χ .
0 χ ( ϕ ) sin θ cos θ ( a sin 2 θ cos 2 ϕ + b sin 2 θ sin 2 ϕ + c cos 2 θ ) 2 d θ = 1 2 c cos 2 α a cos 2 α cos 2 ϕ + b cos 2 α sin 2 ϕ + c sin 2 α cos 2 ( ϕ β ) .
l = a cos 2 α + c sin 2 α cos 2 β , m = b cos 2 α + c sin 2 α sin 2 β , n = c sin 2 α sin β cos β ,
0 2 π 0 χ ( ϕ ) sin θ cos θ d θ d ϕ ( a sin 2 θ cos 2 ϕ + b sin 2 θ sin 2 ϕ + c cos 2 θ ) 2 = cos 2 α 2 c 0 2 π d ϕ l cos 2 ϕ + m sin 2 ϕ + 2 n sin ϕ cos ϕ = ± cos 2 α 2 c 2 π l m n 2 .
[ q ( v ) ] 3 = π cos α c [ a b cos 2 α + c ( a sin 2 β + b cos 2 β ) sin 2 α ] 1 / 2 .
α 0 2 π 0 χ ( ϕ ) ( cos ϕ , sin ϕ ) sin 2 θ d θ d ϕ ( a sin 2 θ cos 2 ϕ + b sin 2 θ sin 2 ϕ + c cos 2 θ ) 2 = 0 2 π ( cos ϕ , sin ϕ ) sin 2 χ ( a sin 2 χ cos 2 ϕ + b sin 2 χ sin 2 ϕ + c cos 2 χ ) 2 χ α d ϕ ,
sin χ χ α = α ( sin α cos ( ϕ β ) [ cos 2 α + sin 2 α cos 2 ( ϕ β ) ] 1 / 2 ) = cos α cos ( ϕ β ) [ cos 2 α + sin 2 α cos 2 ( ϕ β ) ] 3 / 2 .
α 0 2 π 0 χ ( ϕ ) ( cos ϕ , sin ϕ ) sin 2 θ d θ d ϕ ( a sin 2 θ cos 2 ϕ + b sin 2 θ sin 2 ϕ + c cos 2 θ ) 2 = cos 2 α 0 2 π ( cos ϕ , sin ϕ ) cos ( ϕ β ) [ a cos 2 ϕ cos 2 α + b sin 2 ϕ cos 2 α + c sin 2 α cos 2 ( ϕ β ) ] 2 d ϕ = cos 2 α 0 2 π ( sin β sin ϕ cos ϕ + cos β cos 2 ϕ , sin β sin 2 ϕ + cos β sin ϕ cos ϕ ) ( l cos 2 ϕ + m sin 2 ϕ + 2 n sin ϕ cos ϕ ) 2 d ϕ .
0 2 π ( sin 2 ϕ , cos 2 ϕ , sin ϕ cos ϕ ) d ϕ ( l cos 2 ϕ + m sin 2 ϕ + 2 n sin ϕ cos ϕ ) 2 = π ( l , m , n ) ( l m n 2 ) 3 / 2 .
[ q ( v ) ] 1 = π cos 2 α ( m cos β n sin β ) ( l m n 2 ) 3 / 2 d α = a 1 π sin α cos β [ a b cos 2 α + c sin 2 α ( b cos 2 β + a sin 2 β ) ] 1 / 2 ,
[ q ( v ) ] 2 = π cos 2 α ( l sin β n cos β ) ( l m n 2 ) 3 / 2 d α = b 1 π sin α sin β [ a b cos 2 α + c sin 2 α ( b cos 2 β + a sin 2 β ) ] 1 / 2 .
q ( v ) = π Q 1 v a b c ( v Q 1 v ) ,
N T 1 = n · n = 1 ( n T 2 n ) 2 d 2 n = 2 π a b c ( a + b ) ( b + c ) ( c 2 a 2 ) × ( X + Y { b ( c a ) E [ C ] + a ( b + c ) K [ C ] + i b ( c a ) ( E [ A 1 , B ] E [ A 2 , B ] ) + i c ( a + b ) ( F [ A 1 , B ] F [ A 2 , B ] ) } ) ,
A 1 = i arccsch ( a c 2 a 2 ) , A 2 = i ln ( b + c c 2 b 2 ) , B = a 2 ( c 2 b 2 ) b 2 ( c 2 a 2 ) , C = c 2 ( b 2 a 2 ) b 2 ( c 2 a 2 ) , X = c ( c a ) [ ( a + c ) ( b + c ) + a b ] , Y = ( a + b + c ) c 2 a 2 .
C ( v ) A v v · σ A v v · σ .
λ p ( λ ) { S + ( λ ) d 2 v [ α v ( λ ) α ¯ v ] [ n ( λ ) b ] · v + S ( λ ) d 2 v [ α v ( λ ) α ¯ v ] [ n ( λ ) b ] · v } λ p ( λ ) { S + ( λ ) d 2 v [ 1 α ¯ v ] [ n ( λ ) b ] · v S ( λ ) d 2 v [ 1 + α ¯ v ] [ n ( λ ) b ] · v } = λ p ( λ ) d 2 v | [ n ( λ ) b ] · v | λ p ( λ ) d 2 v α ¯ v [ n ( λ ) b ] · v = λ p ( λ ) | n ( λ ) b | d 2 v | v · w ( λ ) | ,
1 4 π d 2 v C ( v ) 1 2 λ p ( λ ) | n ( λ ) b | 1 2 [ λ p ( λ ) | n ( λ ) b | 2 ] 1 / 2 1 2 1 b · b ,
d 2 v C ( v ) = j , k C j k sign ( C j j ) d 2 v v j v k = j , k C j k sign ( C j j ) 4 π 3 δ j k = 4 π 3 j | C j j | .
j | C j j | 3 2 1 b · b .

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