Abstract

Entanglement is known to be an essential resource for many quantum information processes. However, it is now known that some quantum features may be acheived with quantum discord, a generalized measure of quantum correlation. In this paper, we study how quantum discord, or more specifically, the measures of entropic discord and geometric discord are affected by the influence of amplitude damping decoherence. We also show that a protocol deploying weak measurement and quantum measurement reversal can effectively protect quantum discord from amplitude damping decoherence, enabling to distribute quantum correlation between two remote parties in a noisy environment.

© 2015 Optical Society of America

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References

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  1. M. Nielson and I. Chuang, Quantum Computation and Quantum Information (Cambridge Univ. Press, 2000).
  2. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?”; Phys. Rev. 47, 777–780 (1935).
    [Crossref]
  3. H. Ollivier and W. H. Zurek, “Quantum discord: a measure of the quantumness of correlations,” Phy. Rev. Lett. 88, 017901 (2001).
    [Crossref]
  4. D. Cavalcanti, L. Aolita, S. Boixo, K. Modi, M. Piani, and A. Winter, “Operational interpretations of quantum discord,” Phys. Rev. A 83, 032324 (2011).
    [Crossref]
  5. M. Gu, “Observing the operational significance of discord consumption,” Nature Physics 8, 671–675 (2012).
    [Crossref]
  6. A. Datta, A. Shaji, and C. M. Caves, “Quantum discord and the power of one qubit,” Phys. Rev. Lett. 100, 050502 (2008).
    [Crossref] [PubMed]
  7. B. Dakic, Y. O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, C. Brukner, and P. Walther, “Quantum discord as resource for remote state preparation,” Nature Physics 8, 666–670 (2012).
    [Crossref]
  8. S. Pirandola, “Quantum discord as a resource for quantum cryptography,” Sci. Rep. 4, 6956 (2014).
    [Crossref] [PubMed]
  9. C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
    [Crossref] [PubMed]
  10. P. G. Kwiat, S. Barraza-Lopez, A. Stefanov, and N. Gisin, “Experimental entanglement distillation and ‘hidden’ non-locality,” Nature 409, 1014–1017 (2001).
    [Crossref] [PubMed]
  11. J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417–422 (2003).
    [Crossref] [PubMed]
  12. R. Dong, M. Lassen, J. Heersink, C. Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, “Experimental entanglement distillation of mesoscopic quantum states,” Nature Phys. 4, 919–923 (2008).
    [Crossref]
  13. D. A. Lidar, I. Chuang, and K. B. Whaley, “Decoherence-free subspaces for quantum computation,” Phys. Rev. Lett. 81, 2594–2597 (1998).
    [Crossref]
  14. P. G. Kwiat, A. J. Berglund, J. B. Alterpeter, and A. G. White, “Experimental verification of decoherence-free subspaces,” Science 290, 498–501 (2000).
    [Crossref] [PubMed]
  15. S. Maniscalco, F. Francica, R. L. Zaffino, N. Lo Gullo, and F. Plastina, “Protecting entanglement via the quantum Zeno effect,” Phys. Rev. Lett. 100, 090503 (2008).
    [Crossref] [PubMed]
  16. Y.-S. Kim, J-C. Lee, Osung Kwon, and Y-H. Kim, “Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nat. Phys. 8, 117–120 (2012).
    [Crossref]
  17. J.-C. Lee, Y.-C. Jeong, Y.-S. Kim, and Y.-H. Kim, “Experimental demonstration of decoherence suppression via quantum measurement reversal,” Opt. Express 19, 16309–16316 (2011).
    [Crossref] [PubMed]
  18. J.-C. Lee, H.-T. Lim, K.-H. Hong, Y.-C. Jeong, M.S. Kim, and Y.-H. Kim, “Experimental demonstration of delayed-choice decoherence suppression,” Nat. Commun. 5, 4522 (2014).
    [PubMed]
  19. H.-T. Lim, J.-C. Lee, K.-H. Hong, and Y.-H. Kim, “Avoiding entanglement sudden death using single-qubit quantum measurement reversal,” Opt. Express 22, 19055–19068 (2014).
    [Crossref] [PubMed]
  20. L. Henderson and V. Vedral, “Classical, quantum and total correlations,” J. Phys. A 34, 6899 (2001).
    [Crossref]
  21. S. Wu, U. V. Poulsen, and K. Molmer, “Correlations in local measurements on a quantum state, and complementarity as an explanation of nonclassicality,” Phys. Rev. A 80, 032319 (2009).
    [Crossref]
  22. K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: discord and related measures,” Rev. Mod. Phys. 84, 1655–1707 (2012).
    [Crossref]
  23. Y. Huang, “Computing quantum discord is NP-complete,” New J. Phys. 16, 033027 (2014).
    [Crossref]
  24. B. Dakic, V. Vedral, and C. Brukner, “Necessary and sufficient condition for nonzero quantum discord,” Phys. Rev. Lett. 105, 190502 (2010).
    [Crossref]
  25. S. Luo and S. Fu, “Geometric measure of quantum discord,” Phys. Rev. A 82, 034302 (2010).
    [Crossref]
  26. F. M. Paula, T. R. de Oliveira, and M. S. Sarandy, “Geometric quantum discord through Schatten 1-norm,” Phys. Rev. A 87, 064101 (2013).
    [Crossref]
  27. R. A. Bertlmann and P. Krammer, “Bloch vectors for qudits,” J. Phys. A: Math. Theor. 41, 235303 (2008).
    [Crossref]
  28. Y.-S. Kim, Y.-W. Cho, Y.-S. Ra, and Y.-H. Kim, “Reversing the weak quantum measurement for a photonic qubit,” Opt. Express 17, 11978–11985 (2009).
    [Crossref] [PubMed]
  29. C. Benedetti, A. P. Shurupov, M. G. A. Paris, G. Brida, and M. Genovese, “Experimental estimation of quantum discord for a polarization qubit and the use of fidelity to assess quantum correlations,” Phys. Rev. A 87, 052136 (2013).
    [Crossref]
  30. Y. W. Cheong and S.-W. Lee, “Balance between information gain and reversibility in weak measurement,” Phys. Rev. Lett. 109, 150402 (2012).
    [Crossref] [PubMed]
  31. H.-T. Lim, Y.-S. Ra, K.-H. Hong, S.-W. Lee, and Y.-H. Kim, “Fundamental bounds in measurements for estimating quantum states,” Phys. Rev. Lett. 113, 020504 (2014).
    [Crossref] [PubMed]

2014 (5)

S. Pirandola, “Quantum discord as a resource for quantum cryptography,” Sci. Rep. 4, 6956 (2014).
[Crossref] [PubMed]

J.-C. Lee, H.-T. Lim, K.-H. Hong, Y.-C. Jeong, M.S. Kim, and Y.-H. Kim, “Experimental demonstration of delayed-choice decoherence suppression,” Nat. Commun. 5, 4522 (2014).
[PubMed]

H.-T. Lim, J.-C. Lee, K.-H. Hong, and Y.-H. Kim, “Avoiding entanglement sudden death using single-qubit quantum measurement reversal,” Opt. Express 22, 19055–19068 (2014).
[Crossref] [PubMed]

Y. Huang, “Computing quantum discord is NP-complete,” New J. Phys. 16, 033027 (2014).
[Crossref]

H.-T. Lim, Y.-S. Ra, K.-H. Hong, S.-W. Lee, and Y.-H. Kim, “Fundamental bounds in measurements for estimating quantum states,” Phys. Rev. Lett. 113, 020504 (2014).
[Crossref] [PubMed]

2013 (2)

C. Benedetti, A. P. Shurupov, M. G. A. Paris, G. Brida, and M. Genovese, “Experimental estimation of quantum discord for a polarization qubit and the use of fidelity to assess quantum correlations,” Phys. Rev. A 87, 052136 (2013).
[Crossref]

F. M. Paula, T. R. de Oliveira, and M. S. Sarandy, “Geometric quantum discord through Schatten 1-norm,” Phys. Rev. A 87, 064101 (2013).
[Crossref]

2012 (5)

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: discord and related measures,” Rev. Mod. Phys. 84, 1655–1707 (2012).
[Crossref]

Y. W. Cheong and S.-W. Lee, “Balance between information gain and reversibility in weak measurement,” Phys. Rev. Lett. 109, 150402 (2012).
[Crossref] [PubMed]

B. Dakic, Y. O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, C. Brukner, and P. Walther, “Quantum discord as resource for remote state preparation,” Nature Physics 8, 666–670 (2012).
[Crossref]

M. Gu, “Observing the operational significance of discord consumption,” Nature Physics 8, 671–675 (2012).
[Crossref]

Y.-S. Kim, J-C. Lee, Osung Kwon, and Y-H. Kim, “Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nat. Phys. 8, 117–120 (2012).
[Crossref]

2011 (2)

J.-C. Lee, Y.-C. Jeong, Y.-S. Kim, and Y.-H. Kim, “Experimental demonstration of decoherence suppression via quantum measurement reversal,” Opt. Express 19, 16309–16316 (2011).
[Crossref] [PubMed]

D. Cavalcanti, L. Aolita, S. Boixo, K. Modi, M. Piani, and A. Winter, “Operational interpretations of quantum discord,” Phys. Rev. A 83, 032324 (2011).
[Crossref]

2010 (2)

B. Dakic, V. Vedral, and C. Brukner, “Necessary and sufficient condition for nonzero quantum discord,” Phys. Rev. Lett. 105, 190502 (2010).
[Crossref]

S. Luo and S. Fu, “Geometric measure of quantum discord,” Phys. Rev. A 82, 034302 (2010).
[Crossref]

2009 (2)

S. Wu, U. V. Poulsen, and K. Molmer, “Correlations in local measurements on a quantum state, and complementarity as an explanation of nonclassicality,” Phys. Rev. A 80, 032319 (2009).
[Crossref]

Y.-S. Kim, Y.-W. Cho, Y.-S. Ra, and Y.-H. Kim, “Reversing the weak quantum measurement for a photonic qubit,” Opt. Express 17, 11978–11985 (2009).
[Crossref] [PubMed]

2008 (4)

R. A. Bertlmann and P. Krammer, “Bloch vectors for qudits,” J. Phys. A: Math. Theor. 41, 235303 (2008).
[Crossref]

A. Datta, A. Shaji, and C. M. Caves, “Quantum discord and the power of one qubit,” Phys. Rev. Lett. 100, 050502 (2008).
[Crossref] [PubMed]

S. Maniscalco, F. Francica, R. L. Zaffino, N. Lo Gullo, and F. Plastina, “Protecting entanglement via the quantum Zeno effect,” Phys. Rev. Lett. 100, 090503 (2008).
[Crossref] [PubMed]

R. Dong, M. Lassen, J. Heersink, C. Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, “Experimental entanglement distillation of mesoscopic quantum states,” Nature Phys. 4, 919–923 (2008).
[Crossref]

2003 (1)

J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417–422 (2003).
[Crossref] [PubMed]

2001 (3)

P. G. Kwiat, S. Barraza-Lopez, A. Stefanov, and N. Gisin, “Experimental entanglement distillation and ‘hidden’ non-locality,” Nature 409, 1014–1017 (2001).
[Crossref] [PubMed]

H. Ollivier and W. H. Zurek, “Quantum discord: a measure of the quantumness of correlations,” Phy. Rev. Lett. 88, 017901 (2001).
[Crossref]

L. Henderson and V. Vedral, “Classical, quantum and total correlations,” J. Phys. A 34, 6899 (2001).
[Crossref]

2000 (1)

P. G. Kwiat, A. J. Berglund, J. B. Alterpeter, and A. G. White, “Experimental verification of decoherence-free subspaces,” Science 290, 498–501 (2000).
[Crossref] [PubMed]

1998 (1)

D. A. Lidar, I. Chuang, and K. B. Whaley, “Decoherence-free subspaces for quantum computation,” Phys. Rev. Lett. 81, 2594–2597 (1998).
[Crossref]

1996 (1)

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[Crossref] [PubMed]

1935 (1)

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

Alterpeter, J. B.

P. G. Kwiat, A. J. Berglund, J. B. Alterpeter, and A. G. White, “Experimental verification of decoherence-free subspaces,” Science 290, 498–501 (2000).
[Crossref] [PubMed]

Andersen, U. L.

R. Dong, M. Lassen, J. Heersink, C. Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, “Experimental entanglement distillation of mesoscopic quantum states,” Nature Phys. 4, 919–923 (2008).
[Crossref]

Aolita, L.

D. Cavalcanti, L. Aolita, S. Boixo, K. Modi, M. Piani, and A. Winter, “Operational interpretations of quantum discord,” Phys. Rev. A 83, 032324 (2011).
[Crossref]

Barraza-Lopez, S.

P. G. Kwiat, S. Barraza-Lopez, A. Stefanov, and N. Gisin, “Experimental entanglement distillation and ‘hidden’ non-locality,” Nature 409, 1014–1017 (2001).
[Crossref] [PubMed]

Barz, S.

B. Dakic, Y. O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, C. Brukner, and P. Walther, “Quantum discord as resource for remote state preparation,” Nature Physics 8, 666–670 (2012).
[Crossref]

Benedetti, C.

C. Benedetti, A. P. Shurupov, M. G. A. Paris, G. Brida, and M. Genovese, “Experimental estimation of quantum discord for a polarization qubit and the use of fidelity to assess quantum correlations,” Phys. Rev. A 87, 052136 (2013).
[Crossref]

Bennett, C. H.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[Crossref] [PubMed]

Berglund, A. J.

P. G. Kwiat, A. J. Berglund, J. B. Alterpeter, and A. G. White, “Experimental verification of decoherence-free subspaces,” Science 290, 498–501 (2000).
[Crossref] [PubMed]

Bertlmann, R. A.

R. A. Bertlmann and P. Krammer, “Bloch vectors for qudits,” J. Phys. A: Math. Theor. 41, 235303 (2008).
[Crossref]

Boixo, S.

D. Cavalcanti, L. Aolita, S. Boixo, K. Modi, M. Piani, and A. Winter, “Operational interpretations of quantum discord,” Phys. Rev. A 83, 032324 (2011).
[Crossref]

Brassard, G.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[Crossref] [PubMed]

Brida, G.

C. Benedetti, A. P. Shurupov, M. G. A. Paris, G. Brida, and M. Genovese, “Experimental estimation of quantum discord for a polarization qubit and the use of fidelity to assess quantum correlations,” Phys. Rev. A 87, 052136 (2013).
[Crossref]

Brodutch, A.

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: discord and related measures,” Rev. Mod. Phys. 84, 1655–1707 (2012).
[Crossref]

Brukner, C.

B. Dakic, Y. O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, C. Brukner, and P. Walther, “Quantum discord as resource for remote state preparation,” Nature Physics 8, 666–670 (2012).
[Crossref]

B. Dakic, V. Vedral, and C. Brukner, “Necessary and sufficient condition for nonzero quantum discord,” Phys. Rev. Lett. 105, 190502 (2010).
[Crossref]

Cable, H.

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: discord and related measures,” Rev. Mod. Phys. 84, 1655–1707 (2012).
[Crossref]

Cavalcanti, D.

D. Cavalcanti, L. Aolita, S. Boixo, K. Modi, M. Piani, and A. Winter, “Operational interpretations of quantum discord,” Phys. Rev. A 83, 032324 (2011).
[Crossref]

Caves, C. M.

A. Datta, A. Shaji, and C. M. Caves, “Quantum discord and the power of one qubit,” Phys. Rev. Lett. 100, 050502 (2008).
[Crossref] [PubMed]

Cheong, Y. W.

Y. W. Cheong and S.-W. Lee, “Balance between information gain and reversibility in weak measurement,” Phys. Rev. Lett. 109, 150402 (2012).
[Crossref] [PubMed]

Cho, Y.-W.

Chuang, I.

D. A. Lidar, I. Chuang, and K. B. Whaley, “Decoherence-free subspaces for quantum computation,” Phys. Rev. Lett. 81, 2594–2597 (1998).
[Crossref]

M. Nielson and I. Chuang, Quantum Computation and Quantum Information (Cambridge Univ. Press, 2000).

Dakic, B.

B. Dakic, Y. O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, C. Brukner, and P. Walther, “Quantum discord as resource for remote state preparation,” Nature Physics 8, 666–670 (2012).
[Crossref]

B. Dakic, V. Vedral, and C. Brukner, “Necessary and sufficient condition for nonzero quantum discord,” Phys. Rev. Lett. 105, 190502 (2010).
[Crossref]

Datta, A.

A. Datta, A. Shaji, and C. M. Caves, “Quantum discord and the power of one qubit,” Phys. Rev. Lett. 100, 050502 (2008).
[Crossref] [PubMed]

de Oliveira, T. R.

F. M. Paula, T. R. de Oliveira, and M. S. Sarandy, “Geometric quantum discord through Schatten 1-norm,” Phys. Rev. A 87, 064101 (2013).
[Crossref]

Dong, R.

R. Dong, M. Lassen, J. Heersink, C. Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, “Experimental entanglement distillation of mesoscopic quantum states,” Nature Phys. 4, 919–923 (2008).
[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).
[Crossref]

Filip, R.

R. Dong, M. Lassen, J. Heersink, C. Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, “Experimental entanglement distillation of mesoscopic quantum states,” Nature Phys. 4, 919–923 (2008).
[Crossref]

Francica, F.

S. Maniscalco, F. Francica, R. L. Zaffino, N. Lo Gullo, and F. Plastina, “Protecting entanglement via the quantum Zeno effect,” Phys. Rev. Lett. 100, 090503 (2008).
[Crossref] [PubMed]

Fu, S.

S. Luo and S. Fu, “Geometric measure of quantum discord,” Phys. Rev. A 82, 034302 (2010).
[Crossref]

Gasparoni, S.

J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417–422 (2003).
[Crossref] [PubMed]

Genovese, M.

C. Benedetti, A. P. Shurupov, M. G. A. Paris, G. Brida, and M. Genovese, “Experimental estimation of quantum discord for a polarization qubit and the use of fidelity to assess quantum correlations,” Phys. Rev. A 87, 052136 (2013).
[Crossref]

Gisin, N.

P. G. Kwiat, S. Barraza-Lopez, A. Stefanov, and N. Gisin, “Experimental entanglement distillation and ‘hidden’ non-locality,” Nature 409, 1014–1017 (2001).
[Crossref] [PubMed]

Gu, M.

M. Gu, “Observing the operational significance of discord consumption,” Nature Physics 8, 671–675 (2012).
[Crossref]

Gullo, N. Lo

S. Maniscalco, F. Francica, R. L. Zaffino, N. Lo Gullo, and F. Plastina, “Protecting entanglement via the quantum Zeno effect,” Phys. Rev. Lett. 100, 090503 (2008).
[Crossref] [PubMed]

Heersink, J.

R. Dong, M. Lassen, J. Heersink, C. Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, “Experimental entanglement distillation of mesoscopic quantum states,” Nature Phys. 4, 919–923 (2008).
[Crossref]

Henderson, L.

L. Henderson and V. Vedral, “Classical, quantum and total correlations,” J. Phys. A 34, 6899 (2001).
[Crossref]

Hong, K.-H.

H.-T. Lim, J.-C. Lee, K.-H. Hong, and Y.-H. Kim, “Avoiding entanglement sudden death using single-qubit quantum measurement reversal,” Opt. Express 22, 19055–19068 (2014).
[Crossref] [PubMed]

J.-C. Lee, H.-T. Lim, K.-H. Hong, Y.-C. Jeong, M.S. Kim, and Y.-H. Kim, “Experimental demonstration of delayed-choice decoherence suppression,” Nat. Commun. 5, 4522 (2014).
[PubMed]

H.-T. Lim, Y.-S. Ra, K.-H. Hong, S.-W. Lee, and Y.-H. Kim, “Fundamental bounds in measurements for estimating quantum states,” Phys. Rev. Lett. 113, 020504 (2014).
[Crossref] [PubMed]

Huang, Y.

Y. Huang, “Computing quantum discord is NP-complete,” New J. Phys. 16, 033027 (2014).
[Crossref]

Jeong, Y.-C.

J.-C. Lee, H.-T. Lim, K.-H. Hong, Y.-C. Jeong, M.S. Kim, and Y.-H. Kim, “Experimental demonstration of delayed-choice decoherence suppression,” Nat. Commun. 5, 4522 (2014).
[PubMed]

J.-C. Lee, Y.-C. Jeong, Y.-S. Kim, and Y.-H. Kim, “Experimental demonstration of decoherence suppression via quantum measurement reversal,” Opt. Express 19, 16309–16316 (2011).
[Crossref] [PubMed]

Kim, M.S.

J.-C. Lee, H.-T. Lim, K.-H. Hong, Y.-C. Jeong, M.S. Kim, and Y.-H. Kim, “Experimental demonstration of delayed-choice decoherence suppression,” Nat. Commun. 5, 4522 (2014).
[PubMed]

Kim, Y.-H.

Kim, Y.-S.

Kim, Y-H.

Y.-S. Kim, J-C. Lee, Osung Kwon, and Y-H. Kim, “Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nat. Phys. 8, 117–120 (2012).
[Crossref]

Krammer, P.

R. A. Bertlmann and P. Krammer, “Bloch vectors for qudits,” J. Phys. A: Math. Theor. 41, 235303 (2008).
[Crossref]

Kropatschek, S.

B. Dakic, Y. O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, C. Brukner, and P. Walther, “Quantum discord as resource for remote state preparation,” Nature Physics 8, 666–670 (2012).
[Crossref]

Kwiat, P. G.

P. G. Kwiat, S. Barraza-Lopez, A. Stefanov, and N. Gisin, “Experimental entanglement distillation and ‘hidden’ non-locality,” Nature 409, 1014–1017 (2001).
[Crossref] [PubMed]

P. G. Kwiat, A. J. Berglund, J. B. Alterpeter, and A. G. White, “Experimental verification of decoherence-free subspaces,” Science 290, 498–501 (2000).
[Crossref] [PubMed]

Kwon, Osung

Y.-S. Kim, J-C. Lee, Osung Kwon, and Y-H. Kim, “Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nat. Phys. 8, 117–120 (2012).
[Crossref]

Lassen, M.

R. Dong, M. Lassen, J. Heersink, C. Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, “Experimental entanglement distillation of mesoscopic quantum states,” Nature Phys. 4, 919–923 (2008).
[Crossref]

Lee, J.-C.

Lee, J-C.

Y.-S. Kim, J-C. Lee, Osung Kwon, and Y-H. Kim, “Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nat. Phys. 8, 117–120 (2012).
[Crossref]

Lee, S.-W.

H.-T. Lim, Y.-S. Ra, K.-H. Hong, S.-W. Lee, and Y.-H. Kim, “Fundamental bounds in measurements for estimating quantum states,” Phys. Rev. Lett. 113, 020504 (2014).
[Crossref] [PubMed]

Y. W. Cheong and S.-W. Lee, “Balance between information gain and reversibility in weak measurement,” Phys. Rev. Lett. 109, 150402 (2012).
[Crossref] [PubMed]

Leuchs, G.

R. Dong, M. Lassen, J. Heersink, C. Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, “Experimental entanglement distillation of mesoscopic quantum states,” Nature Phys. 4, 919–923 (2008).
[Crossref]

Lidar, D. A.

D. A. Lidar, I. Chuang, and K. B. Whaley, “Decoherence-free subspaces for quantum computation,” Phys. Rev. Lett. 81, 2594–2597 (1998).
[Crossref]

Lim, H.-T.

H.-T. Lim, Y.-S. Ra, K.-H. Hong, S.-W. Lee, and Y.-H. Kim, “Fundamental bounds in measurements for estimating quantum states,” Phys. Rev. Lett. 113, 020504 (2014).
[Crossref] [PubMed]

J.-C. Lee, H.-T. Lim, K.-H. Hong, Y.-C. Jeong, M.S. Kim, and Y.-H. Kim, “Experimental demonstration of delayed-choice decoherence suppression,” Nat. Commun. 5, 4522 (2014).
[PubMed]

H.-T. Lim, J.-C. Lee, K.-H. Hong, and Y.-H. Kim, “Avoiding entanglement sudden death using single-qubit quantum measurement reversal,” Opt. Express 22, 19055–19068 (2014).
[Crossref] [PubMed]

Lipp, Y. O.

B. Dakic, Y. O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, C. Brukner, and P. Walther, “Quantum discord as resource for remote state preparation,” Nature Physics 8, 666–670 (2012).
[Crossref]

Luo, S.

S. Luo and S. Fu, “Geometric measure of quantum discord,” Phys. Rev. A 82, 034302 (2010).
[Crossref]

Ma, X.

B. Dakic, Y. O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, C. Brukner, and P. Walther, “Quantum discord as resource for remote state preparation,” Nature Physics 8, 666–670 (2012).
[Crossref]

Maniscalco, S.

S. Maniscalco, F. Francica, R. L. Zaffino, N. Lo Gullo, and F. Plastina, “Protecting entanglement via the quantum Zeno effect,” Phys. Rev. Lett. 100, 090503 (2008).
[Crossref] [PubMed]

Marquardt, C.

R. Dong, M. Lassen, J. Heersink, C. Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, “Experimental entanglement distillation of mesoscopic quantum states,” Nature Phys. 4, 919–923 (2008).
[Crossref]

Modi, K.

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: discord and related measures,” Rev. Mod. Phys. 84, 1655–1707 (2012).
[Crossref]

D. Cavalcanti, L. Aolita, S. Boixo, K. Modi, M. Piani, and A. Winter, “Operational interpretations of quantum discord,” Phys. Rev. A 83, 032324 (2011).
[Crossref]

Molmer, K.

S. Wu, U. V. Poulsen, and K. Molmer, “Correlations in local measurements on a quantum state, and complementarity as an explanation of nonclassicality,” Phys. Rev. A 80, 032319 (2009).
[Crossref]

Nielson, M.

M. Nielson and I. Chuang, Quantum Computation and Quantum Information (Cambridge Univ. Press, 2000).

Ollivier, H.

H. Ollivier and W. H. Zurek, “Quantum discord: a measure of the quantumness of correlations,” Phy. Rev. Lett. 88, 017901 (2001).
[Crossref]

Pan, J.-W.

J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417–422 (2003).
[Crossref] [PubMed]

Paris, M. G. A.

C. Benedetti, A. P. Shurupov, M. G. A. Paris, G. Brida, and M. Genovese, “Experimental estimation of quantum discord for a polarization qubit and the use of fidelity to assess quantum correlations,” Phys. Rev. A 87, 052136 (2013).
[Crossref]

Paterek, T.

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: discord and related measures,” Rev. Mod. Phys. 84, 1655–1707 (2012).
[Crossref]

B. Dakic, Y. O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, C. Brukner, and P. Walther, “Quantum discord as resource for remote state preparation,” Nature Physics 8, 666–670 (2012).
[Crossref]

Paula, F. M.

F. M. Paula, T. R. de Oliveira, and M. S. Sarandy, “Geometric quantum discord through Schatten 1-norm,” Phys. Rev. A 87, 064101 (2013).
[Crossref]

Piani, M.

D. Cavalcanti, L. Aolita, S. Boixo, K. Modi, M. Piani, and A. Winter, “Operational interpretations of quantum discord,” Phys. Rev. A 83, 032324 (2011).
[Crossref]

Pirandola, S.

S. Pirandola, “Quantum discord as a resource for quantum cryptography,” Sci. Rep. 4, 6956 (2014).
[Crossref] [PubMed]

Plastina, F.

S. Maniscalco, F. Francica, R. L. Zaffino, N. Lo Gullo, and F. Plastina, “Protecting entanglement via the quantum Zeno effect,” Phys. Rev. Lett. 100, 090503 (2008).
[Crossref] [PubMed]

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).
[Crossref]

Popescu, S.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[Crossref] [PubMed]

Poulsen, U. V.

S. Wu, U. V. Poulsen, and K. Molmer, “Correlations in local measurements on a quantum state, and complementarity as an explanation of nonclassicality,” Phys. Rev. A 80, 032319 (2009).
[Crossref]

Ra, Y.-S.

H.-T. Lim, Y.-S. Ra, K.-H. Hong, S.-W. Lee, and Y.-H. Kim, “Fundamental bounds in measurements for estimating quantum states,” Phys. Rev. Lett. 113, 020504 (2014).
[Crossref] [PubMed]

Y.-S. Kim, Y.-W. Cho, Y.-S. Ra, and Y.-H. Kim, “Reversing the weak quantum measurement for a photonic qubit,” Opt. Express 17, 11978–11985 (2009).
[Crossref] [PubMed]

Ringbauer, M.

B. Dakic, Y. O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, C. Brukner, and P. Walther, “Quantum discord as resource for remote state preparation,” Nature Physics 8, 666–670 (2012).
[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).
[Crossref]

Sarandy, M. S.

F. M. Paula, T. R. de Oliveira, and M. S. Sarandy, “Geometric quantum discord through Schatten 1-norm,” Phys. Rev. A 87, 064101 (2013).
[Crossref]

Schumacher, B.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[Crossref] [PubMed]

Shaji, A.

A. Datta, A. Shaji, and C. M. Caves, “Quantum discord and the power of one qubit,” Phys. Rev. Lett. 100, 050502 (2008).
[Crossref] [PubMed]

Shurupov, A. P.

C. Benedetti, A. P. Shurupov, M. G. A. Paris, G. Brida, and M. Genovese, “Experimental estimation of quantum discord for a polarization qubit and the use of fidelity to assess quantum correlations,” Phys. Rev. A 87, 052136 (2013).
[Crossref]

Smolin, J. A.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[Crossref] [PubMed]

Stefanov, A.

P. G. Kwiat, S. Barraza-Lopez, A. Stefanov, and N. Gisin, “Experimental entanglement distillation and ‘hidden’ non-locality,” Nature 409, 1014–1017 (2001).
[Crossref] [PubMed]

Ursin, R.

J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417–422 (2003).
[Crossref] [PubMed]

Vedral, V.

B. Dakic, Y. O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, C. Brukner, and P. Walther, “Quantum discord as resource for remote state preparation,” Nature Physics 8, 666–670 (2012).
[Crossref]

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: discord and related measures,” Rev. Mod. Phys. 84, 1655–1707 (2012).
[Crossref]

B. Dakic, V. Vedral, and C. Brukner, “Necessary and sufficient condition for nonzero quantum discord,” Phys. Rev. Lett. 105, 190502 (2010).
[Crossref]

L. Henderson and V. Vedral, “Classical, quantum and total correlations,” J. Phys. A 34, 6899 (2001).
[Crossref]

Walther, P.

B. Dakic, Y. O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, C. Brukner, and P. Walther, “Quantum discord as resource for remote state preparation,” Nature Physics 8, 666–670 (2012).
[Crossref]

Weihs, G.

J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417–422 (2003).
[Crossref] [PubMed]

Whaley, K. B.

D. A. Lidar, I. Chuang, and K. B. Whaley, “Decoherence-free subspaces for quantum computation,” Phys. Rev. Lett. 81, 2594–2597 (1998).
[Crossref]

White, A. G.

P. G. Kwiat, A. J. Berglund, J. B. Alterpeter, and A. G. White, “Experimental verification of decoherence-free subspaces,” Science 290, 498–501 (2000).
[Crossref] [PubMed]

Winter, A.

D. Cavalcanti, L. Aolita, S. Boixo, K. Modi, M. Piani, and A. Winter, “Operational interpretations of quantum discord,” Phys. Rev. A 83, 032324 (2011).
[Crossref]

Wootters, W. K.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[Crossref] [PubMed]

Wu, S.

S. Wu, U. V. Poulsen, and K. Molmer, “Correlations in local measurements on a quantum state, and complementarity as an explanation of nonclassicality,” Phys. Rev. A 80, 032319 (2009).
[Crossref]

Zaffino, R. L.

S. Maniscalco, F. Francica, R. L. Zaffino, N. Lo Gullo, and F. Plastina, “Protecting entanglement via the quantum Zeno effect,” Phys. Rev. Lett. 100, 090503 (2008).
[Crossref] [PubMed]

Zeilinger, A.

B. Dakic, Y. O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, C. Brukner, and P. Walther, “Quantum discord as resource for remote state preparation,” Nature Physics 8, 666–670 (2012).
[Crossref]

J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417–422 (2003).
[Crossref] [PubMed]

Zurek, W. H.

H. Ollivier and W. H. Zurek, “Quantum discord: a measure of the quantumness of correlations,” Phy. Rev. Lett. 88, 017901 (2001).
[Crossref]

J. Phys. A (1)

L. Henderson and V. Vedral, “Classical, quantum and total correlations,” J. Phys. A 34, 6899 (2001).
[Crossref]

J. Phys. A: Math. Theor. (1)

R. A. Bertlmann and P. Krammer, “Bloch vectors for qudits,” J. Phys. A: Math. Theor. 41, 235303 (2008).
[Crossref]

Nat. Commun. (1)

J.-C. Lee, H.-T. Lim, K.-H. Hong, Y.-C. Jeong, M.S. Kim, and Y.-H. Kim, “Experimental demonstration of delayed-choice decoherence suppression,” Nat. Commun. 5, 4522 (2014).
[PubMed]

Nat. Phys. (1)

Y.-S. Kim, J-C. Lee, Osung Kwon, and Y-H. Kim, “Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nat. Phys. 8, 117–120 (2012).
[Crossref]

Nature (2)

P. G. Kwiat, S. Barraza-Lopez, A. Stefanov, and N. Gisin, “Experimental entanglement distillation and ‘hidden’ non-locality,” Nature 409, 1014–1017 (2001).
[Crossref] [PubMed]

J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417–422 (2003).
[Crossref] [PubMed]

Nature Phys. (1)

R. Dong, M. Lassen, J. Heersink, C. Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, “Experimental entanglement distillation of mesoscopic quantum states,” Nature Phys. 4, 919–923 (2008).
[Crossref]

Nature Physics (2)

M. Gu, “Observing the operational significance of discord consumption,” Nature Physics 8, 671–675 (2012).
[Crossref]

B. Dakic, Y. O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, C. Brukner, and P. Walther, “Quantum discord as resource for remote state preparation,” Nature Physics 8, 666–670 (2012).
[Crossref]

New J. Phys. (1)

Y. Huang, “Computing quantum discord is NP-complete,” New J. Phys. 16, 033027 (2014).
[Crossref]

Opt. Express (3)

Phy. Rev. Lett. (1)

H. Ollivier and W. H. Zurek, “Quantum discord: a measure of the quantumness of correlations,” Phy. Rev. Lett. 88, 017901 (2001).
[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).
[Crossref]

Phys. Rev. A (5)

D. Cavalcanti, L. Aolita, S. Boixo, K. Modi, M. Piani, and A. Winter, “Operational interpretations of quantum discord,” Phys. Rev. A 83, 032324 (2011).
[Crossref]

S. Wu, U. V. Poulsen, and K. Molmer, “Correlations in local measurements on a quantum state, and complementarity as an explanation of nonclassicality,” Phys. Rev. A 80, 032319 (2009).
[Crossref]

S. Luo and S. Fu, “Geometric measure of quantum discord,” Phys. Rev. A 82, 034302 (2010).
[Crossref]

F. M. Paula, T. R. de Oliveira, and M. S. Sarandy, “Geometric quantum discord through Schatten 1-norm,” Phys. Rev. A 87, 064101 (2013).
[Crossref]

C. Benedetti, A. P. Shurupov, M. G. A. Paris, G. Brida, and M. Genovese, “Experimental estimation of quantum discord for a polarization qubit and the use of fidelity to assess quantum correlations,” Phys. Rev. A 87, 052136 (2013).
[Crossref]

Phys. Rev. Lett. (7)

Y. W. Cheong and S.-W. Lee, “Balance between information gain and reversibility in weak measurement,” Phys. Rev. Lett. 109, 150402 (2012).
[Crossref] [PubMed]

H.-T. Lim, Y.-S. Ra, K.-H. Hong, S.-W. Lee, and Y.-H. Kim, “Fundamental bounds in measurements for estimating quantum states,” Phys. Rev. Lett. 113, 020504 (2014).
[Crossref] [PubMed]

B. Dakic, V. Vedral, and C. Brukner, “Necessary and sufficient condition for nonzero quantum discord,” Phys. Rev. Lett. 105, 190502 (2010).
[Crossref]

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[Crossref] [PubMed]

S. Maniscalco, F. Francica, R. L. Zaffino, N. Lo Gullo, and F. Plastina, “Protecting entanglement via the quantum Zeno effect,” Phys. Rev. Lett. 100, 090503 (2008).
[Crossref] [PubMed]

D. A. Lidar, I. Chuang, and K. B. Whaley, “Decoherence-free subspaces for quantum computation,” Phys. Rev. Lett. 81, 2594–2597 (1998).
[Crossref]

A. Datta, A. Shaji, and C. M. Caves, “Quantum discord and the power of one qubit,” Phys. Rev. Lett. 100, 050502 (2008).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: discord and related measures,” Rev. Mod. Phys. 84, 1655–1707 (2012).
[Crossref]

Sci. Rep. (1)

S. Pirandola, “Quantum discord as a resource for quantum cryptography,” Sci. Rep. 4, 6956 (2014).
[Crossref] [PubMed]

Science (1)

P. G. Kwiat, A. J. Berglund, J. B. Alterpeter, and A. G. White, “Experimental verification of decoherence-free subspaces,” Science 290, 498–501 (2000).
[Crossref] [PubMed]

Other (1)

M. Nielson and I. Chuang, Quantum Computation and Quantum Information (Cambridge Univ. Press, 2000).

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Figures (3)

Fig. 1
Fig. 1 Theoretical estimation of entropic discord as functions of decoherence and weak measurement; Plots (a) and (b) are for the maximally correlated state |Φ〉 with |α| = |β|, and plots (c) and (d) are for the non-maximally correlated state |Φ〉 with |α| <|β| with α = 0.42. Entropic quantum discord under the influence of decoherence is shown in plots (a) and (c), whereas the effect of the weak and reversing measurements is shown in plots (b) and (d). Plots (b) and (d) are taken with D1 = 0.6 and D2 = 0.8.
Fig. 2
Fig. 2 Experimental setup. The initial two-qubit state |Φ〉 is the two-photon polarization state. Amplitude-damping decoherence (D) is implemented with the displaced Sagnac interferometers. Brewster-angle glass plates (BPs) and half-wave plates (Hs) are employed to perform weak mesurements (Mwk) and the reversing measurements (Mrev). Waveplates (WPs), polarizers (Pol.), single photon detectors (SPDs), and a coincidence counting unit (CCU) are used for quantum state tomography.
Fig. 3
Fig. 3 Experimental data for protecting entropic (top) and geometric (bottom) quantum discord from decoherence using quantum weak measurement and quantum measurement reversal. The blue curves are for the state |α| = |β|, whereas the red ones are for |α| = 0.42; (a, c) As D increases, the amounts of entropic and geometric quantum discord gradually decrease. (b, d) Even under the effect of strong decoherence (D = 0.6), we can reverse the amounts of quantum discords between Bob and Charlie by performing Mwk(p) and Mrev(p). The error bars represent the statistical error of ±1 standard deviation, and the dashed lines in (a) and (b) represent the corresponding concurrence plots.

Equations (31)

Equations on this page are rendered with MathJax. Learn more.

H ( X ) = i P ( x i ) I ( x i ) = i P ( x i ) log b P ( x i ) ,
I ( A : B ) = H ( A ) + H ( B ) H ( A , B ) ,
S ( ρ ) = Tr ( ρ log b ρ ) .
I ( ρ A B ) = S ( ρ A ) + S ( ρ B ) S ( ρ A B ) ,
D ( ρ A B ) = I ( ρ A B ) J ( ρ A B ) ,
J ( ρ A B ) = sup { B k } I ( ρ A B | { B k } ) ,
D G ( ρ A B ) = inf { B k } ρ A B ρ A B c 1 ,
D G ( 2 ) ( ρ A B ) = inf { B k } ρ A B ρ A B c 2 2 ,
J ( ρ A B ) = sup { B k } ( S ( ρ A ) S ( ρ A B | { B k } ) ) .
F ( ρ A B ) = inf { B k } S ( ρ A B | { B k } ) .
Π 0 = ( 1 0 0 0 ) , Π 0 = ( 0 0 0 1 ) .
V ( θ , ϕ ) = 1 2 ( I i a ^ ( θ , ϕ ) σ ) ,
B k i = V Π i V , i { 0 , 1 } .
a ^ ( θ , ϕ ) = ( sin θ cos ϕ sin θ sin ϕ cos θ ) ,
σ = ( σ 1 σ 2 σ 3 ) ,
σ 1 = ( 0 1 1 0 ) , σ 2 = ( 0 i i 0 ) , σ 3 = ( 1 0 0 1 ) .
ρ A B | { B k } = i { 0 , 1 } 1 p i ( I B k i ) ρ A B ( I B k i ) ,
ρ B | | 0 A = Tr A { [ ( 1 0 0 0 ) I ] ρ A B } ,
ρ B | | 1 A = Tr A { [ ( 0 0 0 1 ) I ] ρ A B } .
ρ A B c = i { 0 , 1 } ( V Π i V ) ρ B | | i A .
Λ s j k = | j k | + | k j | , 1 j < k d ,
Λ a j k = i | j k | + i | k j | , 1 j < k d ,
Λ d l = 2 l ( l + 1 ) ( j = 1 l | j j | + l | l + 1 l + 1 | ) , 1 l d 1.
V = e i k Perm ( { 1 , 2 , , d 2 1 } ) θ k Λ k ,
| 0 S | 0 E | 0 S | 0 E ,
| 1 S | 0 E D ¯ | 1 S | 0 E + D | 0 S | 1 E ,
| Φ = α | 00 S + β | 11 S ,
ρ d = ( | α | 2 + D 1 D 2 | β | 2 0 0 D ¯ 1 D ¯ 2 α β * 0 D 1 D ¯ 2 | β | 2 0 0 0 0 D ¯ 1 D 2 | β | 2 0 D ¯ 1 D ¯ 2 α * β 0 0 D ¯ 1 D ¯ 2 | β | 2 ) ,
M w k ( p 1 , p 2 ) = ( 1 0 0 1 p 1 ) ( 1 0 0 1 p 2 ) ,
M r e v ( p r 1 , p r 2 ) = ( 1 p r 1 0 0 1 ) ( 1 p r 2 0 0 1 ) ,
ρ r = 1 A ( | α | 2 + p ¯ 1 p ¯ 2 D 1 D 2 | β | 2 0 0 α β * 0 p ¯ 1 D 1 | β | 2 0 0 0 0 p ¯ 2 D 2 | β | 2 0 α * β 0 0 | β | 2 ) ,

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