Abstract

We derive a closed-form analytical expression for the linear entropy of a multipartite qutrit state, providing a quantitative measure for quantum entanglement within the class of n-mode nonorthogonal qutrit states with any n. Conditions for enhanced and maximum quantum entanglement of multipartite qutrit states are identified. The usefulness of the introduced multipartite qutrit states as quantum communication channel resources is analyzed. The Hamiltonians allowing for the generation of multipartite qutrit states can be attained by combining optomechanical cavities with sequences of tunable beam splitters.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  1. M. Fujiwara, M. Takeoka, J. Mizuno, and M. Sasaki, “Exceeding the classical capacity limit in a quantum optical channel,” Phys. Rev. Lett. 90, 167906 (2003).
    [Crossref] [PubMed]
  2. M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
    [Crossref]
  3. C. Wang, F.-G. Deng, Y.-S. Li, X.-S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005).
    [Crossref]
  4. N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White,”Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
    [Crossref] [PubMed]
  5. D. Bruß and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88, 127901 (2002).
    [Crossref]
  6. M. L. Almeida, S. Pironio, J. Barrett, G. Tóth, and A. Acín, “Noise robustness of the nonlocality of entangled quantum states,” Phys. Rev. Lett. 99, 040403 (2007).
    [Crossref] [PubMed]
  7. L. Sheridan and V. Scarani, “Security proof for quantum key distribution using qudit systems,” Phys. Rev. A 82, 030301 (2010).
    [Crossref]
  8. T. Vértesi, S. Pironio, and N. Brunner, “Closing the detection loophole in Bell experiments using qudits,” Phys. Rev. Lett. 104, 060401 (2010).
    [Crossref] [PubMed]
  9. B. P. Lanyon, T. J. Weinhold, N. K. Langford, J. L. O’ Brien, K. J. Resch, A. Gilchrist, and A. G. White, “Manipulating biphotonic qutrits,” Phys. Rev. Lett. 100, 060504 (2008).
    [Crossref] [PubMed]
  10. F. Dolde, V. Bergholm, Y. Wang, I. Jakobi, B. Naydenov, S. Pezzagna, J. Meijer, F. Jelezko, P. Neumann, T. Schulte-Herbrüggen, J. Biamonte, and J. Wrachtrup, “High fidelity spin entanglement using optimal control,” Nat. Commun. 5, 3371 (2014).
    [Crossref]
  11. Y. Maleki and A. M. Zheltikov, “Generating maximally-path-entangled number states in two spin ensembles coupled to a superconducting flux qubit,” Phys. Rev. A 26, 012312 (2018).
    [Crossref]
  12. Y. Maleki and A. M. Zheltikov, “Witnessing quantum entanglement in ensembles of nitrogen–vacancy centers coupled to a superconducting resonator,” Opt. Express 97, 17849–17858 (2018).
    [Crossref]
  13. C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
    [Crossref] [PubMed]
  14. A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5, 189–192 (2009).
    [Crossref]
  15. S. L. Braunstein and P. V. Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
    [Crossref]
  16. X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nat. (London) 478, 221–224 (2011).
    [Crossref]
  17. J. Niset, U. L. Andersen, and N. J. Cerf, “Experimentally feasible quantum erasure-correcting code for continuous variables,” Phys. Rev. Lett. 101, 130503 (2008).
    [Crossref] [PubMed]
  18. X. Wang, “Bipartite entangled non-orthogonal states,” J. Phys. A: Math. Gen. 35, 165 (2001).
    [Crossref]
  19. A. Mann, B. C. Sanders, and W. J. Munro, “Bells inequality for an entanglement of nonorthogonal states,” Phys. Rev. A 51, 989 (1995).
    [Crossref] [PubMed]
  20. M. Paternostro and H. Jeong, “Testing nonlocal realism with entangled coherent states,” Phys. Rev. A 81, 032115 (2010).
    [Crossref]
  21. S. Bose and V. Vedral, “Mixedness and teleportation,” Phys. Rev. A 61, 040101 (2000).
    [Crossref]
  22. W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245 (1998).
    [Crossref]
  23. H. Fu, X. Wang, and A. I. Solomon, “Maximal entanglement of nonorthogonal states: classification,” Phys. Lett. A 291, 73–76 (2001).
    [Crossref]
  24. P. Badziag, M. Horodecki, P. Horodecki, and R. Horodecki, “Local environment can enhance fidelity of quantum teleportation,” Phys. Rev. A 62, 012311 (2000).
    [Crossref]
  25. S. Sazim, S. Adhikari, S. Banerjee, and T. Pramanik, “Quantification of entanglement of teleportation in arbitrary dimensions,” Quantum Inf. Process. 13, 863–880 (2014).
    [Crossref]
  26. S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175 (1997).
    [Crossref]
  27. P. v. Loock and S. L. Braunstein, “Multipartite entanglement for continuous variables: a quantum teleportation network,” Phys. Rev. Lett. 84, 3482 (2000).
    [Crossref] [PubMed]
  28. S. L. Braunstein, N. J. Cerf, S. Iblisdir, P. v. Loock, and S. Massar, “Optimal cloning of coherent states with a linear amplifier and beam splitters,” Phys. Rev. Lett. 86, 4938 (2001).
    [Crossref] [PubMed]
  29. S. J. van Enk and O. Hirota, “Entangled coherent states: Teleportation and decoherence,” Phys. Rev. A 67, 022313 (2001).
    [Crossref]
  30. Y. Maleki and A. Maleki, “Entangled multimode spin coherent states of trapped ions,” J. Opt. Soc. Am. B 35, 1211–1217 (2018).
    [Crossref]

2018 (3)

Y. Maleki and A. M. Zheltikov, “Generating maximally-path-entangled number states in two spin ensembles coupled to a superconducting flux qubit,” Phys. Rev. A 26, 012312 (2018).
[Crossref]

Y. Maleki and A. M. Zheltikov, “Witnessing quantum entanglement in ensembles of nitrogen–vacancy centers coupled to a superconducting resonator,” Opt. Express 97, 17849–17858 (2018).
[Crossref]

Y. Maleki and A. Maleki, “Entangled multimode spin coherent states of trapped ions,” J. Opt. Soc. Am. B 35, 1211–1217 (2018).
[Crossref]

2016 (2)

C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
[Crossref]

2014 (2)

S. Sazim, S. Adhikari, S. Banerjee, and T. Pramanik, “Quantification of entanglement of teleportation in arbitrary dimensions,” Quantum Inf. Process. 13, 863–880 (2014).
[Crossref]

F. Dolde, V. Bergholm, Y. Wang, I. Jakobi, B. Naydenov, S. Pezzagna, J. Meijer, F. Jelezko, P. Neumann, T. Schulte-Herbrüggen, J. Biamonte, and J. Wrachtrup, “High fidelity spin entanglement using optimal control,” Nat. Commun. 5, 3371 (2014).
[Crossref]

2011 (1)

X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nat. (London) 478, 221–224 (2011).
[Crossref]

2010 (3)

L. Sheridan and V. Scarani, “Security proof for quantum key distribution using qudit systems,” Phys. Rev. A 82, 030301 (2010).
[Crossref]

T. Vértesi, S. Pironio, and N. Brunner, “Closing the detection loophole in Bell experiments using qudits,” Phys. Rev. Lett. 104, 060401 (2010).
[Crossref] [PubMed]

M. Paternostro and H. Jeong, “Testing nonlocal realism with entangled coherent states,” Phys. Rev. A 81, 032115 (2010).
[Crossref]

2009 (1)

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5, 189–192 (2009).
[Crossref]

2008 (2)

J. Niset, U. L. Andersen, and N. J. Cerf, “Experimentally feasible quantum erasure-correcting code for continuous variables,” Phys. Rev. Lett. 101, 130503 (2008).
[Crossref] [PubMed]

B. P. Lanyon, T. J. Weinhold, N. K. Langford, J. L. O’ Brien, K. J. Resch, A. Gilchrist, and A. G. White, “Manipulating biphotonic qutrits,” Phys. Rev. Lett. 100, 060504 (2008).
[Crossref] [PubMed]

2007 (1)

M. L. Almeida, S. Pironio, J. Barrett, G. Tóth, and A. Acín, “Noise robustness of the nonlocality of entangled quantum states,” Phys. Rev. Lett. 99, 040403 (2007).
[Crossref] [PubMed]

2005 (2)

C. Wang, F.-G. Deng, Y.-S. Li, X.-S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005).
[Crossref]

S. L. Braunstein and P. V. Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
[Crossref]

2004 (1)

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White,”Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

2003 (1)

M. Fujiwara, M. Takeoka, J. Mizuno, and M. Sasaki, “Exceeding the classical capacity limit in a quantum optical channel,” Phys. Rev. Lett. 90, 167906 (2003).
[Crossref] [PubMed]

2002 (1)

D. Bruß and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88, 127901 (2002).
[Crossref]

2001 (4)

X. Wang, “Bipartite entangled non-orthogonal states,” J. Phys. A: Math. Gen. 35, 165 (2001).
[Crossref]

H. Fu, X. Wang, and A. I. Solomon, “Maximal entanglement of nonorthogonal states: classification,” Phys. Lett. A 291, 73–76 (2001).
[Crossref]

S. L. Braunstein, N. J. Cerf, S. Iblisdir, P. v. Loock, and S. Massar, “Optimal cloning of coherent states with a linear amplifier and beam splitters,” Phys. Rev. Lett. 86, 4938 (2001).
[Crossref] [PubMed]

S. J. van Enk and O. Hirota, “Entangled coherent states: Teleportation and decoherence,” Phys. Rev. A 67, 022313 (2001).
[Crossref]

2000 (3)

P. Badziag, M. Horodecki, P. Horodecki, and R. Horodecki, “Local environment can enhance fidelity of quantum teleportation,” Phys. Rev. A 62, 012311 (2000).
[Crossref]

S. Bose and V. Vedral, “Mixedness and teleportation,” Phys. Rev. A 61, 040101 (2000).
[Crossref]

P. v. Loock and S. L. Braunstein, “Multipartite entanglement for continuous variables: a quantum teleportation network,” Phys. Rev. Lett. 84, 3482 (2000).
[Crossref] [PubMed]

1998 (1)

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245 (1998).
[Crossref]

1997 (1)

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175 (1997).
[Crossref]

1995 (1)

A. Mann, B. C. Sanders, and W. J. Munro, “Bells inequality for an entanglement of nonorthogonal states,” Phys. Rev. A 51, 989 (1995).
[Crossref] [PubMed]

Acín, A.

M. L. Almeida, S. Pironio, J. Barrett, G. Tóth, and A. Acín, “Noise robustness of the nonlocality of entangled quantum states,” Phys. Rev. Lett. 99, 040403 (2007).
[Crossref] [PubMed]

Adhikari, S.

S. Sazim, S. Adhikari, S. Banerjee, and T. Pramanik, “Quantification of entanglement of teleportation in arbitrary dimensions,” Quantum Inf. Process. 13, 863–880 (2014).
[Crossref]

Almeida, M. L.

M. L. Almeida, S. Pironio, J. Barrett, G. Tóth, and A. Acín, “Noise robustness of the nonlocality of entangled quantum states,” Phys. Rev. Lett. 99, 040403 (2007).
[Crossref] [PubMed]

Andersen, U. L.

J. Niset, U. L. Andersen, and N. J. Cerf, “Experimentally feasible quantum erasure-correcting code for continuous variables,” Phys. Rev. Lett. 101, 130503 (2008).
[Crossref] [PubMed]

Axline, C.

C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

Badziag, P.

P. Badziag, M. Horodecki, P. Horodecki, and R. Horodecki, “Local environment can enhance fidelity of quantum teleportation,” Phys. Rev. A 62, 012311 (2000).
[Crossref]

Banerjee, S.

S. Sazim, S. Adhikari, S. Banerjee, and T. Pramanik, “Quantification of entanglement of teleportation in arbitrary dimensions,” Quantum Inf. Process. 13, 863–880 (2014).
[Crossref]

Barrett, J.

M. L. Almeida, S. Pironio, J. Barrett, G. Tóth, and A. Acín, “Noise robustness of the nonlocality of entangled quantum states,” Phys. Rev. Lett. 99, 040403 (2007).
[Crossref] [PubMed]

Bartlett, S. D.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White,”Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

Bergholm, V.

F. Dolde, V. Bergholm, Y. Wang, I. Jakobi, B. Naydenov, S. Pezzagna, J. Meijer, F. Jelezko, P. Neumann, T. Schulte-Herbrüggen, J. Biamonte, and J. Wrachtrup, “High fidelity spin entanglement using optimal control,” Nat. Commun. 5, 3371 (2014).
[Crossref]

Biamonte, J.

F. Dolde, V. Bergholm, Y. Wang, I. Jakobi, B. Naydenov, S. Pezzagna, J. Meijer, F. Jelezko, P. Neumann, T. Schulte-Herbrüggen, J. Biamonte, and J. Wrachtrup, “High fidelity spin entanglement using optimal control,” Nat. Commun. 5, 3371 (2014).
[Crossref]

Blumoff, J.

C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

Bose, S.

S. Bose and V. Vedral, “Mixedness and teleportation,” Phys. Rev. A 61, 040101 (2000).
[Crossref]

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175 (1997).
[Crossref]

Braunstein, S. L.

S. L. Braunstein and P. V. Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
[Crossref]

S. L. Braunstein, N. J. Cerf, S. Iblisdir, P. v. Loock, and S. Massar, “Optimal cloning of coherent states with a linear amplifier and beam splitters,” Phys. Rev. Lett. 86, 4938 (2001).
[Crossref] [PubMed]

P. v. Loock and S. L. Braunstein, “Multipartite entanglement for continuous variables: a quantum teleportation network,” Phys. Rev. Lett. 84, 3482 (2000).
[Crossref] [PubMed]

Brunner, N.

T. Vértesi, S. Pironio, and N. Brunner, “Closing the detection loophole in Bell experiments using qudits,” Phys. Rev. Lett. 104, 060401 (2010).
[Crossref] [PubMed]

Bruß, D.

D. Bruß and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88, 127901 (2002).
[Crossref]

Cerf, N. J.

J. Niset, U. L. Andersen, and N. J. Cerf, “Experimentally feasible quantum erasure-correcting code for continuous variables,” Phys. Rev. Lett. 101, 130503 (2008).
[Crossref] [PubMed]

S. L. Braunstein, N. J. Cerf, S. Iblisdir, P. v. Loock, and S. Massar, “Optimal cloning of coherent states with a linear amplifier and beam splitters,” Phys. Rev. Lett. 86, 4938 (2001).
[Crossref] [PubMed]

Chou, K.

C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

Dalton, R. B.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White,”Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

Deng, F.-G.

C. Wang, F.-G. Deng, Y.-S. Li, X.-S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005).
[Crossref]

Devoret, M. H.

C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

Dolde, F.

F. Dolde, V. Bergholm, Y. Wang, I. Jakobi, B. Naydenov, S. Pezzagna, J. Meijer, F. Jelezko, P. Neumann, T. Schulte-Herbrüggen, J. Biamonte, and J. Wrachtrup, “High fidelity spin entanglement using optimal control,” Nat. Commun. 5, 3371 (2014).
[Crossref]

Erhard, M.

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
[Crossref]

Everitt, M. S.

X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nat. (London) 478, 221–224 (2011).
[Crossref]

Ferreyrol, F.

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5, 189–192 (2009).
[Crossref]

Fickler, R.

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
[Crossref]

Frunzio, L.

C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

Fu, H.

H. Fu, X. Wang, and A. I. Solomon, “Maximal entanglement of nonorthogonal states: classification,” Phys. Lett. A 291, 73–76 (2001).
[Crossref]

Fujiwara, M.

M. Fujiwara, M. Takeoka, J. Mizuno, and M. Sasaki, “Exceeding the classical capacity limit in a quantum optical channel,” Phys. Rev. Lett. 90, 167906 (2003).
[Crossref] [PubMed]

Gao, Y. Y.

C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

Gilchrist, A.

B. P. Lanyon, T. J. Weinhold, N. K. Langford, J. L. O’ Brien, K. J. Resch, A. Gilchrist, and A. G. White, “Manipulating biphotonic qutrits,” Phys. Rev. Lett. 100, 060504 (2008).
[Crossref] [PubMed]

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White,”Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

Girvin, S. M.

C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

Grangier, P.

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5, 189–192 (2009).
[Crossref]

Harvey, M. D.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White,”Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

Heeres, R. W.

C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

Hirota, O.

S. J. van Enk and O. Hirota, “Entangled coherent states: Teleportation and decoherence,” Phys. Rev. A 67, 022313 (2001).
[Crossref]

Horodecki, M.

P. Badziag, M. Horodecki, P. Horodecki, and R. Horodecki, “Local environment can enhance fidelity of quantum teleportation,” Phys. Rev. A 62, 012311 (2000).
[Crossref]

Horodecki, P.

P. Badziag, M. Horodecki, P. Horodecki, and R. Horodecki, “Local environment can enhance fidelity of quantum teleportation,” Phys. Rev. A 62, 012311 (2000).
[Crossref]

Horodecki, R.

P. Badziag, M. Horodecki, P. Horodecki, and R. Horodecki, “Local environment can enhance fidelity of quantum teleportation,” Phys. Rev. A 62, 012311 (2000).
[Crossref]

Huber, M.

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
[Crossref]

Iblisdir, S.

S. L. Braunstein, N. J. Cerf, S. Iblisdir, P. v. Loock, and S. Massar, “Optimal cloning of coherent states with a linear amplifier and beam splitters,” Phys. Rev. Lett. 86, 4938 (2001).
[Crossref] [PubMed]

Jacobs, K.

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175 (1997).
[Crossref]

Jakobi, I.

F. Dolde, V. Bergholm, Y. Wang, I. Jakobi, B. Naydenov, S. Pezzagna, J. Meijer, F. Jelezko, P. Neumann, T. Schulte-Herbrüggen, J. Biamonte, and J. Wrachtrup, “High fidelity spin entanglement using optimal control,” Nat. Commun. 5, 3371 (2014).
[Crossref]

Jelezko, F.

F. Dolde, V. Bergholm, Y. Wang, I. Jakobi, B. Naydenov, S. Pezzagna, J. Meijer, F. Jelezko, P. Neumann, T. Schulte-Herbrüggen, J. Biamonte, and J. Wrachtrup, “High fidelity spin entanglement using optimal control,” Nat. Commun. 5, 3371 (2014).
[Crossref]

Jeong, H.

M. Paternostro and H. Jeong, “Testing nonlocal realism with entangled coherent states,” Phys. Rev. A 81, 032115 (2010).
[Crossref]

Jiang, L.

C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

Kakuyanagi, K.

X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nat. (London) 478, 221–224 (2011).
[Crossref]

Karimoto, S.

X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nat. (London) 478, 221–224 (2011).
[Crossref]

Kasu, M.

X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nat. (London) 478, 221–224 (2011).
[Crossref]

Kemp, A.

X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nat. (London) 478, 221–224 (2011).
[Crossref]

Knight, P. L.

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175 (1997).
[Crossref]

Krenn, M.

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
[Crossref]

Langford, N. K.

B. P. Lanyon, T. J. Weinhold, N. K. Langford, J. L. O’ Brien, K. J. Resch, A. Gilchrist, and A. G. White, “Manipulating biphotonic qutrits,” Phys. Rev. Lett. 100, 060504 (2008).
[Crossref] [PubMed]

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White,”Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

Lanyon, B. P.

B. P. Lanyon, T. J. Weinhold, N. K. Langford, J. L. O’ Brien, K. J. Resch, A. Gilchrist, and A. G. White, “Manipulating biphotonic qutrits,” Phys. Rev. Lett. 100, 060504 (2008).
[Crossref] [PubMed]

Li, Y.-S.

C. Wang, F.-G. Deng, Y.-S. Li, X.-S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005).
[Crossref]

Liu, X.-S.

C. Wang, F.-G. Deng, Y.-S. Li, X.-S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005).
[Crossref]

Long, G. L.

C. Wang, F.-G. Deng, Y.-S. Li, X.-S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005).
[Crossref]

Loock, P. V.

S. L. Braunstein and P. V. Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
[Crossref]

S. L. Braunstein, N. J. Cerf, S. Iblisdir, P. v. Loock, and S. Massar, “Optimal cloning of coherent states with a linear amplifier and beam splitters,” Phys. Rev. Lett. 86, 4938 (2001).
[Crossref] [PubMed]

P. v. Loock and S. L. Braunstein, “Multipartite entanglement for continuous variables: a quantum teleportation network,” Phys. Rev. Lett. 84, 3482 (2000).
[Crossref] [PubMed]

Macchiavello, C.

D. Bruß and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88, 127901 (2002).
[Crossref]

Maleki, A.

Y. Maleki and A. Maleki, “Entangled multimode spin coherent states of trapped ions,” J. Opt. Soc. Am. B 35, 1211–1217 (2018).
[Crossref]

Maleki, Y.

Y. Maleki and A. Maleki, “Entangled multimode spin coherent states of trapped ions,” J. Opt. Soc. Am. B 35, 1211–1217 (2018).
[Crossref]

Y. Maleki and A. M. Zheltikov, “Witnessing quantum entanglement in ensembles of nitrogen–vacancy centers coupled to a superconducting resonator,” Opt. Express 97, 17849–17858 (2018).
[Crossref]

Y. Maleki and A. M. Zheltikov, “Generating maximally-path-entangled number states in two spin ensembles coupled to a superconducting flux qubit,” Phys. Rev. A 26, 012312 (2018).
[Crossref]

Malik, M.

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
[Crossref]

Mann, A.

A. Mann, B. C. Sanders, and W. J. Munro, “Bells inequality for an entanglement of nonorthogonal states,” Phys. Rev. A 51, 989 (1995).
[Crossref] [PubMed]

Massar, S.

S. L. Braunstein, N. J. Cerf, S. Iblisdir, P. v. Loock, and S. Massar, “Optimal cloning of coherent states with a linear amplifier and beam splitters,” Phys. Rev. Lett. 86, 4938 (2001).
[Crossref] [PubMed]

Meijer, J.

F. Dolde, V. Bergholm, Y. Wang, I. Jakobi, B. Naydenov, S. Pezzagna, J. Meijer, F. Jelezko, P. Neumann, T. Schulte-Herbrüggen, J. Biamonte, and J. Wrachtrup, “High fidelity spin entanglement using optimal control,” Nat. Commun. 5, 3371 (2014).
[Crossref]

Mirrahimi, M.

C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

Mizuno, J.

M. Fujiwara, M. Takeoka, J. Mizuno, and M. Sasaki, “Exceeding the classical capacity limit in a quantum optical channel,” Phys. Rev. Lett. 90, 167906 (2003).
[Crossref] [PubMed]

Mizuochi, N.

X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nat. (London) 478, 221–224 (2011).
[Crossref]

Munro, W. J.

X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nat. (London) 478, 221–224 (2011).
[Crossref]

A. Mann, B. C. Sanders, and W. J. Munro, “Bells inequality for an entanglement of nonorthogonal states,” Phys. Rev. A 51, 989 (1995).
[Crossref] [PubMed]

Nakano, H.

X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nat. (London) 478, 221–224 (2011).
[Crossref]

Naydenov, B.

F. Dolde, V. Bergholm, Y. Wang, I. Jakobi, B. Naydenov, S. Pezzagna, J. Meijer, F. Jelezko, P. Neumann, T. Schulte-Herbrüggen, J. Biamonte, and J. Wrachtrup, “High fidelity spin entanglement using optimal control,” Nat. Commun. 5, 3371 (2014).
[Crossref]

Nemoto, K.

X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nat. (London) 478, 221–224 (2011).
[Crossref]

Neumann, P.

F. Dolde, V. Bergholm, Y. Wang, I. Jakobi, B. Naydenov, S. Pezzagna, J. Meijer, F. Jelezko, P. Neumann, T. Schulte-Herbrüggen, J. Biamonte, and J. Wrachtrup, “High fidelity spin entanglement using optimal control,” Nat. Commun. 5, 3371 (2014).
[Crossref]

Niset, J.

J. Niset, U. L. Andersen, and N. J. Cerf, “Experimentally feasible quantum erasure-correcting code for continuous variables,” Phys. Rev. Lett. 101, 130503 (2008).
[Crossref] [PubMed]

O’ Brien, J. L.

B. P. Lanyon, T. J. Weinhold, N. K. Langford, J. L. O’ Brien, K. J. Resch, A. Gilchrist, and A. G. White, “Manipulating biphotonic qutrits,” Phys. Rev. Lett. 100, 060504 (2008).
[Crossref] [PubMed]

O’Brien, J. L.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White,”Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

Ofek, N.

C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

Ourjoumtsev, A.

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5, 189–192 (2009).
[Crossref]

Paternostro, M.

M. Paternostro and H. Jeong, “Testing nonlocal realism with entangled coherent states,” Phys. Rev. A 81, 032115 (2010).
[Crossref]

Pezzagna, S.

F. Dolde, V. Bergholm, Y. Wang, I. Jakobi, B. Naydenov, S. Pezzagna, J. Meijer, F. Jelezko, P. Neumann, T. Schulte-Herbrüggen, J. Biamonte, and J. Wrachtrup, “High fidelity spin entanglement using optimal control,” Nat. Commun. 5, 3371 (2014).
[Crossref]

Pironio, S.

T. Vértesi, S. Pironio, and N. Brunner, “Closing the detection loophole in Bell experiments using qudits,” Phys. Rev. Lett. 104, 060401 (2010).
[Crossref] [PubMed]

M. L. Almeida, S. Pironio, J. Barrett, G. Tóth, and A. Acín, “Noise robustness of the nonlocality of entangled quantum states,” Phys. Rev. Lett. 99, 040403 (2007).
[Crossref] [PubMed]

Pramanik, T.

S. Sazim, S. Adhikari, S. Banerjee, and T. Pramanik, “Quantification of entanglement of teleportation in arbitrary dimensions,” Quantum Inf. Process. 13, 863–880 (2014).
[Crossref]

Pryde, G. J.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White,”Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

Reagor, M.

C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

Reinhold, P.

C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

Resch, K. J.

B. P. Lanyon, T. J. Weinhold, N. K. Langford, J. L. O’ Brien, K. J. Resch, A. Gilchrist, and A. G. White, “Manipulating biphotonic qutrits,” Phys. Rev. Lett. 100, 060504 (2008).
[Crossref] [PubMed]

Saito, S.

X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nat. (London) 478, 221–224 (2011).
[Crossref]

Sanders, B. C.

A. Mann, B. C. Sanders, and W. J. Munro, “Bells inequality for an entanglement of nonorthogonal states,” Phys. Rev. A 51, 989 (1995).
[Crossref] [PubMed]

Sasaki, M.

M. Fujiwara, M. Takeoka, J. Mizuno, and M. Sasaki, “Exceeding the classical capacity limit in a quantum optical channel,” Phys. Rev. Lett. 90, 167906 (2003).
[Crossref] [PubMed]

Sazim, S.

S. Sazim, S. Adhikari, S. Banerjee, and T. Pramanik, “Quantification of entanglement of teleportation in arbitrary dimensions,” Quantum Inf. Process. 13, 863–880 (2014).
[Crossref]

Scarani, V.

L. Sheridan and V. Scarani, “Security proof for quantum key distribution using qudit systems,” Phys. Rev. A 82, 030301 (2010).
[Crossref]

Schoelkopf, R. J.

C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

Schulte-Herbrüggen, T.

F. Dolde, V. Bergholm, Y. Wang, I. Jakobi, B. Naydenov, S. Pezzagna, J. Meijer, F. Jelezko, P. Neumann, T. Schulte-Herbrüggen, J. Biamonte, and J. Wrachtrup, “High fidelity spin entanglement using optimal control,” Nat. Commun. 5, 3371 (2014).
[Crossref]

Semba, K.

X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nat. (London) 478, 221–224 (2011).
[Crossref]

Sheridan, L.

L. Sheridan and V. Scarani, “Security proof for quantum key distribution using qudit systems,” Phys. Rev. A 82, 030301 (2010).
[Crossref]

Sliwa, K. M.

C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

Solomon, A. I.

H. Fu, X. Wang, and A. I. Solomon, “Maximal entanglement of nonorthogonal states: classification,” Phys. Lett. A 291, 73–76 (2001).
[Crossref]

Takeoka, M.

M. Fujiwara, M. Takeoka, J. Mizuno, and M. Sasaki, “Exceeding the classical capacity limit in a quantum optical channel,” Phys. Rev. Lett. 90, 167906 (2003).
[Crossref] [PubMed]

Tokura, Y.

X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nat. (London) 478, 221–224 (2011).
[Crossref]

Tóth, G.

M. L. Almeida, S. Pironio, J. Barrett, G. Tóth, and A. Acín, “Noise robustness of the nonlocality of entangled quantum states,” Phys. Rev. Lett. 99, 040403 (2007).
[Crossref] [PubMed]

Tualle-Brouri, R.

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5, 189–192 (2009).
[Crossref]

van Enk, S. J.

S. J. van Enk and O. Hirota, “Entangled coherent states: Teleportation and decoherence,” Phys. Rev. A 67, 022313 (2001).
[Crossref]

Vedral, V.

S. Bose and V. Vedral, “Mixedness and teleportation,” Phys. Rev. A 61, 040101 (2000).
[Crossref]

Vértesi, T.

T. Vértesi, S. Pironio, and N. Brunner, “Closing the detection loophole in Bell experiments using qudits,” Phys. Rev. Lett. 104, 060401 (2010).
[Crossref] [PubMed]

Wang, C.

C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

C. Wang, F.-G. Deng, Y.-S. Li, X.-S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005).
[Crossref]

Wang, X.

X. Wang, “Bipartite entangled non-orthogonal states,” J. Phys. A: Math. Gen. 35, 165 (2001).
[Crossref]

H. Fu, X. Wang, and A. I. Solomon, “Maximal entanglement of nonorthogonal states: classification,” Phys. Lett. A 291, 73–76 (2001).
[Crossref]

Wang, Y.

F. Dolde, V. Bergholm, Y. Wang, I. Jakobi, B. Naydenov, S. Pezzagna, J. Meijer, F. Jelezko, P. Neumann, T. Schulte-Herbrüggen, J. Biamonte, and J. Wrachtrup, “High fidelity spin entanglement using optimal control,” Nat. Commun. 5, 3371 (2014).
[Crossref]

Weinhold, T. J.

B. P. Lanyon, T. J. Weinhold, N. K. Langford, J. L. O’ Brien, K. J. Resch, A. Gilchrist, and A. G. White, “Manipulating biphotonic qutrits,” Phys. Rev. Lett. 100, 060504 (2008).
[Crossref] [PubMed]

White, A. G.

B. P. Lanyon, T. J. Weinhold, N. K. Langford, J. L. O’ Brien, K. J. Resch, A. Gilchrist, and A. G. White, “Manipulating biphotonic qutrits,” Phys. Rev. Lett. 100, 060504 (2008).
[Crossref] [PubMed]

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White,”Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

Wootters, W. K.

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245 (1998).
[Crossref]

Wrachtrup, J.

F. Dolde, V. Bergholm, Y. Wang, I. Jakobi, B. Naydenov, S. Pezzagna, J. Meijer, F. Jelezko, P. Neumann, T. Schulte-Herbrüggen, J. Biamonte, and J. Wrachtrup, “High fidelity spin entanglement using optimal control,” Nat. Commun. 5, 3371 (2014).
[Crossref]

Zeilinger, A.

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
[Crossref]

Zheltikov, A. M.

Y. Maleki and A. M. Zheltikov, “Witnessing quantum entanglement in ensembles of nitrogen–vacancy centers coupled to a superconducting resonator,” Opt. Express 97, 17849–17858 (2018).
[Crossref]

Y. Maleki and A. M. Zheltikov, “Generating maximally-path-entangled number states in two spin ensembles coupled to a superconducting flux qubit,” Phys. Rev. A 26, 012312 (2018).
[Crossref]

Zhu, X.

X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nat. (London) 478, 221–224 (2011).
[Crossref]

J. Opt. Soc. Am. B (1)

Y. Maleki and A. Maleki, “Entangled multimode spin coherent states of trapped ions,” J. Opt. Soc. Am. B 35, 1211–1217 (2018).
[Crossref]

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

X. Wang, “Bipartite entangled non-orthogonal states,” J. Phys. A: Math. Gen. 35, 165 (2001).
[Crossref]

Nat. (London) (1)

X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nat. (London) 478, 221–224 (2011).
[Crossref]

Nat. Commun. (1)

F. Dolde, V. Bergholm, Y. Wang, I. Jakobi, B. Naydenov, S. Pezzagna, J. Meijer, F. Jelezko, P. Neumann, T. Schulte-Herbrüggen, J. Biamonte, and J. Wrachtrup, “High fidelity spin entanglement using optimal control,” Nat. Commun. 5, 3371 (2014).
[Crossref]

Nat. Photonics (1)

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
[Crossref]

Nat. Phys. (1)

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5, 189–192 (2009).
[Crossref]

Opt. Express (1)

Y. Maleki and A. M. Zheltikov, “Witnessing quantum entanglement in ensembles of nitrogen–vacancy centers coupled to a superconducting resonator,” Opt. Express 97, 17849–17858 (2018).
[Crossref]

Phys. Lett. A (1)

H. Fu, X. Wang, and A. I. Solomon, “Maximal entanglement of nonorthogonal states: classification,” Phys. Lett. A 291, 73–76 (2001).
[Crossref]

Phys. Rev. A (9)

P. Badziag, M. Horodecki, P. Horodecki, and R. Horodecki, “Local environment can enhance fidelity of quantum teleportation,” Phys. Rev. A 62, 012311 (2000).
[Crossref]

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175 (1997).
[Crossref]

S. J. van Enk and O. Hirota, “Entangled coherent states: Teleportation and decoherence,” Phys. Rev. A 67, 022313 (2001).
[Crossref]

A. Mann, B. C. Sanders, and W. J. Munro, “Bells inequality for an entanglement of nonorthogonal states,” Phys. Rev. A 51, 989 (1995).
[Crossref] [PubMed]

M. Paternostro and H. Jeong, “Testing nonlocal realism with entangled coherent states,” Phys. Rev. A 81, 032115 (2010).
[Crossref]

S. Bose and V. Vedral, “Mixedness and teleportation,” Phys. Rev. A 61, 040101 (2000).
[Crossref]

C. Wang, F.-G. Deng, Y.-S. Li, X.-S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005).
[Crossref]

Y. Maleki and A. M. Zheltikov, “Generating maximally-path-entangled number states in two spin ensembles coupled to a superconducting flux qubit,” Phys. Rev. A 26, 012312 (2018).
[Crossref]

L. Sheridan and V. Scarani, “Security proof for quantum key distribution using qudit systems,” Phys. Rev. A 82, 030301 (2010).
[Crossref]

Phys. Rev. Lett. (10)

T. Vértesi, S. Pironio, and N. Brunner, “Closing the detection loophole in Bell experiments using qudits,” Phys. Rev. Lett. 104, 060401 (2010).
[Crossref] [PubMed]

B. P. Lanyon, T. J. Weinhold, N. K. Langford, J. L. O’ Brien, K. J. Resch, A. Gilchrist, and A. G. White, “Manipulating biphotonic qutrits,” Phys. Rev. Lett. 100, 060504 (2008).
[Crossref] [PubMed]

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White,”Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

D. Bruß and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88, 127901 (2002).
[Crossref]

M. L. Almeida, S. Pironio, J. Barrett, G. Tóth, and A. Acín, “Noise robustness of the nonlocality of entangled quantum states,” Phys. Rev. Lett. 99, 040403 (2007).
[Crossref] [PubMed]

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245 (1998).
[Crossref]

M. Fujiwara, M. Takeoka, J. Mizuno, and M. Sasaki, “Exceeding the classical capacity limit in a quantum optical channel,” Phys. Rev. Lett. 90, 167906 (2003).
[Crossref] [PubMed]

P. v. Loock and S. L. Braunstein, “Multipartite entanglement for continuous variables: a quantum teleportation network,” Phys. Rev. Lett. 84, 3482 (2000).
[Crossref] [PubMed]

S. L. Braunstein, N. J. Cerf, S. Iblisdir, P. v. Loock, and S. Massar, “Optimal cloning of coherent states with a linear amplifier and beam splitters,” Phys. Rev. Lett. 86, 4938 (2001).
[Crossref] [PubMed]

J. Niset, U. L. Andersen, and N. J. Cerf, “Experimentally feasible quantum erasure-correcting code for continuous variables,” Phys. Rev. Lett. 101, 130503 (2008).
[Crossref] [PubMed]

Quantum Inf. Process. (1)

S. Sazim, S. Adhikari, S. Banerjee, and T. Pramanik, “Quantification of entanglement of teleportation in arbitrary dimensions,” Quantum Inf. Process. 13, 863–880 (2014).
[Crossref]

Rev. Mod. Phys. (1)

S. L. Braunstein and P. V. Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
[Crossref]

Science (1)

C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

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Figures (5)

Fig. 1
Fig. 1 The density of the linear entropy of |Ψ〉 as a function of the overlap parameters p1 and p3 for p2 = 0.05 (a), 0.2 (b), 0.4 (c), and 0.5 (d).
Fig. 2
Fig. 2 The density of the linear entropy of |Ψ〉+ as a function of overlap parameters p1 and p3 for p2 = 0.05 (a), 0.2 (b), 0.4 (c), and 0.5 (d).
Fig. 3
Fig. 3 The linear entropy of two-qutrit coherent states |ψ〉 (a) and |ϕ〉 (b) as a function of α and θ.
Fig. 4
Fig. 4 (a) The linear entropy of the multiqutrit state (37) as a function of p2 for p1 = p3 = 0 and k =1 (dotted line), k =3 (dash–dotted line), k =5 (dashed line), and k = 7 (solid line). (b) The linear entropy of the multiqutrit coherent state (39) as a function of α for k =1 (dotted line), k =2 (dash–dotted line), k =4 (dashed line), and k =8 (solid line). (c) The linear entropy of multiqutrit coherent (solid and dash–dotted lines) and squeezed (dashed and dotted lines) state as a function of α for k =1 (dotted and dash–dotted lines) and k =8 (solid and dashed lines).
Fig. 5
Fig. 5 The concurrence of an amplitude-damped k-mode state as a function of the coherence parameter α with θ = 0 (a, b) and π (c), η = 1 (a) and 0.9 (b, c), k = 1 (black), 2 (red), 4 (green), and 8 (blue).

Equations (91)

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| Ψ = μ | α 1 | β 1 + λ | α 2 | β 2 + ν | α 3 | β 3 .
| 0 | α 1 , | 1 | α 2 p 1 | α 1 N 1 , | 2 ( p 1 p 3 p 2 ) | α 1 + ( p ¯ 1 p 2 p 3 ) | α 2 + N 1 2 | α 3 N N 1 ,
p 1 = α 1 | α 2 , p 2 = α 1 | α 3 , p 3 = α 2 | α 3 ,
N i = 1 | p i | 2 , N = [ 1 | p 1 | 2 | p 2 | 2 | p 3 | 2 + p 1 p ¯ 2 p 3 + p ¯ 1 p 2 p ¯ 3 ] 1 2 ,
| α 1 | 0 , | α 2 N 1 | 1 + p 1 | 0 , | α 3 N | 2 + p 2 N 1 | 0 ( p ¯ 1 p 2 p 3 ) | 1 N 1 ,
| 0 ˜ | β 1 , | 1 ˜ | β 2 q 1 | β 1 M 1 , | 2 ˜ ( q 1 q 3 q 2 ) | β 1 + ( q ¯ 1 q 2 q 3 ) | β 2 + M 1 2 | β 3 M M 1 ,
q 1 = β 1 | β 2 , q 2 = β 1 | β 3 , q 3 = β 2 | β 3 ,
M i = 1 | q i | 2 , M = [ 1 | q 1 | 2 | q 2 | 2 | q 3 | 2 + q 1 q ¯ 2 q 3 + q ¯ 1 q 2 q ¯ 3 ] 1 2 .
| Ψ = ( μ + λ q 1 p 1 + ν p 2 q 2 ) | 0 0 ˜ + ( ν p 2 q 3 q ¯ 1 q 2 M 1 + λ M 1 p 1 ) | 0 1 ˜ + ( ν q 2 p 3 p ¯ 1 p 2 N 1 + λ N 1 q 1 ) | 1 0 ˜ + ( ν ( p 3 p ¯ 1 p 2 ) N 1 ( q 3 q ¯ 1 q 2 ) M 1 + λ N 1 M 1 ) | 1 1 ˜ + ν p 2 M M 1 | 0 2 ˜ + ν q 2 N N 1 | 2 0 ˜ + ( ν p 3 p ¯ 1 p 2 N 1 M M 1 ) | 1 2 ˜ + ν q 3 q ¯ 1 q 2 M 1 N N 1 | 2 1 ˜ + ν N N 1 M M 1 | 2 2 ˜ .
I lin = d d 1 ( 1 Tr ρ 1 2 ) .
I lin = 2 d d 1 Δ 1 + 2 ( Δ 2 + Δ 3 + Δ 4 ) 𝒩 2 ,
Δ 1 = | μ λ | 2 M 1 2 N 1 2 + | μ ν | 2 M 2 2 N 2 2 + | λ ν | 2 M 3 3 N 3 2 , Δ 2 = | μ | 2 Re [ λ ¯ ν ( q ¯ 1 q 2 q 3 ) ( p ¯ 1 p 2 p 3 ) ] , Δ 3 = | λ | 2 Re [ μ ¯ ν ( q ¯ 1 q 3 q 2 ) ( p 1 p 3 p 2 ) ] , Δ 4 = | ν | 2 Re [ μ ¯ λ ( q ¯ 3 q 2 q 1 ) ( p ¯ 3 p 2 p 1 ) ] , 𝒩 = | μ | 2 + | λ | 2 + | ν | 2 + 2 Re ( μ ¯ λ q 1 p 1 + μ ¯ ν q 2 p 2 + λ ¯ ν q 3 p 3 ) .
| Ψ = μ | α 1 | β 1 + ν | α 3 | β 3 ,
I lin = 4 | μ ν | 2 ( 1 | p 2 | 2 ) ( 1 | q 2 | 2 ) ( | μ | 2 + | ν | 2 + 2 Re ( μ ¯ ν q 2 p 2 ) ) 2 .
I lin 4 | μ ν | 2 sin 2 θ 1 sin 2 θ 2 ( | μ | 2 + | ν | 2 2 | μ ν | cos θ 1 cos θ 2 ) 2 .
I lin sin 2 θ 1 sin 2 θ 2 ( 1 cos θ 1 cos θ 2 ) 2 .
I 0 = 2 d d 1 | μ λ | 2 + | μ ν | 2 + | λ ν | 2 ( | μ | 2 + | λ | 2 + | ν | 2 ) 2 .
| Ψ ± = 1 3 ( | 00 ± | 11 + | 22 ) ,
| Ψ = μ | α 1 | 0 + λ | α 2 | 1 + ν | α 3 | 2 .
I lin = 2 d d 1 | μ λ | 2 N 1 2 + | μ ν | 2 N 2 2 + | λ ν | 2 N 3 2 ( | μ | 2 + | λ | 2 + | ν | 2 ) 2 ,
| Ψ ± = | α 1 | β 1 ± | α 2 | β 2 + | α 3 | β 3 .
I lin = 3 ( p 1 2 1 ) 2 + ( p 2 2 1 ) 2 + ( p 3 2 1 ) 2 + 2 ( ( p 2 p 1 p 3 ) 2 ± ( p 1 p 2 p 3 ) 2 ± ( p 3 p 1 p 2 ) 2 ) ( 3 ± 2 p 1 2 + 2 p 2 2 ± 2 p 3 2 ) 2
| ψ = | α | α + e i θ | 2 α | 2 α + | 3 α | 3 α ,
| ϕ = | α | α + e i θ | 2 α | 2 α + e i θ | 3 α | 3 α ,
I lin = 3 2 ( p 2 1 ) 2 + ( p 8 1 ) 2 + cos θ ( p p 5 ) 2 + 6 [ a ( p 4 p 2 ) 2 + b ( p p 5 ) 2 ] ( 3 + 2 ( b + cos θ ) p 2 + 2 a p 8 ) 2 .
| ψ = 1 3 ( | 00 + e i θ | 11 + | 22 )
| Ψ = μ | α 1 | α 1 k | β 1 | β 1 k + λ | α 2 | α 2 | β 2 | β 2 + ν | α 3 | α 3 | β 3 | β 3 .
| Ψ = μ | α 1 | α k k | β 1 | β n k n k + λ | α k + 1 | α 2 k | β n k + 1 | β 2 ( n k ) + ν | α 2 k + 1 | α 3 k | β 2 ( n k ) + 1 | β 3 ( n k ) .
| 0 = | α 1 | α 1 , | 1 = | α 2 | α 2 p 1 k | α 1 | α 1 N 1 ,
| 2 = ( p 1 k p 3 k p 2 k ) | α 1 | α 1 + ( p ¯ 1 k p 2 k p 3 k ) | α 2 | α 2 + N 1 2 | α 3 | α 3 N N 1 ,
p 1 = α 1 | α 2 , p 2 = α 1 | α 3 , p 3 = α 2 | α 3 , N i = 1 | p i | 2 k ,
N = [ 1 | p 1 | 2 k | p 2 | 2 k | p 3 | 2 k + p 1 k p ¯ 2 k p 3 k + p ¯ 1 k p 2 k p ¯ 3 k ] 1 2 .
| 0 ˜ | β 1 | β 1 , | 1 ˜ = | β 2 | β 2 q 1 k | β 1 | β 1 M 1 ,
| 2 ˜ = ( q 1 k q 3 k q 2 k ) | β 1 | β 1 + ( q ¯ 1 k q 2 k q 3 k ) | β 2 | β 2 + M 1 2 | β 3 | β 3 M M 1 ,
q 1 = β 1 | β 2 , q 2 = β 1 | β 3 , q 3 = β 2 | β 3 , M i = 1 | q i | 2 k ,
M = [ 1 | q 1 | 2 k | q 2 | 2 k | q 3 | 2 k + q 1 k q ¯ 2 k q 3 k + q ¯ 1 k q 2 k q p ¯ 3 k ] 1 2 .
I lin = 2 d d 1 Δ 1 k + 2 ( Δ 2 k + Δ 3 k + Δ 4 k ) 𝒩 k 2 ,
Δ 1 k | μ λ | 2 M 1 2 N 1 2 + | μ ν | 2 M 2 2 N 2 2 + | λ ν | 2 M 3 2 N 3 2 Δ 2 k = | μ | 2 Re [ λ ¯ ν ( q ¯ 1 k q 2 k q 3 k ) ( p ¯ 1 k p 2 k p 3 k ) ] Δ 3 k = | λ | 2 Re [ μ ¯ ν ( q 1 k q 3 k q 2 k ) ( p 1 k p 3 k p 2 k ) ] Δ 4 k = | ν | 2 Re [ μ ¯ λ ( q ¯ 3 k q 2 k q 1 k ) ( p ¯ 3 k p 2 k p 1 k ) ] 𝒩 k | μ | 2 + | λ | 2 + | ν | 2 + 2 Re ( μ ¯ λ q 1 k p 1 k + μ ¯ ν q 2 k p 2 k + λ ¯ ν q 3 k p 3 k )
I lin = 3 p 2 4 k + 3 ( 2 p 2 2 k + 3 ) 2 .
| Ψ = | α | α k | α | α k | 2 α | 2 α | 2 α | 2 α + | 3 α | 3 α | 3 α | 3 α ,
I lin = 3 2 ( p 2 k 1 ) 2 + ( p 8 k 1 ) 2 + 2 ( p 4 k p 2 k ) 2 4 ( p k p 5 k ) 2 ( 3 4 p 2 k + 2 p 8 k ) 2 .
| Ψ = | 0 | 0 k | 0 | 0 k | 1 | 1 | 1 | 1 + | 2 | 2 | 2 | 2 .
H = ω 0 a a + ω m b b g a a ( b + b ) ,
U c ( t ) = e i ζ a a t e i κ 2 ( a a ) 2 ( t sin t ) e κ a a ( η b η * b ) t e i b b t ,
| Ψ ( t ) = e | α 1 | 2 2 n = 0 [ α 1 e i ζ t e i κ Im [ β η e i t ] ] n n ! e i κ 2 n 2 ( t sin t ) | n c | β e i t + κ n η m .
| Ψ ( t = 2 π ) c = e | α | 2 2 n = 0 α n n ! e i 2 π κ 2 n 2 | n c ,
| ψ c = 1 2 [ e i π 4 | α c + e i π 4 | α c ] .
| ψ c = μ | α c + λ | α e i π 3 c + ν | α e i π 3 c ,
U BS = B n 1 , n ( sin 1 ( 1 2 ) ) B n 2 , n 1 ( sin 1 ( 1 3 ) ) B 1 , 2 ( sin 1 ( 1 n ) ) ,
| Ψ = U BS | ψ 0 = μ | α | α | α + λ | β | β | β + ν | γ | γ | γ
| Ψ = μ 0 | α A k | α B k + μ 1 | 2 α A k | 2 α B k + μ 2 | 3 α A k | 3 α B k ,
| α | 0 E | η α | 1 η α E ,
| Ψ = ( | η α A k | 1 η α E k | ) ( | η α B k | 1 η α E k ) + e i θ ( | η 3 α A k | 1 η 3 α E k ) ( | η 3 α B k | 1 η 3 α E k ) .
| 0 | η α A k , | 1 | η 3 α A k e 2 η | α | 2 | η α A k 1 e 4 η | α | 2 ,
C = 1 e 4 k η | α | 2 1 + cos ( θ ) e 4 k | α | 2 e 4 k ( 1 η ) | α | 2 .
| Ψ d n = μ 0 | α 1 | α n + μ 1 | β 1 | β n + + μ d 1 | γ 1 | γ n .
| Ψ d n = i = 0 d 1 μ i | α i A | β i B .
| e k | α k A i = 0 k 1 e i | α k A | e i | α k A i = 0 k 1 e i | α k A | e i = | α k A i = 0 k 1 e i | α k A | e i N k ,
| f k = | β k B i = 0 k 1 f i | β k B | f i | β k B i = 0 k 1 f i | β k B | f i = | β k B i = 0 k 1 f i | β k B | f i M k ,
ρ 1 = ( I 0 | ) | Ψ Ψ | ( I 0 | ) + ( I 1 | ) | Ψ Ψ | ( I 1 | ) + ( I 2 | ) | Ψ Ψ | ( I 2 | ) .
ρ ˜ 1 = [ | μ | 2 + | λ | 2 | p 1 | 2 + | ν | 2 | p 2 | 2 + μ λ ¯ q ¯ 1 p ¯ 1 + μ ν ¯ q ¯ 2 p ¯ 2 + λ ν ¯ q ¯ 3 p 1 p ¯ 2 + μ λ ¯ q ¯ 1 p ¯ 1 + μ ¯ ν q 2 p 2 + λ ¯ ν q 3 p ¯ 1 p 2 ] | 0 0 | + [ | λ | 2 p 1 N 1 + μ λ ¯ q ¯ 1 N 1 | ν | 2 p 2 p ¯ 3 p ¯ 2 p 1 N 1 + μ ν ¯ q ¯ 2 ( p ¯ 3 p ¯ 2 p 1 ) N 1 + λ ν ¯ q ¯ 3 p 1 ( p ¯ 3 p ¯ 2 p 1 ) N 1 + λ ¯ ν q 3 N 1 p 2 ] | 0 1 | + [ | λ 2 | p ¯ 1 N 1 + | ν | 2 p ¯ 2 p 3 p 2 p ¯ 1 N 1 μ ¯ λ q 1 N 1 + μ ¯ ν q 2 ( p 3 p 2 p ¯ 1 ) N 1 + λ ¯ ν q 3 p ¯ 1 ( p 3 p 2 p ¯ 1 ) + λ ν ¯ q ¯ 3 N 1 p ¯ 2 ] | 1 0 | + [ | λ | 2 N 1 2 + | ν | 2 | p 3 p 2 p ¯ 1 | 2 N 1 2 + λ ν ¯ q ¯ 3 ( p ¯ 3 p ¯ 2 p 1 ) + λ ¯ ν q 3 ( p 3 p 2 p ¯ 1 ) ] | 1 1 | + [ ( | ν | 2 p 2 N N 1 + ( μ ν ¯ q ¯ 2 + λ ν ¯ q ¯ 3 p 1 ) N N 1 ] | 0 2 | + [ μ ¯ ν q 2 + λ ¯ ν q 3 p ¯ 1 ) N N 1 + | ν | 2 p ¯ 2 N N 1 | 2 0 | + [ | ν | 2 ( p 3 p 2 p ¯ 1 ) N 1 N N 1 ] | 1 2 | + [ | ν | 2 ( p ¯ 3 p ¯ 2 p 1 ) N 1 N N 1 ] | 2 1 | + [ | ν | 2 N 2 N 1 2 ] | 2 2 | .
ρ ˜ 1 = A + B + B ,
A = ( | μ | 2 + | λ | 2 | p 1 | 2 + | ν | 2 | p 2 | 2 ) | 0 0 | + ( | λ | 2 p 1 N 1 + | ν | 2 p 2 p ¯ 3 p ¯ 2 p 1 N 1 ) | 0 1 | + ( | λ | 2 N 1 p ¯ 1 + | ν | 2 p ¯ 2 ( p 3 p 2 p ¯ 1 N 1 ) ) | 1 0 | + [ | λ | 2 N 1 2 + | ν | 2 | p 3 p 2 p ¯ 1 | 2 N 1 2 [ | 1 1 | + | ν | 2 p 2 N N 1 | 0 2 | + | ν | 2 p ¯ 2 N N 1 | 2 0 | + | ν | 2 ( p 3 p 2 p ¯ 1 ) N 1 N N 1 | 1 2 | + | ν | 2 ( p ¯ 3 p ¯ 2 p 1 ) N 1 N N 1 | 2 1 | + | ν | 2 N 2 N 1 2 | 2 2 | ,
B = ( μ λ ¯ q ¯ 1 p ¯ 1 + μ ν ¯ q ¯ 2 p ¯ 2 + λ ν ¯ q ¯ 3 p 1 p ¯ 2 ) | 0 0 | + ( λ ν ¯ q ¯ 3 N 1 p ¯ 2 ) | 1 0 | + ( μ λ ¯ q ¯ 1 N + μ ν ¯ q ¯ 2 ( p ¯ 3 p ¯ 2 p 1 ) N 1 + λ ν ¯ q ¯ 3 p 1 ( p ¯ 3 p ¯ 2 p 1 ) N 1 | 0 1 | + ( λ ν ¯ q ¯ 3 ( p ¯ 3 p ¯ 2 p 1 ) ) | 1 1 | + ( μ ν ¯ q ¯ 2 + λ ν ¯ q ¯ 3 p 1 ) N N 1 ) | 0 2 | .
ρ 1 = ρ ˜ 1 Tr ρ ˜ 1 .
I lin = d d 1 ( 1 Tr ρ ˜ 1 2 Tr 2 ρ ˜ 1 ) = d d 1 Tr 2 ρ ˜ 1 Tr ρ ˜ 1 2 Tr 2 ρ ˜ 1 .
Tr 2 ρ ˜ 1 = Tr 2 A + Tr 2 B + Tr 2 B + 2 Tr A Tr B + 2 Tr A Tr B + 2 Tr B Tr B ,
Tr ρ ˜ 1 2 = Tr A 2 + Tr B 2 + Tr B 2 + 2 Tr ( A B ) + 2 Tr ( A B ) + 2 Tr ( B B ) .
I lin = d ( d 1 ) Tr 2 ρ ˜ 1 [ ( Tr 2 A Tr A 2 ) + ( Tr 2 B Tr B 2 ) + ( Tr 2 B Tr B 2 ) + 2 ( Tr A Tr B Tr ( A B ) ) + 2 ( Tr A Tr B Tr A B ) + 2 ( Tr B Tr B Tr B B ) ] .
Tr 2 A Tr A 2 = 2 [ | μ | 2 | λ | 2 N 1 2 + | μ | 2 | ν | 2 N 2 2 + | λ | 2 | ν | 2 N 3 2 ]
Tr 2 B Tr B 2 = 2 | λ | 2 μ ν ¯ q ¯ 1 q ¯ 3 ( p ¯ 1 p ¯ 3 p ¯ 2 )
Tr 2 B Tr B 2 = 2 | λ | 2 μ ¯ ν q 1 q 3 ( p 1 p 3 p 2 )
Tr A Tr B Tr ( A B ) = | μ | 2 | λ ν ¯ q ¯ 3 ( p 1 p ¯ 2 p ¯ 3 ) | λ | 2 μ ν ¯ q ¯ 2 ( p 1 p ¯ 3 p ¯ 2 ) | ν | 2 μ λ ¯ q ¯ 2 ( p ¯ 2 p 3 p ¯ 1 )
Tr A Tr B Tr ( A B ) = | μ | 2 | λ ¯ ν q 3 ( p ¯ 1 p 2 p 3 ) | λ | 2 μ ¯ ν q 2 ( p 1 p 3 p 2 ) | ν | 2 μ ¯ λ q 2 ( p 2 p ¯ 3 p 1 )
Tr B Tr B Tr ( B B ) = | μ | 2 | λ | 2 | q 1 | 2 N 1 2 | μ | 2 | ν | 2 | q 2 | 2 N 2 2 | λ | 2 | ν | 2 | q 3 | 2 N 3 2 + | μ | 2 λ ¯ ν q ¯ 1 q ¯ 2 ( p ¯ 1 p 2 p 3 ) + | μ | 2 λ ν ¯ q 1 q ¯ 2 ( p ¯ 2 p 1 p ¯ 3 ) + | ν | 2 μ λ ¯ q ¯ 2 q 3 ( p ¯ 2 p 3 p ¯ 1 ) + | ν | 2 μ ¯ λ q 2 q ¯ 3 ( p ¯ 3 p 2 p 1 )
| Ψ = μ | α 1 | α k k | β 1 | β n k n k + λ | α k + 1 | α 2 k | β n k + 1 | β 2 ( n k ) + ν | α 2 k + 1 | α 3 k | β 2 ( n k ) + 1 | β 3 ( n k ) .
| 0 = | α 1 | α k , | 1 = | α k + 1 | α 2 k i = 1 k p i , k + i | α 1 | α k N 1 ,
| 2 = ( i = 1 k p i , k + i i = 1 k p k + i , 2 k + i i = 1 k p i , 2 k + i ) | α 1 | α k N N 1 + ( i = 1 k p ¯ i , k + i i = 1 k p i , 2 k + i p k + i , 2 k + i ) | α k + 1 | α 2 k + N 1 2 | α 2 k + 1 | α 3 k
p i , j = α i | α j , N 1 = 1 | i = 1 k p i , k + i | 2 , N 2 = 1 | i = 1 k p i , 2 k + i | 2 , N 3 = 1 | i = 1 k p k + i , 2 k + i | 2
N = [ 1 | i = 1 k p i , k + i | 2 | i = 1 k p i , 2 k + i | 2 | i = 1 k p k + i , 2 k + i | 2 + i = 1 k p i , k + i i = 1 k p ¯ i , 2 k + i i = 1 k p k + i , 2 k + i + i = 1 k p ¯ i , k + i i = 1 k p i , 2 k + i i = 1 k p ¯ k + i , 2 k + i ] ] 1 2 .
{ | 0 ˜ = | β 1 | β n k , | 1 ˜ = | β n k + 1 | β 2 ( n k ) i = 1 ( n k ) q i , ( n k ) + i | β 1 | β k M 1 ,
| 2 ˜ = ( i = 1 ( n k ) q i , n k + i i = 1 ( n k ) q n k + i , 2 ( n k ) + i i = 1 ( n k ) q i , 2 ( n k ) + i ) | β 1 | β k M M 1 + ( i = 1 ( n k ) q ¯ i , n k + i i = 1 k q i , 2 n k + i i = 1 ( n k ) q n k + i , 2 k + i ) | β n k + 1 | β 2 ( n k ) + N 1 2 | β 2 ( n k ) + 1 | β 3 ( n k ) M M 1 ,
M 1 = 1 | i = 1 ( n k ) q i , n k + i | 2 , M 2 = 1 | i = 1 ( n k ) q i , 2 ( n k ) + i | 2 , M 3 = 1 | i = 1 ( n k ) q n k + i , 2 ( n k ) + i | 2 ,
M = [ 1 | i = 1 ( n k ) q i , n k + i | 2 | i = 1 ( n k ) q i , 2 ( n k ) + i | 2 | i = 1 ( n k ) q k + i , 2 ( n k ) + i | 2 + i = 1 ( n k ) q i , n k + i i = 1 ( n k ) q ¯ i , 2 ( n k ) + i i = 1 k q n k + i , 2 ( n k ) + i i = 1 ( n k ) q ¯ i , ( n k ) + i i = 1 ( n k ) q i , 2 ( n k ) + i i = 1 ( n k ) q ¯ n k + i , 2 ( n k ) + i ] 1 2 .
| Ψ = ( μ + λ i = 1 ( n k ) q i , n k + i i = 1 k p i , k + i + ν i = 1 ( n k ) q i , 2 ( n k ) + i i = 1 k p i , 2 k + i ) | 0 0 ˜ + ( ν M 1 i = 1 k p i , 2 k + i ( i = 1 ( n k ) q n k + i , 2 ( n k ) + i i = 1 ( n k ) q ¯ i , n k + i i = 1 ( n k ) q i , 2 ( n k ) i ) + λ M 1 i = 1 k p i , k + i ) | 0 1 ˜ + ( ν N 1 i = 1 ( n k ) q i , 2 ( n k ) + i ( i = 1 k p k + i , 2 k + i i = 1 k p ¯ i , k + i i = 1 k p i , 2 k + i ) + λ N 1 i = 1 ( n k ) q i , n k + i ) | 1 0 ˜ + ( ν ( i = 1 k p k + i , 2 k + i i = 1 k p ¯ i , k + i i = 1 k p i , 2 k + i ) N 1 ( i = 1 ( n k ) q n k + i , 2 ( n k ) + i i = 1 ( n k ) q ¯ i , n k + i i = 1 ( n k ) q i , 2 ( n k ) + i ) M 1 + λ N 1 M 1 ) | 1 1 ˜ + ν i = 1 k p i , 2 k + i M M 1 | 0 2 ˜ + ν i = 1 ( n k ) q i , 2 ( n k ) + i N N 1 | 2 0 ˜ ( ν i = 1 k p k + i , 2 k + i i = 1 k p ¯ i , k + i i = 1 k p i , 2 k + i N 1 M M 1 ) | 1 2 ˜ + ν i = 1 ( n k ) q n k + i , 2 ( n k ) + i i = 1 ( n k ) q ¯ i , n k + i i = 1 ( n k ) q i , 2 ( n k ) + i M 1 N N 1 | 2 1 ˜ + ν N N 1 M M 1 | 2 2 ˜ .
I l i n = 2 d d 1 Δ 1 n + 2 ( Δ 2 n + Δ 3 n + Δ 4 n ) 𝒩 n 2 ,
Δ 1 n | μ λ | 2 M 1 2 N 1 2 + | μ ν | 2 M 2 2 N 2 2 + | λ ν | 2 M 3 2 N 3 2
Δ 2 n | μ | 2 Re [ λ ¯ ν ( i = 1 ( n k ) q ¯ i , n k + i i = 1 ( n k ) q i , 2 ( n k ) + i i = 1 ( n k ) q n k + i , 2 ( n k ) + i ) ( i = 1 k p ¯ i , k + i i = 1 k p i , 2 k + i i = 1 k p k + i , 2 k + i ) ]
Δ 3 n | λ | 2 Re [ μ ¯ ν ( i = 1 ( n k ) q ¯ i , n k + i i = 1 ( n k ) q n k + i , 2 ( n k ) + i i = 1 ( n k ) q i , 2 ( n k ) + i ) ( i = 1 k p ¯ i , k + i i = 1 k p k + i , 2 k + i i = 1 k p i , 2 k + i ) ]
Δ 4 n | ν | 2 Re [ μ ¯ λ ( i = 1 ( n k ) q ¯ n k + i , 2 ( n k ) + i i = 1 ( n k ) q i , 2 ( n k ) + i i = 1 ( n k ) q i , n k + i ) ( i = 1 k p ¯ k + i , 2 k + i i = 1 k p i , 2 k + i i = 1 k p i , k + i ) ]
𝒩 n | μ | 2 + | λ | 2 + | ν | 2 + 2 Re ( μ ¯ λ ( i = 1 ( n k ) q i , n k + i i = 1 k p i , k + i + μ ¯ ν i = 1 ( n k ) q i , 2 ( n k ) + i i = 1 k p i , 2 k + i + λ ¯ ν i = 1 ( n k ) q n k + i , 2 ( n k ) + i i = 1 k p k + i , 2 k + i )

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