Abstract

Minimal informationally complete positive operator-valued measures (MIC-POVMs) are special kinds of measurement in quantum theory in which the statistics of their $ {d^2} $-outcomes are enough to reconstruct any $ d $-dimensional quantum state. For this reason, MIC-POVMs are referred to as standard measurements for quantum information. Here, we report an experiment with entangled photon pairs that certifies, for what we believe is the first time, a MIC-POVM for qubits following a device-independent protocol (i.e., modeling the state preparation and the measurement devices as black boxes, and using only the statistics of the inputs and outputs). Our certification is achieved under the assumption of freedom of choice, no communication, and fair sampling.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Quantum teleportation protocol with an assistant who prepares amplitude modulated unknown qubits

Sergey A. Podoshvedov
J. Opt. Soc. Am. B 35(4) 861-877 (2018)

Experimental demonstration of Gaussian protocols for one-sided device-independent quantum key distribution

Nathan Walk, Sara Hosseini, Jiao Geng, Oliver Thearle, Jing Yan Haw, Seiji Armstrong, Syed M. Assad, Jiri Janousek, Timothy C. Ralph, Thomas Symul, Howard M. Wiseman, and Ping Koy Lam
Optica 3(6) 634-642 (2016)

Optimality of quantum randomness certification with independent devices

Xin-Wei Fei, Zhen-Qiang Yin, Chao-Han Cui, Wei Huang, Bing-Jie Xu, Shuang Wang, Wei Chen, Yun-Guang Han, Guang-Can Guo, and Zheng-Fu Han
J. Opt. Soc. Am. B 35(9) 2186-2191 (2018)

References

  • View by:
  • |
  • |
  • |

  1. J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements,” J. Math. Phys. 45, 2171 (2004).
    [Crossref]
  2. S. Weigert, “Simple minimal informationally complete measurements for qudits,” Int. J. Mod. Phys. B 20, 1942–1955 (2006).
    [Crossref]
  3. G. Chiribella, G. M. D’Ariano, and D. Schlingemann, “How continuous quantum measurements in finite dimensions are actually discrete,” Phys. Rev. Lett. 98, 190403 (2007).
    [Crossref]
  4. C. M. Caves, C. A. Fuchs, and R. Schack, “Unknown quantum states: the quantum de finetti representation,” J. Math. Phys. 43, 4537–4559 (2002).
    [Crossref]
  5. J. Řeháček, B.-G. Englert, and D. Kaszlikowski, “Minimal qubit tomography,” Phys. Rev. A 70, 052321 (2004).
    [Crossref]
  6. C. A. Fuchs and M. Sasaki, “Squeezing quantum information through a classical channel: measuring the “quantumness” of a set of quantum states,” Quantum Inf. Comput. 3, 377 (2003).
  7. A. Acín, S. Pironio, T. Vértesi, and P. Wittek, “Optimal randomness certification from one entangled bit,” Phys. Rev. A 93, 040102(R) (2016).
    [Crossref]
  8. O. Andersson, P. Badziąg, I. Dumitru, and A. Cabello, “Device-independent certification of two bits of randomness from one entangled bit and Gisin’s elegant Bell inequality,” Phys. Rev. A 97, 012314 (2018).
    [Crossref]
  9. J. Shang, A. Asadian, H. Zhu, and O. Gühne, “Enhanced entanglement criterion via symmetric informationally complete measurements,” Phys. Rev. A 98, 022309 (2018).
    [Crossref]
  10. C. A. Fuchs, “Quantum mechanics as quantum information and only a little more,” 2002, https://arxiv.org/abs/quant-ph/0205039 .
  11. T. Durt, C. Kurtsiefer, A. Lamas-Linares, and A. Ling, “Wigner tomography of two-qubit states and quantum cryptography,” Phys. Rev. A 78, 042338 (2008).
    [Crossref]
  12. Z. E. D. Medendorp, F. A. Torres-Ruiz, L. K. Shalm, G. N. M. Tabia, C. A. Fuchs, and A. M. Steinberg, “Experimental characterization of qutrits using symmetric informationally complete positive operator-valued measurements,” Phys. Rev. A 83, 051801(R) (2011).
    [Crossref]
  13. W. M. Pimenta, B. Marques, T. O. Maciel, R. O. Vianna, A. Delgado, C. Saavedra, and S. Pádua, “Minimum tomography of two entangled qutrits using local measurements of one-qutrit symmetric informationally complete positive operator-valued measure,” Phys. Rev. A 88, 012112 (2013).
    [Crossref]
  14. Z. Hou, J.-F. Tang, J. Shang, H. Zhu, J. Li, Y. Yuan, K.-D. Wu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Deterministic realization of collective measurements via photonic quantum walks,” Nat. Commun. 9, 1414 (2018).
    [Crossref]
  15. Y.-Y. Zhao, N.-K. Yu, P. Kurzyński, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Experimental realization of generalized qubit measurements based on quantum walks,” Phys. Rev. A 91, 042101 (2015).
    [Crossref]
  16. A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
    [Crossref]
  17. A. Acín and L. Masanes, “Certified randomness in quantum physics,” Nature 540, 213–219 (2016).
    [Crossref]
  18. E. S. Gómez, S. Gómez, P. González, G. C. Nas, J. F. Barra, A. Delgado, G. B. Xavier, A. Cabello, M. Kleinmann, T. Vértesi, and G. Lima, “Device-independent certification of a nonprojective qubit measurement,” Phys. Rev. Lett. 117, 260401 (2016).
    [Crossref]
  19. N. Gisin, “Bell inequalities: Many questions, a few answers,” in Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle, W. C. Myrvold and J. Christian, eds. (Springer, 2009), pp. 125–138.
  20. M. Navascués, S. Pironio, and A. Acín, “A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations,” New J. Phys. 10, 073013 (2008).
    [Crossref]
  21. P. Wittek, “Algorithm 950: ncpol2sdpa-sparse semidefinite programming relaxations for polynomial optimization problems of noncommuting variables,” ACM Trans. Math. Softw. 41, 21 (2015).
    [Crossref]
  22. J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308 (1965).
    [Crossref]
  23. T. H. Yang, T. Vertesi, J.-D. Bancal, V. Scarani, and M. Navascues, “Robust and versatile black-box certification of quantum devices,” Phys. Rev. Lett. 113, 040401 (2014).
    [Crossref]
  24. C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 080403 (2015).
    [Crossref]

2018 (3)

O. Andersson, P. Badziąg, I. Dumitru, and A. Cabello, “Device-independent certification of two bits of randomness from one entangled bit and Gisin’s elegant Bell inequality,” Phys. Rev. A 97, 012314 (2018).
[Crossref]

J. Shang, A. Asadian, H. Zhu, and O. Gühne, “Enhanced entanglement criterion via symmetric informationally complete measurements,” Phys. Rev. A 98, 022309 (2018).
[Crossref]

Z. Hou, J.-F. Tang, J. Shang, H. Zhu, J. Li, Y. Yuan, K.-D. Wu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Deterministic realization of collective measurements via photonic quantum walks,” Nat. Commun. 9, 1414 (2018).
[Crossref]

2016 (3)

A. Acín, S. Pironio, T. Vértesi, and P. Wittek, “Optimal randomness certification from one entangled bit,” Phys. Rev. A 93, 040102(R) (2016).
[Crossref]

A. Acín and L. Masanes, “Certified randomness in quantum physics,” Nature 540, 213–219 (2016).
[Crossref]

E. S. Gómez, S. Gómez, P. González, G. C. Nas, J. F. Barra, A. Delgado, G. B. Xavier, A. Cabello, M. Kleinmann, T. Vértesi, and G. Lima, “Device-independent certification of a nonprojective qubit measurement,” Phys. Rev. Lett. 117, 260401 (2016).
[Crossref]

2015 (3)

P. Wittek, “Algorithm 950: ncpol2sdpa-sparse semidefinite programming relaxations for polynomial optimization problems of noncommuting variables,” ACM Trans. Math. Softw. 41, 21 (2015).
[Crossref]

Y.-Y. Zhao, N.-K. Yu, P. Kurzyński, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Experimental realization of generalized qubit measurements based on quantum walks,” Phys. Rev. A 91, 042101 (2015).
[Crossref]

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 080403 (2015).
[Crossref]

2014 (1)

T. H. Yang, T. Vertesi, J.-D. Bancal, V. Scarani, and M. Navascues, “Robust and versatile black-box certification of quantum devices,” Phys. Rev. Lett. 113, 040401 (2014).
[Crossref]

2013 (1)

W. M. Pimenta, B. Marques, T. O. Maciel, R. O. Vianna, A. Delgado, C. Saavedra, and S. Pádua, “Minimum tomography of two entangled qutrits using local measurements of one-qutrit symmetric informationally complete positive operator-valued measure,” Phys. Rev. A 88, 012112 (2013).
[Crossref]

2011 (1)

Z. E. D. Medendorp, F. A. Torres-Ruiz, L. K. Shalm, G. N. M. Tabia, C. A. Fuchs, and A. M. Steinberg, “Experimental characterization of qutrits using symmetric informationally complete positive operator-valued measurements,” Phys. Rev. A 83, 051801(R) (2011).
[Crossref]

2008 (2)

M. Navascués, S. Pironio, and A. Acín, “A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations,” New J. Phys. 10, 073013 (2008).
[Crossref]

T. Durt, C. Kurtsiefer, A. Lamas-Linares, and A. Ling, “Wigner tomography of two-qubit states and quantum cryptography,” Phys. Rev. A 78, 042338 (2008).
[Crossref]

2007 (2)

G. Chiribella, G. M. D’Ariano, and D. Schlingemann, “How continuous quantum measurements in finite dimensions are actually discrete,” Phys. Rev. Lett. 98, 190403 (2007).
[Crossref]

A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
[Crossref]

2006 (1)

S. Weigert, “Simple minimal informationally complete measurements for qudits,” Int. J. Mod. Phys. B 20, 1942–1955 (2006).
[Crossref]

2004 (2)

J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements,” J. Math. Phys. 45, 2171 (2004).
[Crossref]

J. Řeháček, B.-G. Englert, and D. Kaszlikowski, “Minimal qubit tomography,” Phys. Rev. A 70, 052321 (2004).
[Crossref]

2003 (1)

C. A. Fuchs and M. Sasaki, “Squeezing quantum information through a classical channel: measuring the “quantumness” of a set of quantum states,” Quantum Inf. Comput. 3, 377 (2003).

2002 (1)

C. M. Caves, C. A. Fuchs, and R. Schack, “Unknown quantum states: the quantum de finetti representation,” J. Math. Phys. 43, 4537–4559 (2002).
[Crossref]

1965 (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308 (1965).
[Crossref]

Acín, A.

A. Acín, S. Pironio, T. Vértesi, and P. Wittek, “Optimal randomness certification from one entangled bit,” Phys. Rev. A 93, 040102(R) (2016).
[Crossref]

A. Acín and L. Masanes, “Certified randomness in quantum physics,” Nature 540, 213–219 (2016).
[Crossref]

M. Navascués, S. Pironio, and A. Acín, “A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations,” New J. Phys. 10, 073013 (2008).
[Crossref]

A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
[Crossref]

Andersson, O.

O. Andersson, P. Badziąg, I. Dumitru, and A. Cabello, “Device-independent certification of two bits of randomness from one entangled bit and Gisin’s elegant Bell inequality,” Phys. Rev. A 97, 012314 (2018).
[Crossref]

Asadian, A.

J. Shang, A. Asadian, H. Zhu, and O. Gühne, “Enhanced entanglement criterion via symmetric informationally complete measurements,” Phys. Rev. A 98, 022309 (2018).
[Crossref]

Badziag, P.

O. Andersson, P. Badziąg, I. Dumitru, and A. Cabello, “Device-independent certification of two bits of randomness from one entangled bit and Gisin’s elegant Bell inequality,” Phys. Rev. A 97, 012314 (2018).
[Crossref]

Bancal, J.-D.

T. H. Yang, T. Vertesi, J.-D. Bancal, V. Scarani, and M. Navascues, “Robust and versatile black-box certification of quantum devices,” Phys. Rev. Lett. 113, 040401 (2014).
[Crossref]

Barra, J. F.

E. S. Gómez, S. Gómez, P. González, G. C. Nas, J. F. Barra, A. Delgado, G. B. Xavier, A. Cabello, M. Kleinmann, T. Vértesi, and G. Lima, “Device-independent certification of a nonprojective qubit measurement,” Phys. Rev. Lett. 117, 260401 (2016).
[Crossref]

Blume-Kohout, R.

J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements,” J. Math. Phys. 45, 2171 (2004).
[Crossref]

Brunner, N.

A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
[Crossref]

Cabello, A.

O. Andersson, P. Badziąg, I. Dumitru, and A. Cabello, “Device-independent certification of two bits of randomness from one entangled bit and Gisin’s elegant Bell inequality,” Phys. Rev. A 97, 012314 (2018).
[Crossref]

E. S. Gómez, S. Gómez, P. González, G. C. Nas, J. F. Barra, A. Delgado, G. B. Xavier, A. Cabello, M. Kleinmann, T. Vértesi, and G. Lima, “Device-independent certification of a nonprojective qubit measurement,” Phys. Rev. Lett. 117, 260401 (2016).
[Crossref]

Caves, C. M.

J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements,” J. Math. Phys. 45, 2171 (2004).
[Crossref]

C. M. Caves, C. A. Fuchs, and R. Schack, “Unknown quantum states: the quantum de finetti representation,” J. Math. Phys. 43, 4537–4559 (2002).
[Crossref]

Chiribella, G.

G. Chiribella, G. M. D’Ariano, and D. Schlingemann, “How continuous quantum measurements in finite dimensions are actually discrete,” Phys. Rev. Lett. 98, 190403 (2007).
[Crossref]

D’Ariano, G. M.

G. Chiribella, G. M. D’Ariano, and D. Schlingemann, “How continuous quantum measurements in finite dimensions are actually discrete,” Phys. Rev. Lett. 98, 190403 (2007).
[Crossref]

Delgado, A.

E. S. Gómez, S. Gómez, P. González, G. C. Nas, J. F. Barra, A. Delgado, G. B. Xavier, A. Cabello, M. Kleinmann, T. Vértesi, and G. Lima, “Device-independent certification of a nonprojective qubit measurement,” Phys. Rev. Lett. 117, 260401 (2016).
[Crossref]

W. M. Pimenta, B. Marques, T. O. Maciel, R. O. Vianna, A. Delgado, C. Saavedra, and S. Pádua, “Minimum tomography of two entangled qutrits using local measurements of one-qutrit symmetric informationally complete positive operator-valued measure,” Phys. Rev. A 88, 012112 (2013).
[Crossref]

Dumitru, I.

O. Andersson, P. Badziąg, I. Dumitru, and A. Cabello, “Device-independent certification of two bits of randomness from one entangled bit and Gisin’s elegant Bell inequality,” Phys. Rev. A 97, 012314 (2018).
[Crossref]

Durt, T.

T. Durt, C. Kurtsiefer, A. Lamas-Linares, and A. Ling, “Wigner tomography of two-qubit states and quantum cryptography,” Phys. Rev. A 78, 042338 (2008).
[Crossref]

Englert, B.-G.

J. Řeháček, B.-G. Englert, and D. Kaszlikowski, “Minimal qubit tomography,” Phys. Rev. A 70, 052321 (2004).
[Crossref]

Fuchs, C. A.

Z. E. D. Medendorp, F. A. Torres-Ruiz, L. K. Shalm, G. N. M. Tabia, C. A. Fuchs, and A. M. Steinberg, “Experimental characterization of qutrits using symmetric informationally complete positive operator-valued measurements,” Phys. Rev. A 83, 051801(R) (2011).
[Crossref]

C. A. Fuchs and M. Sasaki, “Squeezing quantum information through a classical channel: measuring the “quantumness” of a set of quantum states,” Quantum Inf. Comput. 3, 377 (2003).

C. M. Caves, C. A. Fuchs, and R. Schack, “Unknown quantum states: the quantum de finetti representation,” J. Math. Phys. 43, 4537–4559 (2002).
[Crossref]

Gisin, N.

A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
[Crossref]

N. Gisin, “Bell inequalities: Many questions, a few answers,” in Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle, W. C. Myrvold and J. Christian, eds. (Springer, 2009), pp. 125–138.

Gómez, E. S.

E. S. Gómez, S. Gómez, P. González, G. C. Nas, J. F. Barra, A. Delgado, G. B. Xavier, A. Cabello, M. Kleinmann, T. Vértesi, and G. Lima, “Device-independent certification of a nonprojective qubit measurement,” Phys. Rev. Lett. 117, 260401 (2016).
[Crossref]

Gómez, S.

E. S. Gómez, S. Gómez, P. González, G. C. Nas, J. F. Barra, A. Delgado, G. B. Xavier, A. Cabello, M. Kleinmann, T. Vértesi, and G. Lima, “Device-independent certification of a nonprojective qubit measurement,” Phys. Rev. Lett. 117, 260401 (2016).
[Crossref]

González, P.

E. S. Gómez, S. Gómez, P. González, G. C. Nas, J. F. Barra, A. Delgado, G. B. Xavier, A. Cabello, M. Kleinmann, T. Vértesi, and G. Lima, “Device-independent certification of a nonprojective qubit measurement,” Phys. Rev. Lett. 117, 260401 (2016).
[Crossref]

Gühne, O.

J. Shang, A. Asadian, H. Zhu, and O. Gühne, “Enhanced entanglement criterion via symmetric informationally complete measurements,” Phys. Rev. A 98, 022309 (2018).
[Crossref]

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 080403 (2015).
[Crossref]

Guo, G.-C.

Z. Hou, J.-F. Tang, J. Shang, H. Zhu, J. Li, Y. Yuan, K.-D. Wu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Deterministic realization of collective measurements via photonic quantum walks,” Nat. Commun. 9, 1414 (2018).
[Crossref]

Y.-Y. Zhao, N.-K. Yu, P. Kurzyński, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Experimental realization of generalized qubit measurements based on quantum walks,” Phys. Rev. A 91, 042101 (2015).
[Crossref]

Hou, Z.

Z. Hou, J.-F. Tang, J. Shang, H. Zhu, J. Li, Y. Yuan, K.-D. Wu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Deterministic realization of collective measurements via photonic quantum walks,” Nat. Commun. 9, 1414 (2018).
[Crossref]

Kaszlikowski, D.

J. Řeháček, B.-G. Englert, and D. Kaszlikowski, “Minimal qubit tomography,” Phys. Rev. A 70, 052321 (2004).
[Crossref]

Kleinmann, M.

E. S. Gómez, S. Gómez, P. González, G. C. Nas, J. F. Barra, A. Delgado, G. B. Xavier, A. Cabello, M. Kleinmann, T. Vértesi, and G. Lima, “Device-independent certification of a nonprojective qubit measurement,” Phys. Rev. Lett. 117, 260401 (2016).
[Crossref]

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 080403 (2015).
[Crossref]

Knips, L.

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 080403 (2015).
[Crossref]

Kurtsiefer, C.

T. Durt, C. Kurtsiefer, A. Lamas-Linares, and A. Ling, “Wigner tomography of two-qubit states and quantum cryptography,” Phys. Rev. A 78, 042338 (2008).
[Crossref]

Kurzynski, P.

Y.-Y. Zhao, N.-K. Yu, P. Kurzyński, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Experimental realization of generalized qubit measurements based on quantum walks,” Phys. Rev. A 91, 042101 (2015).
[Crossref]

Lamas-Linares, A.

T. Durt, C. Kurtsiefer, A. Lamas-Linares, and A. Ling, “Wigner tomography of two-qubit states and quantum cryptography,” Phys. Rev. A 78, 042338 (2008).
[Crossref]

Li, C.-F.

Z. Hou, J.-F. Tang, J. Shang, H. Zhu, J. Li, Y. Yuan, K.-D. Wu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Deterministic realization of collective measurements via photonic quantum walks,” Nat. Commun. 9, 1414 (2018).
[Crossref]

Y.-Y. Zhao, N.-K. Yu, P. Kurzyński, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Experimental realization of generalized qubit measurements based on quantum walks,” Phys. Rev. A 91, 042101 (2015).
[Crossref]

Li, J.

Z. Hou, J.-F. Tang, J. Shang, H. Zhu, J. Li, Y. Yuan, K.-D. Wu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Deterministic realization of collective measurements via photonic quantum walks,” Nat. Commun. 9, 1414 (2018).
[Crossref]

Lima, G.

E. S. Gómez, S. Gómez, P. González, G. C. Nas, J. F. Barra, A. Delgado, G. B. Xavier, A. Cabello, M. Kleinmann, T. Vértesi, and G. Lima, “Device-independent certification of a nonprojective qubit measurement,” Phys. Rev. Lett. 117, 260401 (2016).
[Crossref]

Ling, A.

T. Durt, C. Kurtsiefer, A. Lamas-Linares, and A. Ling, “Wigner tomography of two-qubit states and quantum cryptography,” Phys. Rev. A 78, 042338 (2008).
[Crossref]

Maciel, T. O.

W. M. Pimenta, B. Marques, T. O. Maciel, R. O. Vianna, A. Delgado, C. Saavedra, and S. Pádua, “Minimum tomography of two entangled qutrits using local measurements of one-qutrit symmetric informationally complete positive operator-valued measure,” Phys. Rev. A 88, 012112 (2013).
[Crossref]

Marques, B.

W. M. Pimenta, B. Marques, T. O. Maciel, R. O. Vianna, A. Delgado, C. Saavedra, and S. Pádua, “Minimum tomography of two entangled qutrits using local measurements of one-qutrit symmetric informationally complete positive operator-valued measure,” Phys. Rev. A 88, 012112 (2013).
[Crossref]

Masanes, L.

A. Acín and L. Masanes, “Certified randomness in quantum physics,” Nature 540, 213–219 (2016).
[Crossref]

Massar, S.

A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
[Crossref]

Mead, R.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308 (1965).
[Crossref]

Medendorp, Z. E. D.

Z. E. D. Medendorp, F. A. Torres-Ruiz, L. K. Shalm, G. N. M. Tabia, C. A. Fuchs, and A. M. Steinberg, “Experimental characterization of qutrits using symmetric informationally complete positive operator-valued measurements,” Phys. Rev. A 83, 051801(R) (2011).
[Crossref]

Moroder, T.

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 080403 (2015).
[Crossref]

Nas, G. C.

E. S. Gómez, S. Gómez, P. González, G. C. Nas, J. F. Barra, A. Delgado, G. B. Xavier, A. Cabello, M. Kleinmann, T. Vértesi, and G. Lima, “Device-independent certification of a nonprojective qubit measurement,” Phys. Rev. Lett. 117, 260401 (2016).
[Crossref]

Navascues, M.

T. H. Yang, T. Vertesi, J.-D. Bancal, V. Scarani, and M. Navascues, “Robust and versatile black-box certification of quantum devices,” Phys. Rev. Lett. 113, 040401 (2014).
[Crossref]

Navascués, M.

M. Navascués, S. Pironio, and A. Acín, “A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations,” New J. Phys. 10, 073013 (2008).
[Crossref]

Nelder, J. A.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308 (1965).
[Crossref]

Pádua, S.

W. M. Pimenta, B. Marques, T. O. Maciel, R. O. Vianna, A. Delgado, C. Saavedra, and S. Pádua, “Minimum tomography of two entangled qutrits using local measurements of one-qutrit symmetric informationally complete positive operator-valued measure,” Phys. Rev. A 88, 012112 (2013).
[Crossref]

Pimenta, W. M.

W. M. Pimenta, B. Marques, T. O. Maciel, R. O. Vianna, A. Delgado, C. Saavedra, and S. Pádua, “Minimum tomography of two entangled qutrits using local measurements of one-qutrit symmetric informationally complete positive operator-valued measure,” Phys. Rev. A 88, 012112 (2013).
[Crossref]

Pironio, S.

A. Acín, S. Pironio, T. Vértesi, and P. Wittek, “Optimal randomness certification from one entangled bit,” Phys. Rev. A 93, 040102(R) (2016).
[Crossref]

M. Navascués, S. Pironio, and A. Acín, “A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations,” New J. Phys. 10, 073013 (2008).
[Crossref]

A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
[Crossref]

Rehácek, J.

J. Řeháček, B.-G. Englert, and D. Kaszlikowski, “Minimal qubit tomography,” Phys. Rev. A 70, 052321 (2004).
[Crossref]

Renes, J. M.

J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements,” J. Math. Phys. 45, 2171 (2004).
[Crossref]

Richart, D.

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 080403 (2015).
[Crossref]

Saavedra, C.

W. M. Pimenta, B. Marques, T. O. Maciel, R. O. Vianna, A. Delgado, C. Saavedra, and S. Pádua, “Minimum tomography of two entangled qutrits using local measurements of one-qutrit symmetric informationally complete positive operator-valued measure,” Phys. Rev. A 88, 012112 (2013).
[Crossref]

Sasaki, M.

C. A. Fuchs and M. Sasaki, “Squeezing quantum information through a classical channel: measuring the “quantumness” of a set of quantum states,” Quantum Inf. Comput. 3, 377 (2003).

Scarani, V.

T. H. Yang, T. Vertesi, J.-D. Bancal, V. Scarani, and M. Navascues, “Robust and versatile black-box certification of quantum devices,” Phys. Rev. Lett. 113, 040401 (2014).
[Crossref]

A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
[Crossref]

Schack, R.

C. M. Caves, C. A. Fuchs, and R. Schack, “Unknown quantum states: the quantum de finetti representation,” J. Math. Phys. 43, 4537–4559 (2002).
[Crossref]

Schlingemann, D.

G. Chiribella, G. M. D’Ariano, and D. Schlingemann, “How continuous quantum measurements in finite dimensions are actually discrete,” Phys. Rev. Lett. 98, 190403 (2007).
[Crossref]

Schwemmer, C.

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 080403 (2015).
[Crossref]

Scott, A. J.

J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements,” J. Math. Phys. 45, 2171 (2004).
[Crossref]

Shalm, L. K.

Z. E. D. Medendorp, F. A. Torres-Ruiz, L. K. Shalm, G. N. M. Tabia, C. A. Fuchs, and A. M. Steinberg, “Experimental characterization of qutrits using symmetric informationally complete positive operator-valued measurements,” Phys. Rev. A 83, 051801(R) (2011).
[Crossref]

Shang, J.

J. Shang, A. Asadian, H. Zhu, and O. Gühne, “Enhanced entanglement criterion via symmetric informationally complete measurements,” Phys. Rev. A 98, 022309 (2018).
[Crossref]

Z. Hou, J.-F. Tang, J. Shang, H. Zhu, J. Li, Y. Yuan, K.-D. Wu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Deterministic realization of collective measurements via photonic quantum walks,” Nat. Commun. 9, 1414 (2018).
[Crossref]

Steinberg, A. M.

Z. E. D. Medendorp, F. A. Torres-Ruiz, L. K. Shalm, G. N. M. Tabia, C. A. Fuchs, and A. M. Steinberg, “Experimental characterization of qutrits using symmetric informationally complete positive operator-valued measurements,” Phys. Rev. A 83, 051801(R) (2011).
[Crossref]

Tabia, G. N. M.

Z. E. D. Medendorp, F. A. Torres-Ruiz, L. K. Shalm, G. N. M. Tabia, C. A. Fuchs, and A. M. Steinberg, “Experimental characterization of qutrits using symmetric informationally complete positive operator-valued measurements,” Phys. Rev. A 83, 051801(R) (2011).
[Crossref]

Tang, J.-F.

Z. Hou, J.-F. Tang, J. Shang, H. Zhu, J. Li, Y. Yuan, K.-D. Wu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Deterministic realization of collective measurements via photonic quantum walks,” Nat. Commun. 9, 1414 (2018).
[Crossref]

Torres-Ruiz, F. A.

Z. E. D. Medendorp, F. A. Torres-Ruiz, L. K. Shalm, G. N. M. Tabia, C. A. Fuchs, and A. M. Steinberg, “Experimental characterization of qutrits using symmetric informationally complete positive operator-valued measurements,” Phys. Rev. A 83, 051801(R) (2011).
[Crossref]

Vertesi, T.

T. H. Yang, T. Vertesi, J.-D. Bancal, V. Scarani, and M. Navascues, “Robust and versatile black-box certification of quantum devices,” Phys. Rev. Lett. 113, 040401 (2014).
[Crossref]

Vértesi, T.

E. S. Gómez, S. Gómez, P. González, G. C. Nas, J. F. Barra, A. Delgado, G. B. Xavier, A. Cabello, M. Kleinmann, T. Vértesi, and G. Lima, “Device-independent certification of a nonprojective qubit measurement,” Phys. Rev. Lett. 117, 260401 (2016).
[Crossref]

A. Acín, S. Pironio, T. Vértesi, and P. Wittek, “Optimal randomness certification from one entangled bit,” Phys. Rev. A 93, 040102(R) (2016).
[Crossref]

Vianna, R. O.

W. M. Pimenta, B. Marques, T. O. Maciel, R. O. Vianna, A. Delgado, C. Saavedra, and S. Pádua, “Minimum tomography of two entangled qutrits using local measurements of one-qutrit symmetric informationally complete positive operator-valued measure,” Phys. Rev. A 88, 012112 (2013).
[Crossref]

Weigert, S.

S. Weigert, “Simple minimal informationally complete measurements for qudits,” Int. J. Mod. Phys. B 20, 1942–1955 (2006).
[Crossref]

Weinfurter, H.

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 080403 (2015).
[Crossref]

Wittek, P.

A. Acín, S. Pironio, T. Vértesi, and P. Wittek, “Optimal randomness certification from one entangled bit,” Phys. Rev. A 93, 040102(R) (2016).
[Crossref]

P. Wittek, “Algorithm 950: ncpol2sdpa-sparse semidefinite programming relaxations for polynomial optimization problems of noncommuting variables,” ACM Trans. Math. Softw. 41, 21 (2015).
[Crossref]

Wu, K.-D.

Z. Hou, J.-F. Tang, J. Shang, H. Zhu, J. Li, Y. Yuan, K.-D. Wu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Deterministic realization of collective measurements via photonic quantum walks,” Nat. Commun. 9, 1414 (2018).
[Crossref]

Xavier, G. B.

E. S. Gómez, S. Gómez, P. González, G. C. Nas, J. F. Barra, A. Delgado, G. B. Xavier, A. Cabello, M. Kleinmann, T. Vértesi, and G. Lima, “Device-independent certification of a nonprojective qubit measurement,” Phys. Rev. Lett. 117, 260401 (2016).
[Crossref]

Xiang, G.-Y.

Z. Hou, J.-F. Tang, J. Shang, H. Zhu, J. Li, Y. Yuan, K.-D. Wu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Deterministic realization of collective measurements via photonic quantum walks,” Nat. Commun. 9, 1414 (2018).
[Crossref]

Y.-Y. Zhao, N.-K. Yu, P. Kurzyński, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Experimental realization of generalized qubit measurements based on quantum walks,” Phys. Rev. A 91, 042101 (2015).
[Crossref]

Yang, T. H.

T. H. Yang, T. Vertesi, J.-D. Bancal, V. Scarani, and M. Navascues, “Robust and versatile black-box certification of quantum devices,” Phys. Rev. Lett. 113, 040401 (2014).
[Crossref]

Yu, N.-K.

Y.-Y. Zhao, N.-K. Yu, P. Kurzyński, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Experimental realization of generalized qubit measurements based on quantum walks,” Phys. Rev. A 91, 042101 (2015).
[Crossref]

Yuan, Y.

Z. Hou, J.-F. Tang, J. Shang, H. Zhu, J. Li, Y. Yuan, K.-D. Wu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Deterministic realization of collective measurements via photonic quantum walks,” Nat. Commun. 9, 1414 (2018).
[Crossref]

Zhao, Y.-Y.

Y.-Y. Zhao, N.-K. Yu, P. Kurzyński, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Experimental realization of generalized qubit measurements based on quantum walks,” Phys. Rev. A 91, 042101 (2015).
[Crossref]

Zhu, H.

Z. Hou, J.-F. Tang, J. Shang, H. Zhu, J. Li, Y. Yuan, K.-D. Wu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Deterministic realization of collective measurements via photonic quantum walks,” Nat. Commun. 9, 1414 (2018).
[Crossref]

J. Shang, A. Asadian, H. Zhu, and O. Gühne, “Enhanced entanglement criterion via symmetric informationally complete measurements,” Phys. Rev. A 98, 022309 (2018).
[Crossref]

ACM Trans. Math. Softw. (1)

P. Wittek, “Algorithm 950: ncpol2sdpa-sparse semidefinite programming relaxations for polynomial optimization problems of noncommuting variables,” ACM Trans. Math. Softw. 41, 21 (2015).
[Crossref]

Comput. J. (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308 (1965).
[Crossref]

Int. J. Mod. Phys. B (1)

S. Weigert, “Simple minimal informationally complete measurements for qudits,” Int. J. Mod. Phys. B 20, 1942–1955 (2006).
[Crossref]

J. Math. Phys. (2)

C. M. Caves, C. A. Fuchs, and R. Schack, “Unknown quantum states: the quantum de finetti representation,” J. Math. Phys. 43, 4537–4559 (2002).
[Crossref]

J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements,” J. Math. Phys. 45, 2171 (2004).
[Crossref]

Nat. Commun. (1)

Z. Hou, J.-F. Tang, J. Shang, H. Zhu, J. Li, Y. Yuan, K.-D. Wu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Deterministic realization of collective measurements via photonic quantum walks,” Nat. Commun. 9, 1414 (2018).
[Crossref]

Nature (1)

A. Acín and L. Masanes, “Certified randomness in quantum physics,” Nature 540, 213–219 (2016).
[Crossref]

New J. Phys. (1)

M. Navascués, S. Pironio, and A. Acín, “A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations,” New J. Phys. 10, 073013 (2008).
[Crossref]

Phys. Rev. A (8)

Y.-Y. Zhao, N.-K. Yu, P. Kurzyński, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Experimental realization of generalized qubit measurements based on quantum walks,” Phys. Rev. A 91, 042101 (2015).
[Crossref]

T. Durt, C. Kurtsiefer, A. Lamas-Linares, and A. Ling, “Wigner tomography of two-qubit states and quantum cryptography,” Phys. Rev. A 78, 042338 (2008).
[Crossref]

Z. E. D. Medendorp, F. A. Torres-Ruiz, L. K. Shalm, G. N. M. Tabia, C. A. Fuchs, and A. M. Steinberg, “Experimental characterization of qutrits using symmetric informationally complete positive operator-valued measurements,” Phys. Rev. A 83, 051801(R) (2011).
[Crossref]

W. M. Pimenta, B. Marques, T. O. Maciel, R. O. Vianna, A. Delgado, C. Saavedra, and S. Pádua, “Minimum tomography of two entangled qutrits using local measurements of one-qutrit symmetric informationally complete positive operator-valued measure,” Phys. Rev. A 88, 012112 (2013).
[Crossref]

J. Řeháček, B.-G. Englert, and D. Kaszlikowski, “Minimal qubit tomography,” Phys. Rev. A 70, 052321 (2004).
[Crossref]

A. Acín, S. Pironio, T. Vértesi, and P. Wittek, “Optimal randomness certification from one entangled bit,” Phys. Rev. A 93, 040102(R) (2016).
[Crossref]

O. Andersson, P. Badziąg, I. Dumitru, and A. Cabello, “Device-independent certification of two bits of randomness from one entangled bit and Gisin’s elegant Bell inequality,” Phys. Rev. A 97, 012314 (2018).
[Crossref]

J. Shang, A. Asadian, H. Zhu, and O. Gühne, “Enhanced entanglement criterion via symmetric informationally complete measurements,” Phys. Rev. A 98, 022309 (2018).
[Crossref]

Phys. Rev. Lett. (5)

G. Chiribella, G. M. D’Ariano, and D. Schlingemann, “How continuous quantum measurements in finite dimensions are actually discrete,” Phys. Rev. Lett. 98, 190403 (2007).
[Crossref]

A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
[Crossref]

E. S. Gómez, S. Gómez, P. González, G. C. Nas, J. F. Barra, A. Delgado, G. B. Xavier, A. Cabello, M. Kleinmann, T. Vértesi, and G. Lima, “Device-independent certification of a nonprojective qubit measurement,” Phys. Rev. Lett. 117, 260401 (2016).
[Crossref]

T. H. Yang, T. Vertesi, J.-D. Bancal, V. Scarani, and M. Navascues, “Robust and versatile black-box certification of quantum devices,” Phys. Rev. Lett. 113, 040401 (2014).
[Crossref]

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 080403 (2015).
[Crossref]

Quantum Inf. Comput. (1)

C. A. Fuchs and M. Sasaki, “Squeezing quantum information through a classical channel: measuring the “quantumness” of a set of quantum states,” Quantum Inf. Comput. 3, 377 (2003).

Other (2)

C. A. Fuchs, “Quantum mechanics as quantum information and only a little more,” 2002, https://arxiv.org/abs/quant-ph/0205039 .

N. Gisin, “Bell inequalities: Many questions, a few answers,” in Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle, W. C. Myrvold and J. Christian, eds. (Springer, 2009), pp. 125–138.

Supplementary Material (1)

NameDescription
» Supplement 1       Supplementary experimental and theoretical details—Latex source

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1. Scenario considered in our experiment consists of two parties, Alice and Bob, who perform local measurements on maximally entangled pairs of qubits. See further details in the text.
Fig. 2.
Fig. 2. Experimental setup. The following components are used: a beta-barium borate nonlinear crystal (BBO), 3 nm narrow spectral filters (SF), single-mode optical fibers (SMF), phase plates (PP), lambda half-wave plates (HWP), lambda quarter-wave plates (QWP), polarizing beam splitters (PBS), multimode optical fibers (MMF), and single-photon detectors (DET).
Fig. 3.
Fig. 3. Tomography of the prepared maximally entangled state. Real (left) and imaginary (right) parts.
Fig. 4.
Fig. 4. Reconstruction of eight of Alice’s local qubit states, conditioned on Bob’s setting and outcome, as obtained from standard projective tomography (left) and MIC-POVM tomography (right).

Tables (2)

Tables Icon

Table 1. Experimental Values for the Combinations of Settings Needed to Test the Elegant Bell Inequality

Tables Icon

Table 2. Experimental Values for the Probabilities of the Outcomes of the MIC-POVM that are Most Relevant to the DI Certification Protocol [See Eq. (1)]

Equations (10)

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

β e l m = β e l k i = 1 4 P ( a = i , b = + 1 | x = 4 , y = i ) ,
β e l = E 11 + E 12 E 13 E 14 + E 21 E 22 + E 23 E 24 + E 31 E 32 E 33 + E 34 ,
A 1 = σ x , B 1 = 1 3 ( σ x σ y + σ z ) , A 2 = σ y , B 2 = 1 3 ( σ x + σ y σ z ) , A 3 = σ z , B 3 = 1 3 ( σ x σ y σ z ) , B 4 = 1 3 ( σ x + σ y + σ z ) ,
A 4 , 1 = 1 2 ( α β ( 1 + i ) β ( 1 + i ) 1 α ) , A 4 , 2 = 1 2 ( 1 α β ( 1 + i ) β ( 1 + i ) α ) , A 4 , 3 = 1 2 ( 1 α β ( 1 i ) β ( 1 + i ) α ) , A 4 , 4 = 1 2 ( α β ( 1 + i ) β ( 1 i ) 1 α ) ,
x = 1 3 y = 1 4 γ xy E xy k y = 1 4 a = 1 4 b = ± 1 γ bxy P ( a , b | 4 , y ) .
max j = 1 , 2 , 3 , 4 [ x = 1 3 y = 1 4 γ xy E xy k y = 1 4 a j b = ± 1 γ bxy P ( a , b | 4 , y ) ] .
β I C = 0.9541 E 11 + 0.9917 E 12 0.9767 E 13 1.0064 E 14 + 0.9514 E 21 0.9921 E 22 + 0.8211 E 23 1.0237 E 24 + 1.0641 E 31 1.0044 E 32 1.0579 E 33 + 1.1563 E 34 3 [ 1.2068 P ( 1 , 1 | 4 , 1 ) 0.0374 P ( 1 , 2 | 4 , 1 ) 0.0034 P ( 2 , 1 | 4 , 1 ) + 0.0140 P ( 2 , 2 | 4 , 1 ) + 0.0006 P ( 3 , 1 | 4 , 1 ) + 0.0268 P ( 3 , 2 | 4 , 1 ) 0.0163 P ( 4 , 1 | 4 , 1 ) 0.0155 P ( 4 , 2 | 4 , 1 ) 0.0033 P ( 1 , 1 | 4 , 2 ) + 0.0184 P ( 1 , 2 | 4 , 2 ) + 1.1156 P ( 2 , 1 | 4 , 2 ) 0.0046 P ( 2 , 2 | 4 , 2 ) 0.0125 P ( 3 , 1 | 4 , 2 ) + 0.0401 P ( 3 , 2 | 4 , 2 ) 0.0175 P ( 4 , 1 | 4 , 2 ) 0.0240 P ( 4 , 2 | 4 , 2 ) 0.0108 P ( 1 , 1 | 4 , 3 ) + 0.0153 P ( 1 , 2 | 4 , 3 ) 0.1195 P ( 2 , 1 | 4 , 3 ) + 0.1752 P ( 2 , 2 | 4 , 3 ) + 0.6201 P ( 3 , 1 | 4 , 3 ) + 0.0149 P ( 3 , 2 | 4 , 3 ) 0.0399 P ( 4 , 1 | 4 , 3 ) + 0.0527 P ( 4 , 2 | 4 , 3 ) + 0.0058 P ( 1 , 1 | 4 , 4 ) 0.0149 P ( 1 , 2 | 4 , 4 ) + 0.0025 P ( 2 , 1 | 4 , 4 ) + 0.0205 P ( 2 , 2 | 4 , 4 ) + 0.0150 P ( 3 , 1 | 4 , 4 ) + 0.0212 P ( 3 , 2 | 4 , 4 ) + 0.9565 P ( 4 , 1 | 4 , 4 ) 0.0023 P ( 4 , 2 | 4 , 4 ) ] .
β I C 3 o u t c o m e 6.8782 Q u a n t u m 6.9883 ,
β I C e x p = 6.960 ± 0.007
s = 3 j = 1 4 f j A j ,

Metrics