Abstract

A fundamental task in photonics is to characterize an unknown optical process, defined by properties such as birefringence, spectral response, thickness and flatness. Among many ways to achieve this, single-photon probes can be used in a method called quantum process tomography (QPT). However, the precision of QPT is limited by unavoidable shot noise when implemented using single-photon probes or laser light. In situations where measurement resources are limited, for example, where the process (sample) to be probed is very delicate such that the exposure to light has a detrimental effect on the sample, it becomes essential to overcome this precision limit. Here we devise a scheme for process tomography with a quantum-enhanced precision by drawing upon techniques from quantum metrology. We implement a proof-of-principle experiment to demonstrate this scheme—four-photon quantum states are used to probe an unknown arbitrary unitary process realized with an arbitrary polarization rotation. Our results show a substantial reduction of statistical fluctuations compared to traditional QPT methods—in the ideal case, one four-photon probe state yields the same amount of statistical information as twelve single probe photons.

© 2015 Optical Society of America

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  1. J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, A. G. White, “Quantum process tomography of a controlled-NOT gate,” Phys. Rev. Lett. 93, 080502 (2004).
    [Crossref]
  2. J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
    [Crossref]
  3. J. Yuen-Zhou, J. J. Krich, M. Mohseni, A. Aspuru-Guzik, “Quantum state and process tomography of energy transfer systems via ultrafast spectroscopy,” Proc. Natl. Acad. Sci. USA 108, 17615–17620 (2011).
  4. M. Riebe, K. Kim, P. Schindler, T. Monz, P. O. Schmidt, T. K. Körber, W. Hänsel, H. Häffner, C. F. Roos, R. Blatt, “Process tomography of ion trap quantum gates,” Phys. Rev. Lett. 97, 220407 (2006).
    [Crossref]
  5. A. M. Childs, I. L. Chuang, D. W. Leung, “Realization of quantum process tomography in NMR,” Phys. Rev. A 64, 012314 (2001).
    [Crossref]
  6. R. C. Bialczak, M. Ansmann, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, H. Wang, J. Wenner, M. Steffen, A. N. Cleland, J. M. Martinis, “Quantum process tomography of a universal entangling gate implemented with Josephson phase qubits,” Nat. Phys. 6, 409–413 (2010).
    [Crossref]
  7. M. Howard, J. Twamley, C. Wittmann, T. Gaebel, F. Jelezko, J. Wrachtrup, “Quantum process tomography and Linblad estimation of a solid-state qubit,” New J. Phys. 8, 33 (2006).
    [Crossref]
  8. Y. Nambu, K. Usami, A. Tomita, S. Ishizaka, T. Hiroshima, Y. Tsuda, K. Matsumoto, K. Nakamura, “Experimental investigation of pulsed entangled photons and photonic quantum channels,” Proc. SPIE 4917, 13–24 (2002).
  9. F. De Martini, A. Mazzei, M. Ricci, G. M. D’Ariano, “Exploiting quantum parallelism of entanglement for a complete experimental quantum characterization of a single-qubit device,” Phys. Rev. A 67, 062307 (2003).
    [Crossref]
  10. J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, A. G. White, “Ancilla-assisted quantum process tomography,” Phys. Rev. Lett. 90, 193601 (2003).
    [Crossref]
  11. M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, A. I. Lvovsky, “Complete characterization of a quantum-optical processes,” Science 322, 563–566 (2008).
    [Crossref]
  12. M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
    [Crossref]
  13. D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105, 150401 (2010).
    [Crossref]
  14. A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
    [Crossref]
  15. A. Laing, J. L. O’Brien, “Super-stable tomography of any linear optical device,” arXiv:1208.2868 (2012).
  16. S. Rahimi-Keshari, M. A. Broome, R. Fickler, A. Fedrizzi, T. C. Ralph, A. G. White, “Direct characterization of linear-optical networks,” Opt. Express 21, 13450–13458 (2013).
    [Crossref]
  17. T. Sugiyama, P. S. Turner, M. Murao, “Adaptive experimental design for one-qubit state estimation with finite data based on a statistical update criterion,” Phys. Rev. A 85, 052107 (2012).
    [Crossref]
  18. D. H. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohout, A. M. Steinberg, “Adaptive quantum state tomography improves accuracy quadratically,” Phys. Rev. Lett. 111, 183601 (2013).
    [Crossref]
  19. V. Giovannetti, S. Lloyd, L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
    [Crossref]
  20. S. Feng, O. Pfister, “Sub-shot-noise heterodyne polarimetry,” Opt. Lett. 29, 2800–2802 (2004).
    [Crossref]
  21. M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
    [Crossref]
  22. F. Wolfgramm, C. Vitelli, F. A. Beduini, N. Godbout, M. W. Mitchell, “Entanglement-enhanced probing of a delicate material system,” Nat. Photonics 7, 28–32 (2012).
    [Crossref]
  23. P. C. Humphreys, M. Barbieri, A. Datta, I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
    [Crossref]
  24. N. Spagnolo, L. Aparo, C. Vitelli, A. Crespi, R. Ramponi, R. Osellame, P. Mataloni, F. Sciarrino, “Quantum interferometry with three-dimensional geometry,” Sci. Rep. 2, 862 (2012).
    [Crossref]
  25. J. Kahn, “Fast rate estimation of a unitary operation in SU(d),” Phys. Rev. A 75, 022326 (2007).
    [Crossref]
  26. M. A. Ballester, “Optimal estimation of SU(d) using exact and approximate 2-designs,” arXiv:quant-ph/0507073 (2005).
  27. A. Acin, E. Jane, G. Vidal, “Optimal estimation of quantum dynamics,” Phys. Rev. A 64, 050302 (2001).
    [Crossref]
  28. M. Hayashi, “Parallel treatment of estimation of SU(2) and phase estimation,” Phys. Lett. A 354, 183–189 (2006).
    [Crossref]
  29. S. I. Knysh, G. A. Durkin, “Estimation of phase and diffusion: combining quantum statistics and classical noise,” arXiv:1307.0470 (2013).
  30. P. J. Crowley, A. Datta, M. Barbieri, I. A. Walmsley, “Multiparameter quantum metrology,” arXiv:1206.0043 (2012).
  31. L. A. Rozema, D. H. Mahler, R. Blume-Kohout, A. M. Steinberg, “Optimizing the choice of spin-squeezed states for detecting and characterizing quantum processes,” Phys. Rev. X 4, 041025 (2014).
    [Crossref]
  32. M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).
  33. R. Simon, N. Mukunda, “Minimal three-component SU(2) gadget for polarization optics,” Phys. Lett. A 143, 165–169 (1990).
    [Crossref]
  34. S. Barnett, P. M. Radmore, Methods in Theoretical Quantum Optics (Oxford University, 2002).
  35. M. J. Holland, K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355–1358 (1993).
    [Crossref]
  36. J. Schwinger, Quantum Theory of Angular Momentum (Academic, 1965).
  37. J. J. Sakurai, Modern Quantum Mechanics (Addison Wesley, 1994), Revised Edition, Chap. 3.8.
  38. F. W. Olver, NIST Handbook of Mathematical Functions (Cambridge University, 2010).
  39. A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A 83, 063836 (2011).
    [Crossref]
  40. J. C. F. Matthews, X.-Q. Zhou, P. J. Shadbolt, H. Cable, D. J. Saunders, G. A. Durkin, G. J. Pryde, J. L. O’Brien, “Practical quantum metrology,” arXiv:1307.4673 (2013).
  41. F. Mezzadri, “How to generate random matrices from the classical compact groups,” Not. AMS 54, 592–604 (2007).
  42. N. Thomas-Peter, B. J. Smith, A. Datta, L. Zhang, U. Dorner, I. A. Walmsley, “Real-world quantum sensors: Evaluating resources for precision measurement,” Phys. Rev. Lett. 107, 113603 (2011).
    [Crossref]
  43. M. Reck, A. Zeilinger, H. J. Bernstein, P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
    [Crossref]
  44. LIGO Scientific Collaboration, “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photonics 7, 613–619 (2013).
    [Crossref]
  45. J. L. O’Brien, “Optical quantum computing,” Science 318, 1567–1570 (2007).
    [Crossref]
  46. S. D. Bartlett, T. Rudolph, R. W. Spekkens, “Reference frames, superselection rules, and quantum information,” Rev. Mod. Phys. 79, 555–609 (2007).
    [Crossref]
  47. G.-Y. Xiang, B. L. Higgins, D. Berry, H. M. Wiseman, G. Pryde, “Entanglement-enhanced measurement of a completely unknown optical phase,” Nat. Photonics 5, 43–47 (2010).
    [Crossref]

2014 (1)

L. A. Rozema, D. H. Mahler, R. Blume-Kohout, A. M. Steinberg, “Optimizing the choice of spin-squeezed states for detecting and characterizing quantum processes,” Phys. Rev. X 4, 041025 (2014).
[Crossref]

2013 (6)

LIGO Scientific Collaboration, “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photonics 7, 613–619 (2013).
[Crossref]

P. C. Humphreys, M. Barbieri, A. Datta, I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref]

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

S. Rahimi-Keshari, M. A. Broome, R. Fickler, A. Fedrizzi, T. C. Ralph, A. G. White, “Direct characterization of linear-optical networks,” Opt. Express 21, 13450–13458 (2013).
[Crossref]

D. H. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohout, A. M. Steinberg, “Adaptive quantum state tomography improves accuracy quadratically,” Phys. Rev. Lett. 111, 183601 (2013).
[Crossref]

2012 (3)

T. Sugiyama, P. S. Turner, M. Murao, “Adaptive experimental design for one-qubit state estimation with finite data based on a statistical update criterion,” Phys. Rev. A 85, 052107 (2012).
[Crossref]

F. Wolfgramm, C. Vitelli, F. A. Beduini, N. Godbout, M. W. Mitchell, “Entanglement-enhanced probing of a delicate material system,” Nat. Photonics 7, 28–32 (2012).
[Crossref]

N. Spagnolo, L. Aparo, C. Vitelli, A. Crespi, R. Ramponi, R. Osellame, P. Mataloni, F. Sciarrino, “Quantum interferometry with three-dimensional geometry,” Sci. Rep. 2, 862 (2012).
[Crossref]

2011 (5)

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref]

N. Thomas-Peter, B. J. Smith, A. Datta, L. Zhang, U. Dorner, I. A. Walmsley, “Real-world quantum sensors: Evaluating resources for precision measurement,” Phys. Rev. Lett. 107, 113603 (2011).
[Crossref]

A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A 83, 063836 (2011).
[Crossref]

V. Giovannetti, S. Lloyd, L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

J. Yuen-Zhou, J. J. Krich, M. Mohseni, A. Aspuru-Guzik, “Quantum state and process tomography of energy transfer systems via ultrafast spectroscopy,” Proc. Natl. Acad. Sci. USA 108, 17615–17620 (2011).

2010 (4)

R. C. Bialczak, M. Ansmann, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, H. Wang, J. Wenner, M. Steffen, A. N. Cleland, J. M. Martinis, “Quantum process tomography of a universal entangling gate implemented with Josephson phase qubits,” Nat. Phys. 6, 409–413 (2010).
[Crossref]

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105, 150401 (2010).
[Crossref]

G.-Y. Xiang, B. L. Higgins, D. Berry, H. M. Wiseman, G. Pryde, “Entanglement-enhanced measurement of a completely unknown optical phase,” Nat. Photonics 5, 43–47 (2010).
[Crossref]

2008 (1)

M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, A. I. Lvovsky, “Complete characterization of a quantum-optical processes,” Science 322, 563–566 (2008).
[Crossref]

2007 (4)

F. Mezzadri, “How to generate random matrices from the classical compact groups,” Not. AMS 54, 592–604 (2007).

J. L. O’Brien, “Optical quantum computing,” Science 318, 1567–1570 (2007).
[Crossref]

S. D. Bartlett, T. Rudolph, R. W. Spekkens, “Reference frames, superselection rules, and quantum information,” Rev. Mod. Phys. 79, 555–609 (2007).
[Crossref]

J. Kahn, “Fast rate estimation of a unitary operation in SU(d),” Phys. Rev. A 75, 022326 (2007).
[Crossref]

2006 (3)

M. Hayashi, “Parallel treatment of estimation of SU(2) and phase estimation,” Phys. Lett. A 354, 183–189 (2006).
[Crossref]

M. Howard, J. Twamley, C. Wittmann, T. Gaebel, F. Jelezko, J. Wrachtrup, “Quantum process tomography and Linblad estimation of a solid-state qubit,” New J. Phys. 8, 33 (2006).
[Crossref]

M. Riebe, K. Kim, P. Schindler, T. Monz, P. O. Schmidt, T. K. Körber, W. Hänsel, H. Häffner, C. F. Roos, R. Blatt, “Process tomography of ion trap quantum gates,” Phys. Rev. Lett. 97, 220407 (2006).
[Crossref]

2004 (2)

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, A. G. White, “Quantum process tomography of a controlled-NOT gate,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

S. Feng, O. Pfister, “Sub-shot-noise heterodyne polarimetry,” Opt. Lett. 29, 2800–2802 (2004).
[Crossref]

2003 (2)

F. De Martini, A. Mazzei, M. Ricci, G. M. D’Ariano, “Exploiting quantum parallelism of entanglement for a complete experimental quantum characterization of a single-qubit device,” Phys. Rev. A 67, 062307 (2003).
[Crossref]

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, A. G. White, “Ancilla-assisted quantum process tomography,” Phys. Rev. Lett. 90, 193601 (2003).
[Crossref]

2002 (1)

Y. Nambu, K. Usami, A. Tomita, S. Ishizaka, T. Hiroshima, Y. Tsuda, K. Matsumoto, K. Nakamura, “Experimental investigation of pulsed entangled photons and photonic quantum channels,” Proc. SPIE 4917, 13–24 (2002).

2001 (2)

A. M. Childs, I. L. Chuang, D. W. Leung, “Realization of quantum process tomography in NMR,” Phys. Rev. A 64, 012314 (2001).
[Crossref]

A. Acin, E. Jane, G. Vidal, “Optimal estimation of quantum dynamics,” Phys. Rev. A 64, 050302 (2001).
[Crossref]

1994 (1)

M. Reck, A. Zeilinger, H. J. Bernstein, P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

1993 (1)

M. J. Holland, K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355–1358 (1993).
[Crossref]

1990 (1)

R. Simon, N. Mukunda, “Minimal three-component SU(2) gadget for polarization optics,” Phys. Lett. A 143, 165–169 (1990).
[Crossref]

Acin, A.

A. Acin, E. Jane, G. Vidal, “Optimal estimation of quantum dynamics,” Phys. Rev. A 64, 050302 (2001).
[Crossref]

Almeida, M. P.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref]

Altepeter, J. B.

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, A. G. White, “Ancilla-assisted quantum process tomography,” Phys. Rev. Lett. 90, 193601 (2003).
[Crossref]

Ansmann, M.

R. C. Bialczak, M. Ansmann, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, H. Wang, J. Wenner, M. Steffen, A. N. Cleland, J. M. Martinis, “Quantum process tomography of a universal entangling gate implemented with Josephson phase qubits,” Nat. Phys. 6, 409–413 (2010).
[Crossref]

Aparo, L.

N. Spagnolo, L. Aparo, C. Vitelli, A. Crespi, R. Ramponi, R. Osellame, P. Mataloni, F. Sciarrino, “Quantum interferometry with three-dimensional geometry,” Sci. Rep. 2, 862 (2012).
[Crossref]

Aspuru-Guzik, A.

J. Yuen-Zhou, J. J. Krich, M. Mohseni, A. Aspuru-Guzik, “Quantum state and process tomography of energy transfer systems via ultrafast spectroscopy,” Proc. Natl. Acad. Sci. USA 108, 17615–17620 (2011).

Bachor, H.-A.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

Ballester, M. A.

M. A. Ballester, “Optimal estimation of SU(d) using exact and approximate 2-designs,” arXiv:quant-ph/0507073 (2005).

Barbieri, M.

P. C. Humphreys, M. Barbieri, A. Datta, I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref]

P. J. Crowley, A. Datta, M. Barbieri, I. A. Walmsley, “Multiparameter quantum metrology,” arXiv:1206.0043 (2012).

Barnett, S.

S. Barnett, P. M. Radmore, Methods in Theoretical Quantum Optics (Oxford University, 2002).

Bartlett, S. D.

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

S. D. Bartlett, T. Rudolph, R. W. Spekkens, “Reference frames, superselection rules, and quantum information,” Rev. Mod. Phys. 79, 555–609 (2007).
[Crossref]

Becker, S.

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105, 150401 (2010).
[Crossref]

Beduini, F. A.

F. Wolfgramm, C. Vitelli, F. A. Beduini, N. Godbout, M. W. Mitchell, “Entanglement-enhanced probing of a delicate material system,” Nat. Photonics 7, 28–32 (2012).
[Crossref]

Bernstein, H. J.

M. Reck, A. Zeilinger, H. J. Bernstein, P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

Berry, D.

G.-Y. Xiang, B. L. Higgins, D. Berry, H. M. Wiseman, G. Pryde, “Entanglement-enhanced measurement of a completely unknown optical phase,” Nat. Photonics 5, 43–47 (2010).
[Crossref]

Bertani, P.

M. Reck, A. Zeilinger, H. J. Bernstein, P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

Bialczak, R. C.

R. C. Bialczak, M. Ansmann, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, H. Wang, J. Wenner, M. Steffen, A. N. Cleland, J. M. Martinis, “Quantum process tomography of a universal entangling gate implemented with Josephson phase qubits,” Nat. Phys. 6, 409–413 (2010).
[Crossref]

Blatt, R.

M. Riebe, K. Kim, P. Schindler, T. Monz, P. O. Schmidt, T. K. Körber, W. Hänsel, H. Häffner, C. F. Roos, R. Blatt, “Process tomography of ion trap quantum gates,” Phys. Rev. Lett. 97, 220407 (2006).
[Crossref]

Blume-Kohout, R.

L. A. Rozema, D. H. Mahler, R. Blume-Kohout, A. M. Steinberg, “Optimizing the choice of spin-squeezed states for detecting and characterizing quantum processes,” Phys. Rev. X 4, 041025 (2014).
[Crossref]

D. H. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohout, A. M. Steinberg, “Adaptive quantum state tomography improves accuracy quadratically,” Phys. Rev. Lett. 111, 183601 (2013).
[Crossref]

Bowen, W. P.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

Branning, D.

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, A. G. White, “Ancilla-assisted quantum process tomography,” Phys. Rev. Lett. 90, 193601 (2003).
[Crossref]

Broome, M. A.

S. Rahimi-Keshari, M. A. Broome, R. Fickler, A. Fedrizzi, T. C. Ralph, A. G. White, “Direct characterization of linear-optical networks,” Opt. Express 21, 13450–13458 (2013).
[Crossref]

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref]

Burnett, K.

M. J. Holland, K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355–1358 (1993).
[Crossref]

Cable, H.

J. C. F. Matthews, X.-Q. Zhou, P. J. Shadbolt, H. Cable, D. J. Saunders, G. A. Durkin, G. J. Pryde, J. L. O’Brien, “Practical quantum metrology,” arXiv:1307.4673 (2013).

Chen, K.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Chen, Y.-A.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Childs, A. M.

A. M. Childs, I. L. Chuang, D. W. Leung, “Realization of quantum process tomography in NMR,” Phys. Rev. A 64, 012314 (2001).
[Crossref]

Chuang, I. L.

A. M. Childs, I. L. Chuang, D. W. Leung, “Realization of quantum process tomography in NMR,” Phys. Rev. A 64, 012314 (2001).
[Crossref]

M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

Cleland, A. N.

R. C. Bialczak, M. Ansmann, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, H. Wang, J. Wenner, M. Steffen, A. N. Cleland, J. M. Martinis, “Quantum process tomography of a universal entangling gate implemented with Josephson phase qubits,” Nat. Phys. 6, 409–413 (2010).
[Crossref]

Cramer, M.

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Crespi, A.

N. Spagnolo, L. Aparo, C. Vitelli, A. Crespi, R. Ramponi, R. Osellame, P. Mataloni, F. Sciarrino, “Quantum interferometry with three-dimensional geometry,” Sci. Rep. 2, 862 (2012).
[Crossref]

Crowley, P. J.

P. J. Crowley, A. Datta, M. Barbieri, I. A. Walmsley, “Multiparameter quantum metrology,” arXiv:1206.0043 (2012).

D’Ariano, G. M.

F. De Martini, A. Mazzei, M. Ricci, G. M. D’Ariano, “Exploiting quantum parallelism of entanglement for a complete experimental quantum characterization of a single-qubit device,” Phys. Rev. A 67, 062307 (2003).
[Crossref]

Darabi, A.

D. H. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohout, A. M. Steinberg, “Adaptive quantum state tomography improves accuracy quadratically,” Phys. Rev. Lett. 111, 183601 (2013).
[Crossref]

Daria, V.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

Datta, A.

P. C. Humphreys, M. Barbieri, A. Datta, I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref]

A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A 83, 063836 (2011).
[Crossref]

N. Thomas-Peter, B. J. Smith, A. Datta, L. Zhang, U. Dorner, I. A. Walmsley, “Real-world quantum sensors: Evaluating resources for precision measurement,” Phys. Rev. Lett. 107, 113603 (2011).
[Crossref]

P. J. Crowley, A. Datta, M. Barbieri, I. A. Walmsley, “Multiparameter quantum metrology,” arXiv:1206.0043 (2012).

De Martini, F.

F. De Martini, A. Mazzei, M. Ricci, G. M. D’Ariano, “Exploiting quantum parallelism of entanglement for a complete experimental quantum characterization of a single-qubit device,” Phys. Rev. A 67, 062307 (2003).
[Crossref]

Dorner, U.

N. Thomas-Peter, B. J. Smith, A. Datta, L. Zhang, U. Dorner, I. A. Walmsley, “Real-world quantum sensors: Evaluating resources for precision measurement,” Phys. Rev. Lett. 107, 113603 (2011).
[Crossref]

A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A 83, 063836 (2011).
[Crossref]

Durkin, G. A.

J. C. F. Matthews, X.-Q. Zhou, P. J. Shadbolt, H. Cable, D. J. Saunders, G. A. Durkin, G. J. Pryde, J. L. O’Brien, “Practical quantum metrology,” arXiv:1307.4673 (2013).

S. I. Knysh, G. A. Durkin, “Estimation of phase and diffusion: combining quantum statistics and classical noise,” arXiv:1307.0470 (2013).

Eisert, J.

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105, 150401 (2010).
[Crossref]

Fedrizzi, A.

S. Rahimi-Keshari, M. A. Broome, R. Fickler, A. Fedrizzi, T. C. Ralph, A. G. White, “Direct characterization of linear-optical networks,” Opt. Express 21, 13450–13458 (2013).
[Crossref]

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref]

Feng, S.

Ferrie, C.

D. H. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohout, A. M. Steinberg, “Adaptive quantum state tomography improves accuracy quadratically,” Phys. Rev. Lett. 111, 183601 (2013).
[Crossref]

Fickler, R.

Figueroa, E.

M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, A. I. Lvovsky, “Complete characterization of a quantum-optical processes,” Science 322, 563–566 (2008).
[Crossref]

Flammia, S. T.

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105, 150401 (2010).
[Crossref]

Gaebel, T.

M. Howard, J. Twamley, C. Wittmann, T. Gaebel, F. Jelezko, J. Wrachtrup, “Quantum process tomography and Linblad estimation of a solid-state qubit,” New J. Phys. 8, 33 (2006).
[Crossref]

Gilchrist, A.

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, A. G. White, “Quantum process tomography of a controlled-NOT gate,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

Giovannetti, V.

V. Giovannetti, S. Lloyd, L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

Godbout, N.

F. Wolfgramm, C. Vitelli, F. A. Beduini, N. Godbout, M. W. Mitchell, “Entanglement-enhanced probing of a delicate material system,” Nat. Photonics 7, 28–32 (2012).
[Crossref]

Gross, D.

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105, 150401 (2010).
[Crossref]

Häffner, H.

M. Riebe, K. Kim, P. Schindler, T. Monz, P. O. Schmidt, T. K. Körber, W. Hänsel, H. Häffner, C. F. Roos, R. Blatt, “Process tomography of ion trap quantum gates,” Phys. Rev. Lett. 97, 220407 (2006).
[Crossref]

Hage, B.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

Hänsel, W.

M. Riebe, K. Kim, P. Schindler, T. Monz, P. O. Schmidt, T. K. Körber, W. Hänsel, H. Häffner, C. F. Roos, R. Blatt, “Process tomography of ion trap quantum gates,” Phys. Rev. Lett. 97, 220407 (2006).
[Crossref]

Hayashi, M.

M. Hayashi, “Parallel treatment of estimation of SU(2) and phase estimation,” Phys. Lett. A 354, 183–189 (2006).
[Crossref]

Higgins, B. L.

G.-Y. Xiang, B. L. Higgins, D. Berry, H. M. Wiseman, G. Pryde, “Entanglement-enhanced measurement of a completely unknown optical phase,” Nat. Photonics 5, 43–47 (2010).
[Crossref]

Hiroshima, T.

Y. Nambu, K. Usami, A. Tomita, S. Ishizaka, T. Hiroshima, Y. Tsuda, K. Matsumoto, K. Nakamura, “Experimental investigation of pulsed entangled photons and photonic quantum channels,” Proc. SPIE 4917, 13–24 (2002).

Hofheinz, M.

R. C. Bialczak, M. Ansmann, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, H. Wang, J. Wenner, M. Steffen, A. N. Cleland, J. M. Martinis, “Quantum process tomography of a universal entangling gate implemented with Josephson phase qubits,” Nat. Phys. 6, 409–413 (2010).
[Crossref]

Holland, M. J.

M. J. Holland, K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355–1358 (1993).
[Crossref]

Howard, M.

M. Howard, J. Twamley, C. Wittmann, T. Gaebel, F. Jelezko, J. Wrachtrup, “Quantum process tomography and Linblad estimation of a solid-state qubit,” New J. Phys. 8, 33 (2006).
[Crossref]

Hu, X.-F.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Hu, Y.-H.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Huang, Y.-M.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Humphreys, P. C.

P. C. Humphreys, M. Barbieri, A. Datta, I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref]

Ishizaka, S.

Y. Nambu, K. Usami, A. Tomita, S. Ishizaka, T. Hiroshima, Y. Tsuda, K. Matsumoto, K. Nakamura, “Experimental investigation of pulsed entangled photons and photonic quantum channels,” Proc. SPIE 4917, 13–24 (2002).

James, D. F. V.

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, A. G. White, “Quantum process tomography of a controlled-NOT gate,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

Jane, E.

A. Acin, E. Jane, G. Vidal, “Optimal estimation of quantum dynamics,” Phys. Rev. A 64, 050302 (2001).
[Crossref]

Janousek, J.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

Jeffrey, E.

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, A. G. White, “Ancilla-assisted quantum process tomography,” Phys. Rev. Lett. 90, 193601 (2003).
[Crossref]

Jelezko, F.

M. Howard, J. Twamley, C. Wittmann, T. Gaebel, F. Jelezko, J. Wrachtrup, “Quantum process tomography and Linblad estimation of a solid-state qubit,” New J. Phys. 8, 33 (2006).
[Crossref]

Jia, J.-J.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Jiang, H.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Kahn, J.

J. Kahn, “Fast rate estimation of a unitary operation in SU(d),” Phys. Rev. A 75, 022326 (2007).
[Crossref]

Kim, K.

M. Riebe, K. Kim, P. Schindler, T. Monz, P. O. Schmidt, T. K. Körber, W. Hänsel, H. Häffner, C. F. Roos, R. Blatt, “Process tomography of ion trap quantum gates,” Phys. Rev. Lett. 97, 220407 (2006).
[Crossref]

Knittel, J.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

Knysh, S. I.

S. I. Knysh, G. A. Durkin, “Estimation of phase and diffusion: combining quantum statistics and classical noise,” arXiv:1307.0470 (2013).

Körber, T. K.

M. Riebe, K. Kim, P. Schindler, T. Monz, P. O. Schmidt, T. K. Körber, W. Hänsel, H. Häffner, C. F. Roos, R. Blatt, “Process tomography of ion trap quantum gates,” Phys. Rev. Lett. 97, 220407 (2006).
[Crossref]

Korystov, D.

M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, A. I. Lvovsky, “Complete characterization of a quantum-optical processes,” Science 322, 563–566 (2008).
[Crossref]

Kosut, R. L.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref]

Krich, J. J.

J. Yuen-Zhou, J. J. Krich, M. Mohseni, A. Aspuru-Guzik, “Quantum state and process tomography of energy transfer systems via ultrafast spectroscopy,” Proc. Natl. Acad. Sci. USA 108, 17615–17620 (2011).

Kupchak, C.

M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, A. I. Lvovsky, “Complete characterization of a quantum-optical processes,” Science 322, 563–566 (2008).
[Crossref]

Kwiat, P. G.

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, A. G. White, “Ancilla-assisted quantum process tomography,” Phys. Rev. Lett. 90, 193601 (2003).
[Crossref]

Laing, A.

A. Laing, J. L. O’Brien, “Super-stable tomography of any linear optical device,” arXiv:1208.2868 (2012).

Landon-Cardinal, O.

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Langford, N. K.

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, A. G. White, “Quantum process tomography of a controlled-NOT gate,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

Leung, D. W.

A. M. Childs, I. L. Chuang, D. W. Leung, “Realization of quantum process tomography in NMR,” Phys. Rev. A 64, 012314 (2001).
[Crossref]

Liang, H.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Liao, S.-K.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Liu, W.-Y.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Liu, Y.-K.

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105, 150401 (2010).
[Crossref]

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Lloyd, S.

V. Giovannetti, S. Lloyd, L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

Lobino, M.

M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, A. I. Lvovsky, “Complete characterization of a quantum-optical processes,” Science 322, 563–566 (2008).
[Crossref]

Lucero, E.

R. C. Bialczak, M. Ansmann, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, H. Wang, J. Wenner, M. Steffen, A. N. Cleland, J. M. Martinis, “Quantum process tomography of a universal entangling gate implemented with Josephson phase qubits,” Nat. Phys. 6, 409–413 (2010).
[Crossref]

Lvovsky, A. I.

M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, A. I. Lvovsky, “Complete characterization of a quantum-optical processes,” Science 322, 563–566 (2008).
[Crossref]

Maccone, L.

V. Giovannetti, S. Lloyd, L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

Mahler, D. H.

L. A. Rozema, D. H. Mahler, R. Blume-Kohout, A. M. Steinberg, “Optimizing the choice of spin-squeezed states for detecting and characterizing quantum processes,” Phys. Rev. X 4, 041025 (2014).
[Crossref]

D. H. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohout, A. M. Steinberg, “Adaptive quantum state tomography improves accuracy quadratically,” Phys. Rev. Lett. 111, 183601 (2013).
[Crossref]

Martinis, J. M.

R. C. Bialczak, M. Ansmann, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, H. Wang, J. Wenner, M. Steffen, A. N. Cleland, J. M. Martinis, “Quantum process tomography of a universal entangling gate implemented with Josephson phase qubits,” Nat. Phys. 6, 409–413 (2010).
[Crossref]

Mataloni, P.

N. Spagnolo, L. Aparo, C. Vitelli, A. Crespi, R. Ramponi, R. Osellame, P. Mataloni, F. Sciarrino, “Quantum interferometry with three-dimensional geometry,” Sci. Rep. 2, 862 (2012).
[Crossref]

Matsumoto, K.

Y. Nambu, K. Usami, A. Tomita, S. Ishizaka, T. Hiroshima, Y. Tsuda, K. Matsumoto, K. Nakamura, “Experimental investigation of pulsed entangled photons and photonic quantum channels,” Proc. SPIE 4917, 13–24 (2002).

Matthews, J. C. F.

J. C. F. Matthews, X.-Q. Zhou, P. J. Shadbolt, H. Cable, D. J. Saunders, G. A. Durkin, G. J. Pryde, J. L. O’Brien, “Practical quantum metrology,” arXiv:1307.4673 (2013).

Mazzei, A.

F. De Martini, A. Mazzei, M. Ricci, G. M. D’Ariano, “Exploiting quantum parallelism of entanglement for a complete experimental quantum characterization of a single-qubit device,” Phys. Rev. A 67, 062307 (2003).
[Crossref]

Mezzadri, F.

F. Mezzadri, “How to generate random matrices from the classical compact groups,” Not. AMS 54, 592–604 (2007).

Mitchell, M. W.

F. Wolfgramm, C. Vitelli, F. A. Beduini, N. Godbout, M. W. Mitchell, “Entanglement-enhanced probing of a delicate material system,” Nat. Photonics 7, 28–32 (2012).
[Crossref]

Mohseni, M.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref]

J. Yuen-Zhou, J. J. Krich, M. Mohseni, A. Aspuru-Guzik, “Quantum state and process tomography of energy transfer systems via ultrafast spectroscopy,” Proc. Natl. Acad. Sci. USA 108, 17615–17620 (2011).

Monz, T.

M. Riebe, K. Kim, P. Schindler, T. Monz, P. O. Schmidt, T. K. Körber, W. Hänsel, H. Häffner, C. F. Roos, R. Blatt, “Process tomography of ion trap quantum gates,” Phys. Rev. Lett. 97, 220407 (2006).
[Crossref]

Mukunda, N.

R. Simon, N. Mukunda, “Minimal three-component SU(2) gadget for polarization optics,” Phys. Lett. A 143, 165–169 (1990).
[Crossref]

Murao, M.

T. Sugiyama, P. S. Turner, M. Murao, “Adaptive experimental design for one-qubit state estimation with finite data based on a statistical update criterion,” Phys. Rev. A 85, 052107 (2012).
[Crossref]

Nakamura, K.

Y. Nambu, K. Usami, A. Tomita, S. Ishizaka, T. Hiroshima, Y. Tsuda, K. Matsumoto, K. Nakamura, “Experimental investigation of pulsed entangled photons and photonic quantum channels,” Proc. SPIE 4917, 13–24 (2002).

Nambu, Y.

Y. Nambu, K. Usami, A. Tomita, S. Ishizaka, T. Hiroshima, Y. Tsuda, K. Matsumoto, K. Nakamura, “Experimental investigation of pulsed entangled photons and photonic quantum channels,” Proc. SPIE 4917, 13–24 (2002).

Neeley, M.

R. C. Bialczak, M. Ansmann, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, H. Wang, J. Wenner, M. Steffen, A. N. Cleland, J. M. Martinis, “Quantum process tomography of a universal entangling gate implemented with Josephson phase qubits,” Nat. Phys. 6, 409–413 (2010).
[Crossref]

Nielsen, M. A.

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, A. G. White, “Ancilla-assisted quantum process tomography,” Phys. Rev. Lett. 90, 193601 (2003).
[Crossref]

M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

O’Brien, J. L.

J. L. O’Brien, “Optical quantum computing,” Science 318, 1567–1570 (2007).
[Crossref]

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, A. G. White, “Quantum process tomography of a controlled-NOT gate,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, A. G. White, “Ancilla-assisted quantum process tomography,” Phys. Rev. Lett. 90, 193601 (2003).
[Crossref]

A. Laing, J. L. O’Brien, “Super-stable tomography of any linear optical device,” arXiv:1208.2868 (2012).

J. C. F. Matthews, X.-Q. Zhou, P. J. Shadbolt, H. Cable, D. J. Saunders, G. A. Durkin, G. J. Pryde, J. L. O’Brien, “Practical quantum metrology,” arXiv:1307.4673 (2013).

O’Connell, A. D.

R. C. Bialczak, M. Ansmann, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, H. Wang, J. Wenner, M. Steffen, A. N. Cleland, J. M. Martinis, “Quantum process tomography of a universal entangling gate implemented with Josephson phase qubits,” Nat. Phys. 6, 409–413 (2010).
[Crossref]

Olver, F. W.

F. W. Olver, NIST Handbook of Mathematical Functions (Cambridge University, 2010).

Osellame, R.

N. Spagnolo, L. Aparo, C. Vitelli, A. Crespi, R. Ramponi, R. Osellame, P. Mataloni, F. Sciarrino, “Quantum interferometry with three-dimensional geometry,” Sci. Rep. 2, 862 (2012).
[Crossref]

Pan, G.-S.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Pan, J.-W.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Peng, C.-Z.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Pfister, O.

Plenio, M. B.

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Poulin, D.

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Pryde, G.

G.-Y. Xiang, B. L. Higgins, D. Berry, H. M. Wiseman, G. Pryde, “Entanglement-enhanced measurement of a completely unknown optical phase,” Nat. Photonics 5, 43–47 (2010).
[Crossref]

Pryde, G. J.

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, A. G. White, “Quantum process tomography of a controlled-NOT gate,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

J. C. F. Matthews, X.-Q. Zhou, P. J. Shadbolt, H. Cable, D. J. Saunders, G. A. Durkin, G. J. Pryde, J. L. O’Brien, “Practical quantum metrology,” arXiv:1307.4673 (2013).

Qi, B.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Rabitz, H.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref]

Radmore, P. M.

S. Barnett, P. M. Radmore, Methods in Theoretical Quantum Optics (Oxford University, 2002).

Rahimi-Keshari, S.

Ralph, T. C.

S. Rahimi-Keshari, M. A. Broome, R. Fickler, A. Fedrizzi, T. C. Ralph, A. G. White, “Direct characterization of linear-optical networks,” Opt. Express 21, 13450–13458 (2013).
[Crossref]

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, A. G. White, “Quantum process tomography of a controlled-NOT gate,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

Ramponi, R.

N. Spagnolo, L. Aparo, C. Vitelli, A. Crespi, R. Ramponi, R. Osellame, P. Mataloni, F. Sciarrino, “Quantum interferometry with three-dimensional geometry,” Sci. Rep. 2, 862 (2012).
[Crossref]

Reck, M.

M. Reck, A. Zeilinger, H. J. Bernstein, P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

Ren, J.-G.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Ricci, M.

F. De Martini, A. Mazzei, M. Ricci, G. M. D’Ariano, “Exploiting quantum parallelism of entanglement for a complete experimental quantum characterization of a single-qubit device,” Phys. Rev. A 67, 062307 (2003).
[Crossref]

Riebe, M.

M. Riebe, K. Kim, P. Schindler, T. Monz, P. O. Schmidt, T. K. Körber, W. Hänsel, H. Häffner, C. F. Roos, R. Blatt, “Process tomography of ion trap quantum gates,” Phys. Rev. Lett. 97, 220407 (2006).
[Crossref]

Roos, C. F.

M. Riebe, K. Kim, P. Schindler, T. Monz, P. O. Schmidt, T. K. Körber, W. Hänsel, H. Häffner, C. F. Roos, R. Blatt, “Process tomography of ion trap quantum gates,” Phys. Rev. Lett. 97, 220407 (2006).
[Crossref]

Rozema, L. A.

L. A. Rozema, D. H. Mahler, R. Blume-Kohout, A. M. Steinberg, “Optimizing the choice of spin-squeezed states for detecting and characterizing quantum processes,” Phys. Rev. X 4, 041025 (2014).
[Crossref]

D. H. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohout, A. M. Steinberg, “Adaptive quantum state tomography improves accuracy quadratically,” Phys. Rev. Lett. 111, 183601 (2013).
[Crossref]

Rudolph, T.

S. D. Bartlett, T. Rudolph, R. W. Spekkens, “Reference frames, superselection rules, and quantum information,” Rev. Mod. Phys. 79, 555–609 (2007).
[Crossref]

Sakurai, J. J.

J. J. Sakurai, Modern Quantum Mechanics (Addison Wesley, 1994), Revised Edition, Chap. 3.8.

Sanders, B. C.

M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, A. I. Lvovsky, “Complete characterization of a quantum-optical processes,” Science 322, 563–566 (2008).
[Crossref]

Sank, D.

R. C. Bialczak, M. Ansmann, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, H. Wang, J. Wenner, M. Steffen, A. N. Cleland, J. M. Martinis, “Quantum process tomography of a universal entangling gate implemented with Josephson phase qubits,” Nat. Phys. 6, 409–413 (2010).
[Crossref]

Saunders, D. J.

J. C. F. Matthews, X.-Q. Zhou, P. J. Shadbolt, H. Cable, D. J. Saunders, G. A. Durkin, G. J. Pryde, J. L. O’Brien, “Practical quantum metrology,” arXiv:1307.4673 (2013).

Schindler, P.

M. Riebe, K. Kim, P. Schindler, T. Monz, P. O. Schmidt, T. K. Körber, W. Hänsel, H. Häffner, C. F. Roos, R. Blatt, “Process tomography of ion trap quantum gates,” Phys. Rev. Lett. 97, 220407 (2006).
[Crossref]

Schmidt, P. O.

M. Riebe, K. Kim, P. Schindler, T. Monz, P. O. Schmidt, T. K. Körber, W. Hänsel, H. Häffner, C. F. Roos, R. Blatt, “Process tomography of ion trap quantum gates,” Phys. Rev. Lett. 97, 220407 (2006).
[Crossref]

Schwinger, J.

J. Schwinger, Quantum Theory of Angular Momentum (Academic, 1965).

Sciarrino, F.

N. Spagnolo, L. Aparo, C. Vitelli, A. Crespi, R. Ramponi, R. Osellame, P. Mataloni, F. Sciarrino, “Quantum interferometry with three-dimensional geometry,” Sci. Rep. 2, 862 (2012).
[Crossref]

Shabani, A.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref]

Shadbolt, P. J.

J. C. F. Matthews, X.-Q. Zhou, P. J. Shadbolt, H. Cable, D. J. Saunders, G. A. Durkin, G. J. Pryde, J. L. O’Brien, “Practical quantum metrology,” arXiv:1307.4673 (2013).

Shen, Q.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Simon, R.

R. Simon, N. Mukunda, “Minimal three-component SU(2) gadget for polarization optics,” Phys. Lett. A 143, 165–169 (1990).
[Crossref]

Smith, B. J.

A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A 83, 063836 (2011).
[Crossref]

N. Thomas-Peter, B. J. Smith, A. Datta, L. Zhang, U. Dorner, I. A. Walmsley, “Real-world quantum sensors: Evaluating resources for precision measurement,” Phys. Rev. Lett. 107, 113603 (2011).
[Crossref]

Somma, R.

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Spagnolo, N.

N. Spagnolo, L. Aparo, C. Vitelli, A. Crespi, R. Ramponi, R. Osellame, P. Mataloni, F. Sciarrino, “Quantum interferometry with three-dimensional geometry,” Sci. Rep. 2, 862 (2012).
[Crossref]

Spekkens, R. W.

S. D. Bartlett, T. Rudolph, R. W. Spekkens, “Reference frames, superselection rules, and quantum information,” Rev. Mod. Phys. 79, 555–609 (2007).
[Crossref]

Steffen, M.

R. C. Bialczak, M. Ansmann, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, H. Wang, J. Wenner, M. Steffen, A. N. Cleland, J. M. Martinis, “Quantum process tomography of a universal entangling gate implemented with Josephson phase qubits,” Nat. Phys. 6, 409–413 (2010).
[Crossref]

Steinberg, A. M.

L. A. Rozema, D. H. Mahler, R. Blume-Kohout, A. M. Steinberg, “Optimizing the choice of spin-squeezed states for detecting and characterizing quantum processes,” Phys. Rev. X 4, 041025 (2014).
[Crossref]

D. H. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohout, A. M. Steinberg, “Adaptive quantum state tomography improves accuracy quadratically,” Phys. Rev. Lett. 111, 183601 (2013).
[Crossref]

Sugiyama, T.

T. Sugiyama, P. S. Turner, M. Murao, “Adaptive experimental design for one-qubit state estimation with finite data based on a statistical update criterion,” Phys. Rev. A 85, 052107 (2012).
[Crossref]

Tang, Y.-L.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Taylor, M. A.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

Thew, R. T.

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, A. G. White, “Ancilla-assisted quantum process tomography,” Phys. Rev. Lett. 90, 193601 (2003).
[Crossref]

Thomas-Peter, N.

A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A 83, 063836 (2011).
[Crossref]

N. Thomas-Peter, B. J. Smith, A. Datta, L. Zhang, U. Dorner, I. A. Walmsley, “Real-world quantum sensors: Evaluating resources for precision measurement,” Phys. Rev. Lett. 107, 113603 (2011).
[Crossref]

Tomita, A.

Y. Nambu, K. Usami, A. Tomita, S. Ishizaka, T. Hiroshima, Y. Tsuda, K. Matsumoto, K. Nakamura, “Experimental investigation of pulsed entangled photons and photonic quantum channels,” Proc. SPIE 4917, 13–24 (2002).

Tsuda, Y.

Y. Nambu, K. Usami, A. Tomita, S. Ishizaka, T. Hiroshima, Y. Tsuda, K. Matsumoto, K. Nakamura, “Experimental investigation of pulsed entangled photons and photonic quantum channels,” Proc. SPIE 4917, 13–24 (2002).

Turner, P. S.

T. Sugiyama, P. S. Turner, M. Murao, “Adaptive experimental design for one-qubit state estimation with finite data based on a statistical update criterion,” Phys. Rev. A 85, 052107 (2012).
[Crossref]

Twamley, J.

M. Howard, J. Twamley, C. Wittmann, T. Gaebel, F. Jelezko, J. Wrachtrup, “Quantum process tomography and Linblad estimation of a solid-state qubit,” New J. Phys. 8, 33 (2006).
[Crossref]

Usami, K.

Y. Nambu, K. Usami, A. Tomita, S. Ishizaka, T. Hiroshima, Y. Tsuda, K. Matsumoto, K. Nakamura, “Experimental investigation of pulsed entangled photons and photonic quantum channels,” Proc. SPIE 4917, 13–24 (2002).

Vidal, G.

A. Acin, E. Jane, G. Vidal, “Optimal estimation of quantum dynamics,” Phys. Rev. A 64, 050302 (2001).
[Crossref]

Vitelli, C.

N. Spagnolo, L. Aparo, C. Vitelli, A. Crespi, R. Ramponi, R. Osellame, P. Mataloni, F. Sciarrino, “Quantum interferometry with three-dimensional geometry,” Sci. Rep. 2, 862 (2012).
[Crossref]

F. Wolfgramm, C. Vitelli, F. A. Beduini, N. Godbout, M. W. Mitchell, “Entanglement-enhanced probing of a delicate material system,” Nat. Photonics 7, 28–32 (2012).
[Crossref]

Walmsley, I. A.

P. C. Humphreys, M. Barbieri, A. Datta, I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref]

N. Thomas-Peter, B. J. Smith, A. Datta, L. Zhang, U. Dorner, I. A. Walmsley, “Real-world quantum sensors: Evaluating resources for precision measurement,” Phys. Rev. Lett. 107, 113603 (2011).
[Crossref]

A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A 83, 063836 (2011).
[Crossref]

P. J. Crowley, A. Datta, M. Barbieri, I. A. Walmsley, “Multiparameter quantum metrology,” arXiv:1206.0043 (2012).

Wang, H.

R. C. Bialczak, M. Ansmann, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, H. Wang, J. Wenner, M. Steffen, A. N. Cleland, J. M. Martinis, “Quantum process tomography of a universal entangling gate implemented with Josephson phase qubits,” Nat. Phys. 6, 409–413 (2010).
[Crossref]

Wang, J.-Y.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Wei, T. C.

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, A. G. White, “Ancilla-assisted quantum process tomography,” Phys. Rev. Lett. 90, 193601 (2003).
[Crossref]

Wenner, J.

R. C. Bialczak, M. Ansmann, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, H. Wang, J. Wenner, M. Steffen, A. N. Cleland, J. M. Martinis, “Quantum process tomography of a universal entangling gate implemented with Josephson phase qubits,” Nat. Phys. 6, 409–413 (2010).
[Crossref]

White, A. G.

S. Rahimi-Keshari, M. A. Broome, R. Fickler, A. Fedrizzi, T. C. Ralph, A. G. White, “Direct characterization of linear-optical networks,” Opt. Express 21, 13450–13458 (2013).
[Crossref]

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref]

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, A. G. White, “Quantum process tomography of a controlled-NOT gate,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, A. G. White, “Ancilla-assisted quantum process tomography,” Phys. Rev. Lett. 90, 193601 (2003).
[Crossref]

Wiseman, H. M.

G.-Y. Xiang, B. L. Higgins, D. Berry, H. M. Wiseman, G. Pryde, “Entanglement-enhanced measurement of a completely unknown optical phase,” Nat. Photonics 5, 43–47 (2010).
[Crossref]

Wittmann, C.

M. Howard, J. Twamley, C. Wittmann, T. Gaebel, F. Jelezko, J. Wrachtrup, “Quantum process tomography and Linblad estimation of a solid-state qubit,” New J. Phys. 8, 33 (2006).
[Crossref]

Wolfgramm, F.

F. Wolfgramm, C. Vitelli, F. A. Beduini, N. Godbout, M. W. Mitchell, “Entanglement-enhanced probing of a delicate material system,” Nat. Photonics 7, 28–32 (2012).
[Crossref]

Wrachtrup, J.

M. Howard, J. Twamley, C. Wittmann, T. Gaebel, F. Jelezko, J. Wrachtrup, “Quantum process tomography and Linblad estimation of a solid-state qubit,” New J. Phys. 8, 33 (2006).
[Crossref]

Wu, J.-C.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Xiang, G.-Y.

G.-Y. Xiang, B. L. Higgins, D. Berry, H. M. Wiseman, G. Pryde, “Entanglement-enhanced measurement of a completely unknown optical phase,” Nat. Photonics 5, 43–47 (2010).
[Crossref]

Yang, B.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Yang, S.-J.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Yin, J.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Yuen-Zhou, J.

J. Yuen-Zhou, J. J. Krich, M. Mohseni, A. Aspuru-Guzik, “Quantum state and process tomography of energy transfer systems via ultrafast spectroscopy,” Proc. Natl. Acad. Sci. USA 108, 17615–17620 (2011).

Zeilinger, A.

M. Reck, A. Zeilinger, H. J. Bernstein, P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

Zhang, L.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

N. Thomas-Peter, B. J. Smith, A. Datta, L. Zhang, U. Dorner, I. A. Walmsley, “Real-world quantum sensors: Evaluating resources for precision measurement,” Phys. Rev. Lett. 107, 113603 (2011).
[Crossref]

A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A 83, 063836 (2011).
[Crossref]

Zhong, B.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Zhou, X.-Q.

J. C. F. Matthews, X.-Q. Zhou, P. J. Shadbolt, H. Cable, D. J. Saunders, G. A. Durkin, G. J. Pryde, J. L. O’Brien, “Practical quantum metrology,” arXiv:1307.4673 (2013).

Nat. Commun. (1)

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
[Crossref]

Nat. Photonics (6)

V. Giovannetti, S. Lloyd, L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

F. Wolfgramm, C. Vitelli, F. A. Beduini, N. Godbout, M. W. Mitchell, “Entanglement-enhanced probing of a delicate material system,” Nat. Photonics 7, 28–32 (2012).
[Crossref]

LIGO Scientific Collaboration, “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photonics 7, 613–619 (2013).
[Crossref]

G.-Y. Xiang, B. L. Higgins, D. Berry, H. M. Wiseman, G. Pryde, “Entanglement-enhanced measurement of a completely unknown optical phase,” Nat. Photonics 5, 43–47 (2010).
[Crossref]

Nat. Phys. (1)

R. C. Bialczak, M. Ansmann, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, H. Wang, J. Wenner, M. Steffen, A. N. Cleland, J. M. Martinis, “Quantum process tomography of a universal entangling gate implemented with Josephson phase qubits,” Nat. Phys. 6, 409–413 (2010).
[Crossref]

New J. Phys. (1)

M. Howard, J. Twamley, C. Wittmann, T. Gaebel, F. Jelezko, J. Wrachtrup, “Quantum process tomography and Linblad estimation of a solid-state qubit,” New J. Phys. 8, 33 (2006).
[Crossref]

Not. AMS (1)

F. Mezzadri, “How to generate random matrices from the classical compact groups,” Not. AMS 54, 592–604 (2007).

Opt. Express (1)

Opt. Lett. (1)

Phys. Lett. A (2)

M. Hayashi, “Parallel treatment of estimation of SU(2) and phase estimation,” Phys. Lett. A 354, 183–189 (2006).
[Crossref]

R. Simon, N. Mukunda, “Minimal three-component SU(2) gadget for polarization optics,” Phys. Lett. A 143, 165–169 (1990).
[Crossref]

Phys. Rev. A (6)

A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A 83, 063836 (2011).
[Crossref]

J. Kahn, “Fast rate estimation of a unitary operation in SU(d),” Phys. Rev. A 75, 022326 (2007).
[Crossref]

A. M. Childs, I. L. Chuang, D. W. Leung, “Realization of quantum process tomography in NMR,” Phys. Rev. A 64, 012314 (2001).
[Crossref]

A. Acin, E. Jane, G. Vidal, “Optimal estimation of quantum dynamics,” Phys. Rev. A 64, 050302 (2001).
[Crossref]

T. Sugiyama, P. S. Turner, M. Murao, “Adaptive experimental design for one-qubit state estimation with finite data based on a statistical update criterion,” Phys. Rev. A 85, 052107 (2012).
[Crossref]

F. De Martini, A. Mazzei, M. Ricci, G. M. D’Ariano, “Exploiting quantum parallelism of entanglement for a complete experimental quantum characterization of a single-qubit device,” Phys. Rev. A 67, 062307 (2003).
[Crossref]

Phys. Rev. Lett. (10)

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, A. G. White, “Ancilla-assisted quantum process tomography,” Phys. Rev. Lett. 90, 193601 (2003).
[Crossref]

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, A. G. White, “Quantum process tomography of a controlled-NOT gate,” Phys. Rev. Lett. 93, 080502 (2004).
[Crossref]

M. Riebe, K. Kim, P. Schindler, T. Monz, P. O. Schmidt, T. K. Körber, W. Hänsel, H. Häffner, C. F. Roos, R. Blatt, “Process tomography of ion trap quantum gates,” Phys. Rev. Lett. 97, 220407 (2006).
[Crossref]

D. H. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohout, A. M. Steinberg, “Adaptive quantum state tomography improves accuracy quadratically,” Phys. Rev. Lett. 111, 183601 (2013).
[Crossref]

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105, 150401 (2010).
[Crossref]

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref]

P. C. Humphreys, M. Barbieri, A. Datta, I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref]

M. J. Holland, K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355–1358 (1993).
[Crossref]

N. Thomas-Peter, B. J. Smith, A. Datta, L. Zhang, U. Dorner, I. A. Walmsley, “Real-world quantum sensors: Evaluating resources for precision measurement,” Phys. Rev. Lett. 107, 113603 (2011).
[Crossref]

M. Reck, A. Zeilinger, H. J. Bernstein, P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

Phys. Rev. X (1)

L. A. Rozema, D. H. Mahler, R. Blume-Kohout, A. M. Steinberg, “Optimizing the choice of spin-squeezed states for detecting and characterizing quantum processes,” Phys. Rev. X 4, 041025 (2014).
[Crossref]

Proc. Natl. Acad. Sci. USA (1)

J. Yuen-Zhou, J. J. Krich, M. Mohseni, A. Aspuru-Guzik, “Quantum state and process tomography of energy transfer systems via ultrafast spectroscopy,” Proc. Natl. Acad. Sci. USA 108, 17615–17620 (2011).

Proc. SPIE (1)

Y. Nambu, K. Usami, A. Tomita, S. Ishizaka, T. Hiroshima, Y. Tsuda, K. Matsumoto, K. Nakamura, “Experimental investigation of pulsed entangled photons and photonic quantum channels,” Proc. SPIE 4917, 13–24 (2002).

Rev. Mod. Phys. (1)

S. D. Bartlett, T. Rudolph, R. W. Spekkens, “Reference frames, superselection rules, and quantum information,” Rev. Mod. Phys. 79, 555–609 (2007).
[Crossref]

Sci. Rep. (1)

N. Spagnolo, L. Aparo, C. Vitelli, A. Crespi, R. Ramponi, R. Osellame, P. Mataloni, F. Sciarrino, “Quantum interferometry with three-dimensional geometry,” Sci. Rep. 2, 862 (2012).
[Crossref]

Science (2)

M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, A. I. Lvovsky, “Complete characterization of a quantum-optical processes,” Science 322, 563–566 (2008).
[Crossref]

J. L. O’Brien, “Optical quantum computing,” Science 318, 1567–1570 (2007).
[Crossref]

Other (10)

J. C. F. Matthews, X.-Q. Zhou, P. J. Shadbolt, H. Cable, D. J. Saunders, G. A. Durkin, G. J. Pryde, J. L. O’Brien, “Practical quantum metrology,” arXiv:1307.4673 (2013).

S. Barnett, P. M. Radmore, Methods in Theoretical Quantum Optics (Oxford University, 2002).

A. Laing, J. L. O’Brien, “Super-stable tomography of any linear optical device,” arXiv:1208.2868 (2012).

M. A. Ballester, “Optimal estimation of SU(d) using exact and approximate 2-designs,” arXiv:quant-ph/0507073 (2005).

M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

S. I. Knysh, G. A. Durkin, “Estimation of phase and diffusion: combining quantum statistics and classical noise,” arXiv:1307.0470 (2013).

P. J. Crowley, A. Datta, M. Barbieri, I. A. Walmsley, “Multiparameter quantum metrology,” arXiv:1206.0043 (2012).

J. Schwinger, Quantum Theory of Angular Momentum (Academic, 1965).

J. J. Sakurai, Modern Quantum Mechanics (Addison Wesley, 1994), Revised Edition, Chap. 3.8.

F. W. Olver, NIST Handbook of Mathematical Functions (Cambridge University, 2010).

Supplementary Material (1)

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Figures (3)

Fig. 1.
Fig. 1.

(a) Standard procedure for quantum process tomography. This procedure can be considered in three stages: preparation of a series of probe states, interaction of the unknown process with the probe, and detection of the output in a selection of measurement bases. Based on the measurement outcomes, a mathematical map corresponding to the unknown process can be reconstructed. (b) Experimental setup for probing the unknown unitary processes. An 80 MHz pulsed Ti:sapphire laser centered at 808 nm is upconverted to 404 nm and then focused onto a bismuth borate (BiBO) crystal, phase-matched for Type-I SPDC, creating noncollinear degenerate horizontally polarized photon pairs at 808 nm. After converting one arm to vertical polarization using a half-wave plate (HWP), the two arms are combined by a polarizing beam splitter (PBS). The resulting state is passed through the unknown unitary and then measured in the H/V basis. Approximate photon-number counting (see Appendix A) is implemented on each polarization mode using a 1–8 fan-out array onto avalanche photodiode detectors (APDs). By using different wave plate settings before and after the unknown unitary, the basis for the probe state and measurement can be changed to D/A and R/L. The setup is precalibrated using laser light.

Fig. 2.
Fig. 2.

Visual representation of 19 randomly selected unitaries (red) with their experimental reconstructions from four-photon probe states (cyan). Each of these unitaries was experimentally realized with three consecutive wave plates (QWP, HWP, QWP). The four-photon state |2,2HV(DA,RL) is used to probe each unitary and subsequently measured in the H/V (D/A, R/L) basis. The unitaries are then reconstructed from the resulting photon statistics. The representation of each unitary is a point in this unit sphere with coordinates (x,y,z) corresponding to (d,c,b). The precision of two highlighted unitaries, UA and UB, is subject to detailed analysis with a large dataset in Fig. 3.

Fig. 3.
Fig. 3.

Experimental performance of the quantum-enhanced tomography protocol using four-photon states. In this figure, the standard QPT protocol (diagonal fill) using single photons and our method using four photons (unpatterned) are compared, for two unitaries UA and UB. To make a fair comparison between the two protocols, the same total number of probe photons are assumed (1800, 2400, and 3600 for all measurements together). a1 and b1 are derived from experimental data for UA and UB, respectively, and a2 and b2 are based on corresponding theoretical simulations. For each case, the mean infidelity is illustrated, for a series of experimentally derived estimates U˜A or U˜B, where the infidelity is defined relative to central estimates UA or UB derived from the accumulated experimental data. Asymmetric lower- and upper-half error bars are used, reflecting the distribution of infidelity values. The data show that the four-photon version of the protocol achieves greater accuracy and precision than standard QPT with single photons.

Tables (1)

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Table 1. Outcome Probabilities of Four-Photon Events for the H/V Measurementa

Equations (5)

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a2+b2+c2+d2=1.
pHV=a2+b2,pDA=a2+d2,pRL=a2+c2.
P(nH,nV,pHV)=nV!nH!(L(nH+nV)/2(nHnV)/2(2pHV1))2,
(a˜2b˜2c˜2d˜2)=12(1111111111111111)(1p˜DAp˜RLp˜HV),
p˜DA+p˜RL+p˜HV1,p˜DA+p˜RLp˜HV1,p˜DAp˜RL+p˜HV1,p˜DA+p˜RL+p˜HV1.

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