Abstract

Optical sensors based on quantum light need to work even in the presence of imperfections. Here we discuss how to address the presence of noise by measuring multiple parameters at once: those we seek to monitor and those linked to the imperfections. Our method is applied to the investigation of the optical activity of sugary solutions limited by reduced visibility. These studies introduce multiparameter estimation as a viable approach to allow for robust operation of quantum sensors as they are developed from fundamental research to technology.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Experimental multiphase estimation on a chip

Emanuele Polino, Martina Riva, Mauro Valeri, Raffaele Silvestri, Giacomo Corrielli, Andrea Crespi, Nicolò Spagnolo, Roberto Osellame, and Fabio Sciarrino
Optica 6(3) 288-295 (2019)

Super-resolution quantum imaging at the Heisenberg limit

Manuel Unternährer, Bänz Bessire, Leonardo Gasparini, Matteo Perenzoni, and André Stefanov
Optica 5(9) 1150-1154 (2018)

Resurgence of Rayleigh’s curse in the presence of partial coherence

Walker Larson and Bahaa E. A. Saleh
Optica 5(11) 1382-1389 (2018)

References

  • View by:
  • |
  • |
  • |

  1. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330–1336 (2004).
    [Crossref]
  2. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
    [Crossref]
  3. M. G. A. Paris, “Quantum estimation for quantum technology,” Int. J. Quantum Inf. 7, 125–137 (2009).
    [Crossref]
  4. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
    [Crossref]
  5. R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kolodynski, “Quantum limits in optical interferometry,” Prog. Opt. 60, 345–435 (2015).
    [Crossref]
  6. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
    [Crossref]
  7. G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387, 471–475 (1997).
    [Crossref]
  8. L. A. Rozema, D. H. Mahler, R. Blume-Kohout, and A. M. Steinberg, “Optimizing the choice of spin-squeezed states for detecting and characterizing quantum processes,” Phys. Rev. X 4, 041025 (2014).
    [Crossref]
  9. H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49, 2325–2338 (2002).
    [Crossref]
  10. I. Afek, O. Ambar, and Y. Silberberg, “High-noon states by mixing quantum and classical light,” Science 328, 879–881 (2010).
    [Crossref]
  11. J. Joo, W. J. Munro, and T. P. Spiller, “Erratum: quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107, 219902 (2011).
    [Crossref]
  12. U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
    [Crossref]
  13. M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4, 357–360 (2010).
    [Crossref]
  14. N. Thomas-Peter, B. J. Smith, A. Datta, L. Zhang, U. Dorner, and I. A. Walmsley, “Real-world quantum sensors: evaluating resources for precision measurement,” Phys. Rev. Lett. 107, 113603 (2011).
    [Crossref]
  15. A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, and I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A 83, 063836 (2011).
    [Crossref]
  16. S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, “Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state,” Phys. Rev. A 83, 021804 (2011).
    [Crossref]
  17. P. M. Birchall, , J. Sabines-Chesterking, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Beating the shot-noise limit with sources of partially-distinguishable photons,” arXiv:1603.00686 (2016).
  18. B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406–411 (2011).
    [Crossref]
  19. R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
    [Crossref]
  20. R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodyński, and A. Acín, “Noisy metrology beyond the standard quantum limit,” Phys. Rev. Lett. 111, 120401 (2013).
    [Crossref]
  21. J. B. Brask, R. Chaves, and J. Kołodyński, “Improved quantum magnetometry beyond the standard quantum limit,” Phys. Rev. X 5, 031010 (2015).
    [Crossref]
  22. Y. Matsuzaki, S. C. Benjamin, and J. F. Fitzsimons, “Magnetic field sensing beyond the standard quantum limit under the effect of decoherence,” Phys. Rev. A 84, 012103 (2011).
    [Crossref]
  23. A. W. Chin, S. F. Huelga, and M. B. Plenio, “Quantum metrology in non-Markovian environments,” Phys. Rev. Lett. 109, 233601 (2012).
    [Crossref]
  24. A. Smirne, J. Kołodyński, S. F. Huelga, and R. Demkowicz-Dobrzański, “Ultimate precision limits for noisy frequency estimation,” Phys. Rev. Lett. 116, 120801 (2016).
    [Crossref]
  25. J. F. Haase, A. Smirne, J. Kołodyński, R. Demkowicz-Dobrzański, and S. F. Huelga, “Fundamental limits to frequency estimation: a comprehensive microscopic perspective,” New J. Phys. 20, 053009 (2018).
    [Crossref]
  26. A. Górecka, F. A. Pollock, P. Liuzzo-Scorpo, R. Nichols, G. Adesso, and K. Modi, “Noisy frequency estimation with noisy probes,” New J. Phys. (to be published).
    [Crossref]
  27. E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, “Quantum error correction for metrology,” Phys. Rev. Lett. 112, 150802 (2014).
    [Crossref]
  28. G. Arrad, Y. Vinkler, D. Aharonov, and A. Retzker, “Increasing sensing resolution with error correction,” Phys. Rev. Lett. 112, 150801 (2014).
    [Crossref]
  29. W. Dür, M. Skotiniotis, F. Fröwis, and B. Kraus, “Improved quantum metrology using quantum error correction,” Phys. Rev. Lett. 112, 080801 (2014).
    [Crossref]
  30. P. Sekatski, M. Skotiniotis, and W. Dür, “Dynamical decoupling leads to improved scaling in noisy quantum metrology,” New J. Phys. 18, 073034 (2015).
    [Crossref]
  31. M. B. Plenio and S. F. Huelga, “Sensing in the presence of an observed environment,” Phys. Rev. A 93, 032123 (2016).
    [Crossref]
  32. T. Gefen, D. A. Herrera-Martí, and A. Retzker, “Parameter estimation with efficient photodetectors,” Phys. Rev. A 93, 032133 (2016).
    [Crossref]
  33. D. Layden and P. Cappellaro, “Spatial noise filtering through error correction for quantum sensing,” npj Quantum Inf. 4, 30 (2018).
  34. P. Sekatski, M. Skotiniotis, J. Kołodyński, and W. Dür, “Quantum metrology with full and fast quantum control,” Quantum 1, 27 (2017).
    [Crossref]
  35. Y. Matsuzaki and S. Benjamin, “Magnetic-field sensing with quantum error detection under the effect of energy relaxation,” Phys. Rev. A 95, 032303 (2017).
    [Crossref]
  36. S. Zhou, M. Zhang, J. Preskill, and L. Jiang, “Achieving the Heisenberg limit in quantum metrology using quantum error correction,” Nat. Commun. 9, 78 (2018).
    [Crossref]
  37. F. Albarelli, M. A. C. Rossi, D. Tamascelli, and M. G. Genoni, “Restoring Heisenberg scaling in noisy quantum metrology by monitoring the environment,” arXiv:1803.05891 (2018).
  38. A. Orieux, L. Sansoni, M. Persechino, P. Mataloni, M. Rossi, and C. Macchiavello, “Experimental detection of quantum channels,” Phys. Rev. Lett. 111, 220501 (2013).
    [Crossref]
  39. X.-Q. Zhou, H. Cable, R. Whittaker, P. Shadbolt, J. L. O’Brien, and J. C. F. Matthews, “Quantum-enhanced tomography of unitary processes,” Optica 2, 510–516 (2015).
    [Crossref]
  40. R. D. Gill and S. Massar, “State estimation for large ensembles,” Phys. Rev. A 61, 042312 (2000).
    [Crossref]
  41. C. Macchiavello, “Optimal estimation of multiple phases,” Phys. Rev. A 67, 062302 (2003).
    [Crossref]
  42. M. A. Ballester, “Entanglement is not very useful for estimating multiple phases,” Phys. Rev. A 70, 032310 (2004).
    [Crossref]
  43. O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A 88, 040102 (2013).
    [Crossref]
  44. M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, “Optimal estimation of joint parameters in phase space,” Phys. Rev. A 87, 012107 (2013).
    [Crossref]
  45. C. Vaneph, T. Tufarelli, and M. Genoni, “Quantum estimation of a two-phase spin rotation,” Quantum Meas. Quantum Metrol. 1, 12-20 (2013).
    [Crossref]
  46. L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
    [Crossref]
  47. S. I. Knysh and G. A. Durkin, “Estimation of phase and diffusion: combining quantum statistics and classical noise,” arXiv:1307.0470 (2013).
  48. P. J. D. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, “Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry,” Phys. Rev. A 89, 023845 (2014).
    [Crossref]
  49. M. D. Vidrighin, G. Donati, M. G. Genoni, X.-M. Jin, W. S. Kolthammer, A. D. M. S. Kim, M. Barbieri, and I. A. Walmsley, “Joint estimation of phase and phase diffusion for quantum metrology,” Nat. Commun. 5, 3532 (2014).
    [Crossref]
  50. M. Szczykulska, T. Baumgratz, and A. Datta, “Reaching for the quantum limits in the simultaneous estimation of phase and phase diffusion,” Quantum Sci. Technol. 2, 044004 (2017).
    [Crossref]
  51. E. Roccia, M. G. Genoni, L. Mancino, I. Gianani, M. Barbieri, and M. Sbroscia, “Monitoring dispersive samples with single photons: the role of frequency correlations,” Quantum Meas. Quantum Metrol. 4, 64–69 (2017).
    [Crossref]
  52. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
    [Crossref]
  53. B. Gendra, E. Ronco-Bonvehi, J. Calsamiglia, R. M. Tapia, and E. Bagan, “Quantum metrology assisted by abstention,” Phys. Rev. Lett. 110, 100501 (2013).
    [Crossref]
  54. J. Combes, C. Ferrie, Z. Jiang, and C. M. Caves, “Quantum limits on post-selected, probabilistic quantum metrology,” Phys. Rev. A 89, 052117 (2014).
    [Crossref]
  55. N. Tischler, M. Krenn, R. Fickler, X. Vidal, A. Zeilinger, and G. Molina-Terriza, “Quantum optical rotatory dispersion,” Sci. Adv. 2, e1601306 (2016).
    [Crossref]
  56. T. Anderson, An Introduction to Multivariate Statistical Analysis, 3rd ed. (Wiley, 2003)
  57. A. Monras and M. G. A. Paris, “Optimal quantum estimation of loss in bosonic channels,” Phys. Rev. Lett. 98, 160401 (2007).
    [Crossref]
  58. M. G. Genoni, S. Olivares, and M. G. A. Paris, “Optical phase estimation in the presence of phase diffusion,” Phys. Rev. Lett. 106, 153603 (2011).
    [Crossref]
  59. M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355–1358 (1993).
    [Crossref]

2018 (3)

J. F. Haase, A. Smirne, J. Kołodyński, R. Demkowicz-Dobrzański, and S. F. Huelga, “Fundamental limits to frequency estimation: a comprehensive microscopic perspective,” New J. Phys. 20, 053009 (2018).
[Crossref]

D. Layden and P. Cappellaro, “Spatial noise filtering through error correction for quantum sensing,” npj Quantum Inf. 4, 30 (2018).

S. Zhou, M. Zhang, J. Preskill, and L. Jiang, “Achieving the Heisenberg limit in quantum metrology using quantum error correction,” Nat. Commun. 9, 78 (2018).
[Crossref]

2017 (5)

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

M. Szczykulska, T. Baumgratz, and A. Datta, “Reaching for the quantum limits in the simultaneous estimation of phase and phase diffusion,” Quantum Sci. Technol. 2, 044004 (2017).
[Crossref]

E. Roccia, M. G. Genoni, L. Mancino, I. Gianani, M. Barbieri, and M. Sbroscia, “Monitoring dispersive samples with single photons: the role of frequency correlations,” Quantum Meas. Quantum Metrol. 4, 64–69 (2017).
[Crossref]

P. Sekatski, M. Skotiniotis, J. Kołodyński, and W. Dür, “Quantum metrology with full and fast quantum control,” Quantum 1, 27 (2017).
[Crossref]

Y. Matsuzaki and S. Benjamin, “Magnetic-field sensing with quantum error detection under the effect of energy relaxation,” Phys. Rev. A 95, 032303 (2017).
[Crossref]

2016 (4)

M. B. Plenio and S. F. Huelga, “Sensing in the presence of an observed environment,” Phys. Rev. A 93, 032123 (2016).
[Crossref]

T. Gefen, D. A. Herrera-Martí, and A. Retzker, “Parameter estimation with efficient photodetectors,” Phys. Rev. A 93, 032133 (2016).
[Crossref]

A. Smirne, J. Kołodyński, S. F. Huelga, and R. Demkowicz-Dobrzański, “Ultimate precision limits for noisy frequency estimation,” Phys. Rev. Lett. 116, 120801 (2016).
[Crossref]

N. Tischler, M. Krenn, R. Fickler, X. Vidal, A. Zeilinger, and G. Molina-Terriza, “Quantum optical rotatory dispersion,” Sci. Adv. 2, e1601306 (2016).
[Crossref]

2015 (4)

X.-Q. Zhou, H. Cable, R. Whittaker, P. Shadbolt, J. L. O’Brien, and J. C. F. Matthews, “Quantum-enhanced tomography of unitary processes,” Optica 2, 510–516 (2015).
[Crossref]

P. Sekatski, M. Skotiniotis, and W. Dür, “Dynamical decoupling leads to improved scaling in noisy quantum metrology,” New J. Phys. 18, 073034 (2015).
[Crossref]

J. B. Brask, R. Chaves, and J. Kołodyński, “Improved quantum magnetometry beyond the standard quantum limit,” Phys. Rev. X 5, 031010 (2015).
[Crossref]

R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kolodynski, “Quantum limits in optical interferometry,” Prog. Opt. 60, 345–435 (2015).
[Crossref]

2014 (7)

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

E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, “Quantum error correction for metrology,” Phys. Rev. Lett. 112, 150802 (2014).
[Crossref]

G. Arrad, Y. Vinkler, D. Aharonov, and A. Retzker, “Increasing sensing resolution with error correction,” Phys. Rev. Lett. 112, 150801 (2014).
[Crossref]

W. Dür, M. Skotiniotis, F. Fröwis, and B. Kraus, “Improved quantum metrology using quantum error correction,” Phys. Rev. Lett. 112, 080801 (2014).
[Crossref]

J. Combes, C. Ferrie, Z. Jiang, and C. M. Caves, “Quantum limits on post-selected, probabilistic quantum metrology,” Phys. Rev. A 89, 052117 (2014).
[Crossref]

P. J. D. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, “Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry,” Phys. Rev. A 89, 023845 (2014).
[Crossref]

M. D. Vidrighin, G. Donati, M. G. Genoni, X.-M. Jin, W. S. Kolthammer, A. D. M. S. Kim, M. Barbieri, and I. A. Walmsley, “Joint estimation of phase and phase diffusion for quantum metrology,” Nat. Commun. 5, 3532 (2014).
[Crossref]

2013 (6)

B. Gendra, E. Ronco-Bonvehi, J. Calsamiglia, R. M. Tapia, and E. Bagan, “Quantum metrology assisted by abstention,” Phys. Rev. Lett. 110, 100501 (2013).
[Crossref]

A. Orieux, L. Sansoni, M. Persechino, P. Mataloni, M. Rossi, and C. Macchiavello, “Experimental detection of quantum channels,” Phys. Rev. Lett. 111, 220501 (2013).
[Crossref]

O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A 88, 040102 (2013).
[Crossref]

M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, “Optimal estimation of joint parameters in phase space,” Phys. Rev. A 87, 012107 (2013).
[Crossref]

C. Vaneph, T. Tufarelli, and M. Genoni, “Quantum estimation of a two-phase spin rotation,” Quantum Meas. Quantum Metrol. 1, 12-20 (2013).
[Crossref]

R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodyński, and A. Acín, “Noisy metrology beyond the standard quantum limit,” Phys. Rev. Lett. 111, 120401 (2013).
[Crossref]

2012 (2)

A. W. Chin, S. F. Huelga, and M. B. Plenio, “Quantum metrology in non-Markovian environments,” Phys. Rev. Lett. 109, 233601 (2012).
[Crossref]

R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref]

2011 (8)

Y. Matsuzaki, S. C. Benjamin, and J. F. Fitzsimons, “Magnetic field sensing beyond the standard quantum limit under the effect of decoherence,” Phys. Rev. A 84, 012103 (2011).
[Crossref]

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

J. Joo, W. J. Munro, and T. P. Spiller, “Erratum: quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107, 219902 (2011).
[Crossref]

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

S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, “Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state,” Phys. Rev. A 83, 021804 (2011).
[Crossref]

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406–411 (2011).
[Crossref]

M. G. Genoni, S. Olivares, and M. G. A. Paris, “Optical phase estimation in the presence of phase diffusion,” Phys. Rev. Lett. 106, 153603 (2011).
[Crossref]

2010 (2)

I. Afek, O. Ambar, and Y. Silberberg, “High-noon states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[Crossref]

M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4, 357–360 (2010).
[Crossref]

2009 (2)

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref]

M. G. A. Paris, “Quantum estimation for quantum technology,” Int. J. Quantum Inf. 7, 125–137 (2009).
[Crossref]

2007 (1)

A. Monras and M. G. A. Paris, “Optimal quantum estimation of loss in bosonic channels,” Phys. Rev. Lett. 98, 160401 (2007).
[Crossref]

2006 (1)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[Crossref]

2004 (2)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330–1336 (2004).
[Crossref]

M. A. Ballester, “Entanglement is not very useful for estimating multiple phases,” Phys. Rev. A 70, 032310 (2004).
[Crossref]

2003 (1)

C. Macchiavello, “Optimal estimation of multiple phases,” Phys. Rev. A 67, 062302 (2003).
[Crossref]

2002 (1)

H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49, 2325–2338 (2002).
[Crossref]

2000 (1)

R. D. Gill and S. Massar, “State estimation for large ensembles,” Phys. Rev. A 61, 042312 (2000).
[Crossref]

1997 (1)

G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387, 471–475 (1997).
[Crossref]

1993 (1)

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

1987 (1)

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[Crossref]

1981 (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[Crossref]

Acín, A.

R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodyński, and A. Acín, “Noisy metrology beyond the standard quantum limit,” Phys. Rev. Lett. 111, 120401 (2013).
[Crossref]

Adesso, G.

M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, “Optimal estimation of joint parameters in phase space,” Phys. Rev. A 87, 012107 (2013).
[Crossref]

A. Górecka, F. A. Pollock, P. Liuzzo-Scorpo, R. Nichols, G. Adesso, and K. Modi, “Noisy frequency estimation with noisy probes,” New J. Phys. (to be published).
[Crossref]

Afek, I.

I. Afek, O. Ambar, and Y. Silberberg, “High-noon states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[Crossref]

Aharonov, D.

G. Arrad, Y. Vinkler, D. Aharonov, and A. Retzker, “Increasing sensing resolution with error correction,” Phys. Rev. Lett. 112, 150801 (2014).
[Crossref]

Albarelli, F.

F. Albarelli, M. A. C. Rossi, D. Tamascelli, and M. G. Genoni, “Restoring Heisenberg scaling in noisy quantum metrology by monitoring the environment,” arXiv:1803.05891 (2018).

Ambar, O.

I. Afek, O. Ambar, and Y. Silberberg, “High-noon states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[Crossref]

Anderson, T.

T. Anderson, An Introduction to Multivariate Statistical Analysis, 3rd ed. (Wiley, 2003)

Arrad, G.

G. Arrad, Y. Vinkler, D. Aharonov, and A. Retzker, “Increasing sensing resolution with error correction,” Phys. Rev. Lett. 112, 150801 (2014).
[Crossref]

Bagan, E.

B. Gendra, E. Ronco-Bonvehi, J. Calsamiglia, R. M. Tapia, and E. Bagan, “Quantum metrology assisted by abstention,” Phys. Rev. Lett. 110, 100501 (2013).
[Crossref]

Ballester, M. A.

M. A. Ballester, “Entanglement is not very useful for estimating multiple phases,” Phys. Rev. A 70, 032310 (2004).
[Crossref]

Banaszek, K.

M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4, 357–360 (2010).
[Crossref]

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref]

Barbieri, M.

E. Roccia, M. G. Genoni, L. Mancino, I. Gianani, M. Barbieri, and M. Sbroscia, “Monitoring dispersive samples with single photons: the role of frequency correlations,” Quantum Meas. Quantum Metrol. 4, 64–69 (2017).
[Crossref]

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

P. J. D. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, “Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry,” Phys. Rev. A 89, 023845 (2014).
[Crossref]

M. D. Vidrighin, G. Donati, M. G. Genoni, X.-M. Jin, W. S. Kolthammer, A. D. M. S. Kim, M. Barbieri, and I. A. Walmsley, “Joint estimation of phase and phase diffusion for quantum metrology,” Nat. Commun. 5, 3532 (2014).
[Crossref]

Baumgratz, T.

M. Szczykulska, T. Baumgratz, and A. Datta, “Reaching for the quantum limits in the simultaneous estimation of phase and phase diffusion,” Quantum Sci. Technol. 2, 044004 (2017).
[Crossref]

Benjamin, S.

Y. Matsuzaki and S. Benjamin, “Magnetic-field sensing with quantum error detection under the effect of energy relaxation,” Phys. Rev. A 95, 032303 (2017).
[Crossref]

Benjamin, S. C.

Y. Matsuzaki, S. C. Benjamin, and J. F. Fitzsimons, “Magnetic field sensing beyond the standard quantum limit under the effect of decoherence,” Phys. Rev. A 84, 012103 (2011).
[Crossref]

Birchall, P. M.

P. M. Birchall, , J. Sabines-Chesterking, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Beating the shot-noise limit with sources of partially-distinguishable photons,” arXiv:1603.00686 (2016).

Blume-Kohout, R.

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

Brask, J. B.

J. B. Brask, R. Chaves, and J. Kołodyński, “Improved quantum magnetometry beyond the standard quantum limit,” Phys. Rev. X 5, 031010 (2015).
[Crossref]

R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodyński, and A. Acín, “Noisy metrology beyond the standard quantum limit,” Phys. Rev. Lett. 111, 120401 (2013).
[Crossref]

Braun, D.

O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A 88, 040102 (2013).
[Crossref]

Breitenbach, G.

G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387, 471–475 (1997).
[Crossref]

Burnett, K.

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

Cable, H.

X.-Q. Zhou, H. Cable, R. Whittaker, P. Shadbolt, J. L. O’Brien, and J. C. F. Matthews, “Quantum-enhanced tomography of unitary processes,” Optica 2, 510–516 (2015).
[Crossref]

P. M. Birchall, , J. Sabines-Chesterking, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Beating the shot-noise limit with sources of partially-distinguishable photons,” arXiv:1603.00686 (2016).

Calsamiglia, J.

B. Gendra, E. Ronco-Bonvehi, J. Calsamiglia, R. M. Tapia, and E. Bagan, “Quantum metrology assisted by abstention,” Phys. Rev. Lett. 110, 100501 (2013).
[Crossref]

Cappellaro, P.

D. Layden and P. Cappellaro, “Spatial noise filtering through error correction for quantum sensing,” npj Quantum Inf. 4, 30 (2018).

Caves, C. M.

J. Combes, C. Ferrie, Z. Jiang, and C. M. Caves, “Quantum limits on post-selected, probabilistic quantum metrology,” Phys. Rev. A 89, 052117 (2014).
[Crossref]

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[Crossref]

Chaves, R.

J. B. Brask, R. Chaves, and J. Kołodyński, “Improved quantum magnetometry beyond the standard quantum limit,” Phys. Rev. X 5, 031010 (2015).
[Crossref]

R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodyński, and A. Acín, “Noisy metrology beyond the standard quantum limit,” Phys. Rev. Lett. 111, 120401 (2013).
[Crossref]

Chin, A. W.

A. W. Chin, S. F. Huelga, and M. B. Plenio, “Quantum metrology in non-Markovian environments,” Phys. Rev. Lett. 109, 233601 (2012).
[Crossref]

Ciampini, M. A.

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

Combes, J.

J. Combes, C. Ferrie, Z. Jiang, and C. M. Caves, “Quantum limits on post-selected, probabilistic quantum metrology,” Phys. Rev. A 89, 052117 (2014).
[Crossref]

Crowley, P. J. D.

P. J. D. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, “Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry,” Phys. Rev. A 89, 023845 (2014).
[Crossref]

Datta, A.

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

M. Szczykulska, T. Baumgratz, and A. Datta, “Reaching for the quantum limits in the simultaneous estimation of phase and phase diffusion,” Quantum Sci. Technol. 2, 044004 (2017).
[Crossref]

P. J. D. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, “Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry,” Phys. Rev. A 89, 023845 (2014).
[Crossref]

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

Davidovich, L.

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406–411 (2011).
[Crossref]

de Matos Filho, R. L.

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406–411 (2011).
[Crossref]

Demkowicz-Dobrzanski, R.

J. F. Haase, A. Smirne, J. Kołodyński, R. Demkowicz-Dobrzański, and S. F. Huelga, “Fundamental limits to frequency estimation: a comprehensive microscopic perspective,” New J. Phys. 20, 053009 (2018).
[Crossref]

A. Smirne, J. Kołodyński, S. F. Huelga, and R. Demkowicz-Dobrzański, “Ultimate precision limits for noisy frequency estimation,” Phys. Rev. Lett. 116, 120801 (2016).
[Crossref]

R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kolodynski, “Quantum limits in optical interferometry,” Prog. Opt. 60, 345–435 (2015).
[Crossref]

R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref]

M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4, 357–360 (2010).
[Crossref]

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref]

Donati, G.

M. D. Vidrighin, G. Donati, M. G. Genoni, X.-M. Jin, W. S. Kolthammer, A. D. M. S. Kim, M. Barbieri, and I. A. Walmsley, “Joint estimation of phase and phase diffusion for quantum metrology,” Nat. Commun. 5, 3532 (2014).
[Crossref]

Dorner, U.

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

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref]

Dowling, J. P.

H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49, 2325–2338 (2002).
[Crossref]

Dür, W.

P. Sekatski, M. Skotiniotis, J. Kołodyński, and W. Dür, “Quantum metrology with full and fast quantum control,” Quantum 1, 27 (2017).
[Crossref]

P. Sekatski, M. Skotiniotis, and W. Dür, “Dynamical decoupling leads to improved scaling in noisy quantum metrology,” New J. Phys. 18, 073034 (2015).
[Crossref]

W. Dür, M. Skotiniotis, F. Fröwis, and B. Kraus, “Improved quantum metrology using quantum error correction,” Phys. Rev. Lett. 112, 080801 (2014).
[Crossref]

Durkin, G. A.

S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, “Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state,” Phys. Rev. A 83, 021804 (2011).
[Crossref]

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

Escher, B. M.

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406–411 (2011).
[Crossref]

Fabre, C.

O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A 88, 040102 (2013).
[Crossref]

Ferrie, C.

J. Combes, C. Ferrie, Z. Jiang, and C. M. Caves, “Quantum limits on post-selected, probabilistic quantum metrology,” Phys. Rev. A 89, 052117 (2014).
[Crossref]

Fickler, R.

N. Tischler, M. Krenn, R. Fickler, X. Vidal, A. Zeilinger, and G. Molina-Terriza, “Quantum optical rotatory dispersion,” Sci. Adv. 2, e1601306 (2016).
[Crossref]

Fitzsimons, J. F.

Y. Matsuzaki, S. C. Benjamin, and J. F. Fitzsimons, “Magnetic field sensing beyond the standard quantum limit under the effect of decoherence,” Phys. Rev. A 84, 012103 (2011).
[Crossref]

Fröwis, F.

W. Dür, M. Skotiniotis, F. Fröwis, and B. Kraus, “Improved quantum metrology using quantum error correction,” Phys. Rev. Lett. 112, 080801 (2014).
[Crossref]

Gefen, T.

T. Gefen, D. A. Herrera-Martí, and A. Retzker, “Parameter estimation with efficient photodetectors,” Phys. Rev. A 93, 032133 (2016).
[Crossref]

Gendra, B.

B. Gendra, E. Ronco-Bonvehi, J. Calsamiglia, R. M. Tapia, and E. Bagan, “Quantum metrology assisted by abstention,” Phys. Rev. Lett. 110, 100501 (2013).
[Crossref]

Genoni, M.

C. Vaneph, T. Tufarelli, and M. Genoni, “Quantum estimation of a two-phase spin rotation,” Quantum Meas. Quantum Metrol. 1, 12-20 (2013).
[Crossref]

Genoni, M. G.

E. Roccia, M. G. Genoni, L. Mancino, I. Gianani, M. Barbieri, and M. Sbroscia, “Monitoring dispersive samples with single photons: the role of frequency correlations,” Quantum Meas. Quantum Metrol. 4, 64–69 (2017).
[Crossref]

M. D. Vidrighin, G. Donati, M. G. Genoni, X.-M. Jin, W. S. Kolthammer, A. D. M. S. Kim, M. Barbieri, and I. A. Walmsley, “Joint estimation of phase and phase diffusion for quantum metrology,” Nat. Commun. 5, 3532 (2014).
[Crossref]

M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, “Optimal estimation of joint parameters in phase space,” Phys. Rev. A 87, 012107 (2013).
[Crossref]

M. G. Genoni, S. Olivares, and M. G. A. Paris, “Optical phase estimation in the presence of phase diffusion,” Phys. Rev. Lett. 106, 153603 (2011).
[Crossref]

F. Albarelli, M. A. C. Rossi, D. Tamascelli, and M. G. Genoni, “Restoring Heisenberg scaling in noisy quantum metrology by monitoring the environment,” arXiv:1803.05891 (2018).

Gianani, I.

E. Roccia, M. G. Genoni, L. Mancino, I. Gianani, M. Barbieri, and M. Sbroscia, “Monitoring dispersive samples with single photons: the role of frequency correlations,” Quantum Meas. Quantum Metrol. 4, 64–69 (2017).
[Crossref]

Gill, R. D.

R. D. Gill and S. Massar, “State estimation for large ensembles,” Phys. Rev. A 61, 042312 (2000).
[Crossref]

Giovannetti, V.

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

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330–1336 (2004).
[Crossref]

Górecka, A.

A. Górecka, F. A. Pollock, P. Liuzzo-Scorpo, R. Nichols, G. Adesso, and K. Modi, “Noisy frequency estimation with noisy probes,” New J. Phys. (to be published).
[Crossref]

Guta, M.

R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref]

Haase, J. F.

J. F. Haase, A. Smirne, J. Kołodyński, R. Demkowicz-Dobrzański, and S. F. Huelga, “Fundamental limits to frequency estimation: a comprehensive microscopic perspective,” New J. Phys. 20, 053009 (2018).
[Crossref]

Herrera-Martí, D. A.

T. Gefen, D. A. Herrera-Martí, and A. Retzker, “Parameter estimation with efficient photodetectors,” Phys. Rev. A 93, 032133 (2016).
[Crossref]

Holland, M. J.

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

Hong, C. K.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[Crossref]

Huelga, S. F.

J. F. Haase, A. Smirne, J. Kołodyński, R. Demkowicz-Dobrzański, and S. F. Huelga, “Fundamental limits to frequency estimation: a comprehensive microscopic perspective,” New J. Phys. 20, 053009 (2018).
[Crossref]

A. Smirne, J. Kołodyński, S. F. Huelga, and R. Demkowicz-Dobrzański, “Ultimate precision limits for noisy frequency estimation,” Phys. Rev. Lett. 116, 120801 (2016).
[Crossref]

M. B. Plenio and S. F. Huelga, “Sensing in the presence of an observed environment,” Phys. Rev. A 93, 032123 (2016).
[Crossref]

A. W. Chin, S. F. Huelga, and M. B. Plenio, “Quantum metrology in non-Markovian environments,” Phys. Rev. Lett. 109, 233601 (2012).
[Crossref]

Humphreys, P. C.

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

Jarzyna, M.

R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kolodynski, “Quantum limits in optical interferometry,” Prog. Opt. 60, 345–435 (2015).
[Crossref]

Jian, P.

O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A 88, 040102 (2013).
[Crossref]

Jiang, L.

S. Zhou, M. Zhang, J. Preskill, and L. Jiang, “Achieving the Heisenberg limit in quantum metrology using quantum error correction,” Nat. Commun. 9, 78 (2018).
[Crossref]

Jiang, Z.

J. Combes, C. Ferrie, Z. Jiang, and C. M. Caves, “Quantum limits on post-selected, probabilistic quantum metrology,” Phys. Rev. A 89, 052117 (2014).
[Crossref]

Jin, X.-M.

M. D. Vidrighin, G. Donati, M. G. Genoni, X.-M. Jin, W. S. Kolthammer, A. D. M. S. Kim, M. Barbieri, and I. A. Walmsley, “Joint estimation of phase and phase diffusion for quantum metrology,” Nat. Commun. 5, 3532 (2014).
[Crossref]

Joo, J.

J. Joo, W. J. Munro, and T. P. Spiller, “Erratum: quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107, 219902 (2011).
[Crossref]

Kacprowicz, M.

M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4, 357–360 (2010).
[Crossref]

Kessler, E. M.

E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, “Quantum error correction for metrology,” Phys. Rev. Lett. 112, 150802 (2014).
[Crossref]

Kim, A. D. M. S.

M. D. Vidrighin, G. Donati, M. G. Genoni, X.-M. Jin, W. S. Kolthammer, A. D. M. S. Kim, M. Barbieri, and I. A. Walmsley, “Joint estimation of phase and phase diffusion for quantum metrology,” Nat. Commun. 5, 3532 (2014).
[Crossref]

Kim, M. S.

M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, “Optimal estimation of joint parameters in phase space,” Phys. Rev. A 87, 012107 (2013).
[Crossref]

Knight, P. L.

M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, “Optimal estimation of joint parameters in phase space,” Phys. Rev. A 87, 012107 (2013).
[Crossref]

Knysh, S.

S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, “Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state,” Phys. Rev. A 83, 021804 (2011).
[Crossref]

Knysh, S. I.

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

Kok, P.

H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49, 2325–2338 (2002).
[Crossref]

Kolodynski, J.

J. F. Haase, A. Smirne, J. Kołodyński, R. Demkowicz-Dobrzański, and S. F. Huelga, “Fundamental limits to frequency estimation: a comprehensive microscopic perspective,” New J. Phys. 20, 053009 (2018).
[Crossref]

P. Sekatski, M. Skotiniotis, J. Kołodyński, and W. Dür, “Quantum metrology with full and fast quantum control,” Quantum 1, 27 (2017).
[Crossref]

A. Smirne, J. Kołodyński, S. F. Huelga, and R. Demkowicz-Dobrzański, “Ultimate precision limits for noisy frequency estimation,” Phys. Rev. Lett. 116, 120801 (2016).
[Crossref]

R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kolodynski, “Quantum limits in optical interferometry,” Prog. Opt. 60, 345–435 (2015).
[Crossref]

J. B. Brask, R. Chaves, and J. Kołodyński, “Improved quantum magnetometry beyond the standard quantum limit,” Phys. Rev. X 5, 031010 (2015).
[Crossref]

R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodyński, and A. Acín, “Noisy metrology beyond the standard quantum limit,” Phys. Rev. Lett. 111, 120401 (2013).
[Crossref]

R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref]

Kolthammer, W. S.

M. D. Vidrighin, G. Donati, M. G. Genoni, X.-M. Jin, W. S. Kolthammer, A. D. M. S. Kim, M. Barbieri, and I. A. Walmsley, “Joint estimation of phase and phase diffusion for quantum metrology,” Nat. Commun. 5, 3532 (2014).
[Crossref]

Kraus, B.

W. Dür, M. Skotiniotis, F. Fröwis, and B. Kraus, “Improved quantum metrology using quantum error correction,” Phys. Rev. Lett. 112, 080801 (2014).
[Crossref]

Krenn, M.

N. Tischler, M. Krenn, R. Fickler, X. Vidal, A. Zeilinger, and G. Molina-Terriza, “Quantum optical rotatory dispersion,” Sci. Adv. 2, e1601306 (2016).
[Crossref]

Layden, D.

D. Layden and P. Cappellaro, “Spatial noise filtering through error correction for quantum sensing,” npj Quantum Inf. 4, 30 (2018).

Lee, H.

H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49, 2325–2338 (2002).
[Crossref]

Liuzzo-Scorpo, P.

A. Górecka, F. A. Pollock, P. Liuzzo-Scorpo, R. Nichols, G. Adesso, and K. Modi, “Noisy frequency estimation with noisy probes,” New J. Phys. (to be published).
[Crossref]

Lloyd, S.

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

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330–1336 (2004).
[Crossref]

Lovchinsky, I.

E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, “Quantum error correction for metrology,” Phys. Rev. Lett. 112, 150802 (2014).
[Crossref]

Lukin, M. D.

E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, “Quantum error correction for metrology,” Phys. Rev. Lett. 112, 150802 (2014).
[Crossref]

Lundeen, J. S.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref]

Macchiavello, C.

A. Orieux, L. Sansoni, M. Persechino, P. Mataloni, M. Rossi, and C. Macchiavello, “Experimental detection of quantum channels,” Phys. Rev. Lett. 111, 220501 (2013).
[Crossref]

C. Macchiavello, “Optimal estimation of multiple phases,” Phys. Rev. A 67, 062302 (2003).
[Crossref]

Maccone, L.

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

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330–1336 (2004).
[Crossref]

Mahler, D. H.

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

Mancino, L.

E. Roccia, M. G. Genoni, L. Mancino, I. Gianani, M. Barbieri, and M. Sbroscia, “Monitoring dispersive samples with single photons: the role of frequency correlations,” Quantum Meas. Quantum Metrol. 4, 64–69 (2017).
[Crossref]

Mandel, L.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[Crossref]

Markiewicz, M.

R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodyński, and A. Acín, “Noisy metrology beyond the standard quantum limit,” Phys. Rev. Lett. 111, 120401 (2013).
[Crossref]

Massar, S.

R. D. Gill and S. Massar, “State estimation for large ensembles,” Phys. Rev. A 61, 042312 (2000).
[Crossref]

Mataloni, P.

A. Orieux, L. Sansoni, M. Persechino, P. Mataloni, M. Rossi, and C. Macchiavello, “Experimental detection of quantum channels,” Phys. Rev. Lett. 111, 220501 (2013).
[Crossref]

Matsuzaki, Y.

Y. Matsuzaki and S. Benjamin, “Magnetic-field sensing with quantum error detection under the effect of energy relaxation,” Phys. Rev. A 95, 032303 (2017).
[Crossref]

Y. Matsuzaki, S. C. Benjamin, and J. F. Fitzsimons, “Magnetic field sensing beyond the standard quantum limit under the effect of decoherence,” Phys. Rev. A 84, 012103 (2011).
[Crossref]

Matthews, J. C. F.

X.-Q. Zhou, H. Cable, R. Whittaker, P. Shadbolt, J. L. O’Brien, and J. C. F. Matthews, “Quantum-enhanced tomography of unitary processes,” Optica 2, 510–516 (2015).
[Crossref]

P. M. Birchall, , J. Sabines-Chesterking, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Beating the shot-noise limit with sources of partially-distinguishable photons,” arXiv:1603.00686 (2016).

Mlynek, J.

G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387, 471–475 (1997).
[Crossref]

Modi, K.

A. Górecka, F. A. Pollock, P. Liuzzo-Scorpo, R. Nichols, G. Adesso, and K. Modi, “Noisy frequency estimation with noisy probes,” New J. Phys. (to be published).
[Crossref]

Molina-Terriza, G.

N. Tischler, M. Krenn, R. Fickler, X. Vidal, A. Zeilinger, and G. Molina-Terriza, “Quantum optical rotatory dispersion,” Sci. Adv. 2, e1601306 (2016).
[Crossref]

Monras, A.

A. Monras and M. G. A. Paris, “Optimal quantum estimation of loss in bosonic channels,” Phys. Rev. Lett. 98, 160401 (2007).
[Crossref]

Munro, W. J.

J. Joo, W. J. Munro, and T. P. Spiller, “Erratum: quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107, 219902 (2011).
[Crossref]

Nha, H.

M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, “Optimal estimation of joint parameters in phase space,” Phys. Rev. A 87, 012107 (2013).
[Crossref]

Nichols, R.

A. Górecka, F. A. Pollock, P. Liuzzo-Scorpo, R. Nichols, G. Adesso, and K. Modi, “Noisy frequency estimation with noisy probes,” New J. Phys. (to be published).
[Crossref]

O’Brien, J. L.

X.-Q. Zhou, H. Cable, R. Whittaker, P. Shadbolt, J. L. O’Brien, and J. C. F. Matthews, “Quantum-enhanced tomography of unitary processes,” Optica 2, 510–516 (2015).
[Crossref]

P. M. Birchall, , J. Sabines-Chesterking, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Beating the shot-noise limit with sources of partially-distinguishable photons,” arXiv:1603.00686 (2016).

Olivares, S.

M. G. Genoni, S. Olivares, and M. G. A. Paris, “Optical phase estimation in the presence of phase diffusion,” Phys. Rev. Lett. 106, 153603 (2011).
[Crossref]

Orieux, A.

A. Orieux, L. Sansoni, M. Persechino, P. Mataloni, M. Rossi, and C. Macchiavello, “Experimental detection of quantum channels,” Phys. Rev. Lett. 111, 220501 (2013).
[Crossref]

Ou, Z. Y.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[Crossref]

Paris, M. G. A.

M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, “Optimal estimation of joint parameters in phase space,” Phys. Rev. A 87, 012107 (2013).
[Crossref]

M. G. Genoni, S. Olivares, and M. G. A. Paris, “Optical phase estimation in the presence of phase diffusion,” Phys. Rev. Lett. 106, 153603 (2011).
[Crossref]

M. G. A. Paris, “Quantum estimation for quantum technology,” Int. J. Quantum Inf. 7, 125–137 (2009).
[Crossref]

A. Monras and M. G. A. Paris, “Optimal quantum estimation of loss in bosonic channels,” Phys. Rev. Lett. 98, 160401 (2007).
[Crossref]

Persechino, M.

A. Orieux, L. Sansoni, M. Persechino, P. Mataloni, M. Rossi, and C. Macchiavello, “Experimental detection of quantum channels,” Phys. Rev. Lett. 111, 220501 (2013).
[Crossref]

Pezzè, L.

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

Pinel, O.

O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A 88, 040102 (2013).
[Crossref]

Plenio, M. B.

M. B. Plenio and S. F. Huelga, “Sensing in the presence of an observed environment,” Phys. Rev. A 93, 032123 (2016).
[Crossref]

A. W. Chin, S. F. Huelga, and M. B. Plenio, “Quantum metrology in non-Markovian environments,” Phys. Rev. Lett. 109, 233601 (2012).
[Crossref]

Pollock, F. A.

A. Górecka, F. A. Pollock, P. Liuzzo-Scorpo, R. Nichols, G. Adesso, and K. Modi, “Noisy frequency estimation with noisy probes,” New J. Phys. (to be published).
[Crossref]

Preskill, J.

S. Zhou, M. Zhang, J. Preskill, and L. Jiang, “Achieving the Heisenberg limit in quantum metrology using quantum error correction,” Nat. Commun. 9, 78 (2018).
[Crossref]

Retzker, A.

T. Gefen, D. A. Herrera-Martí, and A. Retzker, “Parameter estimation with efficient photodetectors,” Phys. Rev. A 93, 032133 (2016).
[Crossref]

G. Arrad, Y. Vinkler, D. Aharonov, and A. Retzker, “Increasing sensing resolution with error correction,” Phys. Rev. Lett. 112, 150801 (2014).
[Crossref]

Roccia, E.

E. Roccia, M. G. Genoni, L. Mancino, I. Gianani, M. Barbieri, and M. Sbroscia, “Monitoring dispersive samples with single photons: the role of frequency correlations,” Quantum Meas. Quantum Metrol. 4, 64–69 (2017).
[Crossref]

Ronco-Bonvehi, E.

B. Gendra, E. Ronco-Bonvehi, J. Calsamiglia, R. M. Tapia, and E. Bagan, “Quantum metrology assisted by abstention,” Phys. Rev. Lett. 110, 100501 (2013).
[Crossref]

Rossi, M.

A. Orieux, L. Sansoni, M. Persechino, P. Mataloni, M. Rossi, and C. Macchiavello, “Experimental detection of quantum channels,” Phys. Rev. Lett. 111, 220501 (2013).
[Crossref]

Rossi, M. A. C.

F. Albarelli, M. A. C. Rossi, D. Tamascelli, and M. G. Genoni, “Restoring Heisenberg scaling in noisy quantum metrology by monitoring the environment,” arXiv:1803.05891 (2018).

Rozema, L. A.

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

Sabines-Chesterking, J.

P. M. Birchall, , J. Sabines-Chesterking, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Beating the shot-noise limit with sources of partially-distinguishable photons,” arXiv:1603.00686 (2016).

Sansoni, L.

A. Orieux, L. Sansoni, M. Persechino, P. Mataloni, M. Rossi, and C. Macchiavello, “Experimental detection of quantum channels,” Phys. Rev. Lett. 111, 220501 (2013).
[Crossref]

Sbroscia, M.

E. Roccia, M. G. Genoni, L. Mancino, I. Gianani, M. Barbieri, and M. Sbroscia, “Monitoring dispersive samples with single photons: the role of frequency correlations,” Quantum Meas. Quantum Metrol. 4, 64–69 (2017).
[Crossref]

Schiller, S.

G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387, 471–475 (1997).
[Crossref]

Sciarrino, F.

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

Sekatski, P.

P. Sekatski, M. Skotiniotis, J. Kołodyński, and W. Dür, “Quantum metrology with full and fast quantum control,” Quantum 1, 27 (2017).
[Crossref]

P. Sekatski, M. Skotiniotis, and W. Dür, “Dynamical decoupling leads to improved scaling in noisy quantum metrology,” New J. Phys. 18, 073034 (2015).
[Crossref]

Shadbolt, P.

Silberberg, Y.

I. Afek, O. Ambar, and Y. Silberberg, “High-noon states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[Crossref]

Skotiniotis, M.

P. Sekatski, M. Skotiniotis, J. Kołodyński, and W. Dür, “Quantum metrology with full and fast quantum control,” Quantum 1, 27 (2017).
[Crossref]

P. Sekatski, M. Skotiniotis, and W. Dür, “Dynamical decoupling leads to improved scaling in noisy quantum metrology,” New J. Phys. 18, 073034 (2015).
[Crossref]

W. Dür, M. Skotiniotis, F. Fröwis, and B. Kraus, “Improved quantum metrology using quantum error correction,” Phys. Rev. Lett. 112, 080801 (2014).
[Crossref]

Smelyanskiy, V. N.

S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, “Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state,” Phys. Rev. A 83, 021804 (2011).
[Crossref]

Smerzi, A.

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

Smirne, A.

J. F. Haase, A. Smirne, J. Kołodyński, R. Demkowicz-Dobrzański, and S. F. Huelga, “Fundamental limits to frequency estimation: a comprehensive microscopic perspective,” New J. Phys. 20, 053009 (2018).
[Crossref]

A. Smirne, J. Kołodyński, S. F. Huelga, and R. Demkowicz-Dobrzański, “Ultimate precision limits for noisy frequency estimation,” Phys. Rev. Lett. 116, 120801 (2016).
[Crossref]

Smith, B. J.

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

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref]

Spagnolo, N.

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

Spiller, T. P.

J. Joo, W. J. Munro, and T. P. Spiller, “Erratum: quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107, 219902 (2011).
[Crossref]

Steinberg, A. M.

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

Sushkov, A. O.

E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, “Quantum error correction for metrology,” Phys. Rev. Lett. 112, 150802 (2014).
[Crossref]

Szczykulska, M.

M. Szczykulska, T. Baumgratz, and A. Datta, “Reaching for the quantum limits in the simultaneous estimation of phase and phase diffusion,” Quantum Sci. Technol. 2, 044004 (2017).
[Crossref]

Tamascelli, D.

F. Albarelli, M. A. C. Rossi, D. Tamascelli, and M. G. Genoni, “Restoring Heisenberg scaling in noisy quantum metrology by monitoring the environment,” arXiv:1803.05891 (2018).

Tapia, R. M.

B. Gendra, E. Ronco-Bonvehi, J. Calsamiglia, R. M. Tapia, and E. Bagan, “Quantum metrology assisted by abstention,” Phys. Rev. Lett. 110, 100501 (2013).
[Crossref]

Thomas-Peter, N.

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

Tischler, N.

N. Tischler, M. Krenn, R. Fickler, X. Vidal, A. Zeilinger, and G. Molina-Terriza, “Quantum optical rotatory dispersion,” Sci. Adv. 2, e1601306 (2016).
[Crossref]

Treps, N.

O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A 88, 040102 (2013).
[Crossref]

Tufarelli, T.

C. Vaneph, T. Tufarelli, and M. Genoni, “Quantum estimation of a two-phase spin rotation,” Quantum Meas. Quantum Metrol. 1, 12-20 (2013).
[Crossref]

Vaneph, C.

C. Vaneph, T. Tufarelli, and M. Genoni, “Quantum estimation of a two-phase spin rotation,” Quantum Meas. Quantum Metrol. 1, 12-20 (2013).
[Crossref]

Vidal, X.

N. Tischler, M. Krenn, R. Fickler, X. Vidal, A. Zeilinger, and G. Molina-Terriza, “Quantum optical rotatory dispersion,” Sci. Adv. 2, e1601306 (2016).
[Crossref]

Vidrighin, M. D.

M. D. Vidrighin, G. Donati, M. G. Genoni, X.-M. Jin, W. S. Kolthammer, A. D. M. S. Kim, M. Barbieri, and I. A. Walmsley, “Joint estimation of phase and phase diffusion for quantum metrology,” Nat. Commun. 5, 3532 (2014).
[Crossref]

Vinkler, Y.

G. Arrad, Y. Vinkler, D. Aharonov, and A. Retzker, “Increasing sensing resolution with error correction,” Phys. Rev. Lett. 112, 150801 (2014).
[Crossref]

Walmsley, I. A.

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

P. J. D. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, “Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry,” Phys. Rev. A 89, 023845 (2014).
[Crossref]

M. D. Vidrighin, G. Donati, M. G. Genoni, X.-M. Jin, W. S. Kolthammer, A. D. M. S. Kim, M. Barbieri, and I. A. Walmsley, “Joint estimation of phase and phase diffusion for quantum metrology,” Nat. Commun. 5, 3532 (2014).
[Crossref]

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

M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4, 357–360 (2010).
[Crossref]

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref]

Wasilewski, W.

M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4, 357–360 (2010).
[Crossref]

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref]

Whittaker, R.

Zeilinger, A.

N. Tischler, M. Krenn, R. Fickler, X. Vidal, A. Zeilinger, and G. Molina-Terriza, “Quantum optical rotatory dispersion,” Sci. Adv. 2, e1601306 (2016).
[Crossref]

Zhang, L.

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

Zhang, M.

S. Zhou, M. Zhang, J. Preskill, and L. Jiang, “Achieving the Heisenberg limit in quantum metrology using quantum error correction,” Nat. Commun. 9, 78 (2018).
[Crossref]

Zhou, S.

S. Zhou, M. Zhang, J. Preskill, and L. Jiang, “Achieving the Heisenberg limit in quantum metrology using quantum error correction,” Nat. Commun. 9, 78 (2018).
[Crossref]

Zhou, X.-Q.

Int. J. Quantum Inf. (1)

M. G. A. Paris, “Quantum estimation for quantum technology,” Int. J. Quantum Inf. 7, 125–137 (2009).
[Crossref]

J. Mod. Opt. (1)

H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49, 2325–2338 (2002).
[Crossref]

Nat. Commun. (3)

R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref]

S. Zhou, M. Zhang, J. Preskill, and L. Jiang, “Achieving the Heisenberg limit in quantum metrology using quantum error correction,” Nat. Commun. 9, 78 (2018).
[Crossref]

M. D. Vidrighin, G. Donati, M. G. Genoni, X.-M. Jin, W. S. Kolthammer, A. D. M. S. Kim, M. Barbieri, and I. A. Walmsley, “Joint estimation of phase and phase diffusion for quantum metrology,” Nat. Commun. 5, 3532 (2014).
[Crossref]

Nat. Photonics (2)

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

M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4, 357–360 (2010).
[Crossref]

Nat. Phys. (1)

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406–411 (2011).
[Crossref]

Nature (1)

G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387, 471–475 (1997).
[Crossref]

New J. Phys. (2)

P. Sekatski, M. Skotiniotis, and W. Dür, “Dynamical decoupling leads to improved scaling in noisy quantum metrology,” New J. Phys. 18, 073034 (2015).
[Crossref]

J. F. Haase, A. Smirne, J. Kołodyński, R. Demkowicz-Dobrzański, and S. F. Huelga, “Fundamental limits to frequency estimation: a comprehensive microscopic perspective,” New J. Phys. 20, 053009 (2018).
[Crossref]

npj Quantum Inf. (1)

D. Layden and P. Cappellaro, “Spatial noise filtering through error correction for quantum sensing,” npj Quantum Inf. 4, 30 (2018).

Optica (1)

Phys. Rev. A (13)

R. D. Gill and S. Massar, “State estimation for large ensembles,” Phys. Rev. A 61, 042312 (2000).
[Crossref]

C. Macchiavello, “Optimal estimation of multiple phases,” Phys. Rev. A 67, 062302 (2003).
[Crossref]

M. A. Ballester, “Entanglement is not very useful for estimating multiple phases,” Phys. Rev. A 70, 032310 (2004).
[Crossref]

O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A 88, 040102 (2013).
[Crossref]

M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, “Optimal estimation of joint parameters in phase space,” Phys. Rev. A 87, 012107 (2013).
[Crossref]

P. J. D. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, “Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry,” Phys. Rev. A 89, 023845 (2014).
[Crossref]

J. Combes, C. Ferrie, Z. Jiang, and C. M. Caves, “Quantum limits on post-selected, probabilistic quantum metrology,” Phys. Rev. A 89, 052117 (2014).
[Crossref]

M. B. Plenio and S. F. Huelga, “Sensing in the presence of an observed environment,” Phys. Rev. A 93, 032123 (2016).
[Crossref]

T. Gefen, D. A. Herrera-Martí, and A. Retzker, “Parameter estimation with efficient photodetectors,” Phys. Rev. A 93, 032133 (2016).
[Crossref]

Y. Matsuzaki and S. Benjamin, “Magnetic-field sensing with quantum error detection under the effect of energy relaxation,” Phys. Rev. A 95, 032303 (2017).
[Crossref]

Y. Matsuzaki, S. C. Benjamin, and J. F. Fitzsimons, “Magnetic field sensing beyond the standard quantum limit under the effect of decoherence,” Phys. Rev. A 84, 012103 (2011).
[Crossref]

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

S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, “Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state,” Phys. Rev. A 83, 021804 (2011).
[Crossref]

Phys. Rev. D (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[Crossref]

Phys. Rev. Lett. (17)

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

J. Joo, W. J. Munro, and T. P. Spiller, “Erratum: quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107, 219902 (2011).
[Crossref]

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[Crossref]

A. W. Chin, S. F. Huelga, and M. B. Plenio, “Quantum metrology in non-Markovian environments,” Phys. Rev. Lett. 109, 233601 (2012).
[Crossref]

A. Smirne, J. Kołodyński, S. F. Huelga, and R. Demkowicz-Dobrzański, “Ultimate precision limits for noisy frequency estimation,” Phys. Rev. Lett. 116, 120801 (2016).
[Crossref]

R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodyński, and A. Acín, “Noisy metrology beyond the standard quantum limit,” Phys. Rev. Lett. 111, 120401 (2013).
[Crossref]

E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, “Quantum error correction for metrology,” Phys. Rev. Lett. 112, 150802 (2014).
[Crossref]

G. Arrad, Y. Vinkler, D. Aharonov, and A. Retzker, “Increasing sensing resolution with error correction,” Phys. Rev. Lett. 112, 150801 (2014).
[Crossref]

W. Dür, M. Skotiniotis, F. Fröwis, and B. Kraus, “Improved quantum metrology using quantum error correction,” Phys. Rev. Lett. 112, 080801 (2014).
[Crossref]

A. Orieux, L. Sansoni, M. Persechino, P. Mataloni, M. Rossi, and C. Macchiavello, “Experimental detection of quantum channels,” Phys. Rev. Lett. 111, 220501 (2013).
[Crossref]

A. Monras and M. G. A. Paris, “Optimal quantum estimation of loss in bosonic channels,” Phys. Rev. Lett. 98, 160401 (2007).
[Crossref]

M. G. Genoni, S. Olivares, and M. G. A. Paris, “Optical phase estimation in the presence of phase diffusion,” Phys. Rev. Lett. 106, 153603 (2011).
[Crossref]

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

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[Crossref]

B. Gendra, E. Ronco-Bonvehi, J. Calsamiglia, R. M. Tapia, and E. Bagan, “Quantum metrology assisted by abstention,” Phys. Rev. Lett. 110, 100501 (2013).
[Crossref]

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

Phys. Rev. X (2)

J. B. Brask, R. Chaves, and J. Kołodyński, “Improved quantum magnetometry beyond the standard quantum limit,” Phys. Rev. X 5, 031010 (2015).
[Crossref]

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

Prog. Opt. (1)

R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kolodynski, “Quantum limits in optical interferometry,” Prog. Opt. 60, 345–435 (2015).
[Crossref]

Quantum (1)

P. Sekatski, M. Skotiniotis, J. Kołodyński, and W. Dür, “Quantum metrology with full and fast quantum control,” Quantum 1, 27 (2017).
[Crossref]

Quantum Meas. Quantum Metrol. (2)

C. Vaneph, T. Tufarelli, and M. Genoni, “Quantum estimation of a two-phase spin rotation,” Quantum Meas. Quantum Metrol. 1, 12-20 (2013).
[Crossref]

E. Roccia, M. G. Genoni, L. Mancino, I. Gianani, M. Barbieri, and M. Sbroscia, “Monitoring dispersive samples with single photons: the role of frequency correlations,” Quantum Meas. Quantum Metrol. 4, 64–69 (2017).
[Crossref]

Quantum Sci. Technol. (1)

M. Szczykulska, T. Baumgratz, and A. Datta, “Reaching for the quantum limits in the simultaneous estimation of phase and phase diffusion,” Quantum Sci. Technol. 2, 044004 (2017).
[Crossref]

Sci. Adv. (1)

N. Tischler, M. Krenn, R. Fickler, X. Vidal, A. Zeilinger, and G. Molina-Terriza, “Quantum optical rotatory dispersion,” Sci. Adv. 2, e1601306 (2016).
[Crossref]

Science (2)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330–1336 (2004).
[Crossref]

I. Afek, O. Ambar, and Y. Silberberg, “High-noon states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[Crossref]

Other (5)

P. M. Birchall, , J. Sabines-Chesterking, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Beating the shot-noise limit with sources of partially-distinguishable photons,” arXiv:1603.00686 (2016).

F. Albarelli, M. A. C. Rossi, D. Tamascelli, and M. G. Genoni, “Restoring Heisenberg scaling in noisy quantum metrology by monitoring the environment,” arXiv:1803.05891 (2018).

A. Górecka, F. A. Pollock, P. Liuzzo-Scorpo, R. Nichols, G. Adesso, and K. Modi, “Noisy frequency estimation with noisy probes,” New J. Phys. (to be published).
[Crossref]

T. Anderson, An Introduction to Multivariate Statistical Analysis, 3rd ed. (Wiley, 2003)

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

Supplementary Material (1)

NameDescription
» Supplement 1       Supplemental document.

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. Experimental setup: each one of the two single photons (wavelength 810 nm) of the pair generated via type-I SPDC from a beta barium borate (BBO, 3 mm length) nonlinear crystal excited via a continuous-wave (80 mW power) pump laser passes through a half-wave plate (HWP1 at 0° and HWP2 at 45°) before being combined on a polarized beam splitter (PBS1). These photons are used to estimate the birefringent phase imparted by the optical activity of a chiral solution. A wave plate (HWP3) and a second polarizer (PBS2) project the outcoming photons onto different polarizations. In the calibration procedure, an additional HWP, not sketched here, replaces the solution to impart a well-defined phase.
Fig. 2.
Fig. 2. Multiparameter Bayesian estimation for setup calibration. (a) Estimated phase (blue triangles, left scale) and visibility (green circles; right scale) versus calibration phase, ϕ, imparted by an HWP. Dashed lines are linear fit of data. (b) and (c) Estimated variance (times the number of resources M) for (b) visibility and (c) phase as a function of the imparted phase. The dashed line represents the corresponding CRBs. (d) Estimated covariance for the visibility and phase as a function of the imparted phase. The dashed line represents the corresponding CRB. All covariance matrices have been estimated from M70  K repetitions. Error bars are smaller than the marker size for all data.
Fig. 3.
Fig. 3. Upper panels: Bayesian joint probabilities for visibility, V, and phase, ϕ; the joint distribution is normalized to unit. Lower panels: difference of the experimental density probability with respect to the one at the CRB for (a) fructose aqueous solution with 0.3 g/ml nominal concentration and (b) sucrose aqueous solution with 0.3 g/ml nominal concentration. A number M50  K and M75  K of repetitions have been employed for fructose and sucrose, respectively. The expected probabilities are calculated as a two-dimensional Gaussian with covariance matrix F1/M; the observed differences between the experiment and the prediction are partly due to deviations from this shape. The V and ϕ are the same for the upper and lower panels.
Fig. 4.
Fig. 4. Scaling of the Fisher information: (a) effective Fisher information for the distinguishability parameter ε. The points correspond to numerical results, and the solid lines correspond to 2N2 (blue) and N (red). (b) Effective Fisher information for the phase ϕ. The points correspond to numerical results, and the solid lines correspond to 2N(N+1) (black) and N (red). (c) Trade-off in the optimality of individual estimations quantified by ϒ. In all plots: blue circles, ε=0.14; yellow squares, ε=0.23; green diamonds, ε=0.32; red triangles, ε=0.50; purple inverted triangles, ε=1.

Equations (10)

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

a^Ha^V|0=12((a^R)2(a^L)2)|0=12(|2R,0L|0R,2L).
|ψ=cosϕa^Ha^V|0sinϕ(a^H)2(a^V)22|0.
p(1|θ;ϕ,v)=11+v(1+vcos(8θ2ϕ)),p(2|θ;ϕ,v)=v1+vsin2(4θϕ),
p(θ|1;ϕ,v)=14(1+vcos(8θ2ϕ)),
ϕB=ϕPB(ϕ,v|n¯)dϕdv,vB=vPB(ϕ,v|n¯)dϕdv,
Δ2ϕ=Σϕ,ϕ=(ϕϕB)2PB(ϕ,v|n¯)dϕdv,Δ2v=Σv,v=(vvB)2PB(ϕ,v|n¯)dϕdv,Σϕ,v=Σv,ϕ=(ϕϕB)(vvB)PB(ϕ,v|n¯)dϕdv.
Fij=θip(θ|ϕ,v)jp(θ|ϕ,v)p(θ|ϕ,v),
ΣF1/M.
l=M2Tr(F·Σ)M(lndet(Σ)+lndet(MF))2
ϒ=maxγ(F˜ϕ,ϕ(γ)maxαF˜ϕ,ϕ(α)+F˜ε,ε(γ)maxβF˜ε,ε(β)).

Metrics