Abstract

The Cramér-Rao bound plays a central role in both classical and quantum parameter estimation, but finding the observable and the resulting inversion estimator that saturates this bound remains an open issue for general multi-outcome measurements. Here we consider multi-outcome homodyne detection in a coherent-light Mach-Zehnder interferometer and construct a family of inversion estimators that almost saturate the Cramér-Rao bound over the whole range of phase interval. This provides a clue on constructing optimal inversion estimators for phase estimation and other parameter estimation in any multi-outcome measurement.

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

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    [Crossref]

2018 (3)

D. Braun, G. Adesso, F. Benatti, R. Floreanini, U. Marzolino, M. W. Mitchell, and S. Pirandola, “Quantum-enhanced measurements without entanglement,” Rev. Mod. Phys. 90, 035006 (2018).
[Crossref]

L. Ghirardi, I. Siloi, P. Bordone, F. Troiani, and M. G. A. Paris, “Quantum metrology at level anticrossing,” Phys. Rev. A 97, 012120 (2018).
[Crossref]

C. Schafermeier, M. Jezex, L. S. Madsen, T. Gehring, and U. L. Andersen, “Deterministic phase measurements exhibiting super-sensitivity and super-resolution,” Optica 5, 60–64 (2018).
[Crossref]

2017 (3)

G. R. Jin, W. Yang, and C. P. Sun, “Quantum-enhanced microscopy with binary-outcome photon counting,” Phys. Rev. A 95, 013835 (2017).
[Crossref]

J. D. Zhang, Z. J. Zhang, L. Z. Cen, S. Li, Y. Zhao, and F. Wang, “Super-resolution and super-sensitivity of entangled squeezed vacuum state using optimal detection strategy,” Chin. Phys. B 26, 094204 (2017).
[Crossref]

Z. J. Zhang, T. Y. Qiao, L. Z. Cen, J. D. Zhang, F. Wang, and Y. Zhao, “Optimal quantum detection strategy for super-resolving angular-rotation measurement,” Appl. Phys. B 123, 148 (2017).
[Crossref]

2016 (2)

Q. Wang, L. L. Hao, H. X. Tang, Y. Zhang, C. H. Yang, X. Yang, L. Xu, and Y. Zhao, “Super-resolving quantum LiDAR with even coherent states sources in the presence of loss and noise,” Phys. Lett. A 380, 3717 (2016).
[Crossref]

Q. Wang, L. L. Hao, Y. Zhang, L. Xu, C. H. Yang, X. Yang, and Y. Zhao, “Super-resolving quantum lidar: entangled coherent-state sources with binary-outcome photon counting measurement suffice to beat the shot-noise limit,” Opt. Express 24, 5045–5056 (2016).
[Crossref] [PubMed]

2014 (3)

X. M. Feng, G. R. Jin, and W. Yang, “Quantum interferometry with binary-outcome measurements in the presence of phase diffusion,” Phys. Rev. A. 90, 013807 (2014).
[Crossref]

L. Cohen, D. Istrati, L. Dovrat, and H. S. Eisenberg, “Super-resolved phase measurements at the shot noise limit by parity measurement,” Opt. Express 22, 11945–11953 (2014).
[Crossref] [PubMed]

Q. S. Tan, J. Q. Liao, X. G. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89, 053822 (2014).
[Crossref]

2013 (3)

E. Distante, M. Ježek, and U. L. Andersen, “Deterministic Superresolution with Coherent States at the Shot Noise Limit,” Phys. Rev. Lett. 111, 033603 (2013).
[Crossref] [PubMed]

Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88, 043832 (2013).
[Crossref]

K. P. Seshadreesan, S. Kim, J. P. Dowling, and H. Lee, “Phase estimation at the quantum Cram-Rao bound via parity detection,” Phys. Rev. A 87, 043833 (2013).
[Crossref]

2011 (2)

A. Chiruvelli and H. Lee, “Parity measurements in quantum optical metrology,” J. Mod. Opt. 58, 945–953 (2011).
[Crossref]

K. P. Seshadreesan, P. M. Anisimov, H. Lee, and J. P. Dowling, “Parity detection achieves the Heisenberg limit in interferometry with coherent mixed with squeezed vacuum light,” New J. Phys. 13, 083026 (2011).
[Crossref]

2010 (5)

C.C. Gerry and J. Mimih, “The parity operator in quantum optical metrology,” Contemp. Phys. 51, 497–511 (2010).
[Crossref]

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit,” Phys. Rev. Lett. 104, 103602 (2010).
[Crossref] [PubMed]

Y. Gao, P. M. Anisimov, C. F. Wildfeuer, J. Luine, H. Lee, and J. P. Dowling, “Super-resolution at the shot-noise limit with coherent states and photon-number-resolving detectors,” J. Opt. Soc. Am. B 27, A170–A174 (2010).
[Crossref]

Y. A. Chen, X. H. Bao, Z. S. Yuan, S. Chen, B. Zhao, and J. W. Pan, “Heralded Generation of an Atomic NOON State,” Phys. Rev. Lett. 104, 043601 (2010).
[Crossref] [PubMed]

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON States by Mixing Quantum and Classical Ligh,” Science 328, 879–881 (2010).
[Crossref] [PubMed]

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] [PubMed]

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

2008 (1)

J. P. Dowling, “Quantum optical metrology-the lowdown on high-N00N states,” Contemp. Phys. 49, 125–143 (2008).
[Crossref]

2007 (2)

K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-Reversal and Super-Resolving Phase Measurements,” Phys. Rev. Lett. 98, 223601 (2007).
[Crossref] [PubMed]

L. Pezzé, A. Smerzi, G. Khoury, J. F. Hodelin, and D. Bouwmeester, “Phase Detection at the Quantum Limit with Multiphoton Mach-Zehnder Interferometry,” Phys. Rev. Lett. 99, 223602 (2007).
[Crossref]

2004 (3)

D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-Limited Spectroscopy with Multiparticle Entangled States,” Science 304, 1476–1478 (2004).
[Crossref] [PubMed]

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
[Crossref] [PubMed]

P. Walther, J. W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “de Broglie wavelength of a non-local four-photon state,” Nature 429, 158–161 (2004).
[Crossref] [PubMed]

2002 (1)

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Nonlinear interferometer as a resource for maximally entangled photonic states: Application to interferometry,” Phys. Rev. A 66, 013804 (2002).
[Crossref]

2000 (2)

C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61, 043811 (2000).
[Crossref]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit,” Phys. Rev. Lett. 85, 2733 (2000).
[Crossref] [PubMed]

1996 (2)

S. L. Braunstein, C. M. Caves, and G. J. Milburn, “Generalized Uncertainty Relations: Theory, Examples, and Lorentz Invariance,” Ann. Phys. (N.Y.) 247, 135–173 (1996).
[Crossref]

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54, R4649 (1996).
[Crossref] [PubMed]

1994 (2)

D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, “Squeezed atomic states and projection noise in spectroscopy,” Phys. Rev. A. 50, 67 (1994).
[Crossref] [PubMed]

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439 (1994).
[Crossref] [PubMed]

1986 (1)

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033 (1986).
[Crossref]

1981 (1)

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

Abrams, D. S.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit,” Phys. Rev. Lett. 85, 2733 (2000).
[Crossref] [PubMed]

Adesso, G.

D. Braun, G. Adesso, F. Benatti, R. Floreanini, U. Marzolino, M. W. Mitchell, and S. Pirandola, “Quantum-enhanced measurements without entanglement,” Rev. Mod. Phys. 90, 035006 (2018).
[Crossref]

Afek, I.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON States by Mixing Quantum and Classical Ligh,” Science 328, 879–881 (2010).
[Crossref] [PubMed]

Ambar, O.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON States by Mixing Quantum and Classical Ligh,” Science 328, 879–881 (2010).
[Crossref] [PubMed]

Andersen, U. L.

C. Schafermeier, M. Jezex, L. S. Madsen, T. Gehring, and U. L. Andersen, “Deterministic phase measurements exhibiting super-sensitivity and super-resolution,” Optica 5, 60–64 (2018).
[Crossref]

E. Distante, M. Ježek, and U. L. Andersen, “Deterministic Superresolution with Coherent States at the Shot Noise Limit,” Phys. Rev. Lett. 111, 033603 (2013).
[Crossref] [PubMed]

Anisimov, P. M.

K. P. Seshadreesan, P. M. Anisimov, H. Lee, and J. P. Dowling, “Parity detection achieves the Heisenberg limit in interferometry with coherent mixed with squeezed vacuum light,” New J. Phys. 13, 083026 (2011).
[Crossref]

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit,” Phys. Rev. Lett. 104, 103602 (2010).
[Crossref] [PubMed]

Y. Gao, P. M. Anisimov, C. F. Wildfeuer, J. Luine, H. Lee, and J. P. Dowling, “Super-resolution at the shot-noise limit with coherent states and photon-number-resolving detectors,” J. Opt. Soc. Am. B 27, A170–A174 (2010).
[Crossref]

Aspelmeyer, M.

P. Walther, J. W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “de Broglie wavelength of a non-local four-photon state,” Nature 429, 158–161 (2004).
[Crossref] [PubMed]

Banaszek, K.

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] [PubMed]

Bao, X. H.

Y. A. Chen, X. H. Bao, Z. S. Yuan, S. Chen, B. Zhao, and J. W. Pan, “Heralded Generation of an Atomic NOON State,” Phys. Rev. Lett. 104, 043601 (2010).
[Crossref] [PubMed]

Barrett, M. D.

D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-Limited Spectroscopy with Multiparticle Entangled States,” Science 304, 1476–1478 (2004).
[Crossref] [PubMed]

Benatti, F.

D. Braun, G. Adesso, F. Benatti, R. Floreanini, U. Marzolino, M. W. Mitchell, and S. Pirandola, “Quantum-enhanced measurements without entanglement,” Rev. Mod. Phys. 90, 035006 (2018).
[Crossref]

Benmoussa, A.

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Nonlinear interferometer as a resource for maximally entangled photonic states: Application to interferometry,” Phys. Rev. A 66, 013804 (2002).
[Crossref]

Bevington, P. R.

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, 1969).

Bollinger, J. J.

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54, R4649 (1996).
[Crossref] [PubMed]

D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, “Squeezed atomic states and projection noise in spectroscopy,” Phys. Rev. A. 50, 67 (1994).
[Crossref] [PubMed]

Bordone, P.

L. Ghirardi, I. Siloi, P. Bordone, F. Troiani, and M. G. A. Paris, “Quantum metrology at level anticrossing,” Phys. Rev. A 97, 012120 (2018).
[Crossref]

Born, M.

M. Born and E. Wolf, Principle of Optics (Cambridge University, 1999)
[Crossref]

Boto, A. N.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit,” Phys. Rev. Lett. 85, 2733 (2000).
[Crossref] [PubMed]

Bouwmeester, D.

L. Pezzé, A. Smerzi, G. Khoury, J. F. Hodelin, and D. Bouwmeester, “Phase Detection at the Quantum Limit with Multiphoton Mach-Zehnder Interferometry,” Phys. Rev. Lett. 99, 223602 (2007).
[Crossref]

Braun, D.

D. Braun, G. Adesso, F. Benatti, R. Floreanini, U. Marzolino, M. W. Mitchell, and S. Pirandola, “Quantum-enhanced measurements without entanglement,” Rev. Mod. Phys. 90, 035006 (2018).
[Crossref]

Braunstein, S. L.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit,” Phys. Rev. Lett. 85, 2733 (2000).
[Crossref] [PubMed]

S. L. Braunstein, C. M. Caves, and G. J. Milburn, “Generalized Uncertainty Relations: Theory, Examples, and Lorentz Invariance,” Ann. Phys. (N.Y.) 247, 135–173 (1996).
[Crossref]

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439 (1994).
[Crossref] [PubMed]

Britton, J.

D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-Limited Spectroscopy with Multiparticle Entangled States,” Science 304, 1476–1478 (2004).
[Crossref] [PubMed]

Campos, R. A.

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Nonlinear interferometer as a resource for maximally entangled photonic states: Application to interferometry,” Phys. Rev. A 66, 013804 (2002).
[Crossref]

Caves, C. M.

S. L. Braunstein, C. M. Caves, and G. J. Milburn, “Generalized Uncertainty Relations: Theory, Examples, and Lorentz Invariance,” Ann. Phys. (N.Y.) 247, 135–173 (1996).
[Crossref]

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439 (1994).
[Crossref] [PubMed]

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

Cen, L. Z.

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P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit,” Phys. Rev. Lett. 104, 103602 (2010).
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Y. A. Chen, X. H. Bao, Z. S. Yuan, S. Chen, B. Zhao, and J. W. Pan, “Heralded Generation of an Atomic NOON State,” Phys. Rev. Lett. 104, 043601 (2010).
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D. Braun, G. Adesso, F. Benatti, R. Floreanini, U. Marzolino, M. W. Mitchell, and S. Pirandola, “Quantum-enhanced measurements without entanglement,” Rev. Mod. Phys. 90, 035006 (2018).
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P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit,” Phys. Rev. Lett. 104, 103602 (2010).
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P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit,” Phys. Rev. Lett. 104, 103602 (2010).
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Seshadreesan, K. P.

K. P. Seshadreesan, S. Kim, J. P. Dowling, and H. Lee, “Phase estimation at the quantum Cram-Rao bound via parity detection,” Phys. Rev. A 87, 043833 (2013).
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K. P. Seshadreesan, P. M. Anisimov, H. Lee, and J. P. Dowling, “Parity detection achieves the Heisenberg limit in interferometry with coherent mixed with squeezed vacuum light,” New J. Phys. 13, 083026 (2011).
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L. Pezzé, A. Smerzi, G. Khoury, J. F. Hodelin, and D. Bouwmeester, “Phase Detection at the Quantum Limit with Multiphoton Mach-Zehnder Interferometry,” Phys. Rev. Lett. 99, 223602 (2007).
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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).
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M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
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G. R. Jin, W. Yang, and C. P. Sun, “Quantum-enhanced microscopy with binary-outcome photon counting,” Phys. Rev. A 95, 013835 (2017).
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Q. S. Tan, J. Q. Liao, X. G. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89, 053822 (2014).
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Q. Wang, L. L. Hao, H. X. Tang, Y. Zhang, C. H. Yang, X. Yang, L. Xu, and Y. Zhao, “Super-resolving quantum LiDAR with even coherent states sources in the presence of loss and noise,” Phys. Lett. A 380, 3717 (2016).
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L. Ghirardi, I. Siloi, P. Bordone, F. Troiani, and M. G. A. Paris, “Quantum metrology at level anticrossing,” Phys. Rev. A 97, 012120 (2018).
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P. Walther, J. W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “de Broglie wavelength of a non-local four-photon state,” Nature 429, 158–161 (2004).
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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).
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P. Walther, J. W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “de Broglie wavelength of a non-local four-photon state,” Nature 429, 158–161 (2004).
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J. D. Zhang, Z. J. Zhang, L. Z. Cen, S. Li, Y. Zhao, and F. Wang, “Super-resolution and super-sensitivity of entangled squeezed vacuum state using optimal detection strategy,” Chin. Phys. B 26, 094204 (2017).
[Crossref]

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Q. Wang, L. L. Hao, Y. Zhang, L. Xu, C. H. Yang, X. Yang, and Y. Zhao, “Super-resolving quantum lidar: entangled coherent-state sources with binary-outcome photon counting measurement suffice to beat the shot-noise limit,” Opt. Express 24, 5045–5056 (2016).
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Q. S. Tan, J. Q. Liao, X. G. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89, 053822 (2014).
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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).
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Williams, C. P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit,” Phys. Rev. Lett. 85, 2733 (2000).
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D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-Limited Spectroscopy with Multiparticle Entangled States,” Science 304, 1476–1478 (2004).
[Crossref] [PubMed]

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54, R4649 (1996).
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D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, “Squeezed atomic states and projection noise in spectroscopy,” Phys. Rev. A. 50, 67 (1994).
[Crossref] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principle of Optics (Cambridge University, 1999)
[Crossref]

Xu, L.

Q. Wang, L. L. Hao, H. X. Tang, Y. Zhang, C. H. Yang, X. Yang, L. Xu, and Y. Zhao, “Super-resolving quantum LiDAR with even coherent states sources in the presence of loss and noise,” Phys. Lett. A 380, 3717 (2016).
[Crossref]

Q. Wang, L. L. Hao, Y. Zhang, L. Xu, C. H. Yang, X. Yang, and Y. Zhao, “Super-resolving quantum lidar: entangled coherent-state sources with binary-outcome photon counting measurement suffice to beat the shot-noise limit,” Opt. Express 24, 5045–5056 (2016).
[Crossref] [PubMed]

Yang, C. H.

Q. Wang, L. L. Hao, Y. Zhang, L. Xu, C. H. Yang, X. Yang, and Y. Zhao, “Super-resolving quantum lidar: entangled coherent-state sources with binary-outcome photon counting measurement suffice to beat the shot-noise limit,” Opt. Express 24, 5045–5056 (2016).
[Crossref] [PubMed]

Q. Wang, L. L. Hao, H. X. Tang, Y. Zhang, C. H. Yang, X. Yang, L. Xu, and Y. Zhao, “Super-resolving quantum LiDAR with even coherent states sources in the presence of loss and noise,” Phys. Lett. A 380, 3717 (2016).
[Crossref]

Yang, W.

G. R. Jin, W. Yang, and C. P. Sun, “Quantum-enhanced microscopy with binary-outcome photon counting,” Phys. Rev. A 95, 013835 (2017).
[Crossref]

X. M. Feng, G. R. Jin, and W. Yang, “Quantum interferometry with binary-outcome measurements in the presence of phase diffusion,” Phys. Rev. A. 90, 013807 (2014).
[Crossref]

Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88, 043832 (2013).
[Crossref]

Yang, X.

Q. Wang, L. L. Hao, H. X. Tang, Y. Zhang, C. H. Yang, X. Yang, L. Xu, and Y. Zhao, “Super-resolving quantum LiDAR with even coherent states sources in the presence of loss and noise,” Phys. Lett. A 380, 3717 (2016).
[Crossref]

Q. Wang, L. L. Hao, Y. Zhang, L. Xu, C. H. Yang, X. Yang, and Y. Zhao, “Super-resolving quantum lidar: entangled coherent-state sources with binary-outcome photon counting measurement suffice to beat the shot-noise limit,” Opt. Express 24, 5045–5056 (2016).
[Crossref] [PubMed]

Yuan, Z. S.

Y. A. Chen, X. H. Bao, Z. S. Yuan, S. Chen, B. Zhao, and J. W. Pan, “Heralded Generation of an Atomic NOON State,” Phys. Rev. Lett. 104, 043601 (2010).
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B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033 (1986).
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Zeilinger, A.

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Zhang, J. D.

J. D. Zhang, Z. J. Zhang, L. Z. Cen, S. Li, Y. Zhao, and F. Wang, “Super-resolution and super-sensitivity of entangled squeezed vacuum state using optimal detection strategy,” Chin. Phys. B 26, 094204 (2017).
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Z. J. Zhang, T. Y. Qiao, L. Z. Cen, J. D. Zhang, F. Wang, and Y. Zhao, “Optimal quantum detection strategy for super-resolving angular-rotation measurement,” Appl. Phys. B 123, 148 (2017).
[Crossref]

J. D. Zhang, Z. J. Zhang, L. Z. Cen, J. Y. Hu, and Y. Zhao, “Deterministic super-resolved estimation towards angular displacements based upon a Sagnac interferometer and parity measurement,” arXiv:1809.04830 (2018).

Zhang, Y.

Q. Wang, L. L. Hao, Y. Zhang, L. Xu, C. H. Yang, X. Yang, and Y. Zhao, “Super-resolving quantum lidar: entangled coherent-state sources with binary-outcome photon counting measurement suffice to beat the shot-noise limit,” Opt. Express 24, 5045–5056 (2016).
[Crossref] [PubMed]

Q. Wang, L. L. Hao, H. X. Tang, Y. Zhang, C. H. Yang, X. Yang, L. Xu, and Y. Zhao, “Super-resolving quantum LiDAR with even coherent states sources in the presence of loss and noise,” Phys. Lett. A 380, 3717 (2016).
[Crossref]

Zhang, Y. M.

Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88, 043832 (2013).
[Crossref]

Zhang, Z. J.

J. D. Zhang, Z. J. Zhang, L. Z. Cen, S. Li, Y. Zhao, and F. Wang, “Super-resolution and super-sensitivity of entangled squeezed vacuum state using optimal detection strategy,” Chin. Phys. B 26, 094204 (2017).
[Crossref]

Z. J. Zhang, T. Y. Qiao, L. Z. Cen, J. D. Zhang, F. Wang, and Y. Zhao, “Optimal quantum detection strategy for super-resolving angular-rotation measurement,” Appl. Phys. B 123, 148 (2017).
[Crossref]

J. D. Zhang, Z. J. Zhang, L. Z. Cen, J. Y. Hu, and Y. Zhao, “Deterministic super-resolved estimation towards angular displacements based upon a Sagnac interferometer and parity measurement,” arXiv:1809.04830 (2018).

Zhao, B.

Y. A. Chen, X. H. Bao, Z. S. Yuan, S. Chen, B. Zhao, and J. W. Pan, “Heralded Generation of an Atomic NOON State,” Phys. Rev. Lett. 104, 043601 (2010).
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Zhao, Y.

J. D. Zhang, Z. J. Zhang, L. Z. Cen, S. Li, Y. Zhao, and F. Wang, “Super-resolution and super-sensitivity of entangled squeezed vacuum state using optimal detection strategy,” Chin. Phys. B 26, 094204 (2017).
[Crossref]

Z. J. Zhang, T. Y. Qiao, L. Z. Cen, J. D. Zhang, F. Wang, and Y. Zhao, “Optimal quantum detection strategy for super-resolving angular-rotation measurement,” Appl. Phys. B 123, 148 (2017).
[Crossref]

Q. Wang, L. L. Hao, H. X. Tang, Y. Zhang, C. H. Yang, X. Yang, L. Xu, and Y. Zhao, “Super-resolving quantum LiDAR with even coherent states sources in the presence of loss and noise,” Phys. Lett. A 380, 3717 (2016).
[Crossref]

Q. Wang, L. L. Hao, Y. Zhang, L. Xu, C. H. Yang, X. Yang, and Y. Zhao, “Super-resolving quantum lidar: entangled coherent-state sources with binary-outcome photon counting measurement suffice to beat the shot-noise limit,” Opt. Express 24, 5045–5056 (2016).
[Crossref] [PubMed]

J. D. Zhang, Z. J. Zhang, L. Z. Cen, J. Y. Hu, and Y. Zhao, “Deterministic super-resolved estimation towards angular displacements based upon a Sagnac interferometer and parity measurement,” arXiv:1809.04830 (2018).

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Appl. Phys. B (1)

Z. J. Zhang, T. Y. Qiao, L. Z. Cen, J. D. Zhang, F. Wang, and Y. Zhao, “Optimal quantum detection strategy for super-resolving angular-rotation measurement,” Appl. Phys. B 123, 148 (2017).
[Crossref]

Chin. Phys. B (1)

J. D. Zhang, Z. J. Zhang, L. Z. Cen, S. Li, Y. Zhao, and F. Wang, “Super-resolution and super-sensitivity of entangled squeezed vacuum state using optimal detection strategy,” Chin. Phys. B 26, 094204 (2017).
[Crossref]

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C.C. Gerry and J. Mimih, “The parity operator in quantum optical metrology,” Contemp. Phys. 51, 497–511 (2010).
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J. P. Dowling, “Quantum optical metrology-the lowdown on high-N00N states,” Contemp. Phys. 49, 125–143 (2008).
[Crossref]

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M. G. A. Paris, “Quantum estimation for quantum technology, ” Int. J. Quantum. Inform. 7, 125–137 (2009).
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A. Chiruvelli and H. Lee, “Parity measurements in quantum optical metrology,” J. Mod. Opt. 58, 945–953 (2011).
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J. Opt. Soc. Am. B (1)

Nature (2)

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
[Crossref] [PubMed]

P. Walther, J. W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “de Broglie wavelength of a non-local four-photon state,” Nature 429, 158–161 (2004).
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New J. Phys. (1)

K. P. Seshadreesan, P. M. Anisimov, H. Lee, and J. P. Dowling, “Parity detection achieves the Heisenberg limit in interferometry with coherent mixed with squeezed vacuum light,” New J. Phys. 13, 083026 (2011).
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Opt. Express (2)

Optica (1)

Phys. Lett. A (1)

Q. Wang, L. L. Hao, H. X. Tang, Y. Zhang, C. H. Yang, X. Yang, L. Xu, and Y. Zhao, “Super-resolving quantum LiDAR with even coherent states sources in the presence of loss and noise,” Phys. Lett. A 380, 3717 (2016).
[Crossref]

Phys. Rev. A (9)

K. P. Seshadreesan, S. Kim, J. P. Dowling, and H. Lee, “Phase estimation at the quantum Cram-Rao bound via parity detection,” Phys. Rev. A 87, 043833 (2013).
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L. Ghirardi, I. Siloi, P. Bordone, F. Troiani, and M. G. A. Paris, “Quantum metrology at level anticrossing,” Phys. Rev. A 97, 012120 (2018).
[Crossref]

G. R. Jin, W. Yang, and C. P. Sun, “Quantum-enhanced microscopy with binary-outcome photon counting,” Phys. Rev. A 95, 013835 (2017).
[Crossref]

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033 (1986).
[Crossref]

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54, R4649 (1996).
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Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88, 043832 (2013).
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Q. S. Tan, J. Q. Liao, X. G. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89, 053822 (2014).
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Phys. Rev. A. (2)

D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, “Squeezed atomic states and projection noise in spectroscopy,” Phys. Rev. A. 50, 67 (1994).
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X. M. Feng, G. R. Jin, and W. Yang, “Quantum interferometry with binary-outcome measurements in the presence of phase diffusion,” Phys. Rev. A. 90, 013807 (2014).
[Crossref]

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

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Y. A. Chen, X. H. Bao, Z. S. Yuan, S. Chen, B. Zhao, and J. W. Pan, “Heralded Generation of an Atomic NOON State,” Phys. Rev. Lett. 104, 043601 (2010).
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K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-Reversal and Super-Resolving Phase Measurements,” Phys. Rev. Lett. 98, 223601 (2007).
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A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit,” Phys. Rev. Lett. 85, 2733 (2000).
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Rev. Mod. Phys. (1)

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Science (2)

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D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-Limited Spectroscopy with Multiparticle Entangled States,” Science 304, 1476–1478 (2004).
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[Crossref]

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J. D. Zhang, Z. J. Zhang, L. Z. Cen, J. Y. Hu, and Y. Zhao, “Deterministic super-resolved estimation towards angular displacements based upon a Sagnac interferometer and parity measurement,” arXiv:1809.04830 (2018).

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Figures (4)

Fig. 1
Fig. 1 (a) Homodyne detection at one port of the coherent-state interferometer, equivalent to measuring quadrature operator with respect to the output state. (b) Conditional probability P(p|ϕ) against the phase shift ϕ and the measured quadrature p, given by Eq. (11), and the post-processing method by separating the measured data into several bins [17], where the bin’s center is kb (for k = 0, ±1, ...±kf) and the width is 2a, equivalent to a multi-outcome measurement. (c) and (d): Output signal 〈Π̂〉ϕ and phase sensitivity δϕ for kf = 0, equivalent to a binary-outcome measurement. The vertical lines determine the resolution and the best sensitivity, measured respectively by the full width at half maximum (FWHM) and the best sensitivity δϕmin. (e) and (f): Density plots of the ratios 2 π / 3 FWHM and 1 / n ¯ δ ϕ min as functions of the average photon number n ¯ ( = α 0 2 ) and the bin size a. Dashed lines: contours of the two ratios. Solid lines and below: a region that the visibility of the signal ≥ 90%.
Fig. 2
Fig. 2 Conditional probabilities P(k|ϕ) for k = 0, ±1, ..., ±kf, and P ( | ϕ ) = 1 k P ( k | ϕ ). The parameters: = 200, a = 1/2, b = 3.8, and hence the total number of the outcomes 2(kf + 1) = 6, since kf = 2 (see text). Left panel: P (±2|ϕ) for the blue solid lines, and P(±1|ϕ) for the red dashed lines. Numerical simulations: averaged occurrence frequency ��k/�� (the solid circles) and its standard derivation (the bars) of each outcome after M = 10 replicas of �� = 200 independent measurements.
Fig. 3
Fig. 3 Output signal 〈Π̂〉ϕ and phase sensitivity δϕ for μ = 0 and different choices of {μk}. The parameters: = 200, a = 1/2, b = 3.8, and kf = 2. (a) and (b): μk = 1 for all k ’s. (c) and (d): {μ−2, μ−1, μ0, μ1, μ2} = {−0.715, 0.068, 0.839, −0.102, 0.392}. (e) and (f): {μ−2, μ−1, μ0, μ1, μ2} = {1, −1, 1, −1, 1}. The red dashed lines: the CRB 1 / F ( ϕ ). The vertical lines: locations of the first dark points ϕ dark ± b / n ¯. The horizontal lines in right panel: the best sensitivity δ ϕ min 1.37 / n ¯.
Fig. 4
Fig. 4 Output signal 〈Π̂〉ϕ and phase sensitivity δϕ for alternating signs of {μk} and the parameters: = 1000, a = 1/2, b = 3.2, and kf = 5 (i.e., total number of the outcomes is 12). Solid circles: the averaged signal and the phase uncertainty of the inversion estimator ϕinv obtained with �� = 200 and M = 400. The horizontal line in (b): the best sensitivity δ ϕ min 1.37 / n ¯. Inset: Difference between the average value of the inversion estimator ϕ ¯ inv = i = 1 M ϕ inv ( i ) / M and the true value of phase shift ϕ. The bars are the standard deviations of the estimators { ϕ inv ( 1 ) , ϕ inv ( 2 ) , , ϕ inv ( M ) }.

Equations (23)

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δ ϕ = Δ Π ^ | Π ^ ϕ / ϕ | .
Π ^ = k = 0 N μ k Π ^ k ,
Π ^ ϕ = k = 0 N μ k P ( k | ϕ ) k = 0 N μ k 𝒩 k 𝒩 ,
Π ^ ϕ = μ 0 P ( 0 | ϕ ) + μ 1 P ( | ϕ ) , Π ^ 2 ϕ = μ 0 2 P ( 0 | ϕ ) + μ 1 2 P ( | ϕ ) ,
( δ ϕ ) 2 = P ( | ϕ ) P ( 0 | ϕ ) [ P ( | ϕ ) ] 2 ,
F ( ϕ ) k = 0 N [ P ( k | ϕ ) ] 2 P ( k | ϕ ) .
U ^ ( ϕ ) = exp ( i π 2 J ^ y ) exp ( i ϕ a ^ a ^ ) exp ( i π 2 J ^ y ) ,
P ( p | ϕ ) = d x d X d P W out ( α , β ; ϕ ) ,
W out ( α , β ; ϕ ) = W in ( α ˜ ϕ , β ˜ ϕ ) ,
{ α ˜ ϕ = α e i ϕ 1 2 + β e i ϕ + 1 2 β ˜ ϕ = α e i ϕ + 1 2 β e i ϕ 1 2 .
W in ( α , β ) = ( 2 π ) 2 e 2 | α α 0 | 2 e 2 | β | 2 .
P ( p | ϕ ) = 2 π exp [ 2 ( p + α 0 2 sin ϕ ) 2 ] ,
P ( + | ϕ ) = a + a d p P ( p | ϕ ) = 1 2 Erf [ g ( ϕ ) , g + ( ϕ ) ] ,
g ± ( ϕ ) = 2 ( α 0 2 sin ϕ ± a ) .
Π ^ ϕ = μ + P ( + | ϕ ) + μ P ( | ϕ ) ,
δ ϕ = P ( + | ϕ ) P ( | ϕ ) | P ( + | ϕ ) | ,
F ( ϕ ) = k = ± [ P ( k | ϕ ) ] 2 P ( k | ϕ ) = ( δ ϕ ) 2 ,
V = Π ^ ϕ = 0 Π ^ ϕ = π / 2 Π ^ ϕ = 0 + Π ^ ϕ = π / 2 ,
P ( k | ϕ ) = b k a b k + a d p P ( p | ϕ ) = 1 2 Erf [ g ( ϕ ) + 2 b k , g + ( ϕ ) + 2 b k ] ,
Π ^ ϕ = k = k f k f μ k P ( k | ϕ ) + μ P ( | ϕ ) ,
ϕ k = arcsin ( 2 b k α 0 ) ,
δ ϕ = P ( + | ϕ ) P ( | ϕ ) | P ( | ϕ ) | 1 F ( ϕ ) ,
F ( ϕ ) = k = k f k f [ P ( k | ϕ ) ] 2 P ( k | ϕ ) + [ P ( | ϕ ) ] 2 P ( | ϕ ) .

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