Optimal frequency measurements with maximally correlated states
Abstract
We show how maximally correlated states of N two-level particles can be used in spectroscopy to yield a frequency uncertainty equal to (NT${)}^{\mathrm{\ensuremath{-}}1}$, where T is the time of a single measurement. From the time-energy uncertainty relation we show that this is the best precision possible. We rephrase these results in the language of particle interferometry and obtain a state and detection operator which can be used to achieve a phase uncertainty exactly equal to the 1/N Heisenberg limit, where N is the number of particles used in the measurement.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The states that maximize the quantum Fisher information and yield Heisenberg scaling for decoherence-free unitary channels are typically highly entangled: the GHZ state for atomic spectroscopy and the N00N state for optical interferometry.
“The states that maximize the QFI and yield the HS for decoherence-free unitary channels are typically highly entangled: the GHZ state in the case of atomic spectroscopy43, and the N00N state in the case of optical interferometry44.”
From The elusive Heisenberg limit in quantum-enhanced metrology (2012) · cites this paper as [43]
Cites
Quantum Computations with Cold Trapped Ions
Cirac, J. I., & Zoller, P. (1995) · Physical Review Letters · doi:10.1103/physrevlett.74.4091 · Cited by 3,733 · see citing works
Quantum projection noise: Population fluctuations in two-level systems
Itano, W. M., Bergquist, J. C., Bollinger, J. J., Gilligan, J. M., Heinzen, D. J., Moore, F. L., Raizen, M. G., & Wineland, D. J. (1993) · Physical Review A · doi:10.1103/physreva.47.3554 · Cited by 613 · see citing works
Self-citationJacobson, J., Björk, G., Chuang, I., & Yamamoto, Y. (1995) · Physical Review Letters · doi:10.1103/physrevlett.74.4835 · Cited by 296 · see citing works
Mode coupling in a Penning trap:<i>π</i>pulses and a classical avoided crossing
Cornell, E. A., Weisskoff, R. M., Boyce, K. R., & Pritchard, D. E. (1990) · Physical Review A · doi:10.1103/physreva.41.312 · Cited by 181 · see citing works
Flagged first, then by citations
Cited by
Sorted by citations
Degen, C. L., Reinhard, F., & Cappellaro, P. (2017) · Reviews of Modern Physics · doi:10.1103/revmodphys.89.035002 · Cited by 4,045 · see citing works
Giovannetti, V., Lloyd, S., & Maccone, L. (2011) · Nature Photonics · doi:10.1038/nphoton.2011.35 · Cited by 3,894 · see citing works
Giovannetti, V., Lloyd, S., & Maccone, L. (2006) · Physical Review Letters · doi:10.1103/physrevlett.96.010401 · Cited by 2,440 · see citing works
Ludlow, A. D., Boyd, M. M., Ye, J., Peik, E., & Schmidt, P. O. (2015) · Reviews of Modern Physics · doi:10.1103/revmodphys.87.637 · Cited by 2,385 · see citing works
Experimental entanglement of four particles
Sackett, C. A., Kielpinski, D., King, B. E., Langer, C., Meyer, V., Myatt, C. J., Rowe, M., Turchette, Q. A., Itano, W. M., Wineland, D. J., et al. (2000) · Nature · doi:10.1038/35005011 · Cited by 1,436 · see citing works
Self-citationImprovement of Frequency Standards with Quantum Entanglement
Huelga, S. F., Macchiavello, C., Pellizzari, T., Ekert, A. K., Plenio, M. B., & Cirac, J. I. (1997) · Physical Review Letters · doi:10.1103/physrevlett.79.3865 · Cited by 999 · see citing works
General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology
Escher, B. M., de Matos Filho, R. L., & Davidovich, L. (2011) · Nature Physics · doi:10.1038/nphys1958 · Cited by 928 · see citing works
Quantum optical metrology – the lowdown on high-N00N states
Dowling, J. P. (2008) · Contemporary Physics · doi:10.1080/00107510802091298 · Cited by 894 · see citing works
The elusive Heisenberg limit in quantum-enhanced metrology
Demkowicz-Dobrzański, R., Kołodyński, J., & Guţă, M. (2012) · Nature Communications · doi:10.1038/ncomms2067 · Cited by 773 · see citing works
Entanglement-free Heisenberg-limited phase estimation
Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M., & Pryde, G. J. (2007) · Nature · doi:10.1038/nature06257 · Cited by 628 · see citing works
De Broglie wavelength of a non-local four-photon state
Walther, P., Pan, J.-W., Aspelmeyer, M., Ursin, R., Gasparoni, S., & Zeilinger, A. (2004) · Nature · doi:10.1038/nature02552 · Cited by 555 · see citing works
Optimal Quantum Phase Estimation
Dorner, U., Demkowicz-Dobrzanski, R., Smith, B. J., Lundeen, J. S., Wasilewski, W., Banaszek, K., & Walmsley, I. A. (2009) · Physical Review Letters · doi:10.1103/physrevlett.102.040403 · Cited by 494 · see citing works
Quantum-enhanced measurements without entanglement
Braun, D., Adesso, G., Benatti, F., Floreanini, R., Marzolino, U., Mitchell, M. W., & Pirandola, S. (2018) · Reviews of Modern Physics · doi:10.1103/revmodphys.90.035006 · Cited by 434 · see citing works
Efficient Engineering of Multiatom Entanglement through Single-Photon Detections
Duan, L.-M., & Kimble, H. J. (2003) · Physical Review Letters · doi:10.1103/physrevlett.90.253601 · Cited by 406 · see citing works
Quantum enhanced multiple phase estimation
Humphreys, P. C., Barbieri, M., Datta, A., & Walmsley, I. A. (2013) · Physical Review Letters · doi:10.1103/physrevlett.111.070403 · Cited by 379 · see citing works
Quantum phase estimation with lossy interferometers
Demkowicz-Dobrzanski, R., Dorner, U., Smith, B. J., Lundeen, J. S., Wasilewski, W., Banaszek, K., & Walmsley, I. A. (2009) · Physical Review A · doi:10.1103/physreva.80.013825 · Cited by 315 · see citing works
Correlated input-port, matter-wave interferometer: Quantum-noise limits to the atom-laser gyroscope
Dowling, J. P. (1998) · Physical Review A · doi:10.1103/physreva.57.4736 · Cited by 319 · see citing works
Multiparameter Estimation in Networked Quantum Sensors
Proctor, T. J., Knott, P. A., & Dunningham, J. A. (2018) · Physical Review Letters · doi:10.1103/physrevlett.120.080501 · Cited by 318 · see citing works
“conclusions do not necessarily hold if φ1 and φ2 have a known dependence: the extreme case is when we know that φ1 = φ2, in which case estimating φ ≡ φ1 = φ2 is a well-known one-parameter problem, and a global GHZ is optimal [41, 52].”
BackgroundBell States of Atoms with Ultralong Lifetimes and Their Tomographic State Analysis
Roos, C. F., Lancaster, G. P. T., Riebe, M., Häffner, H., Hänsel, W., Gulde, S., Becher, C., Eschner, J., Schmidt-Kaler, F., & Blatt, R. (2004) · Physical Review Letters · doi:10.1103/physrevlett.92.220402 · Cited by 241 · see citing works
Quantum interferometry with and without an external phase reference
Jarzyna, M., & Demkowicz-Dobrzański, R. (2012) · Physical Review A · doi:10.1103/physreva.85.011801 · Cited by 182 · see citing works
Optimal quantum-enhanced interferometry using a laser power source
Lang, M. D., & Caves, C. M. (2013) · Physical Review Letters · doi:10.1103/physrevlett.111.173601 · Cited by 181 · see citing works
Quantum metrology for a general Hamiltonian parameter
Pang, S., & Brun, T. A. (2014) · Physical Review A · doi:10.1103/physreva.90.022117 · Cited by 175 · see citing works
Ziv–Zakai error bounds for quantum parameter estimation
Tsang, M. (2012) · Physical Review Letters · doi:10.1103/physrevlett.108.230401 · Cited by 137 · see citing works
Spin squeezing and decoherence limit in Ramsey spectroscopy
Ulam-Orgikh, D., & Kitagawa, M. (2001) · Physical Review A · doi:10.1103/physreva.64.052106 · Cited by 130 · see citing works
Phase estimation without<i>a priori</i>phase knowledge in the presence of loss
Kołodyński, J., & Demkowicz-Dobrzański, R. (2010) · Physical Review A · doi:10.1103/physreva.82.053804 · Cited by 131 · see citing works
Nonlinear quantum metrology of many-body open systems
Beau, M., & del Campo, A. (2017) · Physical Review Letters · doi:10.1103/physrevlett.119.010403 · Cited by 107 · see citing works
Improved Quantum Magnetometry beyond the Standard Quantum Limit
Brask, J. B., Chaves, R., & Kołodyński, J. (2015) · Physical Review X · doi:10.1103/physrevx.5.031010 · Cited by 99 · see citing works
Creation of maximally entangled photon-number states using optical fiber multiports
Pryde, G. J., & White, A. G. (2003) · Physical Review A · doi:10.1103/physreva.68.052315 · Cited by 91 · see citing works
Experimental phase estimation enhanced by machine learning
Lumino, A., Polino, E., Rab, A. S., Milani, G., Spagnolo, N., Wiebe, N., & Sciarrino, F. (2018) · Physical Review Applied · doi:10.1103/physrevapplied.10.044033 · Cited by 97 · see citing works
Quantum Metrology for Noisy Systems
Escher, B. M., de Matos Filho, R. L., & Davidovich, L. (2011) · Brazilian Journal of Physics · doi:10.1007/s13538-011-0037-y · Cited by 86 · see citing works
Quantum Bell–Ziv–Zakai bounds and Heisenberg limits for waveform estimation
Berry, D. W., Tsang, M., Hall, M. J. W., & Wiseman, H. M. (2015) · Physical Review X · doi:10.1103/physrevx.5.031018 · Cited by 81 · see citing works
Control-enhanced sequential scheme for general quantum parameter estimation at the Heisenberg limit
Hou, Z., Wang, R.-J., Tang, J.-F., Yuan, H., Xiang, G.-Y., Li, C.-F., & Guo, G.-C. (2019) · Physical Review Letters · doi:10.1103/physrevlett.123.040501 · Cited by 68 · see citing works
Efficient Atomic Clocks Operated with Several Atomic Ensembles
Borregaard, J., & Sørensen, A. S. (2013) · Physical Review Letters · doi:10.1103/physrevlett.111.090802 · Cited by 67 · see citing works
Quantum multiparameter estimation with generalized balanced multimode noon-like states
Zhang, L., & Chan, K. W. C. (2017) · Physical Review A · doi:10.1103/physreva.95.032321 · Cited by 54 · see citing works
Quantum metrology of spatial deformation using arrays of classical and quantum light emitters
Sidhu, J. S., & Kok, P. (2017) · Physical Review A · doi:10.1103/physreva.95.063829 · Cited by 31 · see citing works
Estimation of gradients in quantum metrology
Altenburg, S., Oszmaniec, M., Wölk, S., & Gühne, O. (2017) · Physical Review A · doi:10.1103/physreva.96.042319 · Cited by 23 · see citing works
Towards electron-electron entanglement in Penning traps
Lamata, L., Porras, D., Cirac, J. I., Goldman, J., & Gabrielse, G. (2010) · Physical Review A · doi:10.1103/physreva.81.022301 · Cited by 22 · see citing works
- Authors
- Bollinger, J. J., Itano, W. M., Wineland, D. J., & Heinzen, D. J.
- Affiliations
- National Institute of Standards and Technology (US)
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 1996-12-01
- Volume
- 54
- Issue
- 6
- Pages
- R4649-R4652
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W2051618454
- Cited by
- 1,167
- References
- 4
- Citations by year
- 2026: 46 · 2025: 48 · 2024: 58 · 2023: 40 · 2022: 41 · 2021: 62 · 2020: 58 · 2019: 80 · 2018: 64 · 2017: 54 · 2016: 43 · 2015: 52 · 2014: 58 · 2013: 48 · 2012: 55
- FWCI
- 12.90
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Physical Review A (publishedVersion)
- PubMed (publishedVersion)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum optics and atomic interactions
- Keywords
- frequency measurement, time-energy uncertainty relation, Heisenberg limit, phase uncertainty, quantum metrology, spectroscopy, quantum correlations
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
