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Quantum metrology

## Summary

Quantum metrology is the study of making high-resolution and highly sensitive measurements of physical parameters using quantum theory to describe the physical systems,[1][2][3][4][5][6] particularly exploiting quantum entanglement and quantum squeezing. This field promises to develop measurement techniques that give better precision than the same measurement performed in a classical framework. Together with quantum hypothesis testing,[7][8] it represents an important theoretical model at the basis of quantum sensing.[9][10]

## Mathematical foundations

A basic task of quantum metrology is estimating the parameter ${\displaystyle \theta }$ of the unitary dynamics

${\displaystyle \varrho (\theta )=\exp(-iH\theta )\varrho _{0}\exp(+iH\theta ),}$

where ${\displaystyle \varrho _{0}}$ is the initial state of the system and ${\displaystyle H}$ is the Hamiltonian of the system. ${\displaystyle \theta }$ is estimated based on measurements on ${\displaystyle \varrho (\theta ).}$

Typically, the system is composed of many particles, and the Hamiltonian is a sum of single-particle terms

${\displaystyle H=\sum _{k}H_{k},}$

where ${\displaystyle H_{k}}$ acts on the kth particle. In this case, there is no interaction between the particles, and we talk about linear interferometers.

The achievable precision is bounded from below by the quantum Cramér-Rao bound as

${\displaystyle (\Delta \theta )^{2}\geq {\frac {1}{F_{\rm {Q}}[\varrho ,H]}},}$

where ${\displaystyle F_{\rm {Q}}[\varrho ,H]}$ is the quantum Fisher information.[1][11]

## Examples

One example of note is the use of the NOON state in a Mach–Zehnder interferometer to perform accurate phase measurements.[12] A similar effect can be produced using less exotic states such as squeezed states. In atomic ensembles, spin squeezed states can be used for phase measurements.

## Applications

An important application of particular note is the detection of gravitational radiation in projects such as LIGO or the Virgo interferometer, where high-precision measurements must be made for the relative distance between two widely-separated masses. However, the measurements described by quantum metrology are currently not used in this setting, being difficult to implement. Furthermore, there are other sources of noise affecting the detection of gravitational waves which must be overcome first. Nevertheless, plans may call for the use of quantum metrology in LIGO.[13]

## Scaling and the effect of noise

A central question of quantum metrology is how the precision, i.e., the variance of the parameter estimation, scales with the number of particles. Classical interferometers cannot overcome the shot-noise limit

${\displaystyle (\Delta \theta )^{2}\geq {\tfrac {1}{N}},}$

where is ${\displaystyle N}$ the number of particles. Quantum metrology can reach the Heisenberg limit given by

${\displaystyle (\Delta \theta )^{2}\geq {\tfrac {1}{N^{2}}}.}$

However, if uncorrelated local noise is present, then for large particle numbers the scaling of the precision returns to shot-noise scaling ${\displaystyle (\Delta \theta )^{2}\sim {\tfrac {1}{N}}.}$[14][15]

## Relation to quantum information science

There are strong links between quantum metrology and quantum information science. It has been shown that quantum entanglement is needed to outperform classical interferometry in magnetrometry with a fully polarized ensemble of spins.[16] It has been proved that a similar relation is generally valid for any linear interferometer, independent of the details of the scheme.[17] Moreover, higher and higher levels of multipartite entanglement is needed to achieve a better and better accuracy in parameter estimation.[18][19]

## References

1. ^ a b Braunstein, Samuel L.; Caves, Carlton M. (May 30, 1994). "Statistical distance and the geometry of quantum states". Physical Review Letters. American Physical Society (APS). 72 (22): 3439–3443. Bibcode:1994PhRvL..72.3439B. doi:10.1103/physrevlett.72.3439. ISSN 0031-9007. PMID 10056200.
2. ^ Paris, Matteo G. A. (November 21, 2011). "Quantum Estimation for Quantum Technology". International Journal of Quantum Information. 07 (supp01): 125–137. arXiv:0804.2981. doi:10.1142/S0219749909004839. S2CID 2365312.
3. ^ Giovannetti, Vittorio; Lloyd, Seth; Maccone, Lorenzo (March 31, 2011). "Advances in quantum metrology". Nature Photonics. 5 (4): 222–229. arXiv:1102.2318. Bibcode:2011NaPho...5..222G. doi:10.1038/nphoton.2011.35. S2CID 12591819.
4. ^ Tóth, Géza; Apellaniz, Iagoba (October 24, 2014). "Quantum metrology from a quantum information science perspective". Journal of Physics A: Mathematical and Theoretical. 47 (42): 424006. arXiv:1405.4878. Bibcode:2014JPhA...47P4006T. doi:10.1088/1751-8113/47/42/424006.
5. ^ Pezzè, Luca; Smerzi, Augusto; Oberthaler, Markus K.; Schmied, Roman; Treutlein, Philipp (September 5, 2018). "Quantum metrology with nonclassical states of atomic ensembles". Reviews of Modern Physics. 90 (3): 035005. arXiv:1609.01609. Bibcode:2018RvMP...90c5005P. doi:10.1103/RevModPhys.90.035005. S2CID 119250709.
6. ^ Braun, Daniel; Adesso, Gerardo; Benatti, Fabio; Floreanini, Roberto; Marzolino, Ugo; Mitchell, Morgan W.; Pirandola, Stefano (September 5, 2018). "Quantum-enhanced measurements without entanglement". Reviews of Modern Physics. 90 (3): 035006. arXiv:1701.05152. Bibcode:2018RvMP...90c5006B. doi:10.1103/RevModPhys.90.035006. S2CID 119081121.
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8. ^ Holevo, Alexander S (1982). Probabilistic and statistical aspects of quantum theory ([2nd English.] ed.). Scuola Normale Superiore. ISBN 978-88-7642-378-9.
9. ^ Pirandola, S; Bardhan, B. R.; Gehring, T.; Weedbrook, C.; Lloyd, S. (2018). "Advances in photonic quantum sensing". Nature Photonics. 12 (12): 724–733. arXiv:1811.01969. Bibcode:2018NaPho..12..724P. doi:10.1038/s41566-018-0301-6. S2CID 53626745.
10. ^ Kapale, Kishor T.; Didomenico, Leo D.; Kok, Pieter; Dowling, Jonathan P. (July 18, 2005). "Quantum Interferometric Sensors" (PDF). The Old and New Concepts of Physics. 2 (3–4): 225–240.
11. ^ Braunstein, Samuel L.; Caves, Carlton M.; Milburn, G.J. (April 1996). "Generalized Uncertainty Relations: Theory, Examples, and Lorentz Invariance". Annals of Physics. 247 (1): 135–173. arXiv:quant-ph/9507004. Bibcode:1996AnPhy.247..135B. doi:10.1006/aphy.1996.0040. S2CID 358923.
12. ^ Kok, Pieter; Braunstein, Samuel L; Dowling, Jonathan P (July 28, 2004). "Quantum lithography, entanglement and Heisenberg-limited parameter estimation" (PDF). Journal of Optics B: Quantum and Semiclassical Optics. IOP Publishing. 6 (8): S811–S815. arXiv:quant-ph/0402083. Bibcode:2004JOptB...6S.811K. doi:10.1088/1464-4266/6/8/029. ISSN 1464-4266. S2CID 15255876.
13. ^ Kimble, H. J.; Levin, Yuri; Matsko, Andrey B.; Thorne, Kip S.; Vyatchanin, Sergey P. (December 26, 2001). "Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics" (PDF). Physical Review D. American Physical Society (APS). 65 (2): 022002. arXiv:gr-qc/0008026. Bibcode:2002PhRvD..65b2002K. doi:10.1103/physrevd.65.022002. hdl:1969.1/181491. ISSN 0556-2821. S2CID 15339393.
14. ^ Demkowicz-Dobrzański, Rafał; Kołodyński, Jan; Guţă, Mădălin (September 18, 2012). "The elusive Heisenberg limit in quantum-enhanced metrology". Nature Communications. 3: 1063. arXiv:1201.3940. Bibcode:2012NatCo...3.1063D. doi:10.1038/ncomms2067. PMC 3658100. PMID 22990859.
15. ^ Escher, B. M.; Filho, R. L. de Matos; Davidovich, L. (May 2011). "General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology". Nature Physics. 7 (5): 406–411. arXiv:1201.1693. Bibcode:2011NatPh...7..406E. doi:10.1038/nphys1958. ISSN 1745-2481. S2CID 12391055.
16. ^ Sørensen, Anders S. (2001). "Entanglement and Extreme Spin Squeezing". Physical Review Letters. 86 (20): 4431–4434. arXiv:quant-ph/0011035. Bibcode:2001PhRvL..86.4431S. doi:10.1103/physrevlett.86.4431. PMID 11384252. S2CID 206327094.
17. ^ Pezzé, Luca (2009). "Entanglement, Nonlinear Dynamics, and the Heisenberg Limit". Physical Review Letters. 102 (10): 100401. arXiv:0711.4840. Bibcode:2009PhRvL.102j0401P. doi:10.1103/physrevlett.102.100401. PMID 19392092. S2CID 13095638.
18. ^ Hyllus, Philipp (2012). "Fisher information and multiparticle entanglement". Physical Review A. 85 (2): 022321. arXiv:1006.4366. Bibcode:2012PhRvA..85b2321H. doi:10.1103/physreva.85.022321. S2CID 118652590.
19. ^ Tóth, Géza (2012). "Multipartite entanglement and high-precision metrology". Physical Review A. 85 (2): 022322. arXiv:1006.4368. Bibcode:2012PhRvA..85b2322T. doi:10.1103/physreva.85.022322. S2CID 119110009.