Quantum metrology

A basic task of quantum metrology is estimating the parameter $\backslash theta$

of the unitary dynamics

: $\backslash varrho(\backslash theta)=\backslash exp(-iH\backslash theta)\backslash varrho\_0\backslash exp(+iH\backslash theta),$

where $\backslash varrho\_0$ is the initial state of the system and $H$ is the Hamiltonian of the system. $\backslash theta$ is estimated based on measurements on $\backslash varrho(\backslash theta).$

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

: $H=\backslash sum\_k\; H\_k,$

where $H\_k$ 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

: $(\backslash Delta\; \backslash theta)^2\; \backslash ge\; \backslash frac\; 1\; \{m\; F\_\{\backslash rm\; Q\}[\backslash varrho,H]\},$

where $m$ is the number of independent repetitions and $F\_\{\backslash rm\; Q\}[\backslash varrho,H]$ is the quantum Fisher information.{{cite journal |last1=Braunstein |first1=Samuel L. |last2=Caves |first2=Carlton M. |last3=Milburn |first3=G.J. |title=Generalized Uncertainty Relations: Theory, Examples, and Lorentz Invariance |journal=Annals of Physics |date=April 1996 |volume=247 |issue=1 |pages=135–173 |doi=10.1006/aphy.1996.0040|arxiv=quant-ph/9507004 |bibcode=1996AnPhy.247..135B |s2cid=358923 }}

One example of note is the use of the NOON state in a Mach–Zehnder interferometer to perform accurate phase measurements.{{cite journal | last1=Kok | first1=Pieter | last2=Braunstein | first2=Samuel L | last3=Dowling | first3=Jonathan P | title=Quantum lithography, entanglement and Heisenberg-limited parameter estimation | journal=Journal of Optics B: Quantum and Semiclassical Optics | publisher=IOP Publishing | volume=6 | issue=8 | date=2004-07-28 | issn=1464-4266 | doi=10.1088/1464-4266/6/8/029 | pages=S811–S815| arxiv=quant-ph/0402083 | bibcode=2004JOptB...6S.811K | s2cid=15255876 | url=http://www-users.cs.york.ac.uk/~schmuel/papers/kbd04.pdf }} A similar effect can be produced using less exotic states such as squeezed states. In quantum illumination protocols, two-mode squeezed states are widely studied to overcome the limit of classical states represented in coherent states. In atomic ensembles, spin squeezed states can be used for phase measurements.

{{update|section|date=October 2022|reason=Advanced LIGO has already deployed measurements with quantum squeezing.}}

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.{{cite journal | last1=Kimble | first1=H. J. | last2=Levin | first2=Yuri | last3=Matsko | first3=Andrey B. | last4=Thorne | first4=Kip S. | last5=Vyatchanin | first5=Sergey P. | title=Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics | journal=Physical Review D | publisher=American Physical Society (APS) | volume=65 | issue=2 | date=2001-12-26 | issn=0556-2821 | doi=10.1103/physrevd.65.022002 | page=022002| arxiv=gr-qc/0008026 | url=https://authors.library.caltech.edu/4703/1/KIMprd02.pdf | bibcode=2001PhRvD..65b2002K | hdl=1969.1/181491 | s2cid=15339393 }}

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. This limit is also frequently called standard quantum limit (SQL)

: $(\backslash Delta\; \backslash theta)^2\backslash ge\; \backslash tfrac\{1\}\{mN\},$

where is $N$ the number of particles. Shot-noise limit is known to be asymptotically achievable using coherent states and homodyne detection.{{Cite journal|last1=Guha|first1=Saikatł|last2=Erkmen|first2=Baris|date=2009-11-10|title=Gaussian-state quantum-illumination receivers for target detection|journal=Physical Review A|volume=80|issue=5 |pages=052310|doi=10.1103/PhysRevA.80.052310|arxiv=0911.0950 |bibcode=2009PhRvA..80e2310G |s2cid=109058131 }}

Quantum metrology can reach the Heisenberg limit given by

: $(\backslash Delta\; \backslash theta)^2\backslash ge\; \backslash tfrac\{1\}\{mN^2\}.$

However, if uncorrelated local noise is present, then for large particle numbers the scaling of the precision returns to shot-noise scaling $(\backslash Delta\; \backslash theta)^2\backslash propto\; \backslash tfrac\{1\}\{N\}.${{Cite journal|last1=Demkowicz-Dobrzański|first1=Rafał|last2=Kołodyński|first2=Jan|last3=Guţă|first3=Mădălin|date=2012-09-18|title=The elusive Heisenberg limit in quantum-enhanced metrology|journal=Nature Communications|volume=3|pages=1063|doi=10.1038/ncomms2067|pmid=22990859|pmc=3658100|arxiv=1201.3940|bibcode=2012NatCo...3.1063D}}{{Cite journal|last1=Escher|first1=B. M.|last2=Filho|first2=R. L. de Matos|last3=Davidovich|first3=L.|date=May 2011|title=General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology|journal=Nature Physics|volume=7|issue=5|pages=406–411|doi=10.1038/nphys1958|issn=1745-2481|arxiv=1201.1693|bibcode=2011NatPh...7..406E|s2cid=12391055}}

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 magnetometry with a fully polarized ensemble of spins.{{Cite journal|last=Sørensen|first=Anders S.|date=2001|title=Entanglement and Extreme Spin Squeezing|journal=Physical Review Letters|volume=86|issue=20|pages=4431–4434|doi=10.1103/physrevlett.86.4431|arxiv=quant-ph/0011035|bibcode=2001PhRvL..86.4431S|pmid=11384252|s2cid=206327094}} It has been proved that a similar relation is generally valid for any linear interferometer, independent of the details of the scheme.{{Cite journal|last1=Pezzé|first1=Luca|last2=Smerzi|first2=Augusto|date=2009|title=Entanglement, Nonlinear Dynamics, and the Heisenberg Limit|journal=Physical Review Letters|volume=102|issue=10|doi=10.1103/physrevlett.102.100401|arxiv=0711.4840|bibcode=2009PhRvL.102j0401P|pmid=19392092|page=100401|s2cid=13095638}} Moreover, higher and higher levels of multipartite entanglement is needed to achieve a better and better accuracy in parameter estimation.{{Cite journal|last1=Hyllus|first1=Philipp|date=2012|title=Fisher information and multiparticle entanglement|journal=Physical Review A|volume=85|issue=2|pages=022321|doi=10.1103/physreva.85.022321|arxiv=1006.4366|bibcode=2012PhRvA..85b2321H|s2cid=118652590}}{{Cite journal|last=Tóth|first=Géza|date=2012|title=Multipartite entanglement and high-precision metrology|journal=Physical Review A|volume=85|issue=2|pages=022322|doi=10.1103/physreva.85.022322|arxiv=1006.4368|bibcode=2012PhRvA..85b2322T|s2cid=119110009}} Additionally, entanglement in multiple degrees of freedom of quantum systems (known as "hyperentanglement"), can be used to enhance precision, with enhancement arising from entanglement in each degree of freedom.{{Cite journal|last1=Walborn|first1=S. P.|last2=Pimentel|first2=A. H.|last3=Filho|first3=R. L. de Matos|last4=Davidovich|first4=L.|date=January 2018|title=Quantum-enhanced sensing from hyperentanglement|journal=Physical Review A|volume=97|issue=1|pages=010301(R)|doi= 10.1103/PhysRevA.97.010301|arxiv=1709.04513|bibcode=2018PhRvA..97a0301W|s2cid=73689445 }}

{{Main|Outline of metrology and measurement}}

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• Quantum metrological gain
• Dimensional metrology
• Forensic metrology
• Smart Metrology
• Time metrology

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{{Quantum mechanics topics}}

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Category:Quantum information science

Category:Quantum optics

Category:Metrology