Quantum metrology

Quantum metrologyis the study of making high-resolution and highly sensitive measurements of physical parameters using quantum theory to describe the physical systems,particularly exploitingquantum entanglementand quantumsqueezing.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,it represents an important theoretical model at the basis of quantum sensing.

Mathematical foundations

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

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

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

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

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

where$H_{k}$acts on thekth particle. In this case, there is no interaction between the particles, and we talk aboutlinear interferometers.

The achievable precision is bounded from below by thequantum Cramér-Rao boundas

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

where$m$is the number of independent repetitions and$F_{\rm {Q}}[\varrho,H]$is thequantum Fisher information.

Examples

One example of note is the use of theNOON statein aMach–Zehnder interferometerto perform accurate phase measurements.A similar effect can be produced using less exotic states such assqueezed states.In quantum illumination protocols, two-mode squeezed states are widely studied to overcome the limit of classcial states represented incoherent states.In atomic ensembles,spin squeezed statescan be used for phase measurements.

Applications

An important application of particular note is the detection ofgravitational radiationin projects such asLIGOor theVirgo 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.

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

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

where is$N$the number of particles. Shot-noise limit is known to be asymptotically achievable using coherent states and homodyne detection.

Quantum metrology can reach theHeisenberg limitgiven by

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

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

Relation to quantum information science

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

See also

Main article:Outline of metrology and measurement

• Dimensional metrology
• Forensic metrology
• Smart Metrology
• Time metrology