Quantum metrology

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

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

where ${\displaystyle m}$ is the number of independent repetitions and ${\displaystyle F_{\rm {Q}}[\varrho ,H]}$ is the quantum Fisher information.^[1][11]

Examples

One example of note is the use of the

can be used for phase measurements.

Applications

This section needs to be updated. The reason given is: Advanced LIGO has already deployed measurements with quantum squeezing.. Please help update this article to reflect recent events or newly available information. (October 2022)

An important application of particular note is the detection of

where is ${\displaystyle N}$ the number of particles. Shot-noise limit is known to be asymptotically achievable using coherent states and homodyne detection.[14]

Quantum metrology can reach the

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

However, if uncorrelated local noise is present, then for large particle numbers the scaling of the precision returns to shot-noise scaling ${\displaystyle (\Delta \theta )^{2}\propto {\tfrac {1}{N}}.}$^[15][16]

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.^[17] It has been proved that a similar relation is generally valid for any linear interferometer, independent of the details of the scheme.^[18] Moreover, higher and higher levels of multipartite entanglement is needed to achieve a better and better accuracy in parameter estimation.^[19][20] 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.^[21]

See also

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

References