(an ensemble of photons - a type of massless particles) or massive particle
such as atoms.

Usually, an optical gyroscope uses two counterpropagating laser beams to sense
a rotation. If the setup rotates with an angular velocity $\Omega$, the two
laser beams will accumulate a relative phase $\Phi=4M\Omega A_{o}/(\lambda c)$
with $M$ the number of rounds, $A_{o}$ the area enclosed by the gyroscope
parallel to the rotation direction, $\lambda$ the wave length, and $c$ the
speed of light. However, the conventional SQL still places a limit on their
precision, which is proportional to the surface area contained by the optical
path. In a recent theoretical proposal, the utilization of distributed quantum
sensing is envisioned as a means to enhance the performance of fiber
gyroscopes specifically for navigation applications Grace  _et al._ (2020).
The idea is to inject quantum-optical squeezed vacuum light, like what was
utilized in a RF-sensing experiment Xia  _et al._ (2020), into to an array of
fiber gyroscopes to improve the scaling of measurement precisions. The idea is
to inject quantum-optical squeezed vacuum light, like what was utilized in the
RF-sensing experiment Xia  _et al._ (2020), into to an array of fiber
gyroscopes to improve the scaling of the measurement noise. Moreover, such a
configuration can also be applied to improve the readout signal-to-noise ratio
of an array of optomechanical transducers used in inertial sensing Hines  _et
al._ (2020) and atomic force microscopes Sugimoto  _et al._ (2007).

Similar to an optical gyroscope, in an atomic gyroscope, two counter-
propagating atomic beams accumulate a relative phase $\Phi=2m\Omega
A_{a}/\hbar$ with $m$ the atomic mass, the angular velocity $\Omega$, and
$A_{a}$ the area enclosed by the gyroscope parallel to the rotation direction.
Theoretically, the sensitivity of an atom interferometer is about ten orders
of magnitude larger than the optical counterpart due to $m/\hbar$ is much
larger than $1/(\lambda c)$. However, the optical interferometers have much
larger interferometric area than the atom interferometers, i.e. $4MA_{o}\gg
2A_{a}$, practically the sensitivity can be improved by about 3-4 orders of
magnitude gain. Atomic gyroscopes have made significant advancements since
their early demonstrations Kasevich and Chu (1991); Gustavson, Landragin, and
Kasevich (2000), achieving a short-term sensitivity of $6\times 10^{-10}$
(rad/s) $/\sqrt{\textrm{Hz}}$ with a one-second interrogation time Gustavson,
Landragin, and Kasevich (2000). Using two counter-propagating atomic beam
geometries, one can isolate the phase shifts respectively caused by
acceleration and rotation. By removing phase shifts caused by non-inertial
effects such as magnetic field and ac Stark shift, the long-term sensitivity
can be further improved Durfee, Shaham, and Kasevich (2006). To counteract
systematic effects and long-term drift, they employed a technique of regularly
inverting the propagation vector $k_{eff}$, which resulted in an opposite sign
for the inertial phase while preserving the sign of the non-inertial phase. By
utilizing this approach, the system was able to improve its sensitivity to
$3\times 10^{10}$ (rad/s) $\sqrt{\textrm{Hz}}$ during a 5-hour interrogation
duration.

To improve the long-term stability and shorten dead times of atomic
gyroscopes, atomic beam source can be replaced by laser-cooled atoms. By using
two cold atomic clouds launched in curved parabolic trajectories in a counter-
propagating directions, a cold atomic gyroscope has been demonstrated
experimentally Canuel  _et al._ (2006), in which the short-term sensitivities
of acceleration and rotation were $4.7\times 10^{-6}$ (rad/s)
$/\sqrt{\textrm{Hz}}$ and $2.2\times 10^{-6}$ (rad/s) $/\sqrt{\textrm{Hz}}$
respectively. Instead of employing continuously running beams, it utilizes
three pulsed Raman beams oriented in three orthogonal directions. In a similar
experiment Gauguet  _et al._ (2009), the long-term stability after an
integration time of $1000$ s was shown to be limited by the QPN, $1\times
10^{-8}$ (rad/s) $/\sqrt{\textrm{Hz}}$. In recent years, there appear numerous
advancements in cold atomic gyroscopes. A zero dead-time gyroscope was
presented, where continuous operation contributed to enhancing the short-term
sensitivity to $1\times 10^{-7}$ (rad/s) $/\sqrt{\textrm{Hz}}$ Dutta  _et al._
(2016). A Sagnac interferometry with a single-atom clock Stevenson  _et al._
(2015) had been proposed by combining the techniques of state-dependent
manipulations and Ramsey pulses. Based upon an interleaved atom interferometry
with a sampling rate of 3.75 Hz and an interrogation time of 801 ms, in which
three atomic clouds are simultaneously interrogated, the sensitivity has been
improved to $3\times 10^{-10}$ (rad/s) $/\sqrt{\textrm{Hz}}$, which competes
with the best results achieved using fiber-optic gyroscopes Savoie  _et al._
(2018). Savoie  _et al._ (2018). In addition, several compact designs
utilizing atom chips Gersemann  _et al._ (2020); Gebbe  _et al._ (2021) have
shown preliminary rotation sensing capabilities through double Bragg
diffraction. Furthermore, using the nitrogen vacancy center in diamonds
Ledbetter  _et al._ (2012); Soshenko  _et al._ (2021), it is possible to
perform rotational measurements, which in turn paves the way for the
realization of microchip-size gyroscopes.

Beyond the conventional scheme with individual particles, it is interesting to
investigate how to exploit many-body quantum entanglement in measuring
rotation frequency. Ultracold atomic system provides a feasible experimental
platform for achieving this objective Nolan  _et al._ (2016); Luo  _et al._
(2017b); Yao  _et al._ (2019a). A spin-squeezed rotation sensor utilizing the
Sagnac effect in a spin-1 BEC in a ring trap was proposed Nolan  _et al._
(2016). By employing Laguerre-Gauss beams and Raman pulses, the two input
states are initially prepared and subsequently amplified through the bosonic
enhancement of spin-exchange collisions. This process leads to spin-squeezing,
which in turn offers quantum enhancement for rotation sensing. An alternative
multi-particle scheme utilizing two-mode Bose condensed atoms was also
proposed Luo  _et al._ (2017b). In this scheme, an ensemble of entangled two-
state Bose atoms is transported in a ring-shaped path using a state-dependent
rotating potential. Subsequently, the atoms are recombined for interference by
employing Ramsey pulses. The duration of phase accumulation is controlled by
the state-dependent rotating potential. The ultimate sensitivity to rotation
can be enhanced to reach the Heisenberg limit by utilizing entangled initial
internal degrees of freedom. By implementing parity measurement, the
measurement precision can be maximized, and the achieved precisions approach
the Heisenberg limit. However, this scheme requires parity measurement based
upon single-atom resolution detection. The nonlinear detection in Sec. IV.3
may be helpful for removing the need of single-atom resolution detection.

The journey towards practical entanglement-enhanced quantum sensors may be
long and unpredictable. A hard and crucial initial step involves surpassing
the SQL in measurements to demonstrate the feasibility of a specific inertial
quantity. Advancements in entanglement-enhanced clocks provide valuable
insights that inform the development of design strategies to achieve this
goal. Furthermore, quantum sensors for acceleration, rotation, gravity, and
magnetic fields can be integrated into position, navigation, and timing
systems for applications in underwater and underground environments. In such
applications, quantum sensors surpass their classical counterparts by offering
reduced bias, improved precision, and greater stability. This may enable
navigation with meter-level accuracy without relying on global satellite
systems Bongs, Bennett, and Lohmann (2023).

##  VI Summary and outlook

In summary, the utilization of multi-particle entanglement offers promising
potentials to improve measurement precision beyond conventional shot-noise
limit. This has sparked an active and dynamic research field focused on
characterizing, manipulating, and utilizing entangled multi-particle quantum
states Dai  _et al._ (2016); Yu  _et al._ (2021); Guo  _et al._ (2023). With
the rapid progress in creating and detecting multi-particle entangled state,
quantum metrology is emerging as a domain capable of achieving quantum
advantages over conventional measurements and sensing for practical
applications in the near future. The field of entanglement-enhanced quantum
metrology is poised to shift from theoretical studies Zhuang, Huang, and Lee
(2022) to experimental demonstrations and practical applications, specifically
in the realm of precise measurements involving frequency Zhuang  _et al._
(2021), electromagnetic fields Simons  _et al._ (2021); Ji  _et al._ (2024),
and forces. Nevertheless, there remain several challenges in constructing
entanglement-enhanced quantum sensors for real-world applications.

Developing a practical entanglement-enhanced quantum sensor poses a
significant challenge, not in generating entanglement itself, but in
generating entanglement that meets the precise sensing requirements of a
particular application. These requirements, such as particle number and
interrogation time, impose constraints on achievable measurement precision.
One of the difficulties lies in scalability with particle number, as an
increase in particles tends to rapidly amplify errors and complicate the
experiment. If only a limited number of particles can achieve a certain degree
of entanglement, the improvement over the SQL may not be readily apparent.
Similarly, if entanglement can only persist for a short interrogation time or
is susceptible to weak experimental noises, it may not lead to a significant
enhancement in practical sensing tasks. Future research should prioritize
utilizing entanglement to surpass the SQL and focus on developing
entanglement-enhanced quantum metrology protocols specifically tailored for
practical sensing scenarios.

Quantum entanglement is highly susceptible to decoherence Tong  _et al._
(2010); An and Zhang (2007); Wan  _et al._ (2020); Chin, Huelga, and Plenio
(2012); Reilly  _et al._ (2022), leading to a significant reduction in
achievable measurement precision Jeske, Cole, and Huelga (2014); Wang, Chen,
and An (2017); Wan  _et al._ (2020). This fragility is particularly evident
when employing long interrogation times to achieve high precision. Theoretical
investigations have demonstrated the detrimental impact of decoherence on
surpassing the SQL Shaji and Caves (2007); Huelga  _et al._ (1997). Typically,
measurement precision returns to the SQL during an optimal transient
interrogation period but diverges in the long-interrogation-time regime due to
the effects of decoherence. To mitigate the negative effects of decoherence on
quantum metrology, various strategies have been proposed Yuan and Fung (2015);
Berni  _et al._ (2015); Lu, Yu, and Oh (2015); Fallani  _et al._ (2022);
Salvia, Mehboudi, and Perarnau-Llobet (2023); Kessler  _et al._ (2014b); Dür
_et al._ (2014); Unden  _et al._ (2016); Zhou  _et al._ (2018); Nautrup  _et
al._ (2019); Kapourniotis and Datta (2019); Huo and Li (2017); Schlosshauer
(2005); Liu  _et al._ (2023b); Bai and An (2023); Reilly  _et al._ (2022).
These include adaptive Borregaard and Sørensen (2013b); Demkowicz-Dobrzański,
Czajkowski, and Sekatski (2017) and quantum nondemolition measurements Rossi
_et al._ (2020), correlated decoherence Jeske, Cole, and Huelga (2014), and
dynamical control Tan  _et al._ (2013); Dong  _et al._ (2016); Zhou  _et al._
(2020), aiming to restore the Heisenberg limit. With advancements in
strategies such as quantum reservoir engineering Bai  _et al._ (2019), Floquet
engineering Bai and An (2023), quantum error mitigation Huggins  _et al._
(2021); Yamamoto  _et al._ (2022) and other techniques, quantum metrology is
on the verge of transitioning from the laboratory to practical use.
Additionally, quantum feedback Yuan and Fung (2015); Berni  _et al._ (2015);
Fallani  _et al._ (2022); Salvia, Mehboudi, and Perarnau-Llobet (2023),
quantum error correction Kessler  _et al._ (2014b); Dür  _et al._ (2014);
Unden  _et al._ (2016); Zhou  _et al._ (2018); Nautrup  _et al._ (2019); Unden
_et al._ (2016); Zhou  _et al._ (2018); Dür  _et al._ (2014) and other
techniques Kapourniotis and Datta (2019); Huo and Li (2017); Yao  _et al._
(2019b); Liu and Yuan (2017a); Wang  _et al._ (2023); Demkowicz-Dobrzański,
Czajkowski, and Sekatski (2017) can be employed to extend coherence time and
achieve robust high-precision measurements in experiments. Moreover, various
modulation techniques such as periodic modulation Kuo and Lidar (2011); Boss
_et al._ (2017), quantum heterodyne Meinel  _et al._ (2021), quantum mixing
Wang  _et al._ (2022a) and quantum lock-in amplification measurement Kotler
_et al._ (2011); Zhuang, Huang, and Lee (2021) enable high-precision frequency
estimation even in noisy environments. The combination of these quantum
control techniques with quantum entanglement holds the potential for
significant advancements in the field of entanglement-enhanced sensing
technologies.

On the other hand, while enhancing measurement precision, it is crucial to
improve measurement accuracy by minimizing systematic errors. In general,
systematic errors always introduce inaccuracies that impact the overall
performance of measurements Brif, Ruzic, and Biedermann (2020); Len  _et al._
(2022). For instance, in atomic optical clocks, system errors arising from
laser probe frequency or intensity instabilities can lead to systematic
frequency shifts, compromising clock accuracy. To mitigate the effects of
laser intensity variations and reduce corresponding frequency shifts,
composite pulse sequences can be employed Zanon-Willette  _et al._ (2018);
Shuker  _et al._ (2019); Sanner  _et al._ (2018), improving measurement
accuracy. Other sources of systematic errors, such as collision shifts,
Doppler shifts, Stark shifts, and blackbody radiation shifts, can also have
adverse effects Ludlow  _et al._ (2015). Therefore, it is important to assess
systematic errors throughout the entire process of quantum parameter
estimation and develop methods to suppress specific system errors in many-body
sensing systems.

To optimize the performance of a practical quantum metrology, in addition to
conventional optimization tools, machine learning has emerged as a powerful
tool Gray  _et al._ (2018); Cimini  _et al._ (2023). Machine learning may
optimize various aspects of a quantum metrology process Huo  _et al._ (2022),
including generation of entangled states, signal accumulation, entangled state
readout, and parameter estimation. In scenarios with limited resources, such
as limited coherence time or measurement times, machine learning approaches
like reinforcement learning Chen  _et al._ (2019); Qiu  _et al._ (2022a);
Cimini  _et al._ (2023), quantum algorithms Kaubruegger  _et al._ (2019); Li
_et al._ (2017); Marciniak  _et al._ (2022); Kaubruegger  _et al._ (2023), and
Bayesian inference Valeri  _et al._ (2020); Qiu  _et al._ (2022b); Gebhart
_et al._ (2023) can guide us in determining the most effective strategies for
conducting metrology. The exploration of quantum metrology protocols augmented
by machine learning offers an alternative pathway towards achieving practical
high-precision measurements.

Besides the typical entangled states discussed in Sec. III, strong types of
quantum correlations such as EPR steering and Bell nonlocality are also useful
for quantum metrology Frérot, Fadel, and Lewenstein (2023). It has been
demonstrated that EPR-assisted metrology may precisely estimate a local phase
shift and its generating observable Yadin, Fadel, and Gessner (2021); Guo  _et
al._ (2023b). Bell nonlocality seems to be linked to high QFI, implying a
potential connection to metrological advantages Fröwis  _et al._ (2019); Guo
_et al._ (2023c). Moreover, nonclassical states of atomic ensembles can be
spatially distributed in different positions Fadel  _et al._ (2018); Kunkel
_et al._ (2018); Lange  _et al._ (2018); Jing  _et al._ (2019), which allows
for quantum enhanced distributed sensing Fadel and Gessner (2020); Malia  _et
al._ (2022); Fadel  _et al._ (2023).

In addition, associated with quantum entanglement, quantum criticality has
been recognized as a novel quantum resource for enhancing quantum metrology
Frérot and Roscilde (2018); Rams  _et al._ (2018); Montenegro, Mishra, and
Bayat (2021); Zanardi, Paris, and Campos Venuti (2008). On one hand, quantum
systems prepared near the critical point exhibit extreme sensitivity to small
perturbations Invernizzi  _et al._ (2008). By leveraging the criticality
associated with quantum phase transitions, high-precision measurements of
physical quantities can be achieved. Systems near a quantum phase transition
display a divergent susceptibility, indicating the potential for achieving
remarkably high precision by utilizing quantum critical systems as probes for
parameter estimation Salvia, Mehboudi, and Perarnau-Llobet (2023); Liu  _et
al._ (2021b). On the other hand, parameter estimation can be accomplished by
measuring the equilibrium state or dynamical properties of the system when it
is driven close to the critical point of a quantum phase transition
Raghunandan, Wrachtrup, and Weimer (2018); Gietka (2022). Furthermore, the
response of eigenstates in non-Hermitian systems to small perturbations at
singularities can also be utilized to detect specific physical quantities Chen
_et al._ (2017); Lai  _et al._ (2019); Wang  _et al._ (2020); Wiersig (2020);
Chu  _et al._ (2020); Zhang  _et al._ (2019b); Budich and Bergholtz (2020);
Bao  _et al._ (2021); Ou, Huang, and Lee (2021); Li  _et al._ (2023b).

At last, it is important to note that multi-parameter estimation Magdalena
Szczykulska and Datta (2016); Albarelli  _et al._ (2020); Liu and Yuan
(2017b); Lu and Wang (2021); Pezzè  _et al._ (2017); Liu  _et al._ (2019);
Gessner, Pezzè, and Smerzi (2018); Polino  _et al._ (2019) is a crucial
problem in both fundamental research and practical applications Gessner,
Smerzi, and Pezzè (2020); Ren  _et al._ (2021); Pezzè (2021), alongside
single-parameter estimation. It has been extensively employed for various
purposes, including vector field detection Wang  _et al._ (2015, 2021),
imaging and biological sensing Taylor and Bowen (2016); Thiel  _et al._
(2016). Several studies have demonstrated that a multi-parameter approach can
outperform the optimal quantum individual estimation scheme Albarelli  _et
al._ (2020) and offer advantages in the presence of noise Roccia  _et al._
(2018); Vidrighin  _et al._ (2014). Although progress has been made in multi-
parameter estimation scenarios Zhuang, Huang, and Lee (2018); Liu  _et al._
(2016); Xu  _et al._ (2021); Liu  _et al._ (2016); Nichols  _et al._ (2018);
Gessner, Pezzè, and Smerzi (2018); Gessner, Smerzi, and Pezzè (2020); Xu  _et
al._ (2021); Liu  _et al._ (2019); Vidrighin  _et al._ (2014); Roccia  _et
al._ (2018), including finding the optimal measurement strategies Sidhu  _et
al._ (2021); Hayashi and Ouyang (2023, 2024), how to experimentally achieve
the optimal multi-parameter measurement in the presence of noise remains a
challenge. Furthermore, the development of entanglement-enhanced multi-
parameter estimation is crucial for practical quantum sensing, such as vector
magnetometers Wang  _et al._ (2021, 2015) and multimode quantum imaging
Kaubruegger  _et al._ (2023); Albarelli  _et al._ (2020).

###### Acknowledgements.

J. Huang and M. Zhuang contribute equally to this work. The authors
acknowledge Matteo Fadel, Kaoru Yamamoto, and Yingkai Ouyang for their
valuable suggestions. The authors thank Jihao Ma, Jungeng Zhou, Sijie Chen, Yi
Shen, Yuehua Pang, and Jinye Wei for critical reading and helpful suggestions.
This work is supported by the National Key Research and Development Program of
China (2022YFA1404104) and the National Natural Science Foundation of China
(12025509, 12305022).

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