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]
A basic task of quantum metrology is estimating the parameter of the unitary dynamics
where is the initial state of the system and is the Hamiltonian of the system. is estimated based on measurements on
Typically, the system is composed of many particles, and the Hamiltonian is a sum of single-particle terms
where 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
where is the number of independent repetitions and is the quantum Fisher information. [1] [11]
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 quantum illumination protocols, two-mode squeezed states are widely studied to overcome the limit of classcial states represented in coherent states. In atomic ensembles, spin squeezed states can be used for phase measurements.
![]() | This section needs to be updated. The reason given is: Advanced LIGO has already deployed measurements with quantum squeezing..(October 2022) |
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]
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)
where is 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 Heisenberg limit given by
However, if uncorrelated local noise is present, then for large particle numbers the scaling of the precision returns to shot-noise scaling [15] [16]
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]
Quantum entanglement is the phenomenon of a group of particles being generated, interacting, or sharing spatial proximity in such a way that the quantum state of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical and quantum physics: entanglement is a primary feature of quantum mechanics not present in classical mechanics.
The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known. In other words, the more accurately one property is measured, the less accurately the other property can be known.
In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. A fundamental feature of quantum theory is that the predictions it makes are probabilistic. The procedure for finding a probability involves combining a quantum state, which mathematically describes a quantum system, with a mathematical representation of the measurement to be performed on that system. The formula for this calculation is known as the Born rule. For example, a quantum particle like an electron can be described by a quantum state that associates to each point in space a complex number called a probability amplitude. Applying the Born rule to these amplitudes gives the probabilities that the electron will be found in one region or another when an experiment is performed to locate it. This is the best the theory can do; it cannot say for certain where the electron will be found. The same quantum state can also be used to make a prediction of how the electron will be moving, if an experiment is performed to measure its momentum instead of its position. The uncertainty principle implies that, whatever the quantum state, the range of predictions for the electron's position and the range of predictions for its momentum cannot both be narrow. Some quantum states imply a near-certain prediction of the result of a position measurement, but the result of a momentum measurement will be highly unpredictable, and vice versa. Furthermore, the fact that nature violates the statistical conditions known as Bell inequalities indicates that the unpredictability of quantum measurement results cannot be explained away as due to ignorance about "local hidden variables" within quantum systems.
In physics, a squeezed coherent state is a quantum state that is usually described by two non-commuting observables having continuous spectra of eigenvalues. Examples are position and momentum of a particle, and the (dimension-less) electric field in the amplitude and in the mode of a light wave. The product of the standard deviations of two such operators obeys the uncertainty principle:
Quantum tomography or quantum state tomography is the process by which a quantum state is reconstructed using measurements on an ensemble of identical quantum states. The source of these states may be any device or system which prepares quantum states either consistently into quantum pure states or otherwise into general mixed states. To be able to uniquely identify the state, the measurements must be tomographically complete. That is, the measured operators must form an operator basis on the Hilbert space of the system, providing all the information about the state. Such a set of observations is sometimes called a quorum. The term tomography was first used in the quantum physics literature in a 1993 paper introducing experimental optical homodyne tomography.
An optical parametric oscillator (OPO) is a parametric oscillator that oscillates at optical frequencies. It converts an input laser wave with frequency into two output waves of lower frequency by means of second-order nonlinear optical interaction. The sum of the output waves' frequencies is equal to the input wave frequency: . For historical reasons, the two output waves are called "signal" and "idler", where the output wave with higher frequency is the "signal". A special case is the degenerate OPO, when the output frequency is one-half the pump frequency, , which can result in half-harmonic generation when signal and idler have the same polarization.
In quantum optics, a NOON state or N00N state is a quantum-mechanical many-body entangled state:
The one-way or measurement-based quantum computer (MBQC) is a method of quantum computing that first prepares an entangled resource state, usually a cluster state or graph state, then performs single qubit measurements on it. It is "one-way" because the resource state is destroyed by the measurements.
Within quantum technology, a quantum sensor utilizes properties of quantum mechanics, such as quantum entanglement, quantum interference, and quantum state squeezing, which have optimized precision and beat current limits in sensor technology. The field of quantum sensing deals with the design and engineering of quantum sources and quantum measurements that are able to beat the performance of any classical strategy in a number of technological applications. This can be done with photonic systems or solid state systems.
In the case of systems composed of subsystems, the classification of quantum-entangledstates is richer than in the bipartite case. Indeed, in multipartite entanglement apart from fully separable states and fully entangled states, there also exists the notion of partially separable states.
In applied mathematics, the numerical sign problem is the problem of numerically evaluating the integral of a highly oscillatory function of a large number of variables. Numerical methods fail because of the near-cancellation of the positive and negative contributions to the integral. Each has to be integrated to very high precision in order for their difference to be obtained with useful accuracy.
The spin stiffness or spin rigidity is a constant which represents the change in the ground state energy of a spin system as a result of introducing a slow in plane twist of the spins. The importance of this constant is in its use as an indicator of quantum phase transitions—specifically in models with metal-insulator transitions such as Mott insulators. It is also related to other topological invariants such as the Berry phase and Chern numbers as in the Quantum Hall effect.
Spin squeezing is a quantum process that decreases the variance of one of the angular momentum components in an ensemble of particles with a spin. The quantum states obtained are called spin squeezed states. Such states have been proposed for quantum metrology, to allow a better precision for estimating a rotation angle than classical interferometers.
In quantum physics, light is in a squeezed state if its electric field strength Ԑ for some phases has a quantum uncertainty smaller than that of a coherent state. The term squeezing thus refers to a reduced quantum uncertainty. To obey Heisenberg's uncertainty relation, a squeezed state must also have phases at which the electric field uncertainty is anti-squeezed, i.e. larger than that of a coherent state. Since 2019, the gravitational-wave observatories LIGO and Virgo employ squeezed laser light, which has significantly increased the rate of observed gravitational-wave events.
The quantum Cramér–Rao bound is the quantum analogue of the classical Cramér–Rao bound. It bounds the achievable precision in parameter estimation with a quantum system:
The quantum Fisher information is a central quantity in quantum metrology and is the quantum analogue of the classical Fisher information. It is one of the central quantities used to qualify the utility of an input state, especially in Mach–Zehnder interferometer-based phase or parameter estimation. It is shown that the quantum Fisher information can also be a sensitive probe of a quantum phase transition. The quantum Fisher information of a state with respect to the observable is defined as
The symmetric logarithmic derivative is an important quantity in quantum metrology, and is related to the quantum Fisher information.
In quantum physics, entanglement depth characterizes the strength of multiparticle entanglement. An entanglement depth means that the quantum state of a particle ensemble cannot be described under the assumption that particles interacted with each other only in groups having fewer than particles. It has been used to characterize the quantum states created in experiments with cold gases.
SU(1,1) interferometry is a technique that uses parametric amplification for splitting and mixing of electromagnetic waves for precise estimation of phase change and achieves the Heisenberg limit of sensitivity with fewer optical elements than conventional interferometric techniques.
Permutationally invariant quantum state tomography is a method for the partial determination of the state of a quantum system consisting of many subsystems.