Quantum metrology

Last updated

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 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 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 atomic ensembles, spin squeezed states can be used for phase measurements.


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]

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

where is the number of particles. 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 [14] [15]

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

Related Research Articles

Quantum teleportation is a technique for transferring quantum information from a sender at one location to a receiver some distance away. While teleportation is commonly portrayed in science fiction as a means to transfer physical objects from one location to the next, quantum teleportation only transfers quantum information. Moreover, the sender may not know the location of the recipient, and does not know which particular quantum state will be transferred.

Quantum entanglement Correlation between measurements of quantum subsystems, even when spatially separated

Quantum entanglement is a physical phenomenon that occurs when a pair or group of particles is generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the pair or 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 lacking in classical mechanics.

In quantum physics, a measurement is the testing or manipulation of a physical system in order to yield a numerical result. The predictions that quantum physics makes are in general probabilistic. The mathematical tools for making predictions about what measurement outcomes may occur were developed during the 20th century and make use of linear algebra and functional analysis.

Squeezed coherent state type of quantum state

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:

In quantum mechanics, einselections, short for "environment-induced superselection", is a name coined by Wojciech H. Zurek for a process which is claimed to explain the appearance of wavefunction collapse and the emergence of classical descriptions of reality from quantum descriptions. In this approach, classicality is described as an emergent property induced in open quantum systems by their environments. Due to the interaction with the environment, the vast majority of states in the Hilbert space of a quantum open system become highly unstable due to entangling interaction with the environment, which in effect monitors selected observables of the system. After a decoherence time, which for macroscopic objects is typically many orders of magnitude shorter than any other dynamical timescale, a generic quantum state decays into an uncertain state which can be decomposed into a mixture of simple pointer states. In this way the environment induces effective superselection rules. Thus, einselection precludes stable existence of pure superpositions of pointer states. These 'pointer states' are stable despite environmental interaction. The einselected states lack coherence, and therefore do not exhibit the quantum behaviours of entanglement and superposition.

Quantum tomography Reconstruction of quantum states based on measurements

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.

In quantum mechanics, separable quantum states are states without quantum entanglement.

Squashed entanglement, also called CMI entanglement, is an information theoretic measure of quantum entanglement for a bipartite quantum system. If is the density matrix of a system composed of two subsystems and , then the CMI entanglement of system is defined by

A NOON state is a quantum-mechanical many-body entangled state:

Time-bin encoding is a technique used in quantum information science to encode a qubit of information on a photon. Quantum information science makes use of qubits as a basic resource similar to bits in classical computing. Qubits are any two-level quantum mechanical system; there are many different physical implementations of qubits, one of which is time-bin encoding.

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.

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 or helicity modulus or the "superfluid density" 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 can be used for quantum metrology, as they can provide a better precision for estimating a rotation angle than classical interferometers.

Many body localization (MBL) is a dynamical phenomenon occurring in isolated many-body quantum systems. It is characterized by the system failing to reach thermal equilibrium, and retaining a memory of its initial condition in local observables for infinite times.

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. 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.


  1. 1 2 Braunstein, Samuel L.; Caves, Carlton M. (May 30, 1994). "Statistical distance and the geometry of quantum states". Physical Review Letters. American Physical Society (APS). 72 (22): 3439–3443. Bibcode:1994PhRvL..72.3439B. doi:10.1103/physrevlett.72.3439. ISSN   0031-9007. PMID   10056200.
  2. Paris, Matteo G. A. (November 21, 2011). "Quantum Estimation for Quantum Technology". International Journal of Quantum Information. 07 (supp01): 125–137. arXiv: 0804.2981 . doi:10.1142/S0219749909004839. S2CID   2365312.
  3. Giovannetti, Vittorio; Lloyd, Seth; Maccone, Lorenzo (March 31, 2011). "Advances in quantum metrology". Nature Photonics. 5 (4): 222–229. arXiv: 1102.2318 . Bibcode:2011NaPho...5..222G. doi:10.1038/nphoton.2011.35. S2CID   12591819.
  4. Tóth, Géza; Apellaniz, Iagoba (October 24, 2014). "Quantum metrology from a quantum information science perspective". Journal of Physics A: Mathematical and Theoretical. 47 (42): 424006. arXiv: 1405.4878 . Bibcode:2014JPhA...47P4006T. doi: 10.1088/1751-8113/47/42/424006 .
  5. Pezzè, Luca; Smerzi, Augusto; Oberthaler, Markus K.; Schmied, Roman; Treutlein, Philipp (September 5, 2018). "Quantum metrology with nonclassical states of atomic ensembles". Reviews of Modern Physics. 90 (3): 035005. arXiv: 1609.01609 . Bibcode:2018RvMP...90c5005P. doi:10.1103/RevModPhys.90.035005. S2CID   119250709.
  6. Braun, Daniel; Adesso, Gerardo; Benatti, Fabio; Floreanini, Roberto; Marzolino, Ugo; Mitchell, Morgan W.; Pirandola, Stefano (September 5, 2018). "Quantum-enhanced measurements without entanglement". Reviews of Modern Physics. 90 (3): 035006. arXiv: 1701.05152 . Bibcode:2018RvMP...90c5006B. doi:10.1103/RevModPhys.90.035006. S2CID   119081121.
  7. Helstrom, C (1976). Quantum detection and estimation theory. Academic Press. ISBN   0123400503.
  8. Holevo, Alexander S (1982). Probabilistic and statistical aspects of quantum theory ([2nd English.] ed.). Scuola Normale Superiore. ISBN   978-88-7642-378-9.
  9. Pirandola, S; Bardhan, B. R.; Gehring, T.; Weedbrook, C.; Lloyd, S. (2018). "Advances in photonic quantum sensing". Nature Photonics. 12 (12): 724–733. arXiv: 1811.01969 . Bibcode:2018NaPho..12..724P. doi:10.1038/s41566-018-0301-6. S2CID   53626745.
  10. Kapale, Kishor T.; Didomenico, Leo D.; Kok, Pieter; Dowling, Jonathan P. (July 18, 2005). "Quantum Interferometric Sensors" (PDF). The Old and New Concepts of Physics. 2 (3–4): 225–240.
  11. Braunstein, Samuel L.; Caves, Carlton M.; Milburn, G.J. (April 1996). "Generalized Uncertainty Relations: Theory, Examples, and Lorentz Invariance". Annals of Physics. 247 (1): 135–173. arXiv: quant-ph/9507004 . Bibcode:1996AnPhy.247..135B. doi:10.1006/aphy.1996.0040. S2CID   358923.
  12. Kok, Pieter; Braunstein, Samuel L; Dowling, Jonathan P (July 28, 2004). "Quantum lithography, entanglement and Heisenberg-limited parameter estimation" (PDF). Journal of Optics B: Quantum and Semiclassical Optics. IOP Publishing. 6 (8): S811–S815. arXiv: quant-ph/0402083 . Bibcode:2004JOptB...6S.811K. doi:10.1088/1464-4266/6/8/029. ISSN   1464-4266. S2CID   15255876.
  13. Kimble, H. J.; Levin, Yuri; Matsko, Andrey B.; Thorne, Kip S.; Vyatchanin, Sergey P. (December 26, 2001). "Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics" (PDF). Physical Review D. American Physical Society (APS). 65 (2): 022002. arXiv: gr-qc/0008026 . Bibcode:2002PhRvD..65b2002K. doi:10.1103/physrevd.65.022002. hdl:1969.1/181491. ISSN   0556-2821. S2CID   15339393.
  14. Demkowicz-Dobrzański, Rafał; Kołodyński, Jan; Guţă, Mădălin (September 18, 2012). "The elusive Heisenberg limit in quantum-enhanced metrology". Nature Communications. 3: 1063. arXiv: 1201.3940 . Bibcode:2012NatCo...3.1063D. doi:10.1038/ncomms2067. PMC   3658100 . PMID   22990859.
  15. Escher, B. M.; Filho, R. L. de Matos; Davidovich, L. (May 2011). "General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology". Nature Physics. 7 (5): 406–411. arXiv: 1201.1693 . Bibcode:2011NatPh...7..406E. doi:10.1038/nphys1958. ISSN   1745-2481. S2CID   12391055.
  16. Sørensen, Anders S. (2001). "Entanglement and Extreme Spin Squeezing". Physical Review Letters. 86 (20): 4431–4434. arXiv: quant-ph/0011035 . Bibcode:2001PhRvL..86.4431S. doi:10.1103/physrevlett.86.4431. PMID   11384252. S2CID   206327094.
  17. Pezzé, Luca (2009). "Entanglement, Nonlinear Dynamics, and the Heisenberg Limit". Physical Review Letters. 102 (10): 100401. arXiv: 0711.4840 . Bibcode:2009PhRvL.102j0401P. doi:10.1103/physrevlett.102.100401. PMID   19392092. S2CID   13095638.
  18. Hyllus, Philipp (2012). "Fisher information and multiparticle entanglement". Physical Review A. 85 (2): 022321. arXiv: 1006.4366 . Bibcode:2012PhRvA..85b2321H. doi:10.1103/physreva.85.022321. S2CID   118652590.
  19. Tóth, Géza (2012). "Multipartite entanglement and high-precision metrology". Physical Review A. 85 (2): 022322. arXiv: 1006.4368 . Bibcode:2012PhRvA..85b2322T. doi:10.1103/physreva.85.022322. S2CID   119110009.