# Quantum Fisher information

**Quantum Fisher information** (QFI) is a quantity in quantum estimation theory that measures how much information a quantum state carries about an unknown parameter, such as a phase, a magnetic field strength, or a frequency. It is defined as the maximum of the classical [Fisher information](https://www.edgepedia.org/fisher-information) over all possible quantum measurements, and it sets the ultimate precision with which the parameter can be estimated from repeated measurements on the state.<sup>[1](https://arxiv.org/html/2402.03572v3)</sup><sup> • </sup><sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC7513002/)</sup>

| Key fact | Detail |
|---|---|
| Definition | QFI is the classical Fisher information maximized over all POVMs; equivalently the variance of the symmetric logarithmic derivative (SLD) operator, \( F_{Q}(\theta) = \mathrm{Tr}[\hat{\rho}_{\theta}\hat{L}_{\theta}^{2}] \)<sup>[1](https://arxiv.org/html/2402.03572v3)</sup> |
| Precision bound | Quantum Cramér–Rao bound: \( \Delta\theta \ge 1/\sqrt{\nu F_{Q}(\theta)} \) for \( \nu \) independent trials<sup>[1](https://arxiv.org/html/2402.03572v3)</sup> |
| Pure-state form | For a pure state evolved unitarily by \( U_{\theta} = e^{-i\hat{H}_{\theta}} \), \( F_{Q} = 4(\Delta\hat{H}_{\theta})^{2} \), four times the variance of the generator<sup>[1](https://arxiv.org/html/2402.03572v3)</sup> |
| Scaling limits | Separable probes give \( \Delta\theta \propto 1/\sqrt{N} \) (standard quantum limit); entangled probes reach \( \Delta\theta \propto 1/N \) (Heisenberg limit), with maximal \( F_{Q} = N^{2}(\lambda_{\max} - \lambda_{\min})^{2} \)<sup>[1](https://arxiv.org/html/2402.03572v3)</sup> |
| Entanglement witness | A state with \( F_{Q} > N \) is metrologically useful; for \( k \)-particle entanglement depth, \( F_{Q} \le sk^{2} + r^{2} \)<sup>[1](https://arxiv.org/html/2402.03572v3)</sup> |
| Geometry | The Bures distance between neighboring states expands as \( d_{\mathrm{Bures}}^{2} = \frac{1}{4}J\,dx^{2} \), so maximizing QFI is equivalent to maximizing state distinguishability<sup>[3](https://www.nature.com/articles/s41534-017-0014-6)</sup> |
| Multiparameter form | The QFI matrix has elements \( J_{\mu\nu} = \frac{1}{2}\mathrm{Tr}\,\rho\{L_{\mu}, L_{\nu}\} \); the multiparameter bound is not always attainable<sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC7513002/)</sup> |

## Definition and the quantum Cramér–Rao bound

In classical statistics, the Fisher information of a probability distribution \( P(x|\theta) \) bounds the variance of any unbiased estimator through the [Cramér–Rao bound](https://www.edgepedia.org/cramer-rao-bound), \( \Delta\theta \ge 1/\sqrt{\nu F(\theta)} \), where \( \nu \) is the number of independent trials.<sup>[1](https://arxiv.org/html/2402.03572v3)</sup> In the quantum setting the object being measured is not a probability distribution chosen in advance but a [density matrix](https://www.edgepedia.org/density-matrix) \( \hat{\rho}_{\theta} \), and the experimenter must also choose a measurement, described mathematically by a positive operator-valued measure (POVM).<sup>[1](https://arxiv.org/html/2402.03572v3)</sup><sup> • </sup><sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC7513002/)</sup>

The QFI is obtained by maximizing the classical Fisher information of the outcome probabilities over all possible POVMs:<sup>[1](https://arxiv.org/html/2402.03572v3)</sup>

\[ F_{Q}(\theta) = \max_{\hat{E}} F_{C}(\theta; \hat{E}). \]

Because \( F_{Q} \ge F_{C} \) for every measurement, the resulting quantum Cramér–Rao bound,

\[ \Delta\theta \ge \frac{1}{\sqrt{\nu F_{Q}(\theta)}}, \]

is the ultimate achievable limit of the classical bound: no measurement on the given probe state can do better.<sup>[1](https://arxiv.org/html/2402.03572v3)</sup>

**Symmetric logarithmic derivative.** The practical route to computing the QFI runs through the symmetric logarithmic derivative (SLD), the Hermitian operator \( \hat{L}_{\theta} \) implicitly defined by

\[ \partial_{\theta}\hat{\rho}_{\theta} = \frac{1}{2}\left(\hat{\rho}_{\theta}\hat{L}_{\theta} + \hat{L}_{\theta}\hat{\rho}_{\theta}\right). \]

The QFI is then the variance of this operator, \( F_{Q}(\theta) = (\Delta\hat{L}_{\theta})^{2} = \mathrm{Tr}[\hat{\rho}_{\theta}\hat{L}_{\theta}^{2}] \).<sup>[1](https://arxiv.org/html/2402.03572v3)</sup><sup> • </sup><sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC7513002/)</sup> A projective measurement in the eigenbasis of the SLD always yields a classical Fisher information equal to the QFI, so the bound is attainable in the single-parameter case in the limit of many repetitions.<sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC7513002/)</sup>

For a pure state \( \hat{\rho}_{\theta} = |\Psi_{\mathrm{f}}(\theta)\rangle\langle\Psi_{\mathrm{f}}(\theta)| \), the QFI reduces to a geometric quantity involving only the state and its derivative with respect to the parameter, and if the state is produced from a pure input by the unitary \( U_{\theta} = e^{-i\hat{H}_{\theta}} \), it becomes

\[ F_{Q}(\theta) = 4(\Delta\hat{H}_{\theta})^{2}, \]

four times the variance of the generator \( \hat{H}_{\theta} \) in the input state.<sup>[1](https://arxiv.org/html/2402.03572v3)</sup> This formula is the workhorse of quantum metrology: it converts the design of a precise sensor into the preparation of a state with a large spread in the eigenvalues of the parameter-encoding Hamiltonian.

## Scaling limits: standard quantum limit and Heisenberg limit

For \( N \) probes used independently, the QFI is additive and at most \( N(\lambda_{\max} - \lambda_{\min})^{2} \), where \( \lambda_{\max} \) and \( \lambda_{\min} \) are the extreme eigenvalues of the single-particle generator. The best precision then scales as \( \Delta\theta \propto 1/\sqrt{N} \), the standard quantum limit (SQL), also called the shot-noise limit.<sup>[1](https://arxiv.org/html/2402.03572v3)</sup><sup> • </sup><sup>[4](https://www.nature.com/nature-index/topics/l4/quantum-metrology-and-parameter-estimation)</sup>

Entanglement changes the scaling. When the \( N \) particles share an entangled state, the extreme eigenvalues of the collective generator grow linearly with \( N \), and the maximal QFI becomes

\[ F_{Q}(\theta)|_{\max} = N^{2}(\lambda_{\max} - \lambda_{\min})^{2}, \]

which yields \( \Delta\theta \propto 1/N \), the [Heisenberg limit](https://www.edgepedia.org/heisenberg-limit).<sup>[1](https://arxiv.org/html/2402.03572v3)</sup> For \( N \) qubits with generator \( \hat{h}^{(l)} = \frac{1}{2}\sigma_{z}^{(l)} \), this gives \( \Delta\theta_{\mathrm{HL}} = 1/N \), a \( \sqrt{N} \) improvement in precision scaling over the SQL.<sup>[1](https://arxiv.org/html/2402.03572v3)</sup> GHZ-type states, maximally path-entangled superpositions of the form \( (|n,0\rangle + |0,n\rangle)/\sqrt{2} \), attain this quadratic scaling of the Fisher information.<sup>[1](https://arxiv.org/html/2402.03572v3)</sup><sup> • </sup><sup>[5](https://pmc.ncbi.nlm.nih.gov/articles/PMC13512424/)</sup>

## QFI as an entanglement witness

Because the QFI depends only on the state's properties, it doubles as a tool for characterizing entanglement. An entangled state whose QFI exceeds the particle number, \( F_{Q} > N \), is metrologically useful: it can in principle beat the SQL.<sup>[1](https://arxiv.org/html/2402.03572v3)</sup> The QFI also bounds the entanglement depth, the number of particles that are genuinely entangled together. For a state that factorizes into blocks of at most \( k \) entangled particles, with \( s \) the integer part of \( N/k \) and \( r = N - sk \),

\[ F_{Q} \le sk^{2} + r^{2}. \]

Violating this bound certifies \( (k+1) \)-particle entanglement, and at \( k = N \) the bound \( F_{Q} \le N^{2} \) corresponds to the Heisenberg limit.<sup>[1](https://arxiv.org/html/2402.03572v3)</sup> A state such as the GHZ state, which reaches the maximal \( F_{Q} = N^{2} \), is considered ideal for metrological use under a linear generator.<sup>[1](https://arxiv.org/html/2402.03572v3)</sup> Conversely, entanglement is needed to surpass the shot-noise scaling in general metrological tasks with a linear interferometer.<sup>[6](https://doi.org/10.1088/1751-8113/47/42/424006)</sup>

## Multiparameter estimation

When several parameters \( \lambda_{\mu} \) are estimated simultaneously, the QFI generalizes to the quantum Fisher information matrix (QFIM),

\[ J_{\mu\nu} = \frac{1}{2}\mathrm{Tr}\,\rho\{L_{\mu}, L_{\nu}\}, \]

where each \( L_{\mu} \) is the SLD with respect to \( \lambda_{\mu} \).<sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC7513002/)</sup> The multiparameter quantum Cramér–Rao bound reads \( \mathrm{tr}(W\,\mathrm{Cov}(\hat{\boldsymbol{\lambda}})) \ge \mathrm{tr}(W J^{-1}) \) for a cost matrix \( W \), but unlike the single-parameter case this bound cannot always be saturated, because the optimal measurements for different parameters may be incompatible.<sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC7513002/)</sup> A sufficient condition for attainability is that the SLDs commute, \( [L_{\mu}, L_{\nu}] = 0 \), which is not necessary; within the framework of quantum local asymptotic normality, the necessary and sufficient compatibility condition is \( \mathcal{U}_{\mu\nu} = -\frac{i}{4}\mathrm{Tr}\,\rho[L_{\mu}, L_{\nu}] = 0 \) for all parameter pairs.<sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC7513002/)</sup> The SLD bound is tight in the one-parameter case but generally non-tight in multiparameter estimation, where the Holevo bound gives the most general quantum extension of the classical Cramér–Rao bound.<sup>[7](https://link.springer.com/article/10.1007/s00220-019-03433-4)</sup>

## Geometric and broader connections

The QFI has a direct geometric meaning. Expanding the Bures distance between neighboring states \( \rho_{x} \) and \( \rho_{x+dx} \) to second order gives \( d_{\mathrm{Bures}}^{2} = \frac{1}{4}J(\rho_{x})\,dx^{2} \), so maximizing the QFI is equivalent to maximizing the distinguishability, in the Bures-metric sense, of states that differ infinitesimally in the parameter.<sup>[3](https://www.nature.com/articles/s41534-017-0014-6)</sup>

Beyond metrology, the QFI and QFIM appear across quantum mechanics. Studies have connected the QFIM to quantum thermodynamics, quantum phase transitions, entanglement witnesses, the quantum speed limit, and non-Markovianity, indicating that it functions as a fundamental quantity rather than a metrology-specific tool.<sup>[8](https://arxiv.org/abs/1907.08037)</sup> The QFI also provides criteria for macroscopic superpositions, relates to the speed of quantum evolution, and appears in the theory of the quantum Zeno effect.<sup>[6](https://doi.org/10.1088/1751-8113/47/42/424006)</sup>

## Role in quantum sensing and its limits

[Quantum metrology](https://www.edgepedia.org/quantum-metrology) exploits superposition and entanglement to estimate parameters with precision beyond classical limits, and the QFI quantifies the sensitivity of the probe state to the parameter being estimated.<sup>[4](https://www.nature.com/nature-index/topics/l4/quantum-metrology-and-parameter-estimation)</sup> Working implementations include interferometers that use light, atomic clocks, and spin systems in solids, and these are put to use in detecting gravitational waves, sensing magnetic fields, and keeping time.<sup>[4](https://www.nature.com/nature-index/topics/l4/quantum-metrology-and-parameter-estimation)</sup> Entanglement-enhanced sensors based on spin-squeezed states, twin-Fock states, and spin cat states have been demonstrated in cold atoms, trapped ions, and related platforms, targeting atomic clocks, magnetometers, gravimeters, and gyroscopes.<sup>[1](https://arxiv.org/html/2402.03572v3)</sup>

Noise limits these gains. Ubiquitous decoherence degrades the quantum resources and can force the precision back to, or below, the classical limit, a result known as the no-go theorem of noisy quantum metrology.<sup>[9](https://arxiv.org/abs/2307.07701)</sup> GHZ-type entangled states that reach the Heisenberg limit are difficult to create and, in general, susceptible to decoherence and particle loss.<sup>[10](https://www.sciencedirect.com/science/article/abs/pii/S0370157325001565)</sup> How to realize the promised performance in realistic noisy situations is a central concern of current work on quantum-enhanced sensing.<sup>[9](https://arxiv.org/abs/2307.07701)</sup>

## References

1. [Jiahao Huang, Min Zhuang, Chaohong Lee (2024). Entanglement-enhanced quantum metrology: from standard quantum limit to Heisenberg limit. Applied Physics Reviews 11, 031302; arXiv:2402.03572.](https://arxiv.org/html/2402.03572v3)
2. [Angelo Carollo, Bernardo Spagnolo, Davide Valenti (2018). Symmetric Logarithmic Derivative of Fermionic Gaussian States. Entropy 20, 485.](https://pmc.ncbi.nlm.nih.gov/articles/PMC7513002/)
3. [Quantum parameter estimation with general dynamics. npj Quantum Information (2017).](https://www.nature.com/articles/s41534-017-0014-6)
4. [Quantum Metrology and Parameter Estimation, Nature Index topic summary.](https://www.nature.com/nature-index/topics/l4/quantum-metrology-and-parameter-estimation)
5. [Quantum Speed Limits and the Ultimate Scaling of the Quantum Sensors. PMC (2025).](https://pmc.ncbi.nlm.nih.gov/articles/PMC13512424/)
6. [Quantum metrology from a quantum information science perspective. Journal of Physics A 47, 424006 (2014).](https://doi.org/10.1088/1751-8113/47/42/424006)
7. [Yuxiang Yang, Giulio Chiribella, Masahito Hayashi (2019). Attaining the Ultimate Precision Limit in Quantum State Estimation. Communications in Mathematical Physics 368, 223–293.](https://link.springer.com/article/10.1007/s00220-019-03433-4)
8. [Quantum Fisher information matrix and multiparameter estimation. arXiv:1907.08037.](https://arxiv.org/abs/1907.08037)
9. [Lin Jiao, Wei Wu, Si-Yuan Bai, Jun-Hong An (2023). Quantum metrology in the noisy intermediate-scale quantum era. arXiv:2307.07701.](https://arxiv.org/abs/2307.07701)
10. [Review: Quantum metrology and sensing with many-body systems. Physics Reports (2025).](https://www.sciencedirect.com/science/article/abs/pii/S0370157325001565)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Matter and radiation physics › Quantum optics and photonics › Quantum imaging and quantum sensing › Quantum parameter estimation and limits*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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