Quantum Fisher information

The quantum Fisher information is a central quantity in quantum metrology and is the quantum analogue of the classical Fisher information.[1][2][3][4][5] The quantum Fisher information ${\displaystyle F_{\rm {Q}}[\varrho ,A]}$ of a state ${\displaystyle \varrho }$ with respect to the observable ${\displaystyle A}$ is defined as

${\displaystyle F_{\rm {Q}}[\varrho ,A]=2\sum _{k,l}{\frac {(\lambda _{k}-\lambda _{l})^{2}}{(\lambda _{k}+\lambda _{l})}}\vert \langle k\vert A\vert l\rangle \vert ^{2},}$

where ${\displaystyle \lambda _{k}}$ and ${\displaystyle \vert k\rangle }$ are the eigenvalues and eigenvectors of the density matrix ${\displaystyle \varrho ,}$ respectively.

When the observable generates a unitary transformation of the system with a parameter ${\displaystyle \theta }$ from initial state ${\displaystyle \varrho _{0}}$,

${\displaystyle \varrho (\theta )=\exp(-iA\theta )\varrho _{0},}$

the quantum Fisher information constrains the achievable precision in statistical estimation of the parameter ${\displaystyle \theta }$ via the quantum Cramér–Rao bound:

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

It is often desirable to estimate the magnitude of an unknown parameter ${\displaystyle \alpha }$ that controls the strength of a system's Hamiltonian ${\displaystyle H=\alpha A}$ with respect to a known observable ${\displaystyle A}$ during a known dynamical time ${\displaystyle t}$. In this case, defining ${\displaystyle \theta =\alpha t}$, so that ${\displaystyle \theta A=tH}$, means estimates of ${\displaystyle \theta }$ can be directly translated into estimates of ${\displaystyle \alpha }$.

Relation to the Symmetric Logarithmic Derivative

The quantum Fisher information equals the expectation value of ${\displaystyle L_{\varrho }^{2}}$, where ${\displaystyle L_{\varrho }}$ is the Symmetric Logarithmic Derivative.

Convexity properties

The quantum Fisher information equals four times the variance for pure states

${\displaystyle F_{\rm {Q}}[\vert \Psi \rangle ,H]=4(\Delta H)_{\Psi }^{2}}$.

For mixed states it is convex in ${\displaystyle \varrho ,}$ that is,

${\displaystyle F_{\rm {Q}}[p\varrho _{1}+(1-p)\varrho _{2},H]\leq pF_{\rm {Q}}[\varrho _{1},H]+(1-p)F_{\rm {Q}}[\varrho _{2},H].}$

The quantum Fisher information is the largest function that is convex and that equals four times the variance for pure states. That is, it equals four times the convex roof of the variance [6][7]

${\displaystyle F_{\rm {Q}}[\varrho ,H]=4\inf _{\{p_{k},\vert \Psi _{k}\rangle \}}\sum _{k}p_{k}(\Delta H)_{\Psi _{k}}^{2},}$

where the infimum is over all decompositions of the density matrix

${\displaystyle \varrho =\sum _{k}p_{k}\vert \Psi _{k}\rangle \langle \Psi _{k}\vert .}$

Note that ${\displaystyle \vert \Psi _{k}\rangle }$ are not necessarily orthogonal to each other.

Inequalities for composite systems

We need to understand the behavior of quantum Fisher information in composite system in order to study quantum metrology of many-particle systems.[8] For product states,

${\displaystyle F_{\rm {Q}}[\varrho _{1}\otimes \varrho _{2},H_{1}\otimes {\rm {Identity}}+{\rm {Identity}}\otimes H_{2}]=F_{\rm {Q}}[\varrho _{1},H_{1}]+F_{\rm {Q}}[\varrho _{2},H_{2}]}$

holds.

For the reduced state, we have

${\displaystyle F_{\rm {Q}}[\varrho _{12},H_{1}\otimes {\rm {Identity}}_{2}]\geq F_{\rm {Q}}[\varrho _{1},H_{1}],}$

where ${\displaystyle \varrho _{1}={\rm {Tr}}_{2}(\varrho _{12})}$.

Relation to entanglement

There are strong links between quantum metrology and quantum information science. For a multiparticle system of ${\displaystyle N}$ spin-1/2 particles [9]

${\displaystyle F_{\rm {Q}}[\varrho ,J_{z}]\leq N}$

holds for separable states, where

${\displaystyle J_{z}=\sum _{n=1}^{N}j_{z}^{(n)},}$

and ${\displaystyle j_{z}^{(n)}}$ is a single particle angular momentum component. The maximum for general quantum states is given by

${\displaystyle F_{\rm {Q}}[\varrho ,J_{z}]\leq N^{2}.}$ Hence, quantum entanglement is needed to reach the maximum precision in quantum metrology.

Moreover, for quantum states with an entanglement depth ${\displaystyle k}$,

${\displaystyle F_{\rm {Q}}[\varrho ,J_{z}]\leq kN+N_{R}^{2}}$

holds, where ${\displaystyle N_{R}}$ is the remainder from dividing ${\displaystyle N}$ by ${\displaystyle k}$. Hence, a higher and higher levels of multipartite entanglement is needed to achieve a better and better accuracy in parameter estimation.[10][11]

Similar quantities

The Wigner–Yanase skew information is defined as [12]

${\displaystyle I(\varrho ,H)={\rm {Tr}}(H^{2}\varrho )-{\rm {Tr}}(H{\sqrt {\varrho }}H{\sqrt {\varrho }}).}$

It follows that ${\displaystyle I(\varrho ,H)}$ is convex in ${\displaystyle \varrho .}$

For the quantum Fisher information and the Wigner–Yanase skew information, the inequality

${\displaystyle F_{\rm {Q}}[\varrho ,H]\geq 4I(\varrho ,H)}$

holds, where there is an equality for pure states.

References

1. Helstrom, C (1976). Quantum detection and estimation theory. Academic Press. ISBN 0123400503.
2. Holevo, Alexander S (1982). Probabilistic and statistical aspects of quantum theory (2nd English ed.). Scuola Normale Superiore. ISBN 978-88-7642-378-9.
3. Braunstein, Samuel L.; Caves, Carlton M. (1994-05-30). "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.
4. 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.
5. Paris, Matteo G. A. (21 November 2011). "Quantum Estimation for Quantum Technology". International Journal of Quantum Information. 07 (supp01): 125–137. arXiv:0804.2981. doi:10.1142/S0219749909004839.
6. Tóth, Géza; Petz, Dénes (20 March 2013). "Extremal properties of the variance and the quantum Fisher information". Physical Review A. 87 (3): 032324. arXiv:1109.2831. Bibcode:2013PhRvA..87c2324T. doi:10.1103/PhysRevA.87.032324.
7. Yu, Sixia (2013). "Quantum Fisher Information as the Convex Roof of Variance". arXiv:1302.5311 [quant-ph].
8. Tóth, Géza; Apellaniz, Iagoba (24 October 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.
9. Pezzé, Luca; Smerzi, Augusto (10 March 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.
10. 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.
11. 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.
12. Wigner, E. P.; Yanase, M. M. (1 June 1963). "Information Contents of Distributions". Proceedings of the National Academy of Sciences. 49 (6): 910–918. Bibcode:1963PNAS...49..910W. doi:10.1073/pnas.49.6.910.