Quantum Fisher information

Short description: Quantum

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] It is one of the central quantities used to qualify the utility of an input state, especially in Mach–Zehnder (or, equivalently, Ramsey) interferometer-based phase or parameter estimation.[1][3][6] It is shown that the quantum Fisher information can also be a sensitive probe of a quantum phase transition (e.g. recognizing the superradiant quantum phase transition in the Dicke model[6]). The quantum Fisher information FQ[ϱ,A] of a state ϱ with respect to the observable A is defined as

FQ[ϱ,A]=2∑k,l(λk−λl)2(λk+λl)|⟨k|A|l⟩|2,

where λk and |k⟩ are the eigenvalues and eigenvectors of the density matrix ϱ, respectively, and the summation goes over all k and l such that λk+λl>0.

When the observable generates a unitary transformation of the system with a parameter θ from initial state ϱ0,

ϱ(θ)=exp⁡(−iAθ)ϱ0exp⁡(+iAθ),

the quantum Fisher information constrains the achievable precision in statistical estimation of the parameter θ via the quantum Cramér–Rao bound as

(Δθ)2≥1mFQ[ϱ,A],

where m is the number of independent repetitions.

It is often desirable to estimate the magnitude of an unknown parameter α that controls the strength of a system's Hamiltonian H=αA with respect to a known observable A during a known dynamical time t. In this case, defining θ=αt, so that θA=tH, means estimates of θ can be directly translated into estimates of α.

## Contents

* 1 Connection with Fisher information
* 2 Relation to the symmetric logarithmic derivative
* 3 Equivalent expressions
* 4 Generalization and relations to Bures metric and quantum fidelity
* 5 Relation to fidelity susceptibility
* 6 Convexity properties
* 7 Inequalities for composite systems
* 8 Relation to entanglement
* 9 Measuring the Fisher information
* 10 Relation to the Wigner–Yanase skew information
* 11 Relation to the variance
* 12 Uncertainty relations with the quantum Fisher information and the variance
* 13 References

## Connection with Fisher information

Classical Fisher information of measuring observable B on density matrix ϱ(θ) is defined as F[B,θ]=∑b1p(b|θ)(∂p(b|θ)∂θ)2, where p(b|θ)=⟨b|ϱ(θ)|b⟩ is the probability of obtaining outcome b when measuring observable B on the transformed density matrix ϱ(θ). b is the eigenvalue corresponding to eigenvector |b⟩ of observable B.

Quantum Fisher information is the supremum of the classical Fisher information over all such observables,[7]

FQ[ϱ,A]=supBF[B,θ].

## Relation to the symmetric logarithmic derivative

The quantum Fisher information equals the expectation value of Lϱ2, where Lϱ is the symmetric logarithmic derivative

## Equivalent expressions

For a unitary encoding operation ϱ(θ)=exp⁡(−iAθ)ϱ0exp⁡(+iAθ),, the quantum Fisher information can be computed as an integral,[8]

FQ[ϱ,A]=−2∫0∞tr(exp⁡(−ρ0t)[ϱ0,A]exp⁡(−ρ0t)[ϱ0,A]) dt,

where [ , ] on the right hand side denotes commutator. It can be also expressed in terms of Kronecker product and vectorization,[9]

FQ[ϱ,A]=2vec([ϱ0,A])†(ρ0*⊗I+I⊗ρ0)−1vec([ϱ0,A]),

where  * denotes complex conjugate, and  † denotes conjugate transpose. This formula holds for invertible density matrices. For non-invertible density matrices, the inverse above is substituted by the Moore-Penrose pseudoinverse. Alternatively, one can compute the quantum Fisher information for invertible state ρν=(1−ν)ρ0+νπ, where π is any full-rank density matrix, and then perform the limit ν->0+ to obtain the quantum Fisher information for ρ0. Density matrix π can be, for example, Identity/dim⁡ℋ in a finite-dimensional system, or a thermal state in infinite dimensional systems.

## Generalization and relations to Bures metric and quantum fidelity

For any differentiable parametrization of the density matrix ϱ(θ) by a vector of parameters θ=(θ1,…,θn), the quantum Fisher information matrix is defined as

FQij[ϱ(θ)]=2∑k,lRe⁡(⟨k|∂iϱ|l⟩⟨l|∂jϱ|k⟩)λk+λl,

where ∂i denotes partial derivative with respect to parameter θi. The formula also holds without taking the real part Re, because the imaginary part leads to an antisymmetric contribution that disappears under the sum. Note that all eigenvalues λk and eigenvectors |k⟩ of the density matrix potentially depend on the vector of parameters θ.

This definition is identical to four times the Bures metric, up to singular points where the rank of the density matrix changes (those are the points at which λk+λl suddenly becomes zero.) Through this relation, it also connects with quantum fidelity F(ϱ,σ)=(tr[ϱσϱ])2 of two infinitesimally close states,[10]

F(ϱθ,ϱθ+dθ)=1−14∑i,j(FQij[ϱ(θ)]+2∑λk(θ)=0∂i∂jλk)dθidθj+𝒪(dθ3),

where the inner sum goes over all k at which eigenvalues λk(θ)=0. The extra term (which is however zero in most applications) can be avoided by taking a symmetric expansion of fidelity,[11]

F(ϱθ−dθ/2,ϱθ+dθ/2)=1−14∑i,jFQij[ϱ(θ)]dθidθj+𝒪(dθ3).

For n=1 and unitary encoding, the quantum Fisher information matrix reduces to the original definition.

Quantum Fisher information matrix is a part of a wider family of quantum statistical distances.[12]

## Relation to fidelity susceptibility

Assuming that |ψ0(θ)⟩ is a ground state of a parameter-dependent non-degenerate Hamiltonian H(θ), four times the quantum Fisher information of this state is called fidelity susceptibility, and denoted[13]

χF=4FQ(|ψ0(θ)⟩).

Fidelity susceptibility measures the sensitivity of the ground state to the parameter, and its divergence indicates a quantum phase transition. This is because of the aforementioned connection with fidelity: a diverging quantum Fisher information means that |ψ0(θ)⟩ and |ψ0(θ+dθ)⟩ are orthogonal to each other, for any infinitesimal change in parameter dθ, and thus are said to undergo a phase-transition at point θ.

## Convexity properties

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

FQ[|Ψ⟩,H]=4(ΔH)Ψ2.

For mixed states, when the probabilities are parameter independent, i.e., when p(θ)=p, the quantum Fisher information is convex:

FQ[pϱ1(θ)+(1−p)ϱ2(θ),H]≤pFQ[ϱ1(θ),H]+(1−p)FQ[ϱ2(θ),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[14][15]

FQ[ϱ,H]=4inf{pk,|Ψk⟩}∑kpk(ΔH)Ψk2,

where the infimum is over all decompositions of the density matrix

ϱ=∑kpk|Ψk⟩⟨Ψk|.

Note that |Ψk⟩ are not necessarily orthogonal to each other. The above optimization can be rewritten as an optimization over the two-copy space as [16]

FQ[ϱ,H]=minϱ122Tr[(H⊗Identity−Identity⊗H)2ϱ12],

such that ϱ12 is a symmetric separable state and

Tr1(ϱ12)=Tr2(ϱ12)=ϱ.

Later the above statement has been proved even for the case of a minimization over general (not necessarily symmetric) separable states.[17] Thus, the minimum of a correlation term is given as

maxϱ12∈Sep(ϱ)⟨H⊗H⟩ϱ12=⟨H2⟩ϱ−14FQ[ϱ,H],

where Sep(ϱ) is the set of bipartite separable states with a marginal ϱ. Here, H can be a Hermitian operator of any dimension. This can be used to give the minimum for separable states for a ferromagnetic Ising spin chain with a formula containing the quantum Fisher information.[18]

When the probabilities are θ-dependent, an extended-convexity relation has been proved:[19]

FQ[∑ipi(θ)ϱi(θ)]≤∑ipi(θ)FQ[ϱi(θ)]+FC[{pi(θ)}],

where FC[{pi(θ)}]=∑i∂θpi(θ)2pi(θ) is the classical Fisher information associated to the probabilities contributing to the convex decomposition. The first term, in the right hand side of the above inequality, can be considered as the average quantum Fisher information of the density matrices in the convex decomposition.

## 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.[20] For product states,

FQ[ϱ1⊗ϱ2,H1⊗Identity+Identity⊗H2]=FQ[ϱ1,H1]+FQ[ϱ2,H2]

holds.

For the reduced state, we have

FQ[ϱ12,H1⊗Identity2]≥FQ[ϱ1,H1],

where ϱ1=Tr2(ϱ12).

## Relation to entanglement

There are strong links between quantum metrology and quantum information science. For a multiparticle system of N spin-1/2 particles [21]

FQ[ϱ,Jz]≤N

holds for separable states, where

Jz=∑n=1Njz(n),

and jz(n) is a single particle angular momentum component. The maximum for general quantum states is given by

FQ[ϱ,Jz]≤N2.

Hence, quantum entanglement is needed to reach the maximum precision in quantum metrology. Moreover, for quantum states with an entanglement depth k,

FQ[ϱ,Jz]≤sk2+r2

holds, where s=⌊N/k⌋ is the largest integer smaller than or equal to N/k, and r=N−sk is the remainder from dividing N by k. Hence, a higher and higher levels of multipartite entanglement is needed to achieve a better and better accuracy in parameter estimation.[22][23] It is possible to obtain a weaker but simpler bound [24]

FQ[ϱ,Jz]≤Nk.

Hence, a lower bound on the entanglement depth is obtained as

FQ[ϱ,Jz]N≤k.

A related concept is the quantum metrological gain, which for a given Hamiltonian is defined as the ratio of the quantum Fisher information of a state and the maximum of the quantum Fisher information for the same Hamiltonian for separable states

gℋ(ϱ)=ℱQ[ϱ,ℋ]ℱQ(sep)(ℋ),

where the Hamiltonian is

ℋ=h1+h2+...+hN,

and hn acts on the nth spin. The metrological gain is defined by an optimization over all local Hamiltonians as

g(ϱ)=maxℋgℋ(ϱ).

## Measuring the Fisher information

The error propagation formula gives a lower bound on the quantum Fisher information

FQ[ϱ,H]≥⟨i[H,M]⟩ϱ2(ΔM)2,

where M is an operator. This formula can be used to put a lower bound on the quantum Fisher information from experimental results.[25] If M equals the symmetric logarithmic derivative then the inequality is saturated.[26]

For the case of unitary dynamics, the quantum Fisher information is the convex roof of the variance. Based on that, one can obtain lower bounds on it, based on some given operator expectation values using semidefinite programming. The approach considers an optimization of the two-copy space.[27]

There are numerical methods that provide an optimal lower bound for the quantum Fisher information based on the expectation values for some operators, using the theory of Legendre transforms and not semidefinite programming.[28] In some cases, the bounds can even be obtained analytically. For instance, for an N-qubit Greenberger-Horne-Zeilinger (GHZ) state

FQ[ϱ,Jz]N2≥(1−2FGHZ)2,

where for the fidelity with respect to the GHZ state

FGHZ=Tr(ϱ|GHZ⟩⟨GHZ|)≥1/2

holds, otherwise the optimal lower bound is zero.

So far, we discussed bounding the quantum Fisher information for a unitary dynamics. It is also possible to bound the quantum Fisher information for the more general, non-unitary dynamics.[29] The approach is based on the relation between the fidelity and the quantum Fisher information and that the fidelity can be computed based on semidefinite programming.

For systems in thermal equibirum, the quantum Fisher information can be obtained from the dynamic susceptibility.[30]

## Relation to the Wigner–Yanase skew information

The Wigner–Yanase skew information is defined as [31]

I(ϱ,H)=Tr(H2ϱ)−Tr(HϱHϱ).

It follows that I(ϱ,H) is convex in ϱ.

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

FQ[ϱ,H]≥4I(ϱ,H)

holds, where there is an equality for pure states.

## Relation to the variance

For any decomposition of the density matrix given by pk and |Ψk⟩ the relation [14]

(ΔH)2≥∑kpk(ΔH)Ψk2≥14FQ[ϱ,H]

holds, where both inequalities are tight. That is, there is a decomposition for which the second inequality is saturated, which is the same as stating that the quantum Fisher information is the convex roof of the variance over four, discussed above. There is also a decomposition for which the first inequality is saturated, which means that the variance is its own concave roof [14]

(ΔH)2=sup{pk,|Ψk⟩}∑kpk(ΔH)Ψk2.

## Uncertainty relations with the quantum Fisher information and the variance

Knowing that the quantum Fisher information is the convex roof of the variance times four, we obtain the relation [32] (ΔA)2FQ[ϱ,B]≥|⟨i[A,B]⟩|2, which is stronger than the Heisenberg uncertainty relation. For a particle of spin-j, the following uncertainty relation holds (ΔJx)2+(ΔJy)2+(ΔJz)2≥j, where Jl are angular momentum components. The relation can be strengthened as [33][34] (ΔJx)2+(ΔJy)2+FQ[ϱ,Jz]/4≥j.

## References

1. ↑ 1.0 1.1 Helstrom, C (1976). Quantum detection and estimation theory. Academic Press. ISBN 0-12-340050-3.
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. ↑ 3.0 3.1 Braunstein, Samuel L.; Caves, Carlton M. (1994-05-30). "Statistical distance and the geometry of quantum states". Physical Review Letters 72 (22): 3439–3443. doi:10.1103/physrevlett.72.3439. ISSN 0031-9007. PMID 10056200. Bibcode: 1994PhRvL..72.3439B.
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. doi:10.1006/aphy.1996.0040. Bibcode: 1996AnPhy.247..135B.
5. ↑ Paris, Matteo G. A. (21 November 2011). "Quantum Estimation for Quantum Technology". International Journal of Quantum Information 07 (supp01): 125–137. doi:10.1142/S0219749909004839.
6. ↑ 6.0 6.1 Wang, Teng-Long; Wu, Ling-Na; Yang, Wen; Jin, Guang-Ri; Lambert, Neill; Nori, Franco (2014-06-17). "Quantum Fisher information as a signature of the superradiant quantum phase transition". New Journal of Physics 16 (6). doi:10.1088/1367-2630/16/6/063039. ISSN 1367-2630. Bibcode: 2014NJPh...16f3039W. https://iopscience.iop.org/article/10.1088/1367-2630/16/6/063039.
7. ↑ Paris, Matteo G. A. (2009). "Quantum estimation for quantum technology". International Journal of Quantum Information 07 (supp01): 125–137. doi:10.1142/s0219749909004839. ISSN 0219-7499.
8. ↑ PARIS, MATTEO G. A. (2009). "Quantum estimation for quantum technology". International Journal of Quantum Information 07 (supp01): 125–137. doi:10.1142/s0219749909004839. ISSN 0219-7499.
9. ↑ Šafránek, Dominik (2018-04-12). "Simple expression for the quantum Fisher information matrix". Physical Review A 97 (4). doi:10.1103/physreva.97.042322. ISSN 2469-9926. Bibcode: 2018PhRvA..97d2322S.
10. ↑ Šafránek, Dominik (2017-05-11). "Discontinuities of the quantum Fisher information and the Bures metric". Physical Review A 95 (5). doi:10.1103/physreva.95.052320. ISSN 2469-9926. Bibcode: 2017PhRvA..95e2320S.
11. ↑ Zhou, Sisi; Jiang, Liang (18 Oct 2019). "An exact correspondence between the quantum Fisher information and the Bures metric". arXiv:1910.08473 [quant-ph].
12. ↑ Jarzyna, M.; Kołodyński, J. (18 August 2020). "Geometric Approach to Quantum Statistical Inference". IEEE Journal on Selected Areas in Information Theory 1 (2): 367–386. doi:10.1109/JSAIT.2020.3017469. ISSN 2641-8770. Bibcode: 2020IJSAI...1..367J.
13. ↑ Gu, S.-J. (2010). "Fidelity approach to quantum phase transitions". International Journal of Modern Physics B 24 (23): 4371–4458. doi:10.1142/S0217979210056335. Bibcode: 2010IJMPB..24.4371G.
14. ↑ 14.0 14.1 14.2 Tóth, Géza; Petz, Dénes (20 March 2013). "Extremal properties of the variance and the quantum Fisher information". Physical Review A 87 (3). doi:10.1103/PhysRevA.87.032324. Bibcode: 2013PhRvA..87c2324T.
15. ↑ Yu, Sixia (2013). "Quantum Fisher Information as the Convex Roof of Variance". arXiv:1302.5311 [quant-ph].
16. ↑ Tóth, Géza; Moroder, Tobias; Gühne, Otfried (21 April 2015). "Evaluating Convex Roof Entanglement Measures". Physical Review Letters 114 (16). doi:10.1103/PhysRevLett.114.160501. PMID 25955038. Bibcode: 2015PhRvL.114p0501T.
17. ↑ Tóth, Géza; Pitrik, József (16 October 2023). "Quantum Wasserstein distance based on an optimization over separable states". Quantum 7. doi:10.22331/q-2023-10-16-1143. Bibcode: 2023Quant...7.1143T.
18. ↑ Tóth, Géza; Pitrik, József. "General method for obtaining the energy minimum of spin Hamiltonians for separable states". arxiv.
19. ↑ Alipour, S.; Rezakhani, A. T. (2015-04-07). "Extended convexity of quantum Fisher information in quantum metrology" (in en). Physical Review A 91 (4). doi:10.1103/PhysRevA.91.042104. ISSN 1050-2947. Bibcode: 2015PhRvA..91d2104A. https://link.aps.org/doi/10.1103/PhysRevA.91.042104.
20. ↑ Tóth, Géza; Apellaniz, Iagoba (24 October 2014). "Quantum metrology from a quantum information science perspective". Journal of Physics A: Mathematical and Theoretical 47 (42). doi:10.1088/1751-8113/47/42/424006. Bibcode: 2014JPhA...47P4006T.
21. ↑ Pezzé, Luca; Smerzi, Augusto (10 March 2009). "Entanglement, Nonlinear Dynamics, and the Heisenberg Limit". Physical Review Letters 102 (10). doi:10.1103/PhysRevLett.102.100401. PMID 19392092. Bibcode: 2009PhRvL.102j0401P.
22. ↑ Hyllus, Philipp (2012). "Fisher information and multiparticle entanglement". Physical Review A 85 (2). doi:10.1103/physreva.85.022321. Bibcode: 2012PhRvA..85b2321H.
23. ↑ Tóth, Géza (2012). "Multipartite entanglement and high-precision metrology". Physical Review A 85 (2). doi:10.1103/physreva.85.022322. Bibcode: 2012PhRvA..85b2322T.
24. ↑ Tóth, Géza (2021). Entanglement detection and quantum metrology in quantum optical systems. Budapest: Doctoral Dissertation submitted to the Hungarian Academy of Sciences. p. 68. http://real-d.mtak.hu/1230/7/dc_1593_18_doktori_mu.pdf.
25. ↑ Lücke, B.; Scherer, M.; Kruse, J.; Pezzé, L.; Deuretzbacher, F.; Hyllus, P.; Topic, O.; Peise, J. et al. (11 November 2011). "Twin Matter Waves for Interferometry Beyond the Classical Limit". Science 334 (6057): 773–776. doi:10.1126/science.1208798. PMID 21998255. Bibcode: 2011Sci...334..773L.
26. ↑ Escher, B. M. (2012). "Quantum Noise-to-Sensibility Ratio". arXiv:1212.2533 [quant-ph].
27. ↑ Tóth, Géza; Moroder, Tobias; Gühne, Otfried (21 April 2015). "Evaluating Convex Roof Entanglement Measures". Physical Review Letters 114 (16). doi:10.1103/PhysRevLett.114.160501. PMID 25955038. Bibcode: 2015PhRvL.114p0501T.
28. ↑ Apellaniz, Iagoba; Kleinmann, Matthias; Gühne, Otfried; Tóth, Géza (28 March 2017). "Optimal witnessing of the quantum Fisher information with few measurements". Physical Review A 95 (3). doi:10.1103/PhysRevA.95.032330. Bibcode: 2017PhRvA..95c2330A.
29. ↑ Müller-Rigat, Guillem; Srivastava, Anubhav Kumar; Kurdziałek, Stanisław; Rajchel-Mieldzioć, Grzegorz; Lewenstein, Maciej; Frérot, Irénée (24 October 2023). "Certifying the quantum Fisher information from a given set of mean values: a semidefinite programming approach". Quantum 7. doi:10.22331/q-2023-10-24-1152. Bibcode: 2023Quant...7.1152M.
30. ↑ Hauke, Philipp; Heyl, Markus; Tagliacozzo, Luca; Zoller, Peter (August 2016). "Measuring multipartite entanglement through dynamic susceptibilities". Nature Physics 12 (8): 778–782. doi:10.1038/nphys3700. Bibcode: 2016NatPh..12..778H.
31. ↑ Wigner, E. P.; Yanase, M. M. (1 June 1963). "Information Contents of Distributions". Proceedings of the National Academy of Sciences 49 (6): 910–918. doi:10.1073/pnas.49.6.910. PMID 16591109. Bibcode: 1963PNAS...49..910W.
32. ↑ Fröwis, Florian; Schmied, Roman; Gisin, Nicolas (2 July 2015). "Tighter quantum uncertainty relations following from a general probabilistic bound". Physical Review A 92 (1). doi:10.1103/PhysRevA.92.012102. Bibcode: 2015PhRvA..92a2102F.
33. ↑ Tóth, Géza; Fröwis, Florian (31 January 2022). "Uncertainty relations with the variance and the quantum Fisher information based on convex decompositions of density matrices". Physical Review Research 4 (1). doi:10.1103/PhysRevResearch.4.013075. Bibcode: 2022PhRvR...4a3075T.
34. ↑ Chiew, Shao-Hen; Gessner, Manuel (31 January 2022). "Improving sum uncertainty relations with the quantum Fisher information". Physical Review Research 4 (1). doi:10.1103/PhysRevResearch.4.013076. Bibcode: 2022PhRvR...4a3076C.

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