quantum Fisher information

Classical Fisher information of measuring observable $B$ on density matrix $\backslash varrho(\backslash theta)$ is defined as $F[B,\backslash theta]=\backslash sum\_b\backslash frac\{1\}\{p(b|\backslash theta)\}\backslash left(\backslash frac\{\backslash partial\; p(b|\backslash theta)\}\{\backslash partial\; \backslash theta\}\backslash right)^2$, where $p(b|\backslash theta)=\backslash langle\; b\backslash vert\; \backslash varrho(\backslash theta)\backslash vert\; b\; \backslash rangle$ is the probability of obtaining outcome $b$ when measuring observable $B$ on the transformed density matrix $\backslash varrho(\backslash theta)$. $b$ is the eigenvalue corresponding to eigenvector $\backslash vert\; b\; \backslash rangle$ of observable $B$.

Quantum Fisher information is the supremum of the classical Fisher information over all such observables,{{cite journal | last=Paris | first=Matteo G. A. | title=Quantum estimation for quantum technology | journal=International Journal of Quantum Information | volume=07 | issue=supp01 | year=2009 | issn=0219-7499 | doi=10.1142/s0219749909004839 | pages=125–137| arxiv=0804.2981 | s2cid=2365312 }}

:$$

F_{\rm Q}[\varrho,A]=\sup_{B} F[B,\theta].

The quantum Fisher information equals the expectation value of $L\_\{\backslash varrho\}^2$, where $L\_\{\backslash varrho\}$ is the symmetric logarithmic derivative

For a unitary encoding operation $\backslash varrho(\backslash theta)=\backslash exp(-iA\backslash theta)\backslash varrho\_0\backslash exp(+iA\backslash theta),$, the quantum Fisher information can be computed as an integral,{{cite journal | last=PARIS | first=MATTEO G. A. | title=Quantum estimation for quantum technology | journal=International Journal of Quantum Information | volume=07 | issue=supp01 | year=2009 | issn=0219-7499 | doi=10.1142/s0219749909004839 | pages=125–137| arxiv=0804.2981 | s2cid=2365312 }}

:$$

F_{\rm Q}[\varrho,A] = -2\int_0^\infty\text{tr}\left(\exp(-\rho_0 t)[\varrho_0,A] \exp(-\rho_0 t)[\varrho_0,A]\right)\ dt,

where $[\backslash \; ,\backslash \; ]$ on the right hand side denotes commutator.

It can be also expressed in terms of Kronecker product and vectorization,{{cite journal | last=Šafránek | first=Dominik | title=Simple expression for the quantum Fisher information matrix | journal=Physical Review A | volume=97 | issue=4 | date=2018-04-12 | issn=2469-9926 | doi=10.1103/physreva.97.042322 | page=042322| arxiv=1801.00945 | bibcode=2018PhRvA..97d2322S }}

:$$

F_{\rm Q}[\varrho,A] = 2\,\text{vec}([\varrho_0,A])^\dagger\big(\rho_0^*\otimes {\rm I}+{\rm I}\otimes\rho_0\big)^{-1}\text{vec}([\varrho_0,A]),

where $^*$ denotes complex conjugate, and $^\backslash dagger$ 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 $\backslash rho\_\backslash nu=(1-\backslash nu)\backslash rho\_0+\backslash nu\backslash pi$, where $\backslash pi$ is any full-rank density matrix, and then perform the limit $\backslash nu\; \backslash rightarrow\; 0^+$ to obtain the quantum Fisher information for $\backslash rho\_0$. Density matrix $\backslash pi$ can be, for example, $\{\backslash rm\; Identity\}/\backslash dim\{\backslash mathcal\{H\}\}$ in a finite-dimensional system, or a thermal state in infinite dimensional systems.

For any differentiable parametrization of the density matrix $\backslash varrho(\backslash boldsymbol\{\backslash theta\})$ by a vector of parameters $\backslash boldsymbol\{\backslash theta\}=(\backslash theta\_1,\backslash dots,\backslash theta\_n)$, the quantum Fisher information matrix is defined as

:$$

F_{\rm Q}^{ij}[\varrho(\boldsymbol{\theta})]=2\sum_{k,l} \frac{\operatorname{Re}(\langle k \vert \partial_{i}\varrho \vert l\rangle \langle l \vert \partial_{j}\varrho \vert k\rangle )}{\lambda_k+\lambda_l},

where $\backslash partial\_i$ denotes partial derivative with respect to parameter $\backslash theta\_i$. The formula also holds without taking the real part $\backslash operatorname\{Re\}$, because the imaginary part leads to an antisymmetric contribution that disappears under the sum. Note that all eigenvalues $\backslash lambda\_k$ and eigenvectors $\backslash vert\; k\backslash rangle$ of the density matrix potentially depend on the vector of parameters $\backslash boldsymbol\{\backslash theta\}$.

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 $\backslash lambda\_k+\backslash lambda\_l$ suddenly becomes zero.) Through this relation, it also connects with quantum fidelity $F(\backslash varrho,\backslash sigma)=\backslash left(\backslash mathrm\{tr\}\backslash left[\backslash sqrt\{\backslash sqrt\{\backslash varrho\}\backslash sigma\backslash sqrt\{\backslash varrho\}\}\backslash right]\backslash right)^2$ of two infinitesimally close states,{{cite journal | last=Šafránek | first=Dominik | title=Discontinuities of the quantum Fisher information and the Bures metric | journal=Physical Review A | volume=95 | issue=5 | date=2017-05-11 | issn=2469-9926 | doi=10.1103/physreva.95.052320 | page=052320| arxiv=1612.04581 | bibcode=2017PhRvA..95e2320S | s2cid=118962619 }}

:$$

F(\varrho_{\boldsymbol{\theta}},\varrho_{\boldsymbol{\theta}+d\boldsymbol{\theta}})=1-\frac{1}{4}\sum_{i,j}\Big(F_{\rm Q}^{ij}[\varrho(\boldsymbol{\theta})]+2\!\!\sum_{\lambda_k(\boldsymbol{\theta})=0}\!\!\partial_i\partial_j\lambda_k\Big)d\theta_i d\theta_j+\mathcal{O}(d\theta^3),

where the inner sum goes over all $k$ at which eigenvalues $\backslash lambda\_k(\backslash boldsymbol\{\backslash theta\})=0$. The extra term (which is however zero in most applications) can be avoided by taking a symmetric expansion of fidelity,{{cite arXiv |last1=Zhou |first1=Sisi|last2=Jiang |first2=Liang |title=An exact correspondence between the quantum Fisher information and the Bures metric |date=18 Oct 2019 |eprint=1910.08473|class=quant-ph }}

:$$

F\left(\varrho_{\boldsymbol{\theta}-d\boldsymbol{\theta}/2},\varrho_{\boldsymbol{\theta}+d\boldsymbol{\theta}/2}\right)=1-\frac{1}{4}\sum_{i,j}F_{\rm Q}^{ij}[\varrho(\boldsymbol{\theta})]d\theta_i d\theta_j+\mathcal{O}(d\theta^3).

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.{{cite journal |last1=Jarzyna |first1=M. |title=Geometric Approach to Quantum Statistical Inference |last2=Kołodyński |first2=J. |journal=IEEE Journal on Selected Areas in Information Theory |date=18 August 2020 |volume=1 |issue=2 |pages=367–386 | issn=2641-8770 |doi=10.1109/JSAIT.2020.3017469| arxiv=2008.09129 |s2cid=221245983 }}

Assuming that $\backslash vert\; \backslash psi\_0(\backslash theta)\backslash rangle$ is a ground state of a parameter-dependent non-degenerate Hamiltonian $H(\backslash theta)$, four times the quantum Fisher information of this state is called fidelity susceptibility, and denoted{{cite journal |last1=Gu |first1=S.-J. |title=Fidelity approach to quantum phase transitions |journal=International Journal of Modern Physics B |date=2010 |volume=24 |number=23 |pages=4371–4458 |doi=10.1142/S0217979210056335| arxiv=0811.3127 |bibcode=2010IJMPB..24.4371G |s2cid=118375103 }}

:$$

\chi_F=4F_Q(\vert\psi_0(\theta)\rangle).

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 $\backslash vert\backslash psi\_0(\backslash theta)\backslash rangle$ and $\backslash vert\backslash psi\_0(\backslash theta+d\backslash theta)\backslash rangle$ are orthogonal to each other, for any infinitesimal change in parameter $d\backslash theta$, and thus are said to undergo a phase-transition at point $\backslash theta$.

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

:$F\_\{\backslash rm\; Q\}[\backslash vert\; \backslash Psi\; \backslash rangle,H]\; =\; 4\; (\backslash Delta\; H)^2\_\{\backslash Psi\}$.

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

:$F\_\{\backslash rm\; Q\}[p\; \backslash varrho\_1(\backslash theta)\; +\; (1-p)\; \backslash varrho\_2(\backslash theta)\; ,H]\; \backslash le\; p\; F\_\{\backslash rm\; Q\}[\backslash varrho\_1(\backslash theta),H]+(1-p)F\_\{\backslash rm\; Q\}[\backslash varrho\_2(\backslash theta),H].$

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{{cite journal |last1=Tóth |first1=Géza |last2=Petz |first2=Dénes |title=Extremal properties of the variance and the quantum Fisher information |journal=Physical Review A |date=20 March 2013 |volume=87 |issue=3 |pages=032324 |doi=10.1103/PhysRevA.87.032324|bibcode=2013PhRvA..87c2324T |arxiv=1109.2831 |s2cid=55088553 }}{{cite arXiv |last1=Yu |first1=Sixia |title=Quantum Fisher Information as the Convex Roof of Variance |date=2013 |eprint=1302.5311|class=quant-ph }}

:$F\_\{\backslash rm\; Q\}[\backslash varrho,H]\; =\; 4\; \backslash inf\_\{\backslash \{p\_k,\backslash vert\; \backslash Psi\_k\; \backslash rangle\; \backslash \}\}\; \backslash sum\_k\; p\_k\; (\backslash Delta\; H)^2\_\{\backslash Psi\_k\},$

where the infimum is over all decompositions of the density matrix

:$\backslash varrho=\backslash sum\_k\; p\_k\; \backslash vert\; \backslash Psi\_k\backslash rangle\; \backslash langle\; \backslash Psi\_k\; \backslash vert.$

Note that $\backslash vert\; \backslash Psi\_k\backslash rangle$ are not necessarily orthogonal to each other. The above optimization can be rewritten as an optimization over the two-copy space as {{cite journal |last1=Tóth |first1=Géza |last2=Moroder |first2=Tobias |last3=Gühne |first3=Otfried |title=Evaluating Convex Roof Entanglement Measures |journal=Physical Review Letters |date=21 April 2015 |volume=114 |issue=16 |pages=160501 |doi=10.1103/PhysRevLett.114.160501|pmid=25955038 |arxiv=1409.3806 |bibcode=2015PhRvL.114p0501T |s2cid=39578286 }}

:$$

F_Q[\varrho,H]=

\min_{\varrho_{12}} 2{\rm Tr}[(H\otimes {\rm Identity}-{\rm Identity}\otimes H)^2\varrho_{12}],

such that

$\backslash varrho\_\{12\}$ is a symmetric separable state and

:$$

{\rm Tr}_1(\varrho_{12})={\rm Tr}_2(\varrho_{12})=\varrho.

Later the above statement has been proved even for the case of a minimization over general (not necessarily symmetric) separable states.{{cite journal |last1=Tóth |first1=Géza |last2=Pitrik |first2=József |title=Quantum Wasserstein distance based on an optimization over separable states |journal=Quantum |date=16 October 2023 |volume=7 |pages=1143 |doi=10.22331/q-2023-10-16-1143|arxiv=2209.09925|bibcode=2023Quant...7.1143T |s2cid=252408568 }}

When the probabilities are $\backslash theta$-dependent, an extended-convexity relation has been proved:{{Cite journal|last1=Alipour|first1=S.|last2=Rezakhani|first2=A. T.|date=2015-04-07|title=Extended convexity of quantum Fisher information in quantum metrology|url=https://link.aps.org/doi/10.1103/PhysRevA.91.042104|journal=Physical Review A|language=en|volume=91|issue=4|pages=042104|arxiv=1403.8033|doi=10.1103/PhysRevA.91.042104|bibcode=2015PhRvA..91d2104A|s2cid=124094775 |issn=1050-2947|via=}}

:$F\_\{\backslash rm\; Q\}\backslash Big[\backslash sum\_i\; p\_i(\backslash theta)\; \backslash varrho\_i(\backslash theta)\backslash Big]\; \backslash le\; \backslash sum\_i\; p\_i(\backslash theta)\; F\_\{\backslash rm\; Q\}[\backslash varrho\_i(\backslash theta)]+F\_\{\backslash rm\; C\}[\backslash \{p\_i(\backslash theta)\backslash \}],$

where $F\_\{\backslash rm\; C\}[\backslash \{p\_i(\backslash theta)\backslash \}]=\backslash sum\_i\; \backslash frac\{\backslash partial\_\{\backslash theta\}\; p\_i(\backslash theta)^2\}\{p\_i(\backslash theta)\}$ 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.

We need to understand the behavior of quantum Fisher information in composite system in order to study quantum metrology of many-particle systems.{{cite journal |last1=Tóth |first1=Géza |last2=Apellaniz |first2=Iagoba |title=Quantum metrology from a quantum information science perspective |journal=Journal of Physics A: Mathematical and Theoretical |date=24 October 2014 |volume=47 |issue=42 |pages=424006 |doi=10.1088/1751-8113/47/42/424006|bibcode=2014JPhA...47P4006T |arxiv=1405.4878 |s2cid=119261375 }}

For product states,

:$F\_\{\backslash rm\; Q\}[\backslash varrho\_1\; \backslash otimes\; \backslash varrho\_2\; ,\; H\_1\backslash otimes\; \{\backslash rm\; Identity\}+\{\backslash rm\; Identity\}\; \backslash otimes\; H\_2]\; =$

F_{\rm Q}[\varrho_1,H_1]+F_{\rm Q}[\varrho_2,H_2]

holds.

For the reduced state, we have

:$F\_\{\backslash rm\; Q\}[\backslash varrho\_\{12\},\; H\_1\backslash otimes\; \{\backslash rm\; Identity\}\_2]\; \backslash ge$

F_{\rm Q}[\varrho_{1}, H_1],

where $\backslash varrho\_\{1\}=\{\backslash rm\; Tr\}\_2(\backslash varrho\_\{12\})$.

There are strong links between quantum metrology and quantum information science. For a multiparticle system of $N$ spin-1/2 particles {{cite journal |last1=Pezzé |first1=Luca |last2=Smerzi |first2=Augusto |title=Entanglement, Nonlinear Dynamics, and the Heisenberg Limit |journal=Physical Review Letters |date=10 March 2009 |volume=102 |issue=10 |pages=100401 |doi=10.1103/PhysRevLett.102.100401|pmid=19392092 |bibcode=2009PhRvL.102j0401P |arxiv=0711.4840 |s2cid=13095638 }}

:$F\_\{\backslash rm\; Q\}[\backslash varrho,\; J\_z]\; \backslash le\; N$

holds for separable states, where

:$J\_z=\backslash sum\_\{n=1\}^N\; j\_z^\{(n)\},$

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

:$F\_\{\backslash rm\; Q\}[\backslash varrho,\; J\_z]\; \backslash le\; N^2.$

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

:$F\_\{\backslash rm\; Q\}[\backslash varrho,\; J\_z]\; \backslash le\; sk^2\; +\; r^\{2\}$

holds, where $s=\backslash lfloor\; N/k\; \backslash rfloor$ 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.{{Cite journal|last=Hyllus|first=Philipp|date=2012|title=Fisher information and multiparticle entanglement|journal=Physical Review A|volume=85|issue=2|pages=022321|doi=10.1103/physreva.85.022321|arxiv=1006.4366|bibcode=2012PhRvA..85b2321H|s2cid=118652590}}{{Cite journal|last=Tóth|first=Géza|date=2012|title=Multipartite entanglement and high-precision metrology|journal=Physical Review A|volume=85|issue=2|pages=022322|doi=10.1103/physreva.85.022322|arxiv=1006.4368|bibcode=2012PhRvA..85b2322T|s2cid=119110009}} It is possible to obtain a weaker but simpler bound {{cite book |last1=Tóth |first1=Géza |title=Entanglement detection and quantum metrology in quantum optical systems |date=2021 |publisher=Doctoral Dissertation submitted to the Hungarian Academy of Sciences |location=Budapest |page=68 |url=http://real-d.mtak.hu/1230/7/dc_1593_18_doktori_mu.pdf}}

:$F\_\{\backslash rm\; Q\}[\backslash varrho,\; J\_z]\; \backslash le\; Nk.$

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

:$\backslash frac\{F\_\{\backslash rm\; Q\}[\backslash varrho,\; J\_z]\}\{N\}\; \backslash le\; k.$

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

$g\_\{\backslash mathcal\; H\}(\backslash varrho)=\backslash frac\{\backslash mathcal\; F\_Q[\backslash varrho,\{\backslash mathcal\; H\}]\}\{\backslash mathcal\; F\_Q^\{(\{\backslash rm\; sep\})\}(\backslash mathcal\; H)\},$

where the Hamiltonian is

$\backslash mathcal\; H=h\_1+h\_2+...+h\_N,$

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

g(\varrho)=\max_{\mathcal H}g_{\mathcal H}(\varrho).

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

:$$

F_{\rm Q}[\varrho,H]\ge \frac{\langle i[H,M] \rangle_{\varrho}^2}{(\Delta M)^2}

,

where $M$ is an operator.

This formula can be used to put a lower on the quantum Fisher information from experimental results.{{cite journal |last1=Lücke |first1=B. |last2=Scherer |first2=M. |last3=Kruse |first3=J. |last4=Pezzé |first4=L. |last5=Deuretzbacher |first5=F. |last6=Hyllus |first6=P. |last7=Topic |first7=O. |last8=Peise |first8=J. |last9=Ertmer |first9=W. |last10=Arlt |first10=J. |last11=Santos |first11=L. |last12=Smerzi |first12=A. |last13=Klempt |first13=C. |title=Twin Matter Waves for Interferometry Beyond the Classical Limit |journal=Science |date=11 November 2011 |volume=334 |issue=6057 |pages=773–776 |doi=10.1126/science.1208798|pmid=21998255 |arxiv=1204.4102 |bibcode=2011Sci...334..773L }}

If $M$ equals the symmetric logarithmic derivative then the inequality is saturated.{{cite arXiv |last1=Escher |first1=B. M. |title=Quantum Noise-to-Sensibility Ratio |date=2012 |class=quant-ph |eprint=1212.2533}}

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 optimizaton on the two-copy space.{{cite journal |last1=Tóth |first1=Géza |last2=Moroder |first2=Tobias |last3=Gühne |first3=Otfried |title=Evaluating Convex Roof Entanglement Measures |journal=Physical Review Letters |date=21 April 2015 |volume=114 |issue=16 |page=160501 |doi=10.1103/PhysRevLett.114.160501|pmid=25955038 |arxiv=1409.3806 |bibcode=2015PhRvL.114p0501T }}

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.{{cite journal |last1=Apellaniz |first1=Iagoba |last2=Kleinmann |first2=Matthias |last3=Gühne |first3=Otfried |last4=Tóth |first4=Géza |title=Optimal witnessing of the quantum Fisher information with few measurements |journal=Physical Review A |date=28 March 2017 |volume=95 |issue=3 |page=032330 |doi=10.1103/PhysRevA.95.032330|arxiv=1511.05203 |bibcode=2017PhRvA..95c2330A }} In some cases, the bounds can even be obtained analytically. For instance, for an $N$-qubit Greenberger-Horne-Zeilinger (GHZ) state

:$$

\frac{F_{\rm Q}[\varrho,J_z]}{N^2}\ge (1-2F_{\rm GHZ})^2,

where for the fidelity with respect to the GHZ state

:$$

F_{\rm GHZ}={\rm Tr}(\varrho|{\rm GHZ}\rangle\langle{\rm GHZ}|)\ge1/2

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.{{cite journal |last1=Müller-Rigat |first1=Guillem |last2=Srivastava |first2=Anubhav Kumar |last3=Kurdziałek |first3=Stanisław |last4=Rajchel-Mieldzioć |first4=Grzegorz |last5=Lewenstein |first5=Maciej |last6=Frérot |first6=Irénée |title=Certifying the quantum Fisher information from a given set of mean values: a semidefinite programming approach |journal=Quantum |date=24 October 2023 |volume=7 |pages=1152 |doi=10.22331/q-2023-10-24-1152|arxiv=2306.12711 |bibcode=2023Quant...7.1152M }}

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.{{cite journal |last1=Hauke |first1=Philipp |last2=Heyl |first2=Markus |last3=Tagliacozzo |first3=Luca |last4=Zoller |first4=Peter |title=Measuring multipartite entanglement through dynamic susceptibilities |journal=Nature Physics |date=August 2016 |volume=12 |issue=8 |pages=778–782 |doi=10.1038/nphys3700|arxiv=1509.01739 |bibcode=2016NatPh..12..778H }}

The Wigner–Yanase skew information is defined as {{cite journal |last1=Wigner |first1=E. P. |title=Information Contents of Distributions |last2=Yanase |first2=M. M. |journal=Proceedings of the National Academy of Sciences |date=1 June 1963 |volume=49 |issue=6 |pages=910–918 |pmc=300031 |doi=10.1073/pnas.49.6.910|pmid=16591109 |bibcode=1963PNAS...49..910W |doi-access=free }}

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

It follows that $I(\backslash varrho,H)$ is convex in $\backslash varrho.$

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

:$F\_\{\backslash rm\; Q\}[\backslash varrho,H]\; \backslash ge\; 4\; I(\backslash varrho,H)$

holds, where there is an equality for pure states.

For any decomposition of the density matrix given by $p\_k$ and $\backslash vert\; \backslash Psi\_k\backslash rangle$ the relation

:$(\backslash Delta\; H)^2\; \backslash ge\; \backslash sum\_k\; p\_k\; (\backslash Delta\; H)^2\_\{\backslash Psi\_k\}\; \backslash ge\; \backslash frac1\; 4\; F\_\{\backslash rm\; Q\}[\backslash varrho,H]$

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

:$(\backslash Delta\; H)^2\; =\; \backslash sup\_\{\backslash \{p\_k,\backslash vert\; \backslash Psi\_k\; \backslash rangle\; \backslash \}\}\; \backslash sum\_k\; p\_k\; (\backslash Delta\; H)^2\_\{\backslash Psi\_k\}.$

Knowing that the quantum Fisher information is the convex roof of the variance times four, we obtain the relation {{cite journal |last1=Fröwis |first1=Florian |last2=Schmied |first2=Roman |last3=Gisin |first3=Nicolas |title=Tighter quantum uncertainty relations following from a general probabilistic bound |journal=Physical Review A |date=2 July 2015 |volume=92 |issue=1 |pages=012102 |doi=10.1103/PhysRevA.92.012102|arxiv=1409.4440 |bibcode=2015PhRvA..92a2102F |s2cid=58912643 }}

(\Delta A)^2 F_Q[\varrho,B] \geq \vert \langle i[A,B]\rangle\vert^2,

which is stronger than the Heisenberg uncertainty relation. For a particle of spin-$j,$ the following uncertainty relation holds

(\Delta J_x)^2+(\Delta J_y)^2+(\Delta J_z)^2\ge j,

where $J\_l$ are angular momentum components. The relation can be strengthened as {{cite journal |last1=Tóth |first1=Géza |last2=Fröwis |first2=Florian |title=Uncertainty relations with the variance and the quantum Fisher information based on convex decompositions of density matrices |journal=Physical Review Research |date=31 January 2022 |volume=4 |issue=1 |pages=013075 |doi=10.1103/PhysRevResearch.4.013075|arxiv=2109.06893 |bibcode=2022PhRvR...4a3075T |s2cid=237513549 }}{{cite journal |last1=Chiew |first1=Shao-Hen |last2=Gessner |first2=Manuel |title=Improving sum uncertainty relations with the quantum Fisher information |journal=Physical Review Research |date=31 January 2022 |volume=4 |issue=1 |pages=013076 |doi=10.1103/PhysRevResearch.4.013076|arxiv=2109.06900 |bibcode=2022PhRvR...4a3076C |s2cid=237513883 }}

(\Delta J_x)^2+(\Delta J_y)^2+F_Q[\varrho,J_z]/4\ge j.

{{Reflist}}

Category:Quantum information science

Category:Quantum optics