Quantum Fisher Information in QFT

Quantum Fisher information (QFI) measures the local distinguishability of a normalized state family. This page uses the symmetric logarithmic derivative (SLD): it is operationally tied to optimal measurements, but it is not the relative-entropy Hessian. In QFT the distinction becomes practical because a tangent can have finite SLD QFI while failing the stronger modular-energy domain needed by the Bogoliubov–Kubo–Mori response.

Required background. State perturbations and relative-entropy susceptibility provide normalized tangents and the Hessian convention used for relative entropy.

Helpful background. Operational distinguishability bounds provide the measurement and data-processing interpretation.

The chapter’s differentiable-family starting point fixes the tangent and normalization condition. Its metric comparison then fixes the normalization dictionary, while the independent validity gates separate support, domain, and regulator questions.

The SLD Fisher metric

Let $\rho_\lambda$ be a differentiable family of density operators and set $X=\partial_\lambda\rho_\lambda$, so $\operatorname{Tr}X=0$. The SLD $L$ is defined by

$$X=\frac12(\rho L+L\rho), \qquad F_{\rm SLD}(X,X)=\operatorname{Tr}(\rho L^2).$$

For a faithful state $\rho=\sum_n p_n\lvert n\rangle\langle n\rvert$,

$$L_{mn}=\frac{2X_{mn}}{p_m+p_n}, \qquad F_{\rm SLD}(X,X) =\sum_{m,n}\frac{2\lvert X_{mn}\rvert^2}{p_m+p_n}.$$

These formulas use the Petz normalization: a commuting tangent gives the ordinary classical Fisher metric $\sum_n X_{nn}^2/p_n$. At non-full rank, retain only terms with $p_m+p_n>0$ for a fixed-support tangent and analyze any support-changing limit separately. A tangent into a vanishing eigenspace can make the metric divergent or discontinuous.

For a normalized pure vector,

$$F_{\rm SLD} =4\left( \langle\partial_\lambda\psi\mid\partial_\lambda\psi\rangle -\lvert\langle\psi\mid\partial_\lambda\psi\rangle\rvert^2 \right).$$

The subtraction removes the unobservable phase tangent. For a unitary family $\rho_\lambda=e^{-i\lambda G}\rho e^{i\lambda G}$, the spectral formula becomes

$$F_{\rm SLD} =\sum_{m,n} \frac{2(p_m-p_n)^2}{p_m+p_n}\lvert G_{mn}\rvert^2,$$

and reduces to $4\operatorname{Var}_\psi(G)$ on a pure state.

What the measurement statement does and does not say

A POVM $\{M_x\}$ gives $p(x\mid\lambda)=\operatorname{Tr}(\rho_\lambda M_x)$ and

$$F_{\{M_x\}} =\sum_x p(x\mid\lambda) \left[\partial_\lambda\log p(x\mid\lambda)\right]^2.$$

For unrestricted measurements in the regular finite-dimensional problem,

$$F_{\{M_x\}}\leq F_{\rm SLD},$$

and an SLD-adapted measurement attains equality locally Braunstein and Caves 1994, Eqs. (2)–(4) and (14)–(16), pp. 3440–3442. “Locally” matters: the measurement may depend on the unknown parameter, and a QFT experiment may allow only a subalgebra or an energy-bounded class of measurements. In that case the accessible Fisher information can be strictly smaller.

For $N$ independent repetitions and a locally unbiased estimator,

$$\operatorname{Var}(\widehat\lambda)\geq\frac{1}{N F_{\rm SLD}}.$$

Finite samples, bias, nuisance parameters, correlated repetitions, and restricted measurements require their corresponding versions of the bound.

SLD versus relative-entropy Fisher information

For a faithful finite state, define the Bogoliubov–Kubo–Mori (BKM) quadratic form by

$$g_{\rm BKM}(X,X) =\sum_{m,n} \frac{\log p_m-\log p_n}{p_m-p_n} \lvert X_{mn}\rvert^2,$$

where the diagonal quotient is $1/p_n$. It is normalized so that

$$S(\rho+\lambda X\Vert\rho) =\frac{\lambda^2}{2}g_{\rm BKM}(X,X)+O(\lambda^3).$$

The relative-entropy Hessian construction and its monotonicity are developed in Lesniewski and Ruskai 1999, §2.3, Theorems 2.8–2.10. The logarithmic-mean inequality gives, term by term,

$$\frac{2}{p_m+p_n} \leq \frac{\log p_m-\log p_n}{p_m-p_n},$$

so

$$F_{\rm SLD}(X,X)\leq g_{\rm BKM}(X,X).$$

Equality holds for commuting tangents, not for a generic coherent direction. The geometric Bures line element is instead $g_{\rm Bures}=F_{\rm SLD}/4$; forgetting this factor of four is a common source of apparently inconsistent formulas.

QFT application: a Gaussian source amplitude

Consider a finite box and ultraviolet cutoff, with independent real bosonic modes

$$[q_k,p_\ell]=i\delta_{k\ell}, \qquad H_\Lambda=\sum_{k\in\mathcal K_\Lambda} \frac{\omega_k}{2}(q_k^2+p_k^2), \qquad \rho_\beta=Z^{-1}e^{-\beta H_\Lambda}.$$

Here $q_k,p_k$ are dimensionless oscillator quadratures; the real-mode set $\mathcal K_\Lambda$ includes each physical oscillator once. Estimate the amplitude $\lambda$ of the smeared displacement

$$\rho_\lambda=U_\lambda\rho_\beta U_\lambda^\dagger, \qquad U_\lambda=\exp\!\left(-i\lambda\sum_k g_kp_k\right).$$

It shifts $\langle q_k\rangle$ by $\lambda g_k$. A thermal mode has $r_k=e^{-\beta\omega_k}$ and $p_{n,k}=(1-r_k)r_k^n$. Since $p_k$ connects only $n$ to $n+1$, inserting $\lvert\langle n+1\lvert p_k\rvert n\rangle\rvert^2=(n+1)/2$ into the unitary spectral formula yields

$$F_{\rm SLD}^{(\Lambda)} =2\sum_{k\in\mathcal K_\Lambda} g_k^2\tanh\!\left(\frac{\beta\omega_k}{2}\right).$$

The same answer follows from the Gaussian first-moment term in Monras 2013, Eqs. (13) and (16). Because displacement leaves the entropy unchanged,

$$S(\rho_\lambda\Vert\rho_0) =\beta\bigl(\langle H_\Lambda\rangle_\lambda- \langle H_\Lambda\rangle_0\bigr) =\frac{\lambda^2}{2} \underbrace{\beta\sum_k\omega_kg_k^2}_{g_{\rm BKM}^{(\Lambda)}}.$$

Thus the two Petz-normalized metrics are

$$F_{\rm SLD}^{(\Lambda)} =2\sum_k g_k^2\tanh(\beta\omega_k/2), \qquad g_{\rm BKM}^{(\Lambda)} =\beta\sum_k\omega_kg_k^2.$$

The inequality $2\tanh(x/2)\leq x$ checks the general ordering mode by mode. The benchmark is exact at finite volume and cutoff: there is no statistical or truncation error beyond the declared mode regulator. Its controls are $L$, $\Lambda$, $\beta$, the dispersion $\omega_k$, the real-mode counting convention, and the source coefficients $g_k$.

Adversarial test: an energetic-domain failure

Let the high-frequency dispersion be relativistic, $\omega_k\sim\lvert k\rvert$, in $d_s$ spatial dimensions, and let $g(k)\sim C\lvert k\rvert^{-\alpha}$. At fixed positive temperature, the ultraviolet behaviors are

$$F_{\rm SLD}^{(\Lambda)} \sim\int^{\Lambda}dk\;k^{d_s-1-2\alpha}, \qquad g_{\rm BKM}^{(\Lambda)} \sim\beta\int^{\Lambda}dk\;k^{d_s-2\alpha}.$$

Therefore

$$F_{\rm SLD}<\infty\iff 2\alpha>d_s, \qquad g_{\rm BKM}<\infty\iff 2\alpha>d_s+1.$$

Equality at either threshold gives a logarithmic divergence. In the window

$$\frac{d_s}{2}<\alpha\leq\frac{d_s+1}{2},$$

the SLD limit exists but the BKM metric and injected energy

$$\Delta E_\Lambda =\frac{\lambda^2}{2}\sum_k\omega_kg_k^2$$

do not. This is the promised failure injection: finite statistical distinguishability alone does not put the source in the Hamiltonian energy domain. For $\alpha\leq d_s/2$, even the SLD quantity is only a cutoff-dependent family. A continuum claim must specify smearing strong enough for the intended metric, not merely report a finite answer at one cutoff.

For a continuum local algebra, a bounded SLD operator need not exist even when a regulated quadratic form converges. Restricting the state to a subalgebra cannot increase either monotone metric, but it also does not prove ultraviolet finiteness. The strongest justified statement here is the explicit finite-regulator result plus a separately proved limit under the stated source-domain condition.

Common pitfalls

Calling one measurement’s Fisher information “the QFI.” The SLD value is an optimization over the allowed measurement class. State that class and whether the local optimum is implementable.

Equating SLD and BKM because a diagonal example agrees. Their Petz normalizations agree on commuting directions; their noncommuting kernels do not. Geometric Bures normalization adds a further factor of $1/4$.

Removing the cutoff without testing the tangent. A square-summable displacement can have finite SLD QFI but infinite energy and BKM response. Report the cutoff law and the common operator domain.

Exercises

1. Let $\rho_\lambda=\operatorname{diag}(p(\lambda),1-p(\lambda))$. Compute the SLD and show that the SLD and BKM metrics coincide.

Solution

The tangent is $X=\operatorname{diag}(\dot p,-\dot p)$ and commutes with $\rho$. Hence $L=\operatorname{diag}(\dot p/p,-\dot p/(1-p))$, so

$$F_{\rm SLD} =\frac{(\dot p)^2}{p}+\frac{(\dot p)^2}{1-p} =\frac{(\dot p)^2}{p(1-p)}.$$

For diagonal entries the BKM divided difference equals $1/p_n$, giving the same result. The calculation also shows why equality in a classical example says nothing about a noncommuting direction.

2. Derive the single-mode displacement result $F_{\rm SLD}=2g^2\tanh(x/2)$ and $g_{\rm BKM}=xg^2$, where $x=\beta\omega$.

Solution

Write $p_n=(1-r)r^n$ with $r=e^{-x}$ and use $\lvert\langle n+1\lvert p\rvert n\rangle\rvert^2=(n+1)/2$. Combining the two matrix orientations gives

$$F_{\rm SLD} =4g^2\sum_{n\geq0} \frac{(p_n-p_{n+1})^2}{p_n+p_{n+1}} \frac{n+1}{2}.$$

Since $(p_n-p_{n+1})^2/(p_n+p_{n+1})=p_n(1-r)^2/(1+r)$ and $\sum_n p_n(n+1)=1/(1-r)$,

$$F_{\rm SLD}=2g^2\frac{1-r}{1+r}=2g^2\tanh(x/2).$$

The displacement raises the oscillator energy by $\omega\lambda^2g^2/2$. Thermal relative entropy is $\beta$ times that increase, so comparison with $S=\lambda^2g_{\rm BKM}/2$ gives $g_{\rm BKM}=xg^2$.

3. In $d_s=3$, take $g(k)\sim k^{-\alpha}$. Classify the SLD and BKM ultraviolet behavior for $\alpha=3/2$, $7/4$, and $2$.

Solution

The SLD threshold is $2\alpha>3$ and the BKM threshold is $2\alpha>4$. At $\alpha=3/2$, SLD diverges logarithmically and BKM diverges as a positive power. At $\alpha=7/4$, SLD converges but BKM still has a power divergence. At $\alpha=2$, SLD converges while BKM is exactly at its logarithmic threshold. Thus none of the three choices gives a finite BKM continuum metric; one needs $\alpha>2$.

References

• Braunstein, Samuel L., and Carlton M. Caves. “Statistical Distance and the Geometry of Quantum States.” Physical Review Letters 72 (1994): 3439–3443. DOI.
• Lesniewski, Andrew, and Mary Beth Ruskai. “Monotone Riemannian Metrics and Relative Entropy on Non-Commutative Probability Spaces.” Journal of Mathematical Physics 40 (1999): 5702–5724. DOI; arXiv.
• Monras, Alex. “Phase Space Formalism for Quantum Estimation of Gaussian States.” arXiv:1303.3682 (2013). arXiv.