Bures, Kubo–Mori, and Monotone Metrics

Quantum state space admits many contractive Riemannian metrics. Bures geometry is tied to fidelity and the symmetric logarithmic derivative; Bogoliubov–Kubo–Mori geometry is the Hessian of relative entropy and the log partition function. They agree on commuting probability distributions but weight coherent, noncommuting directions differently.

Required background. Quantum Fisher information supplies the SLD normalization and support conditions.

Helpful background. Fidelity and overlap supplies the mixed-state comparison.

Bures distance and SLD metric

Use the root fidelity

f(\rho,\sigma) =\operatorname{Tr} \sqrt{\sqrt\rho\,\sigma\sqrt\rho}

and define the Bures distance by

D_B^2(\rho,\sigma)=2[1-f(\rho,\sigma)].

For $\sigma=\rho+d\rho$ on a smooth faithful family,

D_B^2(\rho,\rho+d\rho) =\frac14 F_Q(d\rho,d\rho)+O(d\rho^3),

where $F_Q$ is the SLD Fisher metric in the convention of the previous page. Some authors call $f^2$ the fidelity; using that convention changes expansion coefficients. Always state whether fidelity is rooted or squared.

In the eigenbasis of $\rho$ ,

ds_B^2 =\frac12\sum_{m,n} \frac{\lvert d\rho_{mn}\rvert^2}{p_m+p_n}.

For a pure-state ray this becomes the Fubini–Study line element.

The structural map places Bures, Kubo–Mori, and Monotone Metrics along the perturbative sequence from normalized state or shape variations to information metrics and modular transport.

A normalized family yields the fixed-reference first law and a positive quadratic response, which branches into state susceptibility, shape kernels, metric choices, and modular holonomy. — At fixed comparison algebra, normalization removes the linear term in relative entropy. The second response supports several inequivalent constructions: state and shape tangents, stress-tensor kernels, monotone metrics, and zero-mode-projected transport. The diagram is schematic and not to scale.

Kubo–Mori metric

The Bogoliubov–Kubo–Mori metric is

g_{\rm KM}(X,Y) =\operatorname{Tr}\!\left[ X\int_0^\infty d\beta\, (\rho+\beta)^{-1}Y(\rho+\beta)^{-1} \right].

Equivalently,

g_{\rm KM}(X,Y) =\sum_{m,n} \frac{\log p_m-\log p_n}{p_m-p_n} X_{nm}Y_{mn}.

It is the Hessian of $S(\rho+\lambda X\Vert\rho)$ and, in exponential coordinates, the Hessian of the log partition function. The corresponding mean is the logarithmic mean

L(x,y)=\frac{x-y}{\log x-\log y}.

By contrast, the SLD kernel uses the arithmetic mean $(x+y)/2$ . Since $L(x,y)\le(x+y)/2$ , the Kubo–Mori quadratic form is at least as large as the SLD Fisher form with matched Petz normalization.

Petz classification of monotone metrics

For faithful finite-dimensional states, Petz 1996, pp. 87–90 shows that every suitably normalized monotone Riemannian metric is associated with a symmetric operator-monotone function $f$ satisfying

f(t)=t f(t^{-1}), \qquad f(1)=1.

Its Morozova–Chentsov kernel is

c_f(x,y)=\frac{1}{y\,f(x/y)},

and the metric has the component form

g_f(X,Y)= \sum_{m,n}c_f(p_m,p_n)X_{nm}Y_{mn}.

Two important choices are

f_{\rm SLD}(t)=\frac{1+t}{2}, \qquad f_{\rm KM}(t)=\frac{t-1}{\log t}.

Contractivity means

g_f\bigl(\Phi(X),\Phi(X)\bigr)_{\Phi(\rho)} \le g_f(X,X)_\rho

for completely positive trace-preserving maps $\Phi$ . Monotonicity narrows the possibilities but does not select one unique metric.

Commuting and pure-state limits

If $[\rho,X]=0$ , all normalized monotone metrics reduce to the classical Fisher metric

g(X,X)=\sum_n\frac{X_{nn}^2}{p_n}.

This explains why metric conventions are easy to miss in classical or diagonal examples. A noncommuting two-level tangent is the simplest diagnostic: compare the denominators $p_m+p_n$ and the logarithmic mean.

At a pure-state boundary, the faithful-state Kubo–Mori metric generally diverges along directions that change support, while the Bures metric has a finite Fubini–Study limit for smooth pure-state rays. This is not a contradiction; the metrics probe different divergences and have different boundary completions.

QFT and regulator dependence

For QFT state families, the density-matrix formulas are regulator models. A continuum definition should specify a local algebra, a family of normal states, and either:

a relative-entropy Hessian for Kubo–Mori geometry;
a fidelity notion available for the chosen algebraic representation;
or a regulated metric with a demonstrated limit under cutoff refinement.

Monotonicity under restriction is robust, but finiteness is not. Boundary area terms, contact terms, and sharp-source divergences can differ among metrics. Equality of universal terms must be shown rather than inferred from the common commuting limit.

Measurement language also differs. SLD/Bures geometry is connected to optimal local estimation. Kubo–Mori geometry is naturally connected to thermodynamic susceptibility and imaginary-time correlations. Neither interpretation makes the other metric incorrect.

Common pitfalls

Using “fidelity” without stating rooted or squared convention. The infinitesimal coefficient changes by a factor. Define it before comparing papers.

Assuming monotonicity makes all metrics equal. Petz monotonicity permits a family of operator-monotone kernels. Noncommuting tangents distinguish them.

Taking a faithful formula directly to a pure boundary. Support-changing directions can diverge in Kubo–Mori geometry even when Bures distance remains finite.

Before interpreting this response coefficient, use the validity map to check normalization, tangent domains, contact and regulator terms, metric choice, and analyticity independently.

A reported susceptibility must pass separate checks for normalized families, common tangent domains, contact and regulator terms, and an explicit metric and analytic strip before receiving a physical interpretation. — Normalization, domain, contact-term, metric, and analyticity checks are independent. A failed gate narrows or withdraws the physical claim; it is not a change of notation. The decision map is schematic and not to scale.

References

Petz, Dénes. “Monotone Metrics on Matrix Spaces.” Linear Algebra and its Applications 244 (1996): 81–96. DOI.