Bogoliubov–Kubo–Mori Manifold

• Bogoliubov–Kubo–Mori geometric manifold is the structured space of quantum states endowed with a metric derived from the second-order expansion of quantum relative entropy.
• It unifies representations across finite-dimensional faithful states, SPD matrices, Gaussian states, and operator algebras using logarithmic coordinates and spectral formulations.
• The manifold offers insights into dual flatness, curvature properties, and practical applications such as quantum reservoir computing and quantum Fisher information.

The Bogoliubov–Kubo–Mori geometric manifold is the manifold of quantum states, or of a structured submanifold of quantum states, equipped with the Bogoliubov–Kubo–Mori metric induced by the second-order expansion of quantum relative entropy. In finite dimensions this geometry appears on the manifold of faithful states on $B(\mathcal H)$; in matrix analysis it appears on $\mathrm{SPD}_n$; in continuous-variable theory it appears on the manifold of faithful, zero-displacement Gaussian states parameterised by covariance matrices; and in operator-algebraic quantum field theory it appears as a local Hessian geometry at modular self-dual points (Naudts, 2018, Thanwerdas et al., 2019, Miller, 2024, Chatterjee, 18 May 2026).

1. Relative-entropy origin of the BKM metric

The foundational definition is entropic. For density operators, the relevant two-point function is the quantum relative entropy

$$S(\hat{\rho}\Vert \hat{\rho}')=\operatorname{tr}\!\left[\hat{\rho}\big(\ln \hat{\rho}-\ln \hat{\rho}'\big)\right].$$

The Kubo–Mori–Bogoliubov inner product, also called the Bogoliubov–Kubo–Mori metric, is defined as the mixed second derivative of relative entropy,

$$g_{\hat{\rho}}(\hat{A},\hat{B}) := -\frac{\partial^2}{\partial s\,\partial t}\bigg|_{s=t=0} S\!\big(\hat{\rho}+t\hat{A}\,\Vert\,\hat{\rho}+s\hat{B}\big),$$

and the same geometry is described in one source by the statement that $ds^2=-d^2S(\hat\rho)$ (Miller, 2024).

In finite dimensions, the same bilinear form is written through the Kubo–Mori map

$$\mathcal K_{\rho}(X)=\int_0^1 ds\, \rho^{\,s}X\rho^{\,1-s},$$

with inverse

$$\mathcal K_{\rho}^{-1}(X)= \int_0^\infty du\,(\rho+uI)^{-1}X(\rho+uI)^{-1},$$

and the BKM bilinear form

$$\gamma_{\rho}^{\mathrm{BKM}}(X,Y) = \operatorname{Tr}\!\left[ X\,\mathcal K_{\rho}^{-1}(Y) \right].$$

In spectral form,

$$\gamma_{\rho}^{\mathrm{BKM}}(X,Y) = \sum_{i,j} c_{\mathrm{BKM}}(p_i,p_j)\, X_{ij}Y_{ji},$$

with

$$c_{\mathrm{BKM}}(x,y) = \frac{\log x-\log y}{x-y}, \qquad c_{\mathrm{BKM}}(x,x)=\frac1x.$$

The same kernel appears in the monotone-metric framework through the operator-monotone function

$\mathrm{SPD}_n$0

and the Morozova–Chentsov function

$\mathrm{SPD}_n$1

These formulas place the BKM manifold inside the standard family of monotone quantum metrics (Ciaglia, 2020).

2. Faithful-state manifolds and logarithmic coordinates

A central realization of the BKM manifold is the manifold $\mathrm{SPD}_n$2 of faithful quantum states on $\mathrm{SPD}_n$3, with $\mathrm{SPD}_n$4 finite-dimensional. Every faithful state is represented by a unique strictly positive density matrix $\mathrm{SPD}_n$5 through

$\mathrm{SPD}_n$6

and the paper on quantum statistical manifolds constructs $\mathrm{SPD}_n$7 as a differentiable Banach manifold. At a base point $\mathrm{SPD}_n$8, tangent vectors are Hermitian linear functionals

$\mathrm{SPD}_n$9

with $$S(\hat{\rho}\Vert \hat{\rho}')=\operatorname{tr}\!\left[\hat{\rho}\big(\ln \hat{\rho}-\ln \hat{\rho}'\big)\right].$$0 in the real Banach space $$S(\hat{\rho}\Vert \hat{\rho}')=\operatorname{tr}\!\left[\hat{\rho}\big(\ln \hat{\rho}-\ln \hat{\rho}'\big)\right].$$1 of centered self-adjoint elements in the commutant $$S(\hat{\rho}\Vert \hat{\rho}')=\operatorname{tr}\!\left[\hat{\rho}\big(\ln \hat{\rho}-\ln \hat{\rho}'\big)\right].$$2. The first atlas $$S(\hat{\rho}\Vert \hat{\rho}')=\operatorname{tr}\!\left[\hat{\rho}\big(\ln \hat{\rho}-\ln \hat{\rho}'\big)\right].$$3 gives the manifold structure, while the second atlas $$S(\hat{\rho}\Vert \hat{\rho}')=\operatorname{tr}\!\left[\hat{\rho}\big(\ln \hat{\rho}-\ln \hat{\rho}'\big)\right].$$4 is adapted to the Bogoliubov inner product and the exponential connection (Naudts, 2018).

The BKM-compatible coordinates are built from the centered logarithmic perturbation

$$S(\hat{\rho}\Vert \hat{\rho}')=\operatorname{tr}\!\left[\hat{\rho}\big(\ln \hat{\rho}-\ln \hat{\rho}'\big)\right].$$5

through

$$S(\hat{\rho}\Vert \hat{\rho}')=\operatorname{tr}\!\left[\hat{\rho}\big(\ln \hat{\rho}-\ln \hat{\rho}'\big)\right].$$6

On tangent spaces one obtains the inner product

$$S(\hat{\rho}\Vert \hat{\rho}')=\operatorname{tr}\!\left[\hat{\rho}\big(\ln \hat{\rho}-\ln \hat{\rho}'\big)\right].$$7

and this coincides with the Hessian metric derived from relative entropy. In the same framework, exponential geodesics satisfy

$$S(\hat{\rho}\Vert \hat{\rho}')=\operatorname{tr}\!\left[\hat{\rho}\big(\ln \hat{\rho}-\ln \hat{\rho}'\big)\right].$$8

and the $$S(\hat{\rho}\Vert \hat{\rho}')=\operatorname{tr}\!\left[\hat{\rho}\big(\ln \hat{\rho}-\ln \hat{\rho}'\big)\right].$$9-coordinates are affine along them: $$g_{\hat{\rho}}(\hat{A},\hat{B}) := -\frac{\partial^2}{\partial s\,\partial t}\bigg|_{s=t=0} S\!\big(\hat{\rho}+t\hat{A}\,\Vert\,\hat{\rho}+s\hat{B}\big),$$0 This gives affine coordinates for the exponential connection (Naudts, 2018).

A complementary realization uses logarithmic coordinates directly on the manifold of faithful states. On

$$g_{\hat{\rho}}(\hat{A},\hat{B}) := -\frac{\partial^2}{\partial s\,\partial t}\bigg|_{s=t=0} S\!\big(\hat{\rho}+t\hat{A}\,\Vert\,\hat{\rho}+s\hat{B}\big),$$1

the logarithmic chart is

$$g_{\hat{\rho}}(\hat{A},\hat{B}) := -\frac{\partial^2}{\partial s\,\partial t}\bigg|_{s=t=0} S\!\big(\hat{\rho}+t\hat{A}\,\Vert\,\hat{\rho}+s\hat{B}\big),$$2

In this chart the BKM geometry is tied to an action of

$$g_{\hat{\rho}}(\hat{A},\hat{B}) := -\frac{\partial^2}{\partial s\,\partial t}\bigg|_{s=t=0} S\!\big(\hat{\rho}+t\hat{A}\,\Vert\,\hat{\rho}+s\hat{B}\big),$$3

on faithful states,

$$g_{\hat{\rho}}(\hat{A},\hat{B}) := -\frac{\partial^2}{\partial s\,\partial t}\bigg|_{s=t=0} S\!\big(\hat{\rho}+t\hat{A}\,\Vert\,\hat{\rho}+s\hat{B}\big),$$4

and the BKM gradient vector field of the expectation $$g_{\hat{\rho}}(\hat{A},\hat{B}) := -\frac{\partial^2}{\partial s\,\partial t}\bigg|_{s=t=0} S\!\big(\hat{\rho}+t\hat{A}\,\Vert\,\hat{\rho}+s\hat{B}\big),$$5 is

$$g_{\hat{\rho}}(\hat{A},\hat{B}) := -\frac{\partial^2}{\partial s\,\partial t}\bigg|_{s=t=0} S\!\big(\hat{\rho}+t\hat{A}\,\Vert\,\hat{\rho}+s\hat{B}\big),$$6

This realizes the BKM manifold as a faithful-state manifold with a logarithmic homogeneous geometry generated by the cotangent extension of the unitary group (Ciaglia, 2020).

3. The BKM manifold on $$g_{\hat{\rho}}(\hat{A},\hat{B}) := -\frac{\partial^2}{\partial s\,\partial t}\bigg|_{s=t=0} S\!\big(\hat{\rho}+t\hat{A}\,\Vert\,\hat{\rho}+s\hat{B}\big),$$7

A second major realization is the manifold

$$g_{\hat{\rho}}(\hat{A},\hat{B}) := -\frac{\partial^2}{\partial s\,\partial t}\bigg|_{s=t=0} S\!\big(\hat{\rho}+t\hat{A}\,\Vert\,\hat{\rho}+s\hat{B}\big),$$8

with tangent spaces canonically identified as

$$g_{\hat{\rho}}(\hat{A},\hat{B}) := -\frac{\partial^2}{\partial s\,\partial t}\bigg|_{s=t=0} S\!\big(\hat{\rho}+t\hat{A}\,\Vert\,\hat{\rho}+s\hat{B}\big),$$9

In this setting the Bogoliubov–Kubo–Mori metric is given in integral form by

$ds^2=-d^2S(\hat\rho)$0

and equivalently by the differential-logarithm form

$ds^2=-d^2S(\hat\rho)$1

The ambient Euclidean metric is

$ds^2=-d^2S(\hat\rho)$2

while the log-Euclidean metric is

$ds^2=-d^2S(\hat\rho)$3

The paper characterizes $ds^2=-d^2S(\hat\rho)$4 as the balanced metric between the Euclidean and log-Euclidean geometries (Thanwerdas et al., 2019).

The structural consequence is information-geometric. Since Euclidean and log-Euclidean metrics are flat, their balanced bilinear form yields a dually flat manifold. The explicit corollary is that

$ds^2=-d^2S(\hat\rho)$5

is a dually flat manifold. In the same paper, BKM is also placed inside the mixed-power-Euclidean family

$ds^2=-d^2S(\hat\rho)$6

with BKM as the case $ds^2=-d^2S(\hat\rho)$7. This formulation makes precise the statement that the BKM manifold on $ds^2=-d^2S(\hat\rho)$8 balances linear and logarithmic flat structures rather than being introduced as an isolated metric space (Thanwerdas et al., 2019).

4. Gaussian-state BKM geometry and curvature

For continuous-variable systems, the BKM manifold is realized on the space of faithful, zero-displacement $ds^2=-d^2S(\hat\rho)$9-mode Gaussian states. The manifold is

$$\mathcal K_{\rho}(X)=\int_0^1 ds\, \rho^{\,s}X\rho^{\,1-s},$$0

with each tangent space identified with the real symmetric $$\mathcal K_{\rho}(X)=\int_0^1 ds\, \rho^{\,s}X\rho^{\,1-s},$$1 matrices. A Gaussian state is written as

$$\mathcal K_{\rho}(X)=\int_0^1 ds\, \rho^{\,s}X\rho^{\,1-s},$$2

and faithfulness means

$$\mathcal K_{\rho}(X)=\int_0^1 ds\, \rho^{\,s}X\rho^{\,1-s},$$3

where the $$\mathcal K_{\rho}(X)=\int_0^1 ds\, \rho^{\,s}X\rho^{\,1-s},$$4 are the symplectic eigenvalues of $$\mathcal K_{\rho}(X)=\int_0^1 ds\, \rho^{\,s}X\rho^{\,1-s},$$5 (Miller, 2024).

The BKM metric is obtained from the relative entropy of two Gaussian states and takes the explicit form

$$\mathcal K_{\rho}(X)=\int_0^1 ds\, \rho^{\,s}X\rho^{\,1-s},$$6

with

$$\mathcal K_{\rho}(X)=\int_0^1 ds\, \rho^{\,s}X\rho^{\,1-s},$$7

The corresponding line element is

$$\mathcal K_{\rho}(X)=\int_0^1 ds\, \rho^{\,s}X\rho^{\,1-s},$$8

This metric is invariant under symplectic transformations,

$$\mathcal K_{\rho}(X)=\int_0^1 ds\, \rho^{\,s}X\rho^{\,1-s},$$9

The Levi-Civita connection and geodesic equation are also explicit: $$\mathcal K_{\rho}^{-1}(X)= \int_0^\infty du\,(\rho+uI)^{-1}X(\rho+uI)^{-1},$$0 and a curve $$\mathcal K_{\rho}^{-1}(X)= \int_0^\infty du\,(\rho+uI)^{-1}X(\rho+uI)^{-1},$$1 is a geodesic iff

$$\mathcal K_{\rho}^{-1}(X)= \int_0^\infty du\,(\rho+uI)^{-1}X(\rho+uI)^{-1},$$2

The paper does not solve these equations in closed form for generic boundary conditions (Miller, 2024).

Curvature is the distinctive feature of this realization. The paper derives explicit formulas for the Riemann tensor, Ricci tensor, and scalar curvature, and reduces the scalar curvature to a symmetric function of the symplectic eigenvalues,

$$\mathcal K_{\rho}^{-1}(X)= \int_0^\infty du\,(\rho+uI)^{-1}X(\rho+uI)^{-1},$$3

In the single-mode case $$\mathcal K_{\rho}^{-1}(X)= \int_0^\infty du\,(\rho+uI)^{-1}X(\rho+uI)^{-1},$$4, the paper proves that $$\mathcal K_{\rho}^{-1}(X)= \int_0^\infty du\,(\rho+uI)^{-1}X(\rho+uI)^{-1},$$5 is strictly increasing in $$\mathcal K_{\rho}^{-1}(X)= \int_0^\infty du\,(\rho+uI)^{-1}X(\rho+uI)^{-1},$$6, hence strictly increasing with entropy, with asymptotics

$$\mathcal K_{\rho}^{-1}(X)= \int_0^\infty du\,(\rho+uI)^{-1}X(\rho+uI)^{-1},$$7

For higher modes, the paper gives analytic high-temperature evidence and numerical support for the same monotonic relation between scalar curvature and entropy (Miller, 2024).

5. Local modular self-duality and type III BKM susceptibility

In operator-algebraic quantum field theory, the BKM manifold is not constructed as a global manifold of all states. Instead, the geometry is local and Hessian. In finite dimensions one fixes an antiunitary involution $$\mathcal K_{\rho}^{-1}(X)= \int_0^\infty du\,(\rho+uI)^{-1}X(\rho+uI)^{-1},$$8 and a reflected family $$\mathcal K_{\rho}^{-1}(X)= \int_0^\infty du\,(\rho+uI)^{-1}X(\rho+uI)^{-1},$$9. At a modularly self-dual point $$\gamma_{\rho}^{\mathrm{BKM}}(X,Y) = \operatorname{Tr}\!\left[ X\,\mathcal K_{\rho}^{-1}(Y) \right].$$0, where $$\gamma_{\rho}^{\mathrm{BKM}}(X,Y) = \operatorname{Tr}\!\left[ X\,\mathcal K_{\rho}^{-1}(Y) \right].$$1, the comparison functional is the symmetrized Umegaki relative entropy

$$\gamma_{\rho}^{\mathrm{BKM}}(X,Y) = \operatorname{Tr}\!\left[ X\,\mathcal K_{\rho}^{-1}(Y) \right].$$2

which vanishes at the fixed point. Its Hessian is governed by the BKM quantum Fisher information: $$\gamma_{\rho}^{\mathrm{BKM}}(X,Y) = \operatorname{Tr}\!\left[ X\,\mathcal K_{\rho}^{-1}(Y) \right].$$3 with

$$\gamma_{\rho}^{\mathrm{BKM}}(X,Y) = \operatorname{Tr}\!\left[ X\,\mathcal K_{\rho}^{-1}(Y) \right].$$4

Here the relevant tangent is the reflected-difference tangent selected by the modular pairing (Chatterjee, 18 May 2026).

The type III extension replaces density matrices by faithful normal states on local von Neumann algebras $$\gamma_{\rho}^{\mathrm{BKM}}(X,Y) = \operatorname{Tr}\!\left[ X\,\mathcal K_{\rho}^{-1}(Y) \right].$$5. For a family $$\gamma_{\rho}^{\mathrm{BKM}}(X,Y) = \operatorname{Tr}\!\left[ X\,\mathcal K_{\rho}^{-1}(Y) \right].$$6, one compares the local restriction

$$\gamma_{\rho}^{\mathrm{BKM}}(X,Y) = \operatorname{Tr}\!\left[ X\,\mathcal K_{\rho}^{-1}(Y) \right].$$7

with the modular pullback of the commutant restriction,

$$\gamma_{\rho}^{\mathrm{BKM}}(X,Y) = \operatorname{Tr}\!\left[ X\,\mathcal K_{\rho}^{-1}(Y) \right].$$8

At a self-dual point $$\gamma_{\rho}^{\mathrm{BKM}}(X,Y) = \operatorname{Tr}\!\left[ X\,\mathcal K_{\rho}^{-1}(Y) \right].$$9, the symmetrized Araki relative entropy is

$$\gamma_{\rho}^{\mathrm{BKM}}(X,Y) = \sum_{i,j} c_{\mathrm{BKM}}(p_i,p_j)\, X_{ij}Y_{ji},$$0

and the local BKM susceptibility is defined by

$$\gamma_{\rho}^{\mathrm{BKM}}(X,Y) = \sum_{i,j} c_{\mathrm{BKM}}(p_i,p_j)\, X_{ij}Y_{ji},$$1

where

$$\gamma_{\rho}^{\mathrm{BKM}}(X,Y) = \sum_{i,j} c_{\mathrm{BKM}}(p_i,p_j)\, X_{ij}Y_{ji},$$2

The Hessian relation is

$$\gamma_{\rho}^{\mathrm{BKM}}(X,Y) = \sum_{i,j} c_{\mathrm{BKM}}(p_i,p_j)\, X_{ij}Y_{ji},$$3

Exact coherent-state realizations are obtained for the free scalar field on wedge algebras and for the chiral $$\gamma_{\rho}^{\mathrm{BKM}}(X,Y) = \sum_{i,j} c_{\mathrm{BKM}}(p_i,p_j)\, X_{ij}Y_{ji},$$4 current on half-line algebras; in both examples the comparison functional is exactly quadratic in the deformation parameter, and the susceptibility coefficients admit explicit boost-energy, stress-tensor, or half-line integral representations (Chatterjee, 18 May 2026).

6. Measure-theoretic and random-state realizations

The BKM manifold also appears through the measure induced by the BKM metric on state space. One paper studies a random mixed-state ensemble induced from von Neumann entropy through the BKM metric. For density matrices $$\gamma_{\rho}^{\mathrm{BKM}}(X,Y) = \sum_{i,j} c_{\mathrm{BKM}}(p_i,p_j)\, X_{ij}Y_{ji},$$5 with eigenvalues $$\gamma_{\rho}^{\mathrm{BKM}}(X,Y) = \sum_{i,j} c_{\mathrm{BKM}}(p_i,p_j)\, X_{ij}Y_{ji},$$6, the spectral density is

$$\gamma_{\rho}^{\mathrm{BKM}}(X,Y) = \sum_{i,j} c_{\mathrm{BKM}}(p_i,p_j)\, X_{ij}Y_{ji},$$7

with

$$\gamma_{\rho}^{\mathrm{BKM}}(X,Y) = \sum_{i,j} c_{\mathrm{BKM}}(p_i,p_j)\, X_{ij}Y_{ji},$$8

The unconstrained lift

$$\gamma_{\rho}^{\mathrm{BKM}}(X,Y) = \sum_{i,j} c_{\mathrm{BKM}}(p_i,p_j)\, X_{ij}Y_{ji},$$9

factorizes under $$c_{\mathrm{BKM}}(x,y) = \frac{\log x-\log y}{x-y}, \qquad c_{\mathrm{BKM}}(x,x)=\frac1x.$$0 into a gamma trace variable and a simplex spectral part. The logarithmic divided-difference factor

$$c_{\mathrm{BKM}}(x,y) = \frac{\log x-\log y}{x-y}, \qquad c_{\mathrm{BKM}}(x,x)=\frac1x.$$1

is the paper’s distinctive fingerprint of the BKM geometry (Sohail et al., 16 Jun 2026).

A qutrit realization uses the unitary-invariant stratified manifold

$$c_{\mathrm{BKM}}(x,y) = \frac{\log x-\log y}{x-y}, \qquad c_{\mathrm{BKM}}(x,x)=\frac1x.$$2

with the BKM monotone metric determined by

$$c_{\mathrm{BKM}}(x,y) = \frac{\log x-\log y}{x-y}, \qquad c_{\mathrm{BKM}}(x,x)=\frac1x.$$3

In this setting the BKM metric is used to define a Riemannian volume form and a unitary-invariant random ensemble, and the main observable is the classicality indicator $$c_{\mathrm{BKM}}(x,y) = \frac{\log x-\log y}{x-y}, \qquad c_{\mathrm{BKM}}(x,x)=\frac1x.$$4, the probability that a random qutrit state has a nonnegative Wigner function everywhere. The global qutrit BKM indicator has

$$c_{\mathrm{BKM}}(x,y) = \frac{\log x-\log y}{x-y}, \qquad c_{\mathrm{BKM}}(x,x)=\frac1x.$$5

Within the comparison HS/Bures/BKM, the BKM ensemble gives the smallest classicality indicator (Khvedelidze et al., 2022).

7. Applications, scope, and common distinctions

An applied realization appears in quantum reservoir computing, where the underlying state space is treated as a BKM manifold near the unconditional state $$c_{\mathrm{BKM}}(x,y) = \frac{\log x-\log y}{x-y}, \qquad c_{\mathrm{BKM}}(x,x)=\frac1x.$$6. For nearby conditional states

$$c_{\mathrm{BKM}}(x,y) = \frac{\log x-\log y}{x-y}, \qquad c_{\mathrm{BKM}}(x,x)=\frac1x.$$7

the Fréchet derivative of the logarithm is

$$c_{\mathrm{BKM}}(x,y) = \frac{\log x-\log y}{x-y}, \qquad c_{\mathrm{BKM}}(x,x)=\frac1x.$$8

and the BKM metric is

$$c_{\mathrm{BKM}}(x,y) = \frac{\log x-\log y}{x-y}, \qquad c_{\mathrm{BKM}}(x,x)=\frac1x.$$9

The quadratic approximation

$\mathrm{SPD}_n$00

allows Holevo memory and predictive capacities to be rewritten as average BKM norms, which are then expanded into explicit spectral-response formulas. In that paper, the BKM manifold is the analytic bridge from relative entropy to spectral resonance, predictive performance, and a generalized Landauer bound for continuous temporal processing (Ding et al., 2 Jul 2026).

The term should be distinguished from several nearby but nonidentical geometries. In a two-band Bose–Einstein condensate, the relevant geometric object is the band quantum metric, explicitly identified as the natural Fubini–Study metric on the Bloch sphere of lower-band Bloch states; that paper states that it does not discuss the Bogoliubov–Kubo–Mori metric (Iskin, 2022). In Hartree–Fock–Bogoliubov theory, the geometric manifold is the orbit of admissible generalized one-particle density matrices under Bogoliubov transformations, yielding reductive homogeneous spaces with invariant symplectic forms and, under spectral conditions, Kähler homogeneous spaces; this is Bogoliubov orbit geometry, not the Kubo–Mori metric manifold (Alvarado et al., 2024). A further neighboring topic is the theory of weighted Kubo–Ando geometric means, which develops operator perspectives and characterization theorems for geometric means but does not define the BKM metric, tangent-space bilinear forms, or a state-space manifold in the information-geometric sense (Frenkel et al., 17 Mar 2025).

Taken together, these constructions show that the Bogoliubov–Kubo–Mori geometric manifold is not a single rigid model but a family of relative-entropy manifolds. Its recurring structural features are the Hessian origin in relative entropy, the logarithmic divided-difference kernel

$\mathrm{SPD}_n$01

faithfulness or strict positivity as the natural domain, and a geometry that is local in some settings, globally homogeneous in others, and explicitly curved in the Gaussian covariance realization.