arXiv:2609.17574

Gaussian Purification Quotients and Fixed Nielsen Penalties

Christian Kerskens

quant-phhep-thmath-phSubmitted Jul 31, 2026• Announced Aug 3, 2026• 0 citations• Revised Sep 17, 2026 (v2)

Abstract

Information distance and circuit complexity are both obtained by minimizing lengths, but they minimize over different objects. We make this distinction explicit for faithful one-mode Gaussian states. First, invariant-form uniqueness implies that no positive-definite quadratic gate cost can be invariant under the full adjoint action of the noncompact symplectic group; a positive Cartan majorant necessarily introduces additional reference data. The Uhlmann purification quotient realizes the Bures metric, and the radial covariance direction requires a system-ancilla coupling because system-only Gaussian unitaries preserve the Williamson eigenvalue. We then minimize fixed right-invariant quadratic norms on the minimal two-mode Gaussian gate algebra $\mathfrak{sp}(4,\mathbb R)$. For the unweighted Frobenius norm, the quotient coefficients for radial and traceless covariance tangents are $G_0=[\hbar^2(u-1)]^{-1}$ and $G_2=[\hbar^2(3u-1)]^{-1}$, where $u=(2ν/\hbar)^2$. Their ratio does not equal the Bures ratio. The radial coefficient, however, reproduces the Bures value exactly at every $u$; the mismatch is confined to the traceless sector. More generally, a constant block-diagonal two-weight schedule gives $G_0/G_2=1+2(β/α)u/(u-1)$; matching Bures throughout the isotropic family would require the state-dependent relation $β/α=1/u$. At the Bures-Fisher determinant crossing $u=\varphi$, pointwise matching is possible only by inserting $β/α=\varphi^{-1}$. Thus the Bures purification quotient is an exact state-geometric cost, but it is neither an unweighted symplectic gate cost nor a member of this fixed two-weight Nielsen family. The existence of a more general fixed positive gate norm realizing the quotient remains open.

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arXiv:2609.17574

Gaussian Purification Quotients and Fixed Nielsen Penalties

Christian Kerskens

quant-phhep-thmath-phSubmitted Jul 31, 2026• Announced Aug 3, 2026• 0 citations• Revised Sep 17, 2026 (v2)

arXiv HTML PDF

Abstract

G_0=[\hbar^2(u-1)]^{-1}

and

G_2=[\hbar^2(3u-1)]^{-1}

, where

u=(2ν/\hbar)^2

. Their ratio does not equal the Bures ratio. The radial coefficient, however, reproduces the Bures value exactly at every

u

; the mismatch is confined to the traceless sector. More generally, a constant block-diagonal two-weight schedule gives

G_0/G_2=1+2(β/α)u/(u-1)

; matching Bures throughout the isotropic family would require the state-dependent relation

β/α=1/u

. At the Bures-Fisher determinant crossing (u=\varphi), pointwise matching is possible only by inserting

β/α=\varphi^{-1}

. Thus the Bures purification quotient is an exact state-geometric cost, but it is neither an unweighted symplectic gate cost nor a member of this fixed two-weight Nielsen family. The existence of a more general fixed positive gate norm realizing the quotient remains open.

Related papers

The Golden Junction of Gaussian Covariance Geometries: Sector Anisotropy, Relative-Entropy Response, and Local Penalties

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G_0=[\hbar^2(u-1)]^{-1}

and

G_2=[\hbar^2(3u-1)]^{-1}

, where

u=(2ν/\hbar)^2

. Their ratio does not equal the Bures ratio. The radial coefficient, however, reproduces the Bures value exactly at every

u

; the mismatch is confined to the traceless sector. More generally, a constant block-diagonal two-weight schedule gives

G_0/G_2=1+2(β/α)u/(u-1)

; matching Bures throughout the isotropic family would require the state-dependent relation

β/α=1/u

. At the Bures-Fisher determinant crossing (u=\varphi), pointwise matching is possible only by inserting

β/α=\varphi^{-1}

x^2=\varphi

. Afterintroducing a constant action-matching parameter

\eta

, theBures--Wasserstein determinant joins them only at

(x,\eta)=(\sqrt\varphi,\sqrt\varphi)

. Equal determinant density doesnot mean tensor equality: relative to Fisher--Rao, the Bures cometric at thejunction is

Q_{\rm gold}=\varphi^{-2}\Pi_0+\varphi\Pi_2, \qquad \det Q_{\rm gold}=1,

so its inverse has the exact scalar-to-weight-two quadratic penalty ratio

\varphi^3

. The normalized Bogoliubov--Kubo--Mori metric supplies anindependent relative-entropy diagnostic. It agrees radially with Bures buthas shape cometric eigenvalue

4\hbar^2x/\log[(x+1)/(x-1)]

; it therefore does not form a fourthdeterminant junction and becomes tangentially stiff at the pure-stateboundary. Operationally, weight-two motion is generated by a system-onlyGaussian unitary, whereas radial motion requires a Gaussian channel orancillary dilation. The known Uhlmann--Ruan purification quotient realizesthe local Bures state metric, but an explicit

\mathfrak{sp}(4,\mathbb R)

test shows that an unweighted right-invariantFrobenius gate norm does not reproduce the golden penalty. Conditionalbranch selection and finite-rate Bures--Wasserstein cost separation arepresented as secondary statements, not as dynamical phase transitions.

\mathfrak{sp}(2N,\mathbb{R})

, this form is the trace form and hence a fixed multiple of the Killing form. We prove that, within the normalized symmetric Petz family, Bures is uniquely selected by requiring such an additive state-independent symplectic correction. We determine the full Williamson-frame spectrum and show that a boundary stratum with

m

pure modes has an

m^2

-dimensional cometric kernel isomorphic to

\mathfrak u(m)

, while the pure Gaussian orbit remains nondegenerate. For radial one-mode estimation, ideal heterodyne detection accesses the exact fraction

(ν-\hbar/2)/(ν+\hbar/2)

of the SLD quantum Fisher information. Its vanishing boundary limit reflects finite heterodyne information relative to a divergent radial quantum Fisher information, not zero measurement information. Finally, a minimal Schur realization fixes the auxiliary inertia and identifies the second Schur complement as the true covariant vertical block.

m^2

-dimensional kernel when

m

modes become pure, yet the boundary remains at finite radial Bures distance. We derive the potential

Φ=-\tfrac12\sum_k\log(ν_k^2-\tfrac14),

which generates fixed-frame covariance dilation and equals

βF

on fixed-Hamiltonian thermal families. At large symplectic eigenvalue, the relative quantum correction is

O(\hbar^2/ν^2)

, recovering covariance Fisher--Rao geometry. This intrinsic classical regime differs from the pure-state boundary, where classical statistics require a specified measurement channel. As a secondary comparison, the normalized Bures, Fisher--Rao, and action-matched Bures--Wasserstein determinant densities possess an exact one-mode junction at

(x,η)=(\sqrt{\varphi},\sqrt{\varphi}),\qquad x=2ν/\hbar.

This junction is kinematic, and its branch interpretation is conditional. At a prescribed finite nonzero compression rate, the Bures action diverges while the Bures--Wasserstein cost remains finite; a bounded Bures budget instead enforces radial deceleration.

We deform the Carlen--Maas--Wirth framework for noncommutative dynamical optimal transport by an Arnold--Nielsen type complexity operator. A positive state-independent operator

G

compatible with the Hilbert bimodule structure of a noncommutative differential calculus $\partial\colon\M\to\Hcal$ can be absorbed into the calculus itself,

\partial_G:=G^{1/2}\partial.

The corresponding complexity-weighted transport problem is exactly the unweighted transport problem generated by

\partial_G

, whenever the deformed quadratic form remains Dirichlet. In finite dimensions we prove existence of minimizers for density-dependent Petz-class metrics and for fixed physical complexity weights, the latter without commutation between

G

and the state-dependent mobility. On unitary orbits we identify the induced distance with a quotient metric coming from a right-invariant complexity geometry. This yields an exact Bell-state preparation result via Clairaut's relation and an exactly computed restricted-path upper bound for GHZ preparation; the Lindblad detailed-balance case is included only as entropy-gradient-flow background.

m^2

-dimensional kernel when

m

Williamson modes are pure, although thefloor remains at finite radial Bures distance. We derive the radial potential

\Phi=-\tfrac12\sum_k\log(\nu_k^2-\tfrac14)

, which generates covariancedilation on fixed-Williamson-frame radial submanifolds and satisfies

\Phi=\beta F

on fixed-Hamiltonian thermal families. At large symplecticeigenvalue the relative symplectic correction is suppressed as

O(\hbar^2/\nu^2)

, yielding an intrinsic high-noise recovery of covarianceFisher--Rao geometry. This limit is distinct from the Williamson floor, whichis a quantum pure-state boundary rather than a classical limit. A classicalmodel at the floor instead requires an explicitly supplied measurement channelor commutative restriction; the boundary geometry constrains the structure onwhich that operational map acts but does not select the map itself. As asecondary comparison, we place the Gaussian Bures geometry alongside theFisher--Rao and Bures--Wasserstein covariance geometries. After introducing aconstant action-matching parameter

\eta

, their normalized determinantdensities possess an exact one-mode junction at

(x,\eta)=(\sqrt{\varphi},\sqrt{\varphi})

. The equality is kinematic: it isneither an additional boundary nor a dynamical transition, and branch selectionrequires an explicitly stated minimum-log-determinant hypothesis. For the sameprescribed finite-rate radial compression toward the floor, the Bures kineticdensity and action diverge while the Bures--Wasserstein cost remains finite;under a bounded Bures kinetic budget, the intrinsic consequence is radialdeceleration rather than a forced change of geometry.