Inverse-Casimir Geometry of Nonunital Multipole Transport in Linear-Spin Lindbladians

Abstract

## Overview This preprint studies the canonical Gorini--Kossakowski--Sudarshan--Lindblad (GKLS) class generated by jump operators linear in the generators of a fixed irreducible spin-$j$ representation. Writing the positive Kossakowski matrix as $C=R+iA$, with $R$ real symmetric and $A_{ab}=\epsilon_{abc}b_c$, separates two sharply different dynamical roles: the real-symmetric sector preserves irreducible tensor rank, while the vector $\mathbf b$ is exactly the nonunitality vector and carries all inter-rank transport. ## Main results For the Hilbert--Schmidt projectors $P_L$ onto irreducible tensor rank $L$, define the neighboring Schrödinger-picture transport blocks $$F_L=P_{L+1}\mathcal D_C P_L,\qquadR_L=P_L\mathcal D_C P_{L+1}.$$ They satisfy the exact scaled-adjoint relation $$R_L=-\frac{L}{L+2}F_L^\dagger,$$ so all nonzero singular values, and more generally all unitarily invariant norms, carry the universal directional factor $(L+2)/L$. This directed imbalance selects the inverse adjoint Casimir $$W=(\mathcal K^2)^{-1}=\sum_{L=1}^{2j}\frac{1}{L(L+1)}P_L$$ on the traceless operator space. The internal nonunital transport is $W$-skew, and multiplicity freeness makes $W$, up to an overall scale, the unique positive $SU(2)$-invariant metric with this property. For the full homogeneous traceless generator, the Hamiltonian and nonunital sectors are $W$-skew, whereas the real-symmetric Kossakowski sector is $W$-self-adjoint and dissipative. This yields an exact Lyapunov identity and contraction in the inverse-Casimir norm. The scalar sector remains exceptional: nonunitality generates a one-way scalar-to-dipole source, which obstructs any positive $SU(2)$-invariant extension of the skew geometry through the scalar mode. Under the standard finite-spin Stratonovich--Weyl correspondence, $$W\longleftrightarrow(-\Delta_{S^2})^{-1}$$ on the mean-zero band-limited sphere. Thus the selected operator metric is exactly the finite-spin $\dot H^{-1}(S^2)$ Green geometry, not merely a semiclassical limit. Complete positivity supplies the sharp resource bound $$2|\mathbf b|\le \operatorname{Tr}C,$$ with equality precisely for rotated rank-one circular $J_\pm$ channels. ## Scope and verification The inverse-Casimir reciprocity, uniqueness, dissipative decomposition, contraction, scalar-boundary obstruction, spherical realization, and complete-positivity extremality are established analytically in the manuscript. Computational checks are used as regression and consistency tests rather than substitutes for proof. Supporting verification code, canonical computational outputs, and figure-generation material are archived separately on Zenodo at DOI **10.5281/zenodo.22096401**. ## Version status This record is **Version v1.0** of the public preprint. It corresponds to the manuscript's final preprint-compatible scientific and notation freeze dated September 3, 2026.