Quantum Michelson Contrast: Logarithmic Parametrisation, Subadditivity, and Geometric Interpretation

Abstract:Building on the operator Michelson contrast introduced in our previous work for positive operators, we investigate its natural logarithmic parametrisation. For a positive invertible operator $A$, the operator Michelson contrast is $\Delta(A)=(\kappa(A)-1)/(\kappa(A)+1)$, where $\kappa(A)=\|A\|\|A^{-1}\|$ is the condition number. Its logarithmic lift is $\operatorname{arctanh}(\Delta(A))=\frac12\ln\kappa(A)$. We call this quantity the \emph{Quantum Logarithmic Contrast} (QLC), and extend it to all invertible operators via the polar decomposition. We prove that QLC is subadditive under multiplication: $\operatorname{QLC}(AB)\le \operatorname{QLC}(A)+\operatorname{QLC}(B)$. For positive invertible operators, we establish an exact geometric identification: $\operatorname{QLC}(A)$ equals the projective Thompson distance from the identity ray to the ray generated by $A$. The main result is a sharp dimensional threshold for equality in the subadditivity law: for $2\times2$ and $3\times3$ positive matrices, equality forces commutativity, while $n=4$ is the smallest dimension in which noncommuting equality examples exist. We also derive contraction properties, sum inequalities, trace estimates for density operators, and limiting behaviour for Cesàro means and infinite products.

Quantum Algebra (math.QA); Operator Algebras (math.OA); Spectral Theory (math.SP); Optics (physics.optics); Quantum Physics (quant-ph)

G.1.3; G.1.0; J.2; G.4

arXiv:2609.06997 [math.QA]

(or arXiv:2609.06997v1 [math.QA] for this version)

https://doi.org/10.48550/arXiv.2609.06997 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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