量子 Michelson 对比度：对数参数化、次可加性与几何解释

中文摘要 AI 辅助

基于我们先前工作中针对正算子引入的算子 Michelson 对比度，我们研究其自然的对数参数化。对于正可逆算子 $A$，算子 Michelson 对比度为 $\Delta(A)=(\kappa(A)-1)/(\kappa(A)+1)$，其中 $\kappa(A)=\\|A\\|\\|A^{-1}\\|$ 是条件数。其对数提升为 $\operatorname{arctanh}(\Delta(A))=\frac12\ln\kappa(A)$。我们将此量称为“量子对数对比度”（QLC），并通过极分解将其扩展到所有可逆算子。我们证明 QLC 在乘法下具有次可加性：$\operatorname{QLC}(AB)\le \operatorname{QLC}(A)+\operatorname{QLC}(B)$。对于正可逆算子，我们建立了精确的几何等同性：$\operatorname{QLC}(A)$ 等于从恒等射线到由 $A$ 生成的射线的投影 Thompson 距离。主要结果是次可加性定律中等式成立的尖锐维度阈值：对于 $2\times2$ 和 $3\times3$ 正矩阵，等式成立迫使可交换性，而 $n=4$ 是存在非交换等式示例的最小维度。我们还推导了压缩性质、和不等式、密度算子的迹估计，以及 Cesàro 均值和无限积的极限行为。

英文摘要

We study the operator Michelson contrast Delta(A) = (kappa(A)-1)/(kappa(A)+1), with kappa(A)=||A|| ||A^{-1}||, for positive invertible operators, and introduce its logarithmic parametrization via arctanh(Delta(A)) = (1/2) ln kappa(A). This defines the Operator Logarithmic Contrast (OLC), extended to all invertible operators via polar decomposition. We prove OLC is subadditive under multiplication: OLC(AB) <= OLC(A)+OLC(B). For positive invertible A, we identify OLC(A) exactly as the projective Thompson distance from the identity ray to the ray of A. We establish a sharp dimensional threshold for equality in subadditivity: for dimensions 2 and 3, equality forces commutativity, while n=4 is the smallest dimension admitting noncommuting equality examples. Moreover, equality still forces commutativity in any total dimension if the operators factor as pure Kronecker products X = tensor X_i and Y = tensor Y_i with each factor dimension <= 3. We derive contraction inequalities, sum bounds, trace estimates for density operators, Lipschitz continuity (including singular limits), and limiting behavior for Cesaro means and infinite products. For density operators, we obtain an exact closed-form bijection between OLC and von Neumann entropy for qubits; prove this bijection does not exist for d >= 3; and establish upper/lower entropy envelopes at fixed condition number, showing the admissible range [S_min(kappa), S_max(kappa)] satisfies ln 2 <= S_max(kappa) < ln 3 for every kappa > 1. The qubit case recovers, as an operator lift, the classical Michelson-Jensen-Shannon equivalence of Bruni, Rossi, and Vitulano, while a counterexample shows this equivalence fails for d > 2 as a direct consequence of the non-bijectivity.

发表机构

• Lean Technologies Laboratory, Kazan（Kazan Lean Technologies Laboratory）
• Novikov Laboratories, Kazan（Kazan Novikov Laboratories）

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