Canonical divergence for flat $\alpha$-connections: Classical and Quantum

Abstract

A recent canonical divergence, which is introduced on a smooth manifold $\mathrm{M}$ endowed with a general dualistic structure $(\mathrm{g},\nabla ,{\nabla }^{\ast })$, is considered for flat $\alpha$-connections. In the classical setting, we compute such a canonical divergence on the manifold of positive measures and prove that it coincides with the classical $\alpha$-divergence. In the quantum framework, the recent canonical divergence is evaluated for the quantum $\alpha$-connections on the manifold of all positive definite Hermitian operators. Also in this case we obtain that the recent canonical divergence is the quantum $\alpha$-divergence.

arXiv comment18 pages

References in corpus (8)

• Quantifying Entanglement
• Fundamental properties of Tsallis relative entropy
• Nonadditive conditional entropy and its significance for local realism
• Information geometric methods for complexity
• Computing complexity measures for quantum states based on exponential families
• Maximizing the divergence from a hierarchical model of quantum states
• Flat connections and Wigner-Yanase-Dyson metrics
• Canonical divergence for measuring classical and quantum complexity