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arXiv 1608.07977quant-ph
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夹心Rényi $α$ -发散的信息几何

Information geometry of sandwiched Rényi $α$-divergence

Kaito Takahashi, Akio Fujiwara

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摘要：

研究了有限量子态空间$\mathcal{S}$上由夹层Rényi $α$ -发散$D_α(ρ\|σ):=\frac{1}{α(α-1)}\log\,{\rm Tr} \left(σ^{\frac{1-α}{2α}}ρ\,σ^{\frac{1-α}{2α}}\right)^α$诱导的信息几何结构$(g^{(D_α)}, \nabla^{(D_α)},\nabla^{(D_α)*})$.证明了黎曼度量$g^{(D_α)}$是单调的当且仅当$α\in(-\infty, -1]\cup [\frac{1}{2},\infty)$，量子统计流形$({\mathcal{S}}, g^{(D_α)}, \nabla^{(D_α)},\nabla^{(D_α)*})$是对偶平坦的当且仅当$α=1$ .

英文摘要：

Information geometrical structure $(g^{(D_α)}, \nabla^{(D_α)},\nabla^{(D_α)*})$ induced from the sandwiched Rényi $α$-divergence $D_α(ρ\|σ):=\frac{1}{α(α-1)}\log\,{\rm Tr} \left(σ^{\frac{1-α}{2α}}ρ\,σ^{\frac{1-α}{2α}}\right)^α$ on a finite quantum state space $\mathcal{S}$ is studied. It is shown that the Riemannian metric $g^{(D_α)}$ is monotone if and only if $α\in(-\infty, -1]\cup [\frac{1}{2},\infty)$, and that the quantum statistical manifold $({\mathcal{S}}, g^{(D_α)}, \nabla^{(D_α)},\nabla^{(D_α)*})$ is dually flat if and only if $α=1$.

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