ある関数族の正定値関数による順序 (作用素論に基づく量子情報理論の幾何学的構造に関する研究と関連する話題)

Abstract

We denote, by fin C(0, infty)_{1, sym}^{+}, f is a continuous function on (0, infty) to (0, infty), unital (f(1)=1), and symmetric (f(t)=tf(1/t)). For f in C(0, infty)_{1, sym}^{+}, we define the function M_{f} : (0, infty) times (0, infty) rightarrow (0, infty) as follows: M_{f}(s, t)=tf(s/t), s, tin(0, infty). Then M_{f} has the following proprties: M_{f}(s, t)>0, M_{f}(1, 1)=1, M_{f}($alpha$ s, $alpha$ t)=$alpha$ M_{f}(s, t) ($alpha$>0), and M_{f}(s, t)=M_{f}(t_{mathcal{S}}). For A be an Ntimes N matrix, we define L_{A} and R_{A} as follows: L_{A} : mathrm{M}_{N}(mathbb{C})ni Xmapsto AXin mathrm{M}_{N}(mathbb{C}), R_{A}:mathrm{M}_{N}(mathbb{C})ni Xmapsto XAin mathrm{M}mathrm{I}_{N}(mathbb{C}). We see (mathrm{M}_{N}(mathbb{C}), Tr) as a Hilbert space. Then L_{A} and R_{A} are bounded linear operator on (mathrm{M}_{N}(mathbb{C}), Tr). We call the function $phi$ : mathbb{R}rightarrow mathbb{R} positive-definite if, for any nin mathrm{N} and x_{1}, x_{2}, . . . , x_{n}in mathbb{R}, the matrix ($phi$(x_{i}-x_{j}))_{i, j=1}^{n} is positive, that ism displaystyle sum{i, j=1}^{n} $phi$(x{i}-x{j})$xi${i}overline{$xi${j}geq 0 forall$xi$_{1}, . . . , $xi$_{n}in mathbb{C}. For f, gin C(0, infty)_{1, sym}^{+}, we denote fpreceq g if displaystyle mathbb{R}ni xmapstofrac{f(e^{x})}{g(e^{x})}in mathbb{R} is positive-definite. Hiai-Kosaki [3] proved that, for f, g in C(0, infty)_{1, sym}^{+}, it is equivalent to hold fpreceq g and to hols the following inequality: |||M_{f}(L_{H}, R_{K})X|||leq|||M_{g}(L_{H}, R_{K})mathrm{X}||| for all H, K, Xin mathrm{M}_{N}(mathbb{C}), H and K are positive and invertible, and |||・||| is any unitarily invariant norm on mathrm{M}_{N}(mathbb{C}). They also proved, for functions f_{a}(t)=displaystyle frac{(a-1)(t^{a}-1)}{a(t^{a-1}-1)}in C(0) infty)_{1, sym}^{+} (ain mathbb{R}), it holds that f_{a} preceq f_{b} if a leq b. Applying this for f_{1/2}(t)=t^{1/2} and f_{2}(t)=displaystyle frac{1+t}{2}, they proved McIntosh 's inequality: |||H^{1/2}XK^{1/2}|||displaystyle leqfrac{1}{2}|||HX+XK||| Let $alpha$=(a_{1}, a_{2}, . . . , a_{n}), $beta$=(b_{1}, b_{2}, . . . , b_{n}) in mathbb{R}^{n}. We consider the function f{$alpha, beta$}(t)=t^{$gamma$($alpha, beta$)}displaystyle prod{i=1}^{n}frac{b{i}(t^{a{i}-1)}{a{i}(t^{b{i}-1)}, where $gamma$($alpha$, $beta$)=displaystyle frac{1-$Sigma$_{i=1}^{n}(a_{i}-b_{i})}{2}. Then we can see that f_{$alpha, beta$}in C(0, infty)_{1, sym}^{+} and f_{$alpha, beta$}(t)=f_{a}(t) if n=1, $alpha$=(a), $beta$=(a-1) Let $alpha$=(a_{1}}ldots, a_{n}), $beta$=(b_{1}, ldots, b_{n}) in mathbb{R}^{n}, $alpha$'=(c_{1}, ldots, c_{m}), $beta$'=(d_{1}, . . . , d_{m}) in mathbb{R}^{m}. Under what condition does it hold f_{$alpha, beta$}preceq f_{$alpha$', $beta$'}? This means whether it holds |||M_{$alpha, beta$}(L_{H}, R_{K})X|||leq|||M_{$alpha$', $beta$'}(L_{H}, R_{K})mathrm{X}||| or not, and here M_{$alpha, beta$} (s, t)=tf_{$alpha, beta$}(s/t). Since displaystyle frac{f{$alpha, beta$}(e^{2x})}{f{$alpha, beta$'}(e^{2x})}=prod{i=1}^{n}frac{b{i}sinh a{i^{X}{a{i}sinh b{i}x}prod{j=1}^{m}frac{c{j}sinh d{j}x}{d{j}sinh c{j}x}, our problem is reduced to the following: for a_{1} geq a_{2} geq. . . geq a_{n}>0, b_{1}geq b_{2}geqcdotsgeq b_{N}>0, under what condition is $phi$(x)=displaystyle prod{i=1}^{N}frac{sinh a{i^{X}{sinh b{i}x} positive definite? Our answer is ・If displaystyle sum_{i=1}^{k}a_{i} leq displaystyle sum_{i=1}^{k}b_{i} for all k=1, 2, . . . , N, then $phi$(x) is positive definite. ・If a_{1}>b_{1} or a_{1}+a_{2}+cdots+a_{N}>b_{1}+b_{2}+cdots+b_{N}, then $phi$(x) is not positive definite. ・Assume that N=2. Then $phi$(x) is positive definite if and only if a_{1}leq b_{1} and a_{1}+a_{2}leq b_{1}+b_{2}.