QuantumInfo.ForMathlib.HayataGroup.TraceInequality.GeneralizedPerspectiveFunction

A function $\Phi : E \to F \to G$ is jointly convex on the sets $s \subseteq E$ and $t \subseteq F$ if for all $A_1, A_2 \in s$, $B_1, B_2 \in t$, and $\theta \in [0, 1]$, the inequality $\Phi((1 - \theta)A_1 + \theta A_2, (1 - \theta)B_1 + \theta B_2) \le (1 - \theta)\Phi(A_1, B_1) + \theta\Phi(A_2, B_2)$ holds.

A function $\Phi : E \to F \to G$ is **jointly convex** if for all $A_1, A_2 \in E$, $B_1, B_2 \in F$, and $\theta \in [0, 1]$, the following inequality holds: $$\Phi((1 - \theta)A_1 + \theta A_2, (1 - \theta)B_1 + \theta B_2) \le (1 - \theta)\Phi(A_1, B_1) + \theta\Phi(A_2, B_2)$$ This definition applies the property of joint convexity to the entire domain of the function, where $E$ and $F$ are vector spaces and $G$ is an ordered vector space.

Let $E$, $F$, and $G$ be vector spaces, where $G$ is an ordered vector space. A function $\Phi : E \to F \to G$ is jointly concave on the sets $s \subseteq E$ and $t \subseteq F$ if for all $A_1, A_2 \in s$, $B_1, B_2 \in t$, and scalar $\theta \in [0, 1]$, the following inequality holds: $$(1 - \theta) \Phi(A_1, B_1) + \theta \Phi(A_2, B_2) \le \Phi((1 - \theta) A_1 + \theta A_2, (1 - \theta) B_1 + \theta B_2)$$ This property indicates that the map is concave with respect to the simultaneous linear interpolation of both its arguments.

Let $E$ and $F$ be vector spaces, and $G$ be an ordered vector space. A function $\Phi : E \to F \to G$ is jointly concave if for all $A_1, A_2 \in E$, $B_1, B_2 \in F$, and any scalar $\theta \in [0, 1]$, the following inequality holds: $$(1 - \theta) \Phi(A_1, B_1) + \theta \Phi(A_2, B_2) \le \Phi((1 - \theta) A_1 + \theta A_2, (1 - \theta) B_1 + \theta B_2)$$ This property represents the joint concavity of the map $\Phi$ over its entire domain $E \times F$, without any restrictions on the sets of the input variables.

For a real-valued function $h: \mathbb{R} \to \mathbb{R}$ and a bounded linear operator $B \in \mathcal{B}(\mathcal{H})$ on a complex Hilbert space $\mathcal{H}$, this definition represents the operator $(h(B))^{1/2}$ obtained via the continuous functional calculus. Specifically, it is the result of applying the function $f(x) = \sqrt{h(x)}$ to the operator $B$.

Let $h: \mathbb{R} \to \mathbb{R}$ be a real-valued function and $B \in \mathcal{B}(\mathcal{H})$ be a bounded linear operator on a complex Hilbert space $\mathcal{H}$. This definition represents the operator $h(B)^{-1/2}$ obtained by applying the function $x \mapsto (h(x))^{-1/2}$ to the operator $B$ via the real continuous functional calculus.

Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ be the space of bounded linear operators on $\mathcal{H}$. Given real-valued functions $f, h: \mathbb{R} \to \mathbb{R}$ and operators $A, B \in \mathcal{B}(\mathcal{H})$, the generalized perspective of $f$ and $h$ with respect to $A$ and $B$ is the operator defined by: $$(f \Delta h)(A, B) = h(B)^{1/2} f\left(h(B)^{-1/2} A h(B)^{-1/2}\right) h(B)^{1/2}$$ where the operators $h(B)^{1/2}$, $h(B)^{-1/2}$, and the evaluation of $f$ on the operator argument are all defined via the continuous functional calculus. This construction is typically applied when $A$ is a Hermitian operator and $h(B)$ is positive and invertible.

The infix operator $\Delta$ represents the generalized perspective operation. For functions $f, h: \mathbb{R} \to \mathbb{R}$ and elements $A, B$ (typically operators in a space $L(\mathcal{H})$), the expression $(f \Delta h) A B$ is defined as the generalized perspective of $f$ and $h$ evaluated at $A$ and $B$, denoted as $\text{GeneralizedPerspective}(f, h, A, B)$.

Let $\mathcal{H}$ be a complex inner product space and $\mathcal{B}(\mathcal{H})$ be the space of bounded linear operators on $\mathcal{H}$. The set `psdSet` consists of all operators $A \in \mathcal{B}(\mathcal{H})$ that are self-adjoint and have a spectrum $\sigma(A)$ contained in the interval $[0, \infty)$. These are known as the positive semidefinite operators.

Let $\mathcal{H}$ be a complex inner product space and $\mathcal{B}(\mathcal{H})$ be the space of bounded linear operators on $\mathcal{H}$. This set consists of all operators $A \in \mathcal{B}(\mathcal{H})$ that are self-adjoint and whose spectrum $\sigma(A)$ is contained within the strictly positive real numbers $(0, \infty)$.

Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ be the space of bounded linear operators on $\mathcal{H}$. Suppose $f: \mathbb{R} \to \mathbb{R}$ is a function that is operator convex on all complex Hilbert spaces and satisfies $f(0) \le 0$. Furthermore, let $h: \mathbb{R} \to \mathbb{R}$ be a function that is continuous, strictly positive, and operator concave on the interval $(0, \infty)$. Then the generalized perspective function $(f \Delta h)(A, B)$, defined by $$(f \Delta h)(A, B) = h(B)^{1/2} f\left(h(B)^{-1/2} A h(B)^{-1/2}\right) h(B)^{1/2},$$ is jointly convex on the set of positive semidefinite operators $A \in \mathcal{B}(\mathcal{H})$ and strictly positive operators $B \in \mathcal{B}(\mathcal{H})$.

Let $f, h: \mathbb{R} \to \mathbb{R}$ be real-valued functions. Suppose that $f$ is continuous and operator convex on the interval $[0, \infty)$ such that $f(0) \le 0$. Further, suppose that $h$ is continuous, operator concave, and satisfies $h(x) > 0$ for all $x \in (0, \infty)$. Then for any complex Hilbert space $\mathcal{H}$, the generalized perspective function $(f \Delta h)(A, B)$ defined by $$(f \Delta h)(A, B) = h(B)^{1/2} f\left(h(B)^{-1/2} A h(B)^{-1/2}\right) h(B)^{1/2}$$ is jointly convex on the set of positive semidefinite operators $A \in \mathcal{B}(\mathcal{H})$ and strictly positive operators $B \in \mathcal{B}(\mathcal{H})$.

Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ be the space of bounded linear operators on $\mathcal{H}$. Let $f, h: \mathbb{R} \to \mathbb{R}$ be real-valued functions. Suppose that $f$ is operator concave on all complex Hilbert spaces and satisfies $f(0) \geq 0$. Further suppose that $h$ is operator concave and continuous on the interval $(0, \infty)$, and $h(x) > 0$ for all $x \in (0, \infty)$. Then the generalized perspective $(f \Delta h)(A, B)$, defined as $$(f \Delta h)(A, B) = h(B)^{1/2} f\left(h(B)^{-1/2} A h(B)^{-1/2}\right) h(B)^{1/2},$$ is jointly concave for positive semidefinite operators $A \in \mathcal{B}(\mathcal{H})$ and strictly positive operators $B \in \mathcal{B}(\mathcal{H})$.

Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ be the space of bounded linear operators on $\mathcal{H}$. Let $f: \mathbb{R} \to \mathbb{R}$ and $h: \mathbb{R} \to \mathbb{R}$ be real-valued functions. Suppose that: 1. $f$ is operator concave on $[0, \infty)$ for all Hilbert spaces, $f$ is continuous on $[0, \infty)$, and $f(0) \ge 0$. 2. $h$ is operator concave on $(0, \infty)$ for the space $\mathcal{B}(\mathcal{H})$, $h$ is continuous on $(0, \infty)$, and $h(x) > 0$ for all $x \in (0, \infty)$. Then the generalized perspective mapping $(A, B) \mapsto (f \Delta h)(A, B)$, defined for a positive semidefinite operator $A \ge 0$ and a strictly positive operator $B > 0$ as $$(f \Delta h)(A, B) = h(B)^{1/2} f\left(h(B)^{-1/2} A h(B)^{-1/2}\right) h(B)^{1/2},$$ is jointly concave on the set of positive semidefinite operators and the set of strictly positive operators. This result is a restricted localized forward form of Corollary 2.6 on the positive cone.