QuantumInfo.ForMathlib.HayataGroup.TraceInequality.JensenOperatorInequalityIImpIV

Let $\mathcal{H}$ be a complex inner product space and let $\mathcal{B}(\mathcal{H})$ denote the space of bounded linear operators on $\mathcal{H}$. The product space $\mathcal{B}(\mathcal{H}) \times \mathcal{B}(\mathcal{H})$ admits a continuous functional calculus over the complex numbers $\mathbb{C}$ for pairs of normal operators (elements satisfying $x^*x = xx^*$).

Let $\mathcal{H}$ be a complex inner product space and let $\mathcal{B}(\mathcal{H})$ denote the space of bounded linear operators on $\mathcal{H}$. Then the product space $\mathcal{B}(\mathcal{H}) \times \mathcal{B}(\mathcal{H})$ admits a continuous functional calculus over $\mathbb{R}$ for self-adjoint elements.

Let $\mathcal{H}$ be a complex Hilbert space and let $\mathcal{H} \oplus \mathcal{H}$ denote the two-fold Hilbert sum (the $\ell^2$ direct sum of two copies of $\mathcal{H}$). The algebra of bounded linear operators on this space, denoted $\mathcal{B}(\mathcal{H} \oplus \mathcal{H})$, is a non-negative spectrum class over $\mathbb{R}$. This implies that for any self-adjoint operator $T \in \mathcal{B}(\mathcal{H} \oplus \mathcal{H})$, its spectrum $\sigma(T)$ is a subset of $[0, \infty)$ if and only if $T$ is a non-negative operator.

Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ be the algebra of bounded linear operators on $\mathcal{H}$. For a function $f: \mathbb{R} \to \mathbb{R}$, this condition states that for every self-adjoint operator $A \in \mathcal{B}(\mathcal{H})$ with non-negative spectrum $\sigma(A) \subseteq [0, \infty)$ and for every operator $X \in \mathcal{B}(\mathcal{H})$ with operator norm $\|X\| \leq 1$, the following inequality holds in the Loewner order: $$f(X^* A X) \leq X^* f(A) X$$ where $f(\cdot)$ is the operator defined via the continuous functional calculus.

Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ be the algebra of bounded linear operators on $\mathcal{H}$. For a function $f: \mathbb{R} \to \mathbb{R}$, this property (Condition (i) in Theorem 2.5.2) holds if $f$ is operator convex on $\mathcal{B}(\mathcal{H})$ and $f(0) \le 0$. Specifically, $f$ is operator convex if for any two self-adjoint operators $A, B \in \mathcal{B}(\mathcal{H})$ and any $t \in [0, 1]$, the inequality $f((1 - t)A + tB) \le (1 - t)f(A) + tf(B)$ holds in the Loewner order.

For a function $f: \mathbb{R} \to \mathbb{R}$, this property holds if $f(0) \leq 0$ and $f$ is operator convex on all non-trivial complex Hilbert spaces $\mathcal{H}$. Specifically, for every such Hilbert space $\mathcal{H}$, given any two bounded self-adjoint operators $A, B \in \mathcal{B}(\mathcal{H})$ and any scalar $t \in [0, 1]$, the following inequality holds in the Loewner partial order: $$f((1 - t)A + tB) \leq (1 - t)f(A) + tf(B)$$ where the function $f$ is applied to the operators via the continuous functional calculus.

Let $f: \mathbb{R} \to \mathbb{R}$ be a real-valued function. This property (a localized version of Condition (i) from the Jensen operator inequality theorem) holds if $f$ satisfies the following conditions: 1. $f$ is operator convex on the interval $[0, \infty)$ for all non-trivial complex Hilbert spaces $\mathcal{H}$. Specifically, for any such $\mathcal{H}$, for all self-adjoint operators $A, B \in \mathcal{B}(\mathcal{H})$ whose spectra $\sigma(A), \sigma(B)$ are contained in $[0, \infty)$, and for any $t \in [0, 1]$, the inequality $f((1 - t)A + tB) \leq (1 - t)f(A) + tf(B)$ holds in the Löwner order. 2. $f$ is continuous on $[0, \infty)$. 3. $f(0) \le 0$.

For a function $f: \mathbb{R} \to \mathbb{R}$, this property (referred to as Condition (v) in Theorem 2.5.2) states that for any complex Hilbert space $\mathcal{H}$ and any bounded self-adjoint operators $A, B \in \mathcal{B}(\mathcal{H})$ whose spectra are contained in $[0, \infty)$, and for any bounded linear operators $X, Y \in \mathcal{B}(\mathcal{H})$ satisfying the contractive condition $X^* X + Y^* Y \le I$, the following operator inequality holds: $$f(X^* A X + Y^* B Y) \le X^* f(A) X + Y^* f(B) Y$$ where $f(\cdot)$ denotes the operator obtained via the continuous functional calculus.

Let $f: \mathbb{R} \to \mathbb{R}$ be a real-valued function. Suppose that for all non-trivial complex Hilbert spaces, $f$ is operator convex on $[0, \infty)$, continuous on $[0, \infty)$, and satisfies $f(0) \leq 0$. Then, for any complex Hilbert space $\mathcal{H}$, the following holds: for every self-adjoint operator $A \in \mathcal{B}(\mathcal{H})$ with spectrum $\sigma(A) \subseteq [0, \infty)$ and for every operator $X \in \mathcal{B}(\mathcal{H})$ with operator norm $\|X\| \leq 1$, the inequality $$f(X^* A X) \leq X^* f(A) X$$ holds in the Löwner order, where $f(\cdot)$ is defined via the continuous functional calculus.

Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that $f(0) \leq 0$. Suppose $f$ is operator convex on all complex Hilbert spaces $\mathcal{H}$; that is, for any $A, B \in \mathcal{B}(\mathcal{H})$ self-adjoint and $t \in [0, 1]$, the inequality $f((1 - t)A + tB) \leq (1 - t)f(A) + tf(B)$ holds in the Loewner partial order (where $f$ is defined via the continuous functional calculus). Then, $f$ satisfies Jensen's operator inequality for contractions: for any complex Hilbert space $\mathcal{H}$, any self-adjoint operator $A \in \mathcal{B}(\mathcal{H})$ with spectrum $\sigma(A) \subseteq [0, \infty)$, and any operator $X \in \mathcal{B}(\mathcal{H})$ with operator norm $\|X\| \leq 1$, it holds that $f(X^* A X) \leq X^* f(A) X$.