QuantumInfo.ForMathlib.HayataGroup.TraceInequality.JensenOperatorInequalityIVtoV

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 algebra $\mathcal{B}(\mathcal{H}) \times \mathcal{B}(\mathcal{H})$ admits a continuous functional calculus over the complex numbers $\mathbb{C}$ for normal elements.

Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ be the space of bounded linear operators on $\mathcal{H}$. There exists a continuous functional calculus for real-valued functions on the product space $\mathcal{B}(\mathcal{H}) \times \mathcal{B}(\mathcal{H})$ for elements $(A, B)$ that are self-adjoint.

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 space of bounded linear operators $\mathcal{B}(\mathcal{H} \oplus \mathcal{H})$ is a non-negative spectrum class over $\mathbb{R}$. This implies that an operator $T \in \mathcal{B}(\mathcal{H} \oplus \mathcal{H})$ is positive if and only if its spectrum $\sigma(T)$ is contained in the set of non-negative real numbers $[0, \infty)$.

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 holds if for every self-adjoint operator $A \in \mathcal{B}(\mathcal{H})$ whose spectrum $\sigma(A)$ is contained in $[0, \infty)$ (i.e., $A$ is a positive operator) and every contraction $X \in \mathcal{B}(\mathcal{H})$ (i.e., $\|X\| \le 1$), the following operator inequality holds: $$f(X^* A X) \le X^* f(A) X$$ where $f(A)$ and $f(X^* A X)$ are defined via the continuous functional calculus for self-adjoint operators, and the inequality $\le$ denotes the Loewner partial order.

For a function $f: \mathbb{R} \to \mathbb{R}$, this property holds if for every non-trivial complex Hilbert space $\mathcal{H}$, the Jensen operator inequality is satisfied. That is, for every self-adjoint operator $A \in \mathcal{B}(\mathcal{H})$ whose spectrum $\sigma(A)$ is contained in $[0, \infty)$ and every contraction $X \in \mathcal{B}(\mathcal{H})$ (such that $\|X\| \le 1$), the inequality $$f(X^* A X) \le X^* f(A) X$$ holds, where $f(\cdot)$ denotes the continuous functional calculus and $\le$ denotes the Loewner partial order.

For a real-valued function $f: \mathbb{R} \to \mathbb{R}$, this property holds if for any self-adjoint operators $A, B \in \mathcal{B}(\mathcal{H})$ whose spectra are contained in $[0, \infty)$, and any bounded linear operators $X, Y \in \mathcal{B}(\mathcal{H})$ satisfying $X^* X + Y^* Y \le I$ (where $I$ is the identity operator), the following operator inequality is satisfied: $$f(X^* A X + Y^* B Y) \le X^* f(A) X + Y^* f(B) Y$$ Here, $f(\cdot)$ denotes the continuous functional calculus for self-adjoint operators, and $X^*$ denotes the adjoint of $X$.

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function. Suppose that for every complex Hilbert space $\mathcal{H}$, every self-adjoint operator $A \in \mathcal{B}(\mathcal{H})$ with spectrum $\sigma(A) \subseteq [0, \infty)$, and every contraction $X \in \mathcal{B}(\mathcal{H})$ (such that $\|X\| \le 1$), the inequality $$f(X^* A X) \le X^* f(A) X$$ holds, where $f(\cdot)$ is defined via the continuous functional calculus and $\le$ denotes the Loewner partial order. Then, for any self-adjoint operators $A, B \in \mathcal{B}(\mathcal{H})$ with spectra contained in $[0, \infty)$ and any bounded linear operators $X, Y \in \mathcal{B}(\mathcal{H})$ satisfying $X^* X + Y^* Y \le I$, the following inequality holds: $$f(X^* A X + Y^* B Y) \le X^* f(A) X + Y^* f(B) Y.$$