QuantumInfo.ForMathlib.HermitianMat.Peierls

Let $d$ be a finite type and $A$ be a $d \times d$ Hermitian matrix over the complex numbers $\mathbb{C}$. For any function $g : \mathbb{R} \to \mathbb{R}$ and any unitary matrix $U$ of size $d$, the trace of the matrix obtained by applying the continuous functional calculus of $g$ to the unitarily conjugated matrix $U A U^\dagger$ is equal to the trace of the matrix obtained by applying the continuous functional calculus of $g$ directly to $A$. That is, $\text{Tr}(g(U A U^\dagger)) = \text{Tr}(g(A))$.

Let $d$ be a finite index set and $A$ be a $d \times d$ Hermitian matrix with complex entries. If $g: \mathbb{R} \to \mathbb{R}$ is a convex function, then the sum of $g$ applied to the real parts of the diagonal entries of $A$ is less than or equal to the trace of the matrix $g(A)$ (the result of the continuous functional calculus of $g$ applied to $A$): $$\sum_{i \in d} g(\text{Re}(A_{ii})) \le \text{Tr}(g(A))$$

Let $A$ be a $d \times d$ complex Hermitian matrix that is positive semidefinite ($A \ge 0$). Let $g: \mathbb{R} \to \mathbb{R}$ be a function that is convex on the interval $[0, \infty)$. Then the sum of $g$ applied to the real parts of the diagonal entries of $A$ is less than or equal to the trace of the matrix $g(A)$ obtained via continuous functional calculus: $$\sum_{i} g(\text{Re}(A_{ii})) \le \text{Tr}(g(A))$$ Note that since $A$ is Hermitian, its diagonal entries $A_{ii}$ are already real, so $\text{Re}(A_{ii}) = A_{ii}$.

Let $g: \mathbb{R} \to \mathbb{R}$ be a convex function. The mapping $A \mapsto \text{tr}(g(A))$ is convex on the space of $d \times d$ complex Hermitian matrices, where $g(A)$ is defined by the continuous functional calculus of $g$ applied to $A$.

Let $g : \mathbb{R} \to \mathbb{R}$ be a function that is convex on the interval $[0, \infty)$. Then the map $A \mapsto \text{Tr}(g(A))$, which assigns to each positive semidefinite Hermitian matrix $A \in \text{Herm}_d(\mathbb{C})$ the trace of the matrix obtained by applying the functional calculus of $g$ to $A$, is convex on the set of positive semidefinite matrices $\{A \in \text{Herm}_d(\mathbb{C}) \mid A \ge 0\}$.