QuantumInfo.ForMathlib.HermitianMat.LiebConcavity

Let $d$ be a natural number. $\Phi$ is the $\mathbb{C}$-star-algebra isomorphism between the space of $d \times d$ complex matrices $\text{Mat}_d(\mathbb{C})$ and the space of continuous linear operators $\mathcal{L}(\mathbb{C}^d)$ on $d$-dimensional Euclidean space. This isomorphism identifies each matrix with its corresponding linear operator while preserving the addition, multiplication, scalar multiplication, and the star operation (Hermitian adjoint).

Let $d$ be a natural number. The star-algebra isomorphism $\Phi : \text{Mat}_d(\mathbb{C}) \to \mathcal{L}(\mathbb{C}^d)$, which identifies a complex $d \times d$ matrix with its corresponding continuous linear operator on the $d$-dimensional Euclidean space $\mathbb{C}^d$, is a continuous function.

Let $d$ be a natural number. Let $A$ be a $d \times d$ Hermitian matrix over $\mathbb{C}$ and $A_{\text{mat}}$ be its underlying matrix representation. Let $\Phi: \text{Mat}_d(\mathbb{C}) \cong \mathcal{L}(\mathbb{C}^d)$ be the star-algebra isomorphism that identifies a complex $d \times d$ matrix with its corresponding continuous linear operator on the $d$-dimensional Euclidean space $\mathbb{C}^d$. Then the linear operator $\Phi(A_{\text{mat}})$ is self-adjoint.

Let $d$ be a type and $A$ be a $d \times d$ complex Hermitian matrix. If $A$ is positive semidefinite (denoted as $0 \le A$ in the Loewner order), then the corresponding continuous linear operator $\Phi(A)$ acting on the $d$-dimensional Euclidean space $\mathbb{C}^d$ is also positive semidefinite (denoted as $0 \le \Phi(A)$).

Let $d$ be a non-empty index set and $\mathbb{C}^d$ denote the corresponding $d$-dimensional Euclidean space. Let $A$ be a $d \times d$ complex Hermitian matrix. If the underlying matrix of $A$ is positive definite (i.e., $A \succ 0$), then its image under the star-algebra isomorphism $\Phi$ is a strictly positive operator on $\mathbb{C}^d$, belonging to the set of operators with a strictly positive spectrum $(0, \infty)$.

Let $A$ be a $d \times d$ complex Hermitian matrix and $f: \mathbb{R} \to \mathbb{R}$ be a function. Let $\Phi: \text{Mat}_d(\mathbb{C}) \cong \mathcal{L}(\mathbb{C}^d)$ be the $\mathbb{C}$-star-algebra isomorphism that identifies a complex matrix with its corresponding bounded linear operator on $d$-dimensional Euclidean space. Then the continuous functional calculus commutes with $\Phi$, such that $$\Phi(f(A)) = f(\Phi(A)).$$

Let $A$ be a $d \times d$ complex Hermitian matrix such that $A$ is positive semidefinite ($A \ge 0$). For any real number $r \in \mathbb{R}$, let $A^r$ denote the real power of $A$ defined via functional calculus. Let $\Phi : \text{Mat}_d(\mathbb{C}) \cong \mathcal{L}(\mathbb{C}^d)$ be the star-algebra isomorphism identifying a complex matrix with its corresponding continuous linear operator on the $d$-dimensional Euclidean space. Then the following identity holds: $$\Phi(A^r) = (\Phi(A))^r$$

For any complex $d \times d$ matrix $M$, the trace of the linear operator $\Phi(M)$ on the $d$-dimensional Euclidean space $\mathbb{C}^d$ is equal to the matrix trace of $M$, that is: $$\text{Tr}(\Phi(M)) = \text{tr}(M)$$ where $\Phi: \text{Mat}_d(\mathbb{C}) \to \mathcal{L}(\mathbb{C}^d)$ is the star-algebra isomorphism between the space of complex matrices and continuous linear operators.

Let $M \in \text{Mat}_d(\mathbb{C})$ be a complex $d \times d$ matrix. Let $\Phi : \text{Mat}_d(\mathbb{C}) \cong \mathcal{L}(\mathbb{C}^d)$ be the star-algebra isomorphism that identifies a matrix with its corresponding continuous linear operator on $d$-dimensional Euclidean space. Then the real part of the trace of the operator $\Phi(M)$ is equal to the real part of the trace of the matrix $M$: $$\text{Re}(\text{Tr}(\Phi(M))) = \text{Re}(\text{Tr}(M))$$

Let $H$ be a $d \times d$ positive semidefinite Hermitian matrix over the complex numbers $\mathbb{C}$ (i.e., $H \ge 0$). For any real number $\alpha > 1$, the function mapping a positive semidefinite Hermitian matrix $\sigma$ to the trace functional $$\sigma \mapsto \text{Tr} \left[ \left( \sigma^{\frac{\alpha - 1}{2\alpha}} H \sigma^{\frac{\alpha - 1}{2\alpha}} \right)^{\frac{\alpha}{\alpha - 1}} \right]$$ is concave on the set of positive semidefinite matrices $\{\sigma \in \text{Herm}_d(\mathbb{C}) \mid \sigma \ge 0\}$.