QuantumInfo.ForMathlib.HayataGroup.TraceInequality.OperatorGeometricMean

Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ be the space of bounded linear operators on $\mathcal{H}$. For real numbers $\alpha, \beta$ and operators $A, B \in \mathcal{B}(\mathcal{H})$, the $(\alpha, \beta)$-power mean of $A$ and $B$ is defined as the generalized perspective of the functions $f(x) = x^\alpha$ and $h(x) = x^\beta$. Specifically, it is the operator: $$B^{\beta/2} \left( B^{-\beta/2} A B^{-\beta/2} \right)^\alpha B^{\beta/2}$$ where the operator powers are defined via the continuous functional calculus. This is typically applied when $B$ is a positive invertible operator and $A$ is Hermitian.

Let $\mathcal{H}$ be a complex inner product space and $L(\mathcal{H})$ be the space of bounded linear operators on $\mathcal{H}$. Let $\mathcal{P}(\mathcal{H}) \subset L(\mathcal{H})$ denote the set of strictly positive operators (self-adjoint operators with strictly positive spectrum). For any real numbers $\alpha$ and $\beta$ such that $0 \le \alpha \le 1$ and $0 \le \beta \le 1$, the operator $(\alpha, \beta)$-power mean $\Phi(A, B) = B^{\beta/2} (B^{-\beta/2} A B^{-\beta/2})^\alpha B^{\beta/2}$ is jointly concave on $\mathcal{P}(\mathcal{H}) \times \mathcal{P}(\mathcal{H})$. That is, for all $A_1, A_2, B_1, B_2 \in \mathcal{P}(\mathcal{H})$ and any scalar $\theta \in [0, 1]$, the following inequality holds in the Löwner order: $$(1 - \theta) \Phi(A_1, B_1) + \theta \Phi(A_2, B_2) \le \Phi((1 - \theta) A_1 + \theta A_2, (1 - \theta) B_1 + \theta B_2)$$

Let $\mathcal{H}$ be a complex inner product space and $\mathcal{B}(\mathcal{H})$ be the space of bounded linear operators on $\mathcal{H}$. Let $\mathcal{P} \subset \mathcal{B}(\mathcal{H})$ denote the set of strictly positive operators (self-adjoint operators with spectrum contained in $(0, \infty)$). For real numbers $\alpha$ and $\beta$ such that $1 \le \alpha \le 2$ and $0 \le \beta \le 1$, the $(\alpha, \beta)$-power mean of $A, B \in \mathcal{P}$, defined by $$\Phi(A, B) = B^{\beta/2} \left( B^{-\beta/2} A B^{-\beta/2} \right)^\alpha B^{\beta/2},$$ is jointly convex on $\mathcal{P} \times \mathcal{P}$. That is, for all $A_1, A_2, B_1, B_2 \in \mathcal{P}$ and $\theta \in [0, 1]$, the following operator inequality holds: $$\Phi((1 - \theta)A_1 + \theta A_2, (1 - \theta)B_1 + \theta B_2) \le (1 - \theta)\Phi(A_1, B_1) + \theta\Phi(A_2, B_2).$$