PhyslibAlpha.ProbabilisticTheory.HilbertSpace.TraceClass.Pairing

Let $H$ be a complex Hilbert space and let $T: H \to H$ be a trace-class bounded linear operator. Then the absolute value of the trace of $T$ is bounded by its trace norm: $$|\operatorname{Tr}(T)| \le \|T\|_1$$ where $\operatorname{Tr}(T)$ is the trace of $T$ and $\|T\|_1$ is its trace norm.

Let $H$ be a complex Hilbert space and let $X$ and $Y$ be bounded linear operators on $H$. If $X = Y$ and $X$ is a trace-class operator (meaning the sum of the diagonal entries of its absolute value converges), then $Y$ is also a trace-class operator and their traces are equal: $$\operatorname{Tr}(X) = \operatorname{Tr}(Y)$$ where the trace $\operatorname{Tr}(T)$ is defined as the sum of the diagonal entries $\sum_{i \in I} \langle e_i, T e_i \rangle$ for a Hilbert basis $\{e_i\}_{i \in I}$.

Let $H$ be a complex Hilbert space. For any bounded linear operator $A \in \mathcal{B}(H)$ and any trace-class operator $T \in \mathcal{S}_1(H)$, the composition $AT$ is also a trace-class operator. This demonstrates that the set of trace-class operators forms a left ideal within the algebra of bounded linear operators.

Let $H$ be a complex Hilbert space. For a fixed bounded linear operator $A \in \mathcal{B}(H)$, the **trace pairing linear map** is the $\mathbb{C}$-linear functional on the trace-class Banach space $\mathcal{S}_1(H)$ defined by: $$T \mapsto \operatorname{Tr}(AT)$$ where $T \in \mathcal{S}_1(H)$ and $\operatorname{Tr}$ denotes the trace of the operator product.

Let $H$ be a complex Hilbert space. For any bounded linear operator $A \in \mathcal{B}(H)$ and any trace-class operator $T \in \mathcal{S}_1(H)$, the value of the trace pairing linear functional $\text{tracePairingLinearMap}(A)$ applied to $T$ is equal to the trace of the operator product $AT$: $$\text{tracePairingLinearMap}(A)(T) = \operatorname{Tr}(AT)$$

Let $H$ be a complex Hilbert space. For any bounded linear operator $A \in \mathcal{B}(H)$ and any trace-class operator $T \in \mathcal{S}_1(H)$, the absolute value of the trace of their product is bounded by the product of the operator norm of $A$ and the trace norm of $T$: $$|\operatorname{Tr}(AT)| \le \|A\| \|T\|_1$$

Let $H$ be a complex Hilbert space and let $\mathcal{B}(H)$ denote the space of bounded linear operators on $H$. For a fixed operator $A \in \mathcal{B}(H)$, the **trace pairing** is the continuous (bounded) linear functional on the trace-class Banach space $\mathcal{S}_1(H)$ defined by the mapping: $$T \mapsto \operatorname{Tr}(AT)$$ where $T \in \mathcal{S}_1(H)$ and $\operatorname{Tr}$ denotes the operator trace. This functional is bounded with respect to the trace norm $\|\cdot\|_1$.

Let $H$ be a complex Hilbert space. For any bounded linear operator $A \in \mathcal{B}(H)$ and any trace-class operator $T \in \mathcal{S}_1(H)$, the value of the trace pairing functional associated with $A$ evaluated at $T$ is equal to the trace of the product $AT$: $$\text{tracePairing}(A)(T) = \operatorname{Tr}(AT)$$ where $\operatorname{Tr}$ denotes the operator trace.

Let $H$ be a complex Hilbert space and let $A \in \mathcal{B}(H)$ be a bounded linear operator. The operator norm of the continuous linear functional on the trace-class Banach space $\mathcal{S}_1(H)$ defined by the trace pairing $T \mapsto \operatorname{Tr}(AT)$ is bounded by the operator norm of $A$: $$\|\operatorname{Tr}(A \cdot)\| \leq \|A\|$$ where $\|\cdot\|$ on the left denotes the norm in the dual space $\mathcal{S}_1(H)^*$.

Let $H$ be a complex Hilbert space. Let $\mathcal{B}(H)$ denote the space of bounded linear operators on $H$, and let $\mathcal{S}_1(H)$ be the trace-class Banach space. This definition is the $\mathbb{C}$-linear map from $\mathcal{B}(H)$ to the space of continuous linear functionals on $\mathcal{S}_1(H)$, which sends an operator $A \in \mathcal{B}(H)$ to the trace pairing functional $T \mapsto \operatorname{Tr}(AT)$.

Let $H$ be a complex Hilbert space. This definition is the continuous linear map from the space of bounded linear operators $\mathcal{B}(H)$ to the continuous dual space of trace-class operators $\mathcal{S}_1(H)^*$, which sends each operator $A \in \mathcal{B}(H)$ to the trace pairing functional $T \mapsto \text{Tr}(AT)$ for $T \in \mathcal{S}_1(H)$.

Let $H$ be a complex Hilbert space. For any bounded linear operator $A \in \mathcal{B}(H)$, the value of the continuous linear map $\mathcal{B}(H) \to \mathcal{S}_1(H)^*$ (which maps $A \mapsto \operatorname{Tr}(A \cdot)$) evaluated at $A$ is equal to the trace pairing functional $T \mapsto \operatorname{Tr}(AT)$.