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### 2.2.1: Quantum basic structure$[{\mathcal C}(H) \subseteq B(H) \subseteq B(H)]$: Compact operator, Trace operator

In quantum system, the basic structure$[{\mathcal A} \subseteq \overline{\mathcal A} \subseteq B(H)]$ is characterized as

\begin{align} [{\mathcal C}(H) \subseteq B(H) \subseteq B(H)] \tag{2.7} \end{align} That is, we see:
Quantum basic structure:$[{\mathcal C}(H) \subseteq B(H) \subseteq B(H)]$
$\require{AMScd}$ \begin{align} \begin{CD} {\mathcal Tr}(H) @. {} @. {} \\ @AA{\mbox{ dual}}A @. \\ \quad \fbox{${\mathcal C}(H)$} \quad @>{\subseteq}>\mbox{ subalgebra$\cdot$weak-closure}> \quad \fbox{$B(H)$}\quad @>{\subseteq}>\mbox{ subalgebra}> \quad \fbox{${B(H)}$}\quad \\ @. @VV{\mbox{ pre-dual}}V \\ {} @. {\mathcal Tr}(H) @. {} \\ \label{eq2.8} \end{CD} \end{align}

where "compact operators class ${\mathcal C}(H)$" and "trace class ${\mathcal Tr}(H)$"

Before we explain "compact operators class ${\mathcal C}(H)$" and "trace class ${\mathcal Tr}(H)$" in Theorem 2.6 and 2.7, we have to prepare "Dirac notation" and "CONS" as follows.

Definition 2.5 [(i): Dirac notation]

Let $H$ be a Hilbert space. For any $u, v \in H$, define $| u \rangle \langle v | \in B(H)$ such that

\begin{align} (| u \rangle \langle v |)w = \langle v , w \rangle u \quad (\forall w \in H) \end{align}

Here, $\langle v |$ $\big[$ resp. $| u \rangle$ $\big]$ is called the "Bra-vector" $\big[$ resp. "Ket-vector"$\big]$.

[(ii):ONS(orthonormal system), CONS(complete orthonormal system)] The sequence $\{e_k\}_{k=1}^\infty$ in a Hilbert space $H$ is called an orthonormal system (i.e., ONS), if it satisfies
 $(\sharp)$ $\langle e_k, e_j \rangle = \left\{\begin{array}{ll} 1 \quad & (k = j ) \\ 0 \quad & (k \not= j ) \end{array}\right.$

In addition, an ONS $\{e_k\}_{k=1}^\infty$ is called a complete orthonormal system (i.e., CONS), if it satisfies

 $(\sharp)$ $\langle x, e_k \rangle =0 \; (\forall k=1,2,...)$ implies that $x=0$.

Theorem 2.6 [The properties of compact operators class ${\mathcal C}(H)$]
Let ${\mathcal C}(H) (\subseteq B(H))$ be the compact operators class. Then, we see the following (C$_1$)-(C$_4$) $\Big($ particularly, "(C$_1$)$\leftrightarrow$ (C$_2$)" may be regarded as the definition of the compact operators class ${\mathcal C}(H) (\subseteq B(H))$ $\Big)$.
 $(C_1)$: $T \in {\mathcal C}(H)$.That is, for any bounded sequence $\{u_n \}_{n=1}^\infty$ in Hilbert space $H$, $\{Tu_n \}_{n=1}^\infty$ has the subsequence which converges in the sense of the norm topology. $(C_2)$: There exist two ONSs $\{e_k\}_{k=1}^\infty$ and $\{f_k\}_{k=1}^\infty$ in the Hilbert space $H$ and a positive real sequence $\{\lambda_k \}_{k=1}^\infty$ (where, $\lim_{k \to \infty } \lambda_k =0$ ) such that \begin{align} T=\sum_{k=1}^\infty \lambda_k |e_k \rangle \langle f_k| \qquad (\mbox{in the sense of weak topology}) \tag{2.10} \end{align} $(C_3)$: ${\mathcal C}(H)( \subseteq B(H))$ is a $C^*$-algebra. When $T (\in {\mathcal C}(H))$ is represented as in (C$_2$), the following equality holds \begin{align} \| T \|_{B(H)}= \max_{k=1,2, \cdots } \lambda_k \tag{2.11} \end{align} $(C_4)$: The weak closure of ${\mathcal C}(H)$ is equal to $B(H)$. That is, \begin{align} \overline{{\mathcal C}(H)}=B(H) \label{2.12} \end{align}
Theorem 2.7 [The properties of trace class ${\mathcal Tr}(H)$]
Let ${\mathcal Tr}(H) (\subseteq B(H))$ be the trace class. Then, we see the following (D$_1$)-(D$_4$)( particularly, "(D$_1$)$\leftrightarrow$ (D$_2$)" may be regarded as the definition of the trace class ${\mathcal Tr}(H) (\subseteq B(H))$).
 $(D1)$: $T \in {\mathcal Tr}(H) (\subseteq {\mathcal C}(H) \subseteq B(H))$. $(D2)$: There exist two ONSs $\{e_k\}_{k=1}^\infty$ and $\{ f_k \}_{ k=1}^\infty$ in the Hilbert space $H$ and a positive real sequence $\{\lambda_k \}_{k=1}^\infty$ (where, $\sum_{k=1}^{ \infty } \lambda_k < \infty$) such that \begin{align*} T=\sum_{k=1}^\infty \lambda_k |e_k \rangle \langle f_k| \qquad (\mbox{in the sense of weak topology}) \end{align*} $(D3)$: It holds that \begin{align} {\mathcal C}(H)^*={\mathcal Tr}(H) \tag{2.13} \end{align} Here, the dual norm $\| \cdot \|_{{\mathcal C}(H)^*}$ is characterized as the trace norm $\| \cdot \|_{Tr}$ such as \begin{align} \|T\|_{Tr}= \sum_{k=1}^\infty \lambda_k \tag{2.14} \end{align} when $T (\in {\mathcal Tr}(H))$ is represented as in (D$_2$), $(D4)$: Also, it holds that \begin{align} {\mathcal Tr}(H)^*=B(H) \qquad \mbox{in the same sense,} \qquad {\mathcal Tr}(H)=B(H)_* \label{2.15} \end{align}
Remark 2.8.[finite dimensional Hilbert space] Assume that a Hilbert space $H$ is finite dimensional, i.e., $H={\mathbb C}^n$, i.e., \begin{align} {\mathbb C}^n=\{ z= \left[\begin{array}{l} z_1 \\ z_2 \\ \vdots \\ z_n \end{array}\right] \;|\; z_k \in {\mathbb C}, k=1,2,...,n \} \end{align} Put \begin{align*} M({\mathbb C},n)=\mbox{The set of all $(n\times n)$-complex matrices} \end{align*} and thus, \begin{align} {\mathcal A}=\overline{\mathcal A}=B({\mathbb C}^n)={\mathcal C}(H)= {\mathcal Tr}(H)=M({\mathbb C},n) \tag{2.16} \end{align} However, it should be noted that the norms are different as mentioned in (C$_3$) and (D$_3$).

### 2.2.2 Quantum basic structure$[{\mathcal C}(H) \subseteq B(H) \subseteq B(H)]$ and State space

Consider the quantum basic structure: \begin{align} [{\mathcal C}(H) \subseteq B(H) \subseteq B(H)] \end{align} and see the following diagram:
(E): Quantum basic structure and state space
\begin{align} & \begin{array}{rlrlll} \underset{{{\mbox{ $C^*$-pure state}}}}{ {{\frak S}^p({\mathcal Tr}(H))}} \subset \underset{{\mbox{ $C^*$-mixed state}}}{ {{\frak S}^m({\mathcal Tr}(H))}} \subset & {\mathcal Tr}(H) &&&& \\ & \Big\uparrow \mbox{ dual} &&&& \\ & \fbox{${\mathcal C}(H)$} & \xrightarrow[\mbox{ subalgebra$\cdot$weak-closure}]{\subseteq} & \fbox{${B(H)}$} & \xrightarrow[\mbox{ subalgebra}]{\subseteq} \fbox{${B(H)}$} & \\ & && \Big\downarrow \;\mbox{ pre-dual} && \end{array} \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \underset{{\mbox{ $W^*$-mixed state}}}{\overline{\frak S}^m({\mathcal Tr}(H))} \subset {\mathcal Tr}(H) \nonumber \end{align}
In what follows, we shall explain the above diagram. Firstly, we note that \begin{align} {\mathcal C}(H)^*= {\mathcal Tr}(H), \qquad {\mathcal Tr}(H)^* = B(H) \label{2.18} \end{align} and \begin{align} & { {\frak S}^m({\mathcal Tr}(H))=\overline{\frak S}^m({\mathcal Tr}(H)) } \nonumber \\ = & \{ \rho = \sum_{n=1}^\infty \lambda_n | e_n \rangle \langle e_n | \;\;: \;\; \{e_n \}_{n=1}^\infty \mbox{ is ONS ,{\;} }\sum_{n=1}^\infty \lambda_n=1, \lambda_n > 0 \} \nonumber \\ =: & {\mathcal Tr}_{+1}(H) \tag{2.19} \end{align} Also, concerning the pure state space, we see: \begin{align} & {\frak S}^p({\mathcal Tr}(H)) \nonumber \\ = & \{ \rho= | e \rangle \langle e | \;\; : \;\; \|e\|_H=1 \} =:{\mathcal Tr}_{+1}^p(H) \tag{2.20} \end{align} Therefore, under the following identification: \begin{align} {\frak S}^p({\mathcal Tr}(H)) \ni |u \rangle \langle u | \underset{\mbox{ identification}}{\longleftrightarrow} u \in H \qquad(\|u\|=1) \tag{2.21} \end{align} we see,{\;} \begin{align} {\frak S}^p({\mathcal Tr}(H)) = \{ u \in H \;:\; \|u\|=1 \} \tag{2.22} \end{align} where we assume the equivalence: $u \approx e^{i\theta}u (\theta \in {\mathbb R})$.
Definition 2.9 [Tr: trace]. Define the trace ${\mbox{Tr}}: {\mathcal Tr}(H) \to {\mathbb C}$ such that \begin{align} {\mbox{Tr}}(T) = \sum_{n=1}^\infty \langle e_n, T e_n \rangle \qquad (\forall T \in {\mathcal Tr}(H) ) \end{align} where $\{e_n\}_{n=1}^\infty$ is a CONS in $H$. It is well known that the ${\mbox{Tr}}(T)$ does not depend on the choice of CONS $\{e_n\}_{n=1}^\infty$. Thus, clearly we see that \begin{align} {}_{{}_{{\mathcal Tr}{H}}}\Big( |u \rangle \langle u |, F \Big){}_{{}_{B(H)}} = {\mbox{Tr}} ( |u \rangle \langle u | \cdot F ) = \langle u, F u \rangle \quad (\forall ||u||_H=1, F \in B(H) ) \end{align}
Remark 2.10 Assume that a Hilbert space $H$ is finite dimensional, i.e., $H={\mathbb C}^n$. Then, \begin{align*} M({\mathbb C},n)=\mbox{The set of all $(n\times n)$-complex matrices} \end{align*} That is, \begin{align} F= \left[\begin{array}{llll} f_{11} & f_{12} & \cdots & f_{1n}\\ f_{21} & f_{22} & \cdots & f_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ f_{n1} & f_{n2} & \cdots & f_{nn} \end{array}\right] \in M({\mathbb C},n) \label{eq2.25} \end{align} As mentioned before, we see \begin{align} {\mathcal A}=\overline{\mathcal A}=B({\mathbb C}^n)={\mathcal C}(H)= {\mathcal Tr}(H)=M({\mathbb C},n) \label{eq2.26} \end{align} and further, under the following notations: \begin{align*} & {\mathcal Tr}_{+1}^D({\mathbb C}^n) = \Big\{ \mbox{diagonal matrix}F= \left[\begin{array}{llll} f_{11} & 0 & \cdots & 0\\ 0 & f_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & f_{nn} \end{array}\right] \;\; \Big| \;\; f_{kk} \ge 0, \;\; \sum_{k=1}^n f_{kk}=1 \Big\} \\ & {\mathcal Tr}_{+1}^{DP}({\mathbb C}^n) = \Big\{ F= \left[\begin{array}{llll} f_{11} & 0 & \cdots & 0\\ 0 & f_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & f_{nn} \end{array}\right] \in {\mathcal Tr}_{+1}^D({\mathbb C}^n) \;\; \Big| \;\; f_{kk} = 1 \; (\mbox{for some } k=j ), =0 \; ( k \not=j ) \Big\} \end{align*} We see, \begin{align} & \mbox{mixed state space: }{\mathcal Tr}_{+1}({\mathbb C}^n) = \Big\{ U F U^\ast \;\; : \;\; F \in {\mathcal Tr}_{+1}^D ({\mathbb C}^n), \; \mbox{$U$ is a unitary matrix} \Big\} \\ & \mbox{pure state space: }{\mathcal Tr}_{+1}^p({\mathbb C}^n) = \Big\{ U F U^\ast \;\; : \;\; F \in {\mathcal Tr}_{+1}^{DP} ({\mathbb C}^n), \; \mbox{$U$ is a unitary matrix} \Big\} \end{align}