An infinite dimensional Birhkoff's Theorem, a majorization relation for two density matrices and LOCC-convertibility (The research of geometric structures in quantum information based on Operator Theory and related topics)

全文

(1)142. 数理解析研究所講究録第2033巻 2017年 142-146. An infinite dimensional Birhkoffs relation for two. density. matrices. Theorem, a majorization and LOCC‐convertibility. Daiki Asakura1. 1Graduate School of Information Systems, University of Electro‐Communications, 1‐5‐1. Chofugaoka, Chofu‐shi, Tokyo, 182‐8585, Japan.. Introduction. 1. In quantum mechanics, entanglement between bipartite system is known as quantum correlations which do not arise in classical systems. With entanglement, we can consider useful tasks that can never be. accomplished by classical systems, such as quantum teleportation and quantum dense coding. For this reason, entanglement has been regarded as a resource in quantum information theory. If a state | $\psi$\rangle\{ $\psi$| \in S(\mathcal{H}\otimes \mathcal{K}) on a bipartite system \mathcal{H}\otimes \mathcal{K} is incomplete as an entanglement resource, one may want to convert it into a more entangled form | $\phi$\rangle( $\phi$|\in S(\mathcal{H}\otimes \mathcal{K}) However, if two particles are far apart from each other, it is difficult to apply full quantum operations that is allowed theoretically in the composite system. Instead, as a practical class of quantum operations, local operations and classical communications (LOCC) play an important role in this situation. For the LOCC‐convertibility states, Nielsen proved [11, 12] in 1999 that the following statements are equivalent (the Nielsens theorem). .. (i). (ii). The initial state. | $\psi$\rangle. can. be converted to the target state. The Schmidt coefficients of the initial state. Mathematically,. the Nielsens theorem. Theorem 1.1.. (Nielsenlll, 121). be unit vectors. Then, the \bullet. One. can. convert. | $\psi$\rangle. can. are. | $\phi$\rangle by. the. to. unitary operators \{U_{i}\}_{i}. on. is. be written. Let \mathcal{H} and \mathcal{K} be. following. | $\psi$ ). majorized by. as. finite equivalent.. | $\phi$) by. LOCC. those of the target state. follows.. dimensional Hilbert spaces, and let. following LOCC:. there exist. a. POVM. \mathcal{K} such that. | $\phi$\displaystyle \rangle\langle $\phi$|=\sum_{i}(M_{i}\otimes U_{i})| $\psi$\rangle\langle $\psi$|(M_{i}^{*}\otimes U_{i}^{*}) where the .. The. sum. is. finite. | $\phi$\rangle.. ,. \{M_{i}\}_{i}. on. $\psi$, $\phi$\in \mathcal{H}\otimes \mathcal{K}. \mathcal{H} and. a. set. of. (1). sum.. followzng majorization. relation holds:. \mathrm{T}\mathrm{r}_{\mathcal{K} | $\psi$\rangle\langle $\psi$|\prec \mathrm{T}\upar ow $\kap a$| $\phi$\rangle\langle $\phi$|. majorization condition (ii) fully characterizes the LOCC‐convertibility of pure states in systems. Subsequently, in 2006, Owari et at. [13] extended the Nielsens theorem to infinite dimensional systems. They proved that the implication (i) \Rightarrow (ii) (necessary condition for the LOCC‐convertibility) holds in the same form as finite dimensional systems. Moreover, Owari et al. [13] introduced a notion of $\epsilon$convertibility by LOCC in infinite dimensional systems and proved that $\epsilon$ ‐convertibility for LOCC gives. Namely,. the. finite dimensional. a. characterization of the sufficient condition.. implication (ii) \Rightarrow (i) (the sufficient condition for LOCC‐ systems. To solve this problem, in [1], we develop an infinite dimensional analogue of Birkhoffs theorem [Theorem 2.2] and use this to prove the following theorem. However, it has been. convertibility). holds. or. open whether the. not in infinite dimensional.

(2) 143. (Asakura Ỉll). Theorem 1.2.. $\phi$\in \mathcal{H}\otimes \mathcal{K} be hll rank. Let \mathcal{H} and \mathcal{K} be. unit vectors.. infinite dimensional separable Habert Then, the following are equivalent.. spaces, and let. $\psi$,. There evist a Borel set I of a certain of metrec space, a probability mesure $\mu$ on I a set of densely defined (not necessarily bounded) operators \{M_{i}\}_{i\in I} on \mathcal{H} a dense subspace \mathcal{H}_{0}\subset \mathcal{H} and a set of unitary operators \{U_{$\iota$'}\}_{i\in I} on \mathcal{K} such that. \bullet. ,. ,. ,. | $\psi$\rangle\in D(M_{i}\otimes U_{i}) for i\in I (\mathrm{T}\mathrm{n}_{\mathcal{K} | $\psi$\rangle\langle $\psi$|)\mathcal{H}_{0}\subset \mathcal{H}0 i e \{(\mathrm{T}\mathrm{r}_{\mathcal{K} | $\psi$\rangle\langle $\psi$|)| $\eta$\rangle : $\eta$\in \mathcal{H}_{0}\}\subset \mathcal{H}_{0} D(M_{i})\supset \mathcal{H}_{0} for any i\in I ,. .. ,. (2) (3) (4). ,. .,. ,. ,. l\langle $\eta$|M_{i}^{*}M_{i}| $\xi$\rangle d $\mu$(i)=\langle $\eta$| $\xi$). ,. for. $\eta$,. $\xi$\in \mathcal{H}_{0}. (5). ,. I\ni i\mapsto(M_{i}\otimes U_{i})| $\psi$\rangle\langle $\psi$|(M_{i}^{*}\otimes U_{i}^{*})\in \mathfrak{C}_{1}(\mathcal{H}). | $\phi$)\displaystyle \langle $\phi$|=\int_{I}(M_{i}\otimes U_{i})| $\psi$\rangle\langle $\psi$|(M_{i}^{*}\otimes U_{i}^{*})d $\mu$(i) \mathrm{H}_{\mathcal{K} | $\psi$\rangle\langle $\psi$|\prec \mathrm{T}\mathrm{r}_{\mathcal{K} | $\phi$)\langle $\phi$|. \bullet. Remark 1.3. Note that in the. case. I is. finite. (7). ,. is. integrable, and. (6). in. \mathfrak{C}_{1}(H). (7). .. holds.. becomes. set and. M_{i}. are. (1). and. (5). becomes. equality for. an. a. POVM with. finite cardinality. all bounded.. In this paper, we introduce a new characterization for majorization relation between two density ma‐ is the characterization derived from our infinite dimensional analogue of Birkhoffs theorem. trices, which. [Theorem 2.2].. This paper is organized as follows. In Section 2, we introduce an infinite dimensional analogue of Birkhoffs theorem. In Section 3, we give a sketch of the proof of the sufficient condition of Theorem 1.1. In Section 4, we give a new characterization for majorization relation between two density matrices.. Infinite dimensional Birkhoffs theorem with WOT. 2. Let \mathcal{H} be. a. separable Hilbert. \mathcal{P}(\mathcal{H}^{(|i\rangle)}). D(\mathcal{H}^{(|i\rangle)}) Remark 2.1.. space and. (|i\rangle)_{i=1}^{\infty}. { \displaystyle\sum_{i,j=1}^{\infty}a_{ij}|\rangle\langlej|\in\mathfrak{B}(\mathcal{H}) :=\displaystyle\{ sum_{i J'=}1^{\infty}a_{ij}|)\langlej|\in\mathfrak{B}(\mathcal{H}) :=. When \mathcal{H}=\mathbb{C}^{n} and. (|i\rangle)_{i}. be. CONS in \mathcal{H} We. a. a_{ij}=0. .. a_{ij}\in[0 1], ,. u a. following. \displayst le\sum_{\mathrm{j}=1}^{\infty}a_{ij}=1, \displaystyle\sum_{i=1}^{\infty}a_{ij}=1 \displaystyle\sum_{j=1}^{\infty}a_{ij}=1, \displaystyle\sum_{i=1}^{\infty}a_{ij}=1. (for. (for. standard basis. n\times n. the. 1,. or. permutation matrices and D(\mathcal{H}^{(|i\rangle)}) is equal In the sequel, we abbreviate D(\mathbb{C}^{n(1e_{*}\rangle)}) as D(\mathbb{C}^{n}) all. use. notation.. }, i,j)\}.. any. any. i,j). in \mathbb{C}^{n}, \mathcal{P}(\mathcal{H}^{(|i\rangle)}) is equal to the set of of all n\times n doubly stochastic matrices.. (e_{i})_{i}. to the set. .. Using. (1). ex. (2). Let. the notations in the. D(\mathbb{C}^{n})=\prime P(\mathbb{C}^{n}). previous section, Birkhoffs theorem [5]. can. be written. as. follows:. ,. \{P_{i}\}_{i=1}^{n!} :=\mathcal{P}(\mathbb{C}^{n}). such that. .. Then for any. D\in D(\mathbb{C}^{ $\gamma$}). ,. there exists. a. probability. mass. function. \{p_{i}\}_{i=1}^{n!}. D=\displaystyle\sum_{i=1}^{n!}p_{i}P_{i}, (3) D(\mathbb{C}^{n})=\mathrm{c}\mathrm{o}P(\mathbb{C}^{n}). .. Note that it is known that the three assertions see. [4,. Section. II.2].. For the property. (2),. we. are. equivalent. proved the following theorem.. to each. other, by Carathéodry theorem;.

(3) 144. (Asakura Ĩll). Theorem 2.2.. D\in \mathcal{D}(\mathcal{H}^{(|i\rangle)}). For any. such that. ,. there exists. D=w-\displaystyle \int_{\mathcal{P}(\mathcal{H}(|i\rangle) }Xd$\mu$_{D}(X) where. probabihty. ( \mathfrak{B}(\mathcal{H})_{1} WOT).. measure. $\mu$_{D}. on. P(\mathcal{H}^{(|i\rangle)}) (8). .. P(\mathcal{H}^{(|i\rangle)}). of the weak operator topology (WOT) and. the convergence. w‐ means. metnc space. a. us a. Borel set. of. a. ,. Remark 2.3. An infinite dimensional analogue of Birkhoffs theorem is known For Birkhoffs Problemlll, see l8, Section 20l and Ỉ9, Sectionl4.81. We remark that. This theorem. no one. treated in any. study (ii). in. infinite dimensional. immediately implies the following theorem, which. is the. as. Birkhoffs Problemlll.. space with. key. operator topologies.. tool to prove the sufficient. condition of Theorem 1.2.. (Asakura Í1J). Theorem 2.4.. $\rho$\prec $\sigma$ , there exist that. a. Let $\rho$ and $\sigma$ be density matrices on \mathcal{H} and a probability measure $\mu$_{D} on. D\in D(\mathcal{H}^{(|i\rangle)}). $\rho$=\displaystyle \int_{\mathcal{P}(?\{(1:) }X $\sigma$ X^{*}d$\mu$_{D}(X) where in. \mathbb{C}_{1}(H). the convergence. means. of the. trace. Proof. From [16, only 3], assumption, there exist a=(a_{n})_{r $\iota$=1}^{\infty}, b=(b_{n})_{n=1}^{\infty} Theorem. such that a\prec b and. \mathrm{C}_{1}(H). eigenbasis (|i\rangle)_{i=1}^{\infty} If corresponding to D such. same. .. (9). ,. \Vert\cdot\Vert_{1}.. norm. have to show that the. we. in. ,. having. \mathcal{P}(\mathcal{H}^{(|i) }). integral. (9). in. converges to $\rho$ in WOT.. By. \displaystyle \in\{(a_{i})_{i=1}^{\infty} \in\ell^{1}|a_{i}\geq 0, \sum_{i=1}^{\infty}a_{i}=1, a_{i}\geq a_{i+1}(i\in \mathrm{N})\}. $\rho$:=\displaystyle\sum_{n=1}^{\infty}a_{n}|i_{n}\rangle\langlei_{n}|, $\sigma$:=\sum_{n=1}^{\infty}b_{n}|i_{n}\rangle\langlei_{n}|, where the infinite. sums. converge in the trace. norm.. By [7, Theorem 4.7, Corolloary 6.1], there. |a)=\tilde{D}|b\rangle. in. exists. \ell^{2} From Theorem 2.4, there exists. a. .. a_{n}=\displaystyle \int_{P(l^{2}) \langle e_{n}|X|b\rangle d$\mu$_{D}^{-}(X) where. (|e_{n}))_{n=1}^{\infty}. standard basis in l^{2}. is. a. a. probability. 2.4, there exists. measure. a_{7l}=\displaystyle \int_{P(\mathcal{H}(|i\rangle) }(n|X $\sigma$ X^{*}|n\rangle d$\mu$_{D}(X) 3 We. Sketch of the assume. that. \tilde{$\phi$} :=(U_{tt}\otimes U_{\mathcal{K}}) $\phi$. Let on. ,. measure. $\mu$_{D^{-}. =. on. \prec. a same. .. \in. \mathcal{D}(\ell^{2}). such that. such that. (10). D\in \mathcal{D}(\mathcal{H}^{(|i\rangle)}) be D=\displaystyle \sum_{i,j}d_{ij}|i\rangle\langle j| Then, from Theorem P(\mathcal{H}^{(|i\rangle)}) such that (8) holds. Thus, we have \langle n| $\rho$|n\rangle .. =. This implies that the. \mathrm{T}\mathrm{r}_{\mathcal{K} | $\phi$\rangle\{ $\phi$|. (d_{ij}) \mathcal{P}(l^{2}). for any n\in \mathrm{N} ,. integral. in. (9). converges to $\rho$ in WOT.. proof of the sufficient condition. \mathrm{T}\mathrm{r}_{\mathcal{K} | $\psi$\rangle\{ $\psi$| have. .. .. $\mu$_{D}. \tilde{D}. infinite matrix. an. probability. Then, there. exist unitaries. \square. of Theorem 1.2.. U_{\mathcal{H} and U_{\mathcal{K} such that $\psi$ and. Schmidt basis, i.e.,. $\psi$=\displaystyle\sum_{i=1}^{\infty}\sqrt{a_{i}|\rangle_{\mathcal{H}|i\rangle_{\mathcal{K},\tilde{$\phi$}=\sum_{i=1}^{\infty}\sqrt{b_{i}|\rangle_{?i}|\rangle_{\mathcal{K} ( |i\rangle_{\mathcal{H} )_{i=1}^{\infty}\mathrm{a}\mathrm{n}\mathrm{d}(|i) \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}_{\mathrm{S}\mathrm{D\iO} \nmD(at\hmatrm{hmcal}\{mHa}t^h{(r|m$\i{eo}ata=$'(a)_}{)i\})mat\prhecrmb{=a(}b\_mat{i})\hirnm\{enm}\pmattysehtr_m{1{d}2}\.4mat, hrthatand m{a} $\mu$_{D}\mathrm{o}\mathrm{n}P(\mathcal{H}^{(|i}\hslah^{)_i=1}^{\infty}.\mathrm{M}\mathrm{o}\mathrm{}\mathrm{e}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{},\mathrm{f}x\mathrm{o}\mathrm{ }\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}$\Gam a$\mathrm{e}\mathrm{ })\mathrm{c}\mathrm{o}$\iota$ s \mathrm{o}\mathrm{m}\mathrm{e}\mathrm{C} ONSs. there exist. probability. a. $\rho\psi$=\displaystyle\int_{P(\mathcal{H}^{(|i\rangle)} X$\rho$_{\overline{$\phi$} X^{*}d$\mu$_{D}(X) For any. X\in P(\mathcal{H}^{(|i\rangle)}). ,. let define. a. responding t \mathrm{o}D such. measure. ,. in ￠ 1(H) ,. densely defined operator M_{X}. \left\{ begin{ar y}{l M_{X}:=U_{H^*}\sqrt{$\rho$_{\overline{$\phi$} X^{*}($\rho\psi$^{-1}\mathrm{z}),\ D(M_{X}):=D($\rho$_{ \psi$}^{-\mathrm{z})1, \end{ar y}\right.. on. \mathcal{H} by. (11).

(4) 145. :=U_{\mathcal{K}}^{*}X^{*} Then,. and let U_{X}. we. .. have. (M_{X}\otimes U_{X})| $\psi$\rangle\langle $\psi$|(M_{X^{*}}\otimes U_{X^{*}}) is U_{X^{*} )=| $\phi$\rangle\langle $\phi$| holds. Let \mathcal{H}_{0} :=1\mathrm{i}\mathrm{n}\{|i)\}_{i=1}^{\infty} then the. dense. ,. for any X. ,. we. (M_{X}\otimes U_{X})| $\psi$) =| $\phi$\rangle. constant. function. In. subspace \mathcal{H}_{0}. \subset. for any X , and then the function. P(\mathcal{H})\ni X\mapsto. \displaystyle \int_{\mathrm{P}(H^{(1:\rangle)})}(M_{X}\otimes U_{X})| $\psi$\rangle\{ $\psi$|(M_{X^{*} \otimes. particular,. (3). satisfies the conditions. (4). Moreover,. and. have. M_{X^{*} M_{X}= $\rho \psi$^{-\frac{1}{2} (X$\rho$_{ $\phi$}X^{*}) $\rho \psi$^{-\frac{1}{2}. D(M_{X})\supset \mathcal{H}_{0}.. on. Thus, \{M_{X}\}_{X} (5). Putting it all together, the conditions (2), (3), (4), (5), (6) and (7) for satisfies the condition. (\mathfrak{B}(\mathcal{H})_{1} WOT). metric space. Remark S.l.. $\Delta$_{ $\rho \phi,\beta$_{ $\psi$}. is. a. densely defined operator M_{X} in of relative modular operator Ĩ2, 31.. the above. relation between two. majorization relation between. For. I=\mathcal{P}(\mathcal{H}). Borel set. a. of. a. satisfied. \square. The. kind. Majorization. 4. are. ,. two. density matrices,. proof. equal. is. density. U_{\mathcal{H} $\Delta$_{$\rho$_{ $\phi$}, $\rho \psi$}(X^{*}). to. ,. where. matrices following. it has been known that the. theorems. hold.. (Ỉ14,. Theorem 4.1.. following. (a) (B). Section. equivalent.. are. 4\cdot SJ). Let \mathcal{H} be. an. finite. the. ,. $\rho$\prec $\sigma$. There exists. { U $\sigma$ U^{*}|U. (C). mixed unitary map $\Phi$ such that is the unitary orbit of $\sigma$.. a. unitary}. is. There exists. a. unital and. $\Phi$( $\sigma$). Theorem 4.3. For $\rho$,. $\sigma$\in \mathcal{S}(\mathcal{H}). (Ĩ10,. the. ,. (a). $\rho$\prec $\sigma$.. (b). $\rho$\in. (c). There exists. (d). There exist. co. U( $\sigma$). unitary channel. any mưed. By definition,. Theorem. 3.31,. following. are. us. i. e.,. $\rho$ \in. U( $\sigma$). co. .. Here, U( $\sigma$). map $\Phi$ such that. :=. $\Phi$( $\sigma$)= $\rho$.. =\displaystyle \sum_{i=1}^{n}p_{i}U_{i}^{*}XU_{i},. unitary channel, if $\Phi$(X). \displaystyle \sum_{i=1}^{n}p_{1}=1.. unital and CP‐TP.. l6, Theorem. 2.5(1)1). Let \mathcal{H} be. infinite dimensional Hilbert. an. space.. equivalent.. .. an. a. unital CP‐TP map $\Phi$ such that. of. sequence. $\Phi$( $\sigma$)= $\rho$.. mixed unitary channels. \Vert$\Phi$_{n}(X)- $\Phi$(X)\Vert_{1}\rightarrow 0 for Using Theorem 2.4, Theorem 4.4.. $\rho$ ,. =. completely positive‐trace preserving (CP‐TP). Remark 4.2. A linear map $\Phi$ on \mathrm{C}_{1}(\mathcal{H}) is called mixed where n<\infty , the U_{i} are all unitary operators and p_{i}>0,. (e). $\sigma$\in S(\mathcal{H}). dimensional Hilbert space. For $\rho$,. \{$\Phi$_{n}\}_{n=1}^{\infty} all. X\in \mathrm{C}_{1}(\mathcal{H}). add. a new. (Asakura). For. density. matrices $\rho$,. There exist a Borel set I of a certain of partial isometry operators \{Vx\}x\in I such. $\sigma$\in S(\mathcal{H}). metric space,. .. ,. unital CP‐TP map $\Phi$ such that. $\Phi$( $\sigma$)= $\rho$.. ,. as. follows.. the conditions a. probability. (a)\sim(e). measure. are. $\mu$. on. equivalent. I and. a. set. of. that. $\rho$=\displaystyle \int_{I}V_{X} $\sigma$ V_{X}^{*}d $\mu$(X) .. an. characterization to Theorem 4.3. we. Proof. (a) \Rightarrow (e) eigenbasis (|i))_{i=1}^{\infty}. and. ,. in. \mathrm{C}_{1}(H). (12). .. There exist two partial isometry V, W such that V^{*} $\rho$ V and W $\sigma$ W^{*} have same Thus, by Theorem 2.4, letting I := P(\mathcal{H}^{(|i))}) and V_{X} :=VXW, the equality (12). holds.. (e)\Rightarrow (a) Ftom Weyls eigenvalue theorem [15], we only have to show that nonnegative operator P ; see also [10, page 8]. Since Tr $\rho$ P $\sigma$\geq \mathrm{T}\mathrm{r} $\rho$ PV_{X} $\sigma$ V_{X^{*} for .. TJ. where several. Tr. $\rho$ P\geq. Tr $\sigma$ P for any we have. any X\in I ,. $\rho$ P\displaystyle \geq\int_{I}\mathrm{T}\mathrm{r}[V_{X} $\sigma$ V_{X^{*} P]d $\mu$(X)=\mathrm{T}\mathrm{r}[(lV_{X} $\sigma$ V_{X^{*} d $\mu$(X) P]=\mathrm{T}\mathrm{r} $\sigma$ P,. interchanges. in the. equalities. are. all. legitimate from [17, V.5.].. \square.

(5) 146. Remark 4.5. From this exist. (I, $\mu$). above and. as. proof, for full‐rank density matrices $\rho$, $\sigma$ \in S(\mathcal{H}) set of unitary operators \{U_{X}\}_{X\in I} such that. if. and. in Theorem. 4.1.. ,. $\rho$ \prec. $\sigma$. only if. there. a. $\rho$=\displaystyle \int_{I}U_{X} $\sigma$ U_{X^{*} d $\mu$(X) Note that this characterization is. a. natural. ,. in. \mathrm{C}_{1}(H). generalization of (a). .. \Leftarrow\Rightarrow. (C). References [1] Asakura, D.,. An infinite dimensional Birkhoffs theorem with the weak operator in infinite dimensional systems in preparation. ]. topology and. LOCC−convertibility. [2] Araki, H.,. Relative entropy of states of. von. Neumann. algebras. Publ. Res. Inst. Math. Sci. Kyoto. Univ.11, 809 (1975/76).. [3] Araki, H.,. Relative entropy of states of. Kyoto Univ. 13,. [4] Bhatia, R.,. 173. Matrix. Analysis, Springer,. [5] Birkhoff, G., Three :147‐151(1946). observations. [6] Hiai, F., Majorization and. analysis. von. Neumann. algebras II. Publ. Res. Inst. Math. Sci.. (1977/78). New. linear. on. York, (1997).. algebra.(Spanish). and stochastic maps in 127.1 : 18-48(1987). applications. von. Neumann. Univ. Nac. Tucuman. Revista A. 5. algebras. Journal. of mathematical. .. [7] Kaftal,. V. and Weiss, G., An infinite dimensional Schur‐Horn Theorem and majorization theory Journal of Functional Analysis, Vol. 259, No. 12, ;3115-3162(2010) .. [8] Krinik,. A. C. and Swift, R. J., Stochastic Processes and Functional Analysis, A Dekker Series of Applied Mathematics, 238. CRC Press, New York, (2004).. [9] Lax,. D., Functional Analysis, Wiley, (2002).. Lecture Notes in Pure and. [10] Li,. P.. Y. and. and. Busch, P., Von Neumann entropy Applications, Vol. 408, :384-393(2013). [11] Nielsen,. M. A., Condition for Vol.83, Issue2 :436-439(1999). a. class of. and. Journal of Mathematical. majorization. Analysis. entanglement transformations, Physical Review Letters,. [12] Nielsen,. M. A. and Chuang, I. L., Quantum Computation and Quantum Information, Cambrige University Press, Cambrige, (2000).. [13] Owari, M., Braunstein, Vol.8. L., Nemoto,. K. and. . Murao, M., $\epsilon$ ‐convertibility of entangled states and extension of Schmidt rank in infinite dimensional systems, Quantum Information and Computation, S.. :30-52(2008). [14] Watrous, J., Theory. of quantum. . information, University of Waterloo. [15] Weyl, H., Inequalities between the of the National. [16] Wherl, A., ematical. Academy. Fall 128. two kinds of. of Sciences of the. eigenvalues of a linear transformation USA, Vol. 35, No. 7, : 408-411(1949). Three theorems about entropy and convergence of Vol. 10, : 159-163(1976). density matrices. Physics,. [17] Yosida, K.,. Functional. Analysis (6th. ed. (2011).. Springer‐Verlag,. New. York, (1980).. Proceedings. Report. on. Math‐.

(6)