양자 상태 판별

양자 정보 과학에서 양자 상태 판별은 관측된 측정 확률을 생성한 양자 상태를 추론하는 작업을 말한다.

보다 정확히 말하면, 표준 공식에서 문제는 수신된 상태가 { $}$ $\{\sigma _{i}\}_{i}$ 집합의 요소라는 약속 하에 특정 미지의 상태에서 $\rho$ $\{\sigma _{i}\}_{i}$ $(E_{i})_{i}$ i(\ $displaystyle$ $($ $E_$ ${i$ $\rho$ ${$ $i$ _ ${i}$ 를 $(E_{i})_{i}$ 실행하는 $것$ 입니다. $\sigma _{i}$ $i$ i $（$ $display$ $style$ ） $_$ { i $\sigma _{i}$ $p_{i}$ } 。 $\rho =\sum _{i}p_{i}\sigma _{i}$ , $\rho =\sum _{i}p_{i}\sigma _{i}$ $style$ = \ $sum _$ { i } p _ { i $}$ \ display style $_$ { i } $with$ . = \ $sum$ _ { i } \ display style _ { $\rho =\sum _{i}p_{i}\sigma _{i}$ i } 。다음으로 POVM $(E_{i})_{i}$ i $(E_{i})_{i}$ (\ $displaystyle(E_{i}_{$ i})_ ${i$ })가 $(E_{i})_{i}$ 수신된 상태를 올바르게 추측할 가능성을 찾는 것이 과제입니다. $지정$ 된 상태가 $\sigma _{j}$ j일 때 POVM이 i $(\$ $displaystyle$ i $)$ $i$ $\sigma _{j}$ 결과를 반환할 확률은 ${\text{Prob}}(i|j)=\operatorname {tr} (E_{i}\sigma _{j})$ ( ${\text{Prob}}(i|j)=\operatorname {tr} (E_{i}\sigma _{j})$ i ${\text{Prob}}(i|j)=\operatorname {tr} (E_{i}\sigma _{j})$ j ) ${\text{Prob}}(i|j)=\operatorname {tr} (E_{i}\sigma _{j})$ ${\text{Prob}}(i|j)=\operatorname {tr} (E_{i}\sigma _{j})$ ${\text{Prob}}(i|j)=\operatorname {tr} (E_{i}\sigma _{j})$ ${\text{Prob}}(i|j)=\operatorname {tr} (E_{i}\sigma _{j})$ i ${\text{Prob}}(i|j)=\operatorname {tr} (E_{i}\sigma _{j})$ j ${\text{Prob}}(i|j)=\operatorname {tr} (E_{i}\sigma _{j})$ ） { $displaystyle$ { Prob } ( i j ) = \ $operatorname {tr$ } ( $E$ _ { $i$ $\sigma _{j}$ $}$ { s $igma$ ) ${\text{Prob}}(i|j)=\operatorname {tr} (E_{i}\sigma _{j})$ 。올바른 상태는 $P_{\rm {success}}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}E_{i})$ $P_{\rm {success}}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}E_{i})$ $P_{\rm {success}}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}E_{i})$ $P_{\rm {success}}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}E_{i})$ $P_{\rm {success}}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}E_{i})$ $=$ $P_{\rm {success}}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}E_{i})$ $P_{\rm {success}}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}E_{i})$ $P_{\rm {success}}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}E_{i})$ $P_{\rm {success}}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}E_{i})$ $P_{\rm {success}}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}E_{i})$ $P_{\rm {success}}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}E_{i})$ i $P_{\rm {success}}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}E_{i})$ ）{ $display style$ P_ ${\rm$ { $success$ }=\ $sum$ _ ${i}p_{i}\operatorname {tr}(\sum$ _ { $i}E_{i$ ^[1]입니다.

헬스트롬 측정

두 상태의 구별은 헬스트롬 ^[2]측정을 사용하여 최적으로 해결할 수 있습니다. $\{\sigma _{0},\sigma _{1}\}$ 의 상태 { $},$ {\ $displaystyle \{\displaystyle$ \{ $0},$ \ $displaystyle$ $\{p_{0},p_{1}\}$ { $1}\},$ {{ $displaystyle$ \{ $p_{0},$ p_ $\{p_{0},p_{1}\}$ ${$ $1$ }\}}}, $\{E_{0},E_{1}\}$ { $\{E_{0},E_{1}\}$ , $\{E_{0},E_{1}\}$ 1 $}$ 의 2개의 확률 $\{p_{0},p_{1}\}$ {\ $styledisplaystyle$ \{ $0}이후$ }, $E_0$ }이 $\{\sigma _{0},\sigma _{1}\}$ $\{p_{0},p_{1}\}$ $\{E_{0},E_{1}\}$ $모든$ POVM에 대해 $E_{1}=I-E_{0}$ $},$ $E_{1}=I-E_{0}$ $=$ I - $E_{1}=I-E_{0}$ 0 (\ $displaystyle E_$ {1 $}=$ I-E_ ${0$ 따라서 성공 확률은 다음과 같습니다.

$P_{success}=p_{0}\operatorname {tr}(\display_{0}E_{0})+p_{1}\operatorname {tr}(\display_{0}E_{1}=p_{0}\operatorname {tr})E_{0}]$

성공 확률을 최대화하려면 추적을 최소화해야 합니다. $p_{0}\sigma _{0}-p_{1}\sigma _{1}$ 는 E $E_{0}$ 0(\ $displaystyle E_{0})$ 이 $E_{0}$ p 0 $p_{0}\sigma _{0}-p_{1}\sigma _{1}$ 0 - $p_{0}\sigma _{0}-p_{1}\sigma _{1}$ $p_{0}\sigma _{0}-p_{1}\sigma _{1}$ 1(\ $display p_{0}\sigma$ _{ $1}\sigma$ _{1 $p_{0}\sigma _{0}-p_{1}\sigma _{1}$ ^[2]sigma _{1})의 $p_{0}\sigma _{0}-p_{1}\sigma _{1}$ 의 $Eigenspace$ 상에 프로젝터일 때 가능합니다.

꽤 좋은 측정

3개 이상의 상태를 구별하기 위해 PGM(Pretty Good Measurement)은 제곱근 측정이라고도 하며 최적은 아니지만 ^[3]상당히 잘 작동합니다.PGM에서 $\{E_{i}\}=\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\}$ { $\{E_{i}\}=\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\}$ $\{E_{i}\}=\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\}$ $\{E_{i}\}=\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\}$ { $\{E_{i}\}=\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\}$ - $\{E_{i}\}=\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\}$ $\{E_{i}\}=\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\}$ $\{E_{i}\}=\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\}$ $\{E_{i}\}=\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\}$ $\{E_{i}\}=\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\}$ i $\{E_{i}\}=\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\}$ - $\{E_{i}\}=\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\}$ 2 $}$ { $displaystyle$ \ { $E _$ { i } \ } = \ { $S$ ^ { - { \ $frac$ { } { $}$ $p _$ { $i$ } \ $frac$ { $}$ { i } 、 [ $S=\sum _{i}p_{i}\sigma _{i}$ $}$ } 。 $P_{success}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\})$ 를 통해 $P_{success}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\})$ s $P_{success}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\})$ $P_{success}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\})$ e $P_{success}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\})$ $P_{success}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\})$ $P_{success}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\})$ i $P_{success}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\})$ tr （ $P_{success}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\})$ $P_{success}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\})$ { $P_{success}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\})$ - $P_{success}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\})$ 2 $P_{success}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\})$ $P_{success}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\})$ $P_{success}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\})$ i $P_{success}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\})$ - 1 $P_{success}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\})$ $P_{success}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\})$ ） { $displaystyle P_{$ success } = \ $sum$ _ { $i$ } $p_{$ i $P_{success}=\sum _{i}p_{i}\operatorname {tr} (\sigma _{i}\{S^{-{\frac {1}{2}}}p_{i}\sigma _{i}S^{-{\frac {1}{2}}}\})$ } \ $operatorname { tr$ （ \ $p sigma$ _ $i } }$ ）。

$\sigma _{0}=|+\rangle \langle +|$ 를 들어, $display$ 0 $\sigma _{0}=|+\rangle \langle +|$ + $display$ $\sigma _{0}=|+\rangle \langle +|$ \ $displaystyle$ $_$ { 0 } $=$ + \ $rangle$ \ rangle + } , $\sigma _{2}={\frac {1}{2}}(|+\rangle \langle +|+|-\rangle \langle -|$ 1 $\sigma _{1}=|-\rangle \langle -|$ - $displaydisplay$ - \ $\sigma _{2}={\frac {1}{2}}(|+\rangle \langle +|+|-\rangle \langle -|$ style $\sigma _{2}={\frac {1}{2}}(|+\rangle \langle +|+|-\rangle \langle -|$ _ ${$ 1} $=$ - \ $rangle$ \ $langle$ - $\sigma _{2}={\frac {1}{2}}(|+\rangle \langle +|+|-\rangle \langle -|$ $\sigma _{1}=|-\rangle \langle -|$ , = $=$ 1 ( + $\sigma _{2}={\frac {1}{2}}(|+\rangle \langle +|+|-\rangle \langle -|$ display + display + $\sigma _{2}={\frac {1}{2}}(|+\rangle \langle +|+|-\rangle \langle -|$ display + $\sigma _{2}={\frac {1}{2}}(|+\rangle \langle +|+|-\rangle \langle -|$ display ) - 2 $displaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplay$ - { 2 } display style + 2 $\sigma _{2}={\frac {1}{2}}(|+\rangle \langle +|+|-\rangle \langle -|$ } display style + displaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaylities는 $p_{0}={\frac {1}{6}}$ $p_{0}={\frac {1}{6}}$ $p_{0}={\frac {1}{6}}$ $p_{0}={\frac {1}{6}}$ { $style$ p $_$ { 0 } = $specfrac$ $p_{1}={\frac {1}{3}}$ { $1$ $p_{0}={\frac {1}{6}}$ } { $6$ } $p_{0}={\frac {1}{6}}$ 、 $p_{1}={\frac {1}{3}}$ $p_{1}={\frac {1}{3}}$ 3 { $p_{2}={\frac {1}{2}}$ $p_{2}={\frac {1}{2}}$ _ {1} = $specfrac$ ${$ 1} ${3$ } $p_{2}={\frac {1}{2}}$ } 、 $p_{2}={\frac {1}{2}}$ 2 $p_{2}={\frac {1}{2}}$ 1 . $displaystyle p_{2} = spec$ frac { 1 $S={\frac {1}{6}}\sigma _{0}+{\frac {1}{3}}\sigma _{1}+{\frac {1}{2}}\sigma _{2}={\begin{bmatrix}{\frac {1}{12}}&{\frac {1}{12}}\\{\frac {1}{12}}&{\frac {1}{12}}\end{bmatrix}}+{\begin{bmatrix}{\frac {1}{6}}&-{\frac {1}{6}}\\-{\frac {1}{6}}&{\frac {1}{6}}\end{bmatrix}}+{\begin{bmatrix}{\frac {1}{4}}&0\\0&{\frac {1}{4}}\end{bmatrix}}={\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}$ $S={\frac {1}{6}}\sigma _{0}+{\frac {1}{3}}\sigma _{1}+{\frac {1}{2}}\sigma _{2}={\begin{bmatrix}{\frac {1}{12}}&{\frac {1}{12}}\\{\frac {1}{12}}&{\frac {1}{12}}\end{bmatrix}}+{\begin{bmatrix}{\frac {1}{6}}&-{\frac {1}{6}}\\-{\frac {1}{6}}&{\frac {1}{6}}\end{bmatrix}}+{\begin{bmatrix}{\frac {1}{4}}&0\\0&{\frac {1}{4}}\end{bmatrix}}={\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}$ $S={\frac {1}{6}}\sigma _{0}+{\frac {1}{3}}\sigma _{1}+{\frac {1}{2}}\sigma _{2}={\begin{bmatrix}{\frac {1}{12}}&{\frac {1}{12}}\\{\frac {1}{12}}&{\frac {1}{12}}\end{bmatrix}}+{\begin{bmatrix}{\frac {1}{6}}&-{\frac {1}{6}}\\-{\frac {1}{6}}&{\frac {1}{6}}\end{bmatrix}}+{\begin{bmatrix}{\frac {1}{4}}&0\\0&{\frac {1}{4}}\end{bmatrix}}={\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}$ 1 $S={\frac {1}{6}}\sigma _{0}+{\frac {1}{3}}\sigma _{1}+{\frac {1}{2}}\sigma _{2}={\begin{bmatrix}{\frac {1}{12}}&{\frac {1}{12}}\\{\frac {1}{12}}&{\frac {1}{12}}\end{bmatrix}}+{\begin{bmatrix}{\frac {1}{6}}&-{\frac {1}{6}}\\-{\frac {1}{6}}&{\frac {1}{6}}\end{bmatrix}}+{\begin{bmatrix}{\frac {1}{4}}&0\\0&{\frac {1}{4}}\end{bmatrix}}={\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}$ $S={\frac {1}{6}}\sigma _{0}+{\frac {1}{3}}\sigma _{1}+{\frac {1}{2}}\sigma _{2}={\begin{bmatrix}{\frac {1}{12}}&{\frac {1}{12}}\\{\frac {1}{12}}&{\frac {1}{12}}\end{bmatrix}}+{\begin{bmatrix}{\frac {1}{6}}&-{\frac {1}{6}}\\-{\frac {1}{6}}&{\frac {1}{6}}\end{bmatrix}}+{\begin{bmatrix}{\frac {1}{4}}&0\\0&{\frac {1}{4}}\end{bmatrix}}={\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}$ $S={\frac {1}{6}}\sigma _{0}+{\frac {1}{3}}\sigma _{1}+{\frac {1}{2}}\sigma _{2}={\begin{bmatrix}{\frac {1}{12}}&{\frac {1}{12}}\\{\frac {1}{12}}&{\frac {1}{12}}\end{bmatrix}}+{\begin{bmatrix}{\frac {1}{6}}&-{\frac {1}{6}}\\-{\frac {1}{6}}&{\frac {1}{6}}\end{bmatrix}}+{\begin{bmatrix}{\frac {1}{4}}&0\\0&{\frac {1}{4}}\end{bmatrix}}={\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}$ 1 、 0 、 3 、 3 、 3 σ1 $S={\frac {1}{6}}\sigma _{0}+{\frac {1}{3}}\sigma _{1}+{\frac {1}{2}}\sigma _{2}={\begin{bmatrix}{\frac {1}{12}}&{\frac {1}{12}}\\{\frac {1}{12}}&{\frac {1}{12}}\end{bmatrix}}+{\begin{bmatrix}{\frac {1}{6}}&-{\frac {1}{6}}\\-{\frac {1}{6}}&{\frac {1}{6}}\end{bmatrix}}+{\begin{bmatrix}{\frac {1}{4}}&0\\0&{\frac {1}{4}}\end{bmatrix}}={\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}$ 1 、 3 。6− 16− 1616-RSB-+[140014])[120012]{\displaystyle S={\frac{1}{6}}\sigma _{0}+{\frac{1}{3}}\sigma _{1}+{\frac{1}{2}}\sigma _{2}={\begin{bmatrix}{\frac{1}{12}}&{\frac{1}{12}}\\{\frac{1}{12}}&{\frac{1}{12}}\end{bmatrix}}와{\begin{bmatrix.}{\frac {1}{6}}&-{\frac{1}{6}}\\-{\frac{1}{6}}&{\frac{1}{6}}\end{bmatrix}}+{\begin{bmatrix}{\frac{1}{4}}&0\\0&,{\frac{1}{4}}\end{bmatrix}}={\begin{bmatrix}{\frac{1}{2}}&0\\0&,{\frac{1}{2}}\end{bmatrix}}}과 S12−)[2002]{\displaystyle S^{-{\frac{1}{2}}}={\begin{bmatrix}{\sqrt.{2}}&0\\0&,{\sqrt{2}}\e $nd {bmatrix$

$\displaystyle E_{0}=bgin{bmatrix}{\sqrt {2}}{\0&{\sqrt {2}}\end {bmatrix}{\display {bmatrix}{\frac {1}{12}\frac {1}{12}&{\frac}{\frac}{\frac}$

$E_{1}={\begin{bmatrix}{\sqrt{2}}&0\\0&,{\sqrt{2}}\end{bmatrix}}{\begin{bmatrix}{\frac{1}{6}}&-{\frac{1}{6}}\\-{\frac{1}{6}}&{\frac{1}{6}}\end{bmatrix}}{\begin{bmatrix}{\sqrt{2}}&0\\0&,{\sqrt{2}}\end{bmatrix}}={\begin{bmatrix}{\frac{1}{3}}&-{\frac{1}{3}}\\-{\frac{1}{3}}&{\frac{1}{3}}\end{bmatr.ix}}$

E2)[2002][120012][2002])[120012]{\displaystyle E_{2}={\begin{bmatrix}{\sqrt{2}}&0\\0&,{\sqrt{2}}\end{bmatrix}}{\begin{bmatrix}{\frac{1}{2}}&0\\0&,{\frac{1}{2}}\end{bmatrix}}{\begin{bmatr.ix}{\sqrt{2}}&0 $\\0&{\fracrt {2}}\end{bmatrix}=param{bmatrix}{\frac{$ 1}{2 $}}&0&{\frac{1}{2}}\end{bmatrix$

이러한 프로젝터의 경우 $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ c $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ tr $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ [ $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ 1 $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ ][ $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ 8 $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ 3 $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ + $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ tr $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ [ $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ - 1 $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ 2 $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ ]+ $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ tr $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ [ $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ - $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ - $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ 3 - 1 $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ - $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ 3 1 1 1 $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ 1 3 $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ ]+ $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ tr $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ [ $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ tr ][ 1 0 $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ $P_{success}={\frac {1}{6}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {3}{8}}&1\\{\frac {3}{8}}&{\frac {3}{8}}\end{bmatrix}}]+{\frac {1}{3}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\-{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{3}}&-{\frac {1}{3}}\\-{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}]+{\frac {1}{2}}\operatorname {tr} [{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac {1}{2}}&0\\0&{\frac {1}{2}}\end{bmatrix}}]=0.65$ 1 ]2][120012]]=0.65{\displaystyle P_{성공}={\frac{1}{6}}\operatorname{tr}[{\begin{bmatrix}{\frac{1}{2}}&,{\frac{1}{2}}\\{\frac{1}{2}}&,{\frac{1}{2}}\end{bmatrix}}{\begin{bmatrix}{\frac{3}{8}}&1\\{\frac{3}{8}}&,{\frac{3}{8}}\end{bmatrix}}]+{\frac{1}{3}}\operatorname{tr}[{\beg.in{bmatrix}{\frabmatrix}}=0.65 -RSB-}

레퍼런스

^ Bae, Joonwoo; Kwek, Leong-Chuan (2015). "Quantum state discrimination and its applications". Journal of Physics A: Mathematical and Theoretical. 48 (8): 083001. arXiv:1707.02571. Bibcode:2015JPhA...48h3001B. doi:10.1088/1751-8113/48/8/083001. S2CID 119199057.
^ ^a ^b Barnett, Stephen M.; Croke, Sarah (2009). "Quantum state discrimination". Adv. Opt. Photon. 1 (8): 238–278. arXiv:0810.1970. Bibcode:2009AdOP....1..238B. doi:10.1364/AOP.1.000238. S2CID 15398601.
^ ^a ^b Montanaro, Ashley (2007). "On the distinguishability of random quantum states". Commun. Math. Phys. 273 (3): 619–636. arXiv:quant-ph/0607011. Bibcode:2007CMaPh.273..619M. doi:10.1007/s00220-007-0221-7. S2CID 12516161.

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[1] Bae, Joonwoo; Kwek, Leong-Chuan (2015). "Quantum state discrimination and its applications". Journal of Physics A: Mathematical and Theoretical. 48 (8): 083001. arXiv:1707.02571. Bibcode:2015JPhA...48h3001B. doi:10.1088/1751-8113/48/8/083001. S2CID 119199057.

[BC09-2] Barnett, Stephen M.; Croke, Sarah (2009). "Quantum state discrimination". Adv. Opt. Photon. 1 (8): 238–278. arXiv:0810.1970. Bibcode:2009AdOP....1..238B. doi:10.1364/AOP.1.000238. S2CID 15398601.

[AM06-3] Montanaro, Ashley (2007). "On the distinguishability of random quantum states". Commun. Math. Phys. 273 (3): 619–636. arXiv:quant-ph/0607011. Bibcode:2007CMaPh.273..619M. doi:10.1007/s00220-007-0221-7. S2CID 12516161.

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