Formula Report

Source: /home/wkolbe/Downloads/2004.05631v1.lines.json  |  MathExpressions: 1116  |  Equations: 254

Inline Math — MathExpression tiddlers (1116)

Display Equations (254)

Tiddler	LaTeX source	Rendered (KaTeX)	MathPix image
2004.05631v1_EQ0001_p012(1.1)	a(x):=\{y \in Y \mid R(x, y)=1\} \quad b(y):=\{x \in X \mid R(x, y)=1\}
2004.05631v1_EQ0002_p012(\1.2\)	f(A):=\bigcap_{x \in A} a(x) .
2004.05631v1_EQ0003_p012(\1.3\)	g(B):=\bigcap_{y \in B} b(y)
2004.05631v1_EQ0004_p012(1.4)	A \subseteq g(B) \quad \text { if and only if } \quad B \subseteq f(A)
2004.05631v1_EQ0005_p013	X=\{\text { orange, green, purple }\} \quad Y=\{\text { fruit, vegetable }\}
2004.05631v1_EQ0006_p013	\begin{aligned} R(\text { orange, fruit }) & =1 \\ R(\text { green, fruit }) & =1 \\ R(\text { purple, vegetable }) & =1 \end{aligned}
2004.05631v1_EQ0007_p013	\begin{aligned} a(\text { orange }) & =\{\text { fruit }\} \\ a(\text { green }) & =\{\text { fruit }\} \\ a(\text { purple }) & =\{\text { vegetable }\} \end{aligned}
2004.05631v1_EQ0008_p013	\begin{aligned} b(\text { fruit }) & =\{\text { orange }, \text { green }\} \\ b(\text { vegetable }) & =\{\text { purple }\} \end{aligned}
2004.05631v1_EQ0009_p015	\text { formal concepts of } R \cong \text { Fix } f g \cong \text { Fix } g f \text {. }
2004.05631v1_EQ0010_p015	f(g B)=f A=B \quad g(f A)=g B=A .
2004.05631v1_EQ0011_p015	A \subseteq g(B) \text { if and only if } B \subseteq f(A),
2004.05631v1_EQ0012_p016	\left[\begin{array}{c} \mid \\ \alpha(x) \\ \mid \end{array}\right]=M=\left[\begin{array}{lll} - & \beta(y) & - \end{array}\right]
2004.05631v1_EQ0013_p016(\1.5\)	\operatorname{hom}_{\mathrm{C}}(F A, B) \cong \operatorname{hom}_{\mathrm{D}}(A, G B)
2004.05631v1_EQ0014_p017(1.6)	\langle M v, w\rangle=\left\langle v, M^{\dagger} w\right\rangle .
2004.05631v1_EQ0015_p017	\langle M v, w\rangle=\left\langle v, M^{\dagger} w\right\rangle
2004.05631v1_EQ0016_p022	\sum_{s \in S} \pi(s)=1, \quad \pi(s) \geq 0 \text { for all } s \in S
2004.05631v1_EQ0017_p022	\sum_{(x, y) \in X \times Y} \pi(x, y)=1
2004.05631v1_EQ0018_p022	\pi_{X}(x)=\sum_{y \in Y} \pi(x, y) \quad \pi_{Y}(y)=\sum_{x \in X} \pi(x, y) .
2004.05631v1_EQ0019_p022	X=\{\text { orange, green, purple }\} \quad Y=\{\text { fruit, vegetable }\}
2004.05631v1_EQ0020_p023	\begin{array}{lll} \pi_{X}=\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right) & \leftrightarrow & \begin{array}{l} \text { orange } \\ \text { green } \\ \text { purple } \end{array} \\ \hline \pi_{Y}=\left(\frac{2}{3}, \frac{1}{3}\right) & \leftrightarrow & \begin{array}{l} \text { fruit } \\ \text { fruit } \\ \text { vegetable } \end{array} \end{array}
2004.05631v1_EQ0021_p024	M=\left[\begin{array}{ccc} \sqrt{\frac{1}{3}} & \sqrt{\frac{1}{3}} & 0 \\ 0 & 0 & \sqrt{\frac{1}{3}} \end{array}\right]
2004.05631v1_EQ0022_p024	M^{\dagger} M=\left[\begin{array}{lll} \frac{1}{3} & \frac{1}{3} & 0 \\ \frac{1}{3} & \frac{1}{3} & 0 \\ 0 & 0 & \frac{1}{3} \end{array}\right]
2004.05631v1_EQ0023_p024	\text { orange }\left[\begin{array}{ccc} \text { orange } & \text { green } & \text { purple } \\ \frac{1}{3} & \frac{1}{3} & 0 \\ \frac{1}{3} & \frac{1}{3} & 0 \\ 0 & 0 & \frac{1}{3} \end{array}\right] \quad \leftrightarrow \quad \pi_{X}=\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right)
2004.05631v1_EQ0024_p025	\begin{array}{ccc} \pi \text { (orange|fruit) } \\ \pi \text { (green|fruit) } & {\left[\begin{array}{c} \sqrt{\frac{1}{2}} \\ \sqrt{\frac{1}{2}} \\ 0 \end{array}\right]} & \text { fruit } \\ \pi \text { (purple|fruit) } & \begin{array}{c} \text { Orange fruit } \\ \text { green fruit } \end{array} \\ \pi \text { (orange|vegetable) } & {\left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right]} & \begin{array}{c} \text { vegetable } \\ \text { r(green|vegetable) } \end{array} \end{array} \text { purple vegetable }
2004.05631v1_EQ0025_p025	M_{i j}:=\sqrt{\pi\left(x_{j}, y_{i}\right)}
2004.05631v1_EQ0026_p027	\langle v \mid w\rangle=\sum_{s \in S} \overline{v(s)} w(s) .
2004.05631v1_EQ0027_p027	\left|s_{i}\right\rangle=\left[\begin{array}{c} 0 \\ \vdots \\ 1 \\ \vdots \\ 0 \end{array}\right] \leftarrow i \text { th entry }
2004.05631v1_EQ0028_p027	|v\rangle=\sum_{s \in S} v(s)|s\rangle=\left[\begin{array}{c} v\left(s_{1}\right) \\ \vdots \\ v\left(s_{n}\right) \end{array}\right] \quad v(s) \in \mathbb{C}
2004.05631v1_EQ0029_p028	\begin{gathered} V \xrightarrow{\langle v|} \mathbb{C} \\ \left|v^{\prime}\right\rangle \longmapsto\left\langle v \mid v^{\prime}\right\rangle \end{gathered}
2004.05631v1_EQ0030_p028	\langle v|=\sum_{s \in S} \overline{v(s)}\langle s|=\left[\overline{\overline{v\left(s_{1}\right)}} \quad \cdots \quad \overline{v\left(s_{n}\right)}\right] \quad v(s) \in \mathbb{C}
2004.05631v1_EQ0031_p028	|v\rangle \longleftrightarrow\langle v|
2004.05631v1_EQ0032_p028	\begin{aligned} & V \xrightarrow{|w\rangle\langle v|} W \\ & \left|v^{\prime}\right\rangle \longmapsto|w\rangle\left\langle v \mid v^{\prime}\right\rangle \end{aligned}
2004.05631v1_EQ0033_p028(2.1)	|w\rangle \otimes|v\rangle=|w\rangle\langle v| .
2004.05631v1_EQ0034_p028	F(X)=X \times Y
2004.05631v1_EQ0035_p028	\begin{aligned} & \text { Set } \xrightarrow{X \times-} \text { Set } \\ & Y \longmapsto X \times Y \end{aligned}
2004.05631v1_EQ0036_p028	\begin{aligned} & V \xrightarrow{\langle v \mid-\rangle} \mathbb{C} \\ & v^{\prime} \longmapsto\left\langle v \mid v^{\prime}\right\rangle \end{aligned}
2004.05631v1_EQ0037_p029	|w\rangle=\left[\begin{array}{c} 1 \\ 2 \\ 3 i \end{array}\right] \quad|v\rangle=\left[\begin{array}{c} 4 i \\ 5 \end{array}\right]
2004.05631v1_EQ0038_p029	|w\rangle \otimes|v\rangle=|w\rangle\langle v|=\left[\begin{array}{c} 1 \\ 2 \\ 3 i \end{array}\right]\left[\begin{array}{ll} -4 i & 5 \end{array}\right]=\left[\begin{array}{cc} -4 i & 5 \\ -8 i & 10 \\ 12 & 15 i \end{array}\right]
2004.05631v1_EQ0039_p029	|w\rangle \otimes|v\rangle=\left[\begin{array}{c} -4 i \\ 5 \\ -8 i \\ 10 \\ 12 \\ 15 i \end{array}\right]
2004.05631v1_EQ0040_p029	\operatorname{tr}\left|v^{\prime}\right\rangle\langle v|=\left\langle v \mid v^{\prime}\right\rangle
2004.05631v1_EQ0041_p030	\left|v^{\prime}\right\rangle\langle v|=\left[\begin{array}{c} \vdots \\ v^{\prime}(s) \\ \vdots \end{array}\right]\left[\begin{array}{lll} \cdots & \overline{v(s)} & \cdots \end{array}\right]
2004.05631v1_EQ0042_p030(2.2)	|v w\rangle\left\langle v^{\prime} w^{\prime}\right|=|v\rangle\left\langle v^{\prime}\right| \otimes|w\rangle\left\langle w^{\prime}\right| .
2004.05631v1_EQ0043_p030(\2.3\)	\left(f \otimes f^{\prime}\right)|a\rangle \otimes\left|a^{\prime}\right\rangle:=f|a\rangle \otimes f\left|a^{\prime}\right\rangle .
2004.05631v1_EQ0044_p030	\begin{aligned} \left\langle v^{\prime} w^{\prime} \mid s t\right\rangle & =\left(\left\langle v^{\prime}\right| \otimes\left\langle w^{\prime}\right|\right)|s\rangle \otimes|t\rangle \\ & =\left\langle v^{\prime} \mid s\right\rangle\left\langle w^{\prime} \mid t\right\rangle \end{aligned}
2004.05631v1_EQ0045_p030	|v w\rangle \overbrace{\left\langle v^{\prime} \mid s\right\rangle\left\langle w^{\prime} \mid t\right\rangle}^{\text {a number }} .
2004.05631v1_EQ0046_p031	\begin{aligned} \left(|v\rangle\left\langle v^{\prime}\right| \otimes|w\rangle\left\langle w^{\prime}\right|\right)|s\rangle \otimes|t\rangle & =|v\rangle\left\langle v^{\prime} \mid s\right\rangle \otimes|w\rangle\left\langle w^{\prime} \mid t\right\rangle \\ & =|v\rangle \otimes|w\rangle\left\langle v^{\prime} \mid s\right\rangle\left\langle w^{\prime} \mid t\right\rangle \\ & =|v w\rangle \overbrace{\left\langle v^{\prime} \mid s\right\rangle\left\langle w^{\prime} \mid t\right\rangle}^{\text {a number }} . \end{aligned}
2004.05631v1_EQ0047_p031	i-\bigcirc-j
2004.05631v1_EQ0048_p035	\sum_{i} M_{i i}
2004.05631v1_EQ0049_p036	|\psi\rangle=\sum_{\left(x_{i_{1}}, \cdots, x_{i_{N}}\right) \in X^{N}} \psi_{i_{1} i_{2} \cdots i_{N}}\left|x_{i_{1}}\right\rangle \otimes\left|x_{i_{2}}\right\rangle \otimes \cdots \otimes\left|x_{i_{N}}\right\rangle .
2004.05631v1_EQ0050_p038(\2.4\)	\pi_{\rho}(s)=\langle s| \rho|s\rangle \quad s \in S .
2004.05631v1_EQ0051_p038(\2.5\)	\pi=\pi_{\rho}
2004.05631v1_EQ0052_p038	\rho_{\operatorname{diag}}=\left[\begin{array}{cccc} \pi\left(s_{1}\right) & & & \\ & \pi\left(s_{2}\right) & & \\ & & \ddots & \\ & & & \pi\left(s_{n}\right) \end{array}\right] \quad \pi=\left(\frac{3}{5}, \frac{1}{5}, \frac{1}{5}\right) \quad \text { तm }\left[\begin{array}{ccc} \frac{3}{5} & 0 & 0 \\ 0 & \frac{1}{5} & 0 \\ 0 & 0 & \frac{1}{5} \end{array}\right]=\rho_{\text {diag }}
2004.05631v1_EQ0053_p039(2.6)	|\psi\rangle=\sum_{s \in S} \sqrt{\pi(s)}|s\rangle
2004.05631v1_EQ0054_p039(\2.7\)	\rho_{\pi}:=|\psi\rangle\langle\psi| .
2004.05631v1_EQ0055_p039	\pi_{\rho_{\pi}}(s)=\langle s \mid \psi\rangle\langle\psi \mid s\rangle=(\sqrt{\pi(s)})^{2}=\pi(s) .
2004.05631v1_EQ0056_p039	\rho_{\pi}=\left[\begin{array}{c} \vdots \\ \sqrt{\pi(s)} \\ \vdots \end{array}\right]\left[\begin{array}{lll} \cdots & \sqrt{\pi(s)} & \cdots \end{array}\right]
2004.05631v1_EQ0057_p039	|\psi\rangle=\square|\psi\rangle\langle\psi|=\square
2004.05631v1_EQ0058_p039	\begin{aligned} \pi=\left(\frac{3}{5}, \frac{1}{5}, \frac{1}{5}\right) & \leadsto\left[\begin{array}{l} \sqrt{\frac{3}{5}} \\ \sqrt{\frac{1}{5}} \\ \sqrt{\frac{1}{5}} \end{array}\right]\left[\begin{array}{lll} \sqrt{\frac{3}{5}} & \sqrt{\frac{1}{5}} & \sqrt{\frac{1}{5}} \end{array}\right] \\ & =\left[\begin{array}{ccc} \frac{3}{5} & \frac{\sqrt{3}}{5} & \frac{\sqrt{3}}{5} \\ \frac{\sqrt{3}}{5} & \frac{1}{5} & \frac{1}{5} \\ \frac{\sqrt{3}}{5} & \frac{1}{5} & \frac{1}{5} \end{array}\right]=\rho_{\pi} \end{aligned}
2004.05631v1_EQ0059_p040(2.8)	\rho=\sum_{i=1}^{r} \lambda_{i}\left|e_{i}\right\rangle\left\langle e_{i}\right|
2004.05631v1_EQ0060_p041	\rho=\left[\begin{array}{lll} \frac{1}{3} & \frac{1}{3} & 0 \\ \frac{1}{3} & \frac{1}{3} & 0 \\ 0 & 0 & \frac{1}{3} \end{array}\right]
2004.05631v1_EQ0061_p041	\left|e_{1}\right\rangle=\left[\begin{array}{c} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \\ 0 \end{array}\right] \quad\left|e_{2}\right\rangle=\left[\begin{array}{l} 0 \\ 0 \\ 1 \end{array}\right]
2004.05631v1_EQ0062_p041	\begin{aligned} & s_{1} \text { has probability }\left|\left\langle s_{1} \mid e_{1}\right\rangle\right|^{2}=\left(\frac{1}{\sqrt{2}}\right)=\frac{1}{2}, \\ & s_{2} \text { has probability }\left|\left\langle s_{2} \mid e_{1}\right\rangle\right|^{2}=\left(\frac{1}{\sqrt{2}}\right)=\frac{1}{2}, \\ & s_{3} \text { has probability }\left|\left\langle s_{3} \mid e_{1}\right\rangle\right|^{2}=0 . \end{aligned}
2004.05631v1_EQ0063_p041	\begin{aligned} & s_{1} \text { has probability }\left|\left\langle s_{1} \mid e_{2}\right\rangle\right|^{2}=0, \\ & s_{2} \text { has probability }\left|\left\langle s_{2} \mid e_{2}\right\rangle\right|^{2}=0, \\ & s_{3} \text { has probability }\left|\left\langle s_{3} \mid e_{2}\right\rangle\right|^{2}=1 . \end{aligned}
2004.05631v1_EQ0064_p042	p(|v\rangle,|w\rangle+|z\rangle)=|v\rangle \neq 2|v\rangle=p(|v\rangle,|w\rangle)+p(|v\rangle,|z\rangle)
2004.05631v1_EQ0065_p043	\operatorname{End}(V \otimes W) \cong \operatorname{End} V \otimes \operatorname{End} W .
2004.05631v1_EQ0066_p043	\begin{aligned} \operatorname{End}(V \otimes W) & \cong(V \otimes W) \otimes(V \otimes W)^{*} \\ & \cong V \otimes W \otimes V^{*} \otimes W^{*} \\ & \cong V \otimes V^{*} \otimes W \otimes W^{*} \\ & \cong \operatorname{End}(V) \otimes \operatorname{End}(W) \end{aligned}
2004.05631v1_EQ0067_p044	\operatorname{tr}_{W}(f \otimes g):=f \operatorname{tr}(g) \quad \operatorname{tr}_{V}(f \otimes g):=g \operatorname{tr}(f)
2004.05631v1_EQ0068_p044(2.9)	f=\sum_{\substack{i, \alpha \\ j, \beta}} f_{i \alpha, j \beta}\left|x_{i} y_{\alpha}\right\rangle\left\langle x_{j} y_{\beta}\right|
2004.05631v1_EQ0069_p044(2.9)	\operatorname{tr}_{W} f=\sum_{\substack{i, j \\ \alpha}} f_{i \alpha, j \alpha}\left|x_{i}\right\rangle\left\langle x_{j}\right| \quad \operatorname{tr}_{V} f=\sum_{\substack{\alpha, \beta \\ i}} f_{i \alpha, i \beta}\left|y_{\alpha}\right\rangle\left\langle y_{\beta}\right| .
2004.05631v1_EQ0070_p044	\begin{aligned} \operatorname{tr}_{W} & :=\operatorname{id}_{\operatorname{End}(V)} \otimes \operatorname{tr} \\ \operatorname{tr}_{V} & :=\operatorname{tr} \otimes \operatorname{id}_{\operatorname{End}(W)} \end{aligned}
2004.05631v1_EQ0071_p044	\operatorname{tr} M=\sum_{i} M_{i i}
2004.05631v1_EQ0072_p044	f=\left[\begin{array}{lll|lll} f_{1 \alpha, 1 \alpha} & f_{1 \alpha, 1 \beta} & f_{1 \alpha, 1 \gamma} & f_{1 \alpha, 2 \alpha} & f_{1 \alpha, 2 \beta} & f_{1 \alpha, 2 \gamma} \\ f_{1 \beta, 1 \alpha} & f_{1 \beta, 1 \beta} & f_{1 \beta, 1 \gamma} & f_{1 \beta, 2 \alpha} & f_{1 \beta, 2 \beta} & f_{1 \beta, 2 \gamma} \\ f_{1 \gamma, 1 \alpha} & f_{1 \gamma, 1 \beta} & f_{1 \gamma, 1 \gamma} & f_{1 \gamma, 2 \alpha} & f_{1 \gamma, 2 \beta} & f_{1 \gamma, 2 \gamma} \\ \hdashline f_{2 \alpha, 1 \alpha} & f_{2 \alpha, 1 \beta} & f_{2 \alpha, 1 \gamma} & f_{2 \alpha, 2 \alpha} & f_{2 \alpha, 2 \beta} & f_{2 \alpha, 2 \gamma} \\ f_{2 \beta, 1 \alpha} & f_{2 \beta, 1 \beta} & f_{2 \beta, 1 \gamma} & f_{2 \beta, 2 \alpha} & f_{2 \beta, 2 \beta} & f_{2 \beta, 2 \gamma} \\ f_{2 \gamma, 1 \alpha} & f_{2 \gamma, 1 \beta} & f_{2 \gamma, 1 \gamma} & f_{2 \gamma, 2 \alpha} & f_{2 \gamma, 2 \beta} & f_{2 \gamma, 2 \gamma} \end{array}\right] \quad \begin{gathered} \left(\operatorname{tr}_{W} f\right)_{11}=\sum_{\alpha} f_{1 \alpha, 1 \alpha} \\ \operatorname{tr}_{W} f= \end{gathered} \quad \begin{gathered} \\ \square \end{gathered} f_{12}=\sum_{\alpha} f_{1 \alpha, 2 \alpha}
2004.05631v1_EQ0073_p045	f=\sum_{\substack{i, \alpha \\ j, \beta}} f_{i \alpha, j \beta}\left|x_{i} y_{\alpha}\right\rangle\left\langle x_{j} y_{\beta}\right|
2004.05631v1_EQ0074_p045	\left(f_{V}\right)_{i j}=\sum_{\alpha} f_{i \alpha, j \alpha}=\sum_{\alpha} \overline{f_{j \alpha, i \alpha}}=\overline{\left(f_{V}\right)_{j i}}
2004.05631v1_EQ0075_p045	\sum_{\substack{i, \alpha \\ j, \beta}} \overline{\phi_{i \alpha}} \phi_{j \beta} f_{i \alpha, j \beta} \geq 0 \quad \text { for any } \phi_{i \alpha}, \phi_{j \beta} \in \mathbb{C} .
2004.05631v1_EQ0076_p045	\begin{aligned} \operatorname{tr}_{W} f & =\sum_{\substack{i, \alpha \\ j, \beta}} f_{i \alpha, j \beta} \operatorname{tr}_{W}\left|x_{i} y_{\alpha}\right\rangle\left\langle x_{j} y_{\beta}\right| \\ & =\sum_{\substack{i, \alpha \\ j, \beta}} f_{i \alpha, j \beta} \operatorname{tr}_{W}\left(\left|x_{i}\right\rangle\left\langle x_{j}\right| \otimes\left|y_{\alpha}\right\rangle\left\langle y_{\beta}\right|\right) \\ & =\sum_{\substack{i, \alpha \\ j, \beta}} f_{i \alpha, j \beta}\left|x_{i}\right\rangle\left\langle x_{j}\right| \otimes \operatorname{tr}\left|y_{\alpha}\right\rangle\left\langle y_{\beta}\right| \\ & =\sum_{\substack{i, \alpha \\ j, \beta}} f_{i \alpha, j \beta}\left|x_{i}\right\rangle\left\langle x_{j}\right| \otimes\left\langle y_{\alpha} \mid y_{\beta}\right\rangle \\ & =\sum_{\substack{i, j \\ \alpha}} f_{i \alpha, j \alpha}\left|x_{i}\right\rangle\left\langle x_{j}\right| . \end{aligned}
2004.05631v1_EQ0077_p046	\rho_{V}:=\operatorname{tr}_{W} \rho \quad \text { and } \quad \rho_{W}:=\operatorname{tr}_{V} \rho
2004.05631v1_EQ0078_p046	\pi_{\rho_{X}}=\left(\pi_{\rho}\right)_{X} \quad \text { and } \quad \pi_{\rho_{Y}}=\left(\pi_{\rho}\right)_{Y}
2004.05631v1_EQ0079_p046	\rho=\sum_{\substack{x y \\ x^{\prime} y^{\prime}}} \rho_{x y, x^{\prime} y^{\prime}}|x y\rangle\left\langle x^{\prime} y^{\prime}\right|
2004.05631v1_EQ0080_p046	\pi_{\rho}(x, y):=\langle x y| \rho|x y\rangle=\rho_{x y, x y}
2004.05631v1_EQ0081_p046	\langle x| \rho_{X}\left|x^{\prime}\right\rangle=\sum_{y} \rho_{x y, x^{\prime} y}
2004.05631v1_EQ0082_p046	\pi_{\rho_{X}}(x):=\langle x| \rho_{X}|x\rangle=\sum_{y} \rho_{x y, x y}=\sum_{y} \pi_{\rho}(x, y)=\left(\pi_{\rho}\right)_{X}(x)
2004.05631v1_EQ0083_p047	\rho_{V}=\sum_{\alpha} B_{\alpha} \rho B_{\alpha}^{\dagger} \quad \rho_{W}=\sum_{i} A_{i} \rho A_{i}^{\dagger}
2004.05631v1_EQ0084_p047	B_{\alpha}\left(\left|x_{j}\right\rangle \otimes\left|y_{\beta}\right\rangle\right)= \begin{cases}\left|x_{j}\right\rangle & \text { if } \beta=\alpha \\ 0 & \text { otherwise }\end{cases}
2004.05631v1_EQ0085_p047	B_{\alpha}^{\dagger}|x\rangle=|x\rangle \otimes\left|y_{\alpha}\right\rangle .
2004.05631v1_EQ0086_p047(2.10)	\left\langle x_{i}\right| \sum_{\gamma} B_{\gamma} \rho B_{\gamma}^{\dagger}\left|x_{j}\right\rangle=\sum_{\gamma} \rho_{i \gamma, j \gamma} .
2004.05631v1_EQ0087_p047(2.11)	\left\langle x_{i}\right| B_{\gamma} \rho B_{\gamma}^{\dagger}\left|x_{j}\right\rangle
2004.05631v1_EQ0088_p048	\rho\left|x_{j} y_{\gamma}\right\rangle=\sum_{\substack{i^{\prime}, \alpha \\ j^{\prime}, \beta}} \rho_{i^{\prime} \alpha, j^{\prime} \beta}\left|x_{i} y_{\alpha}\right\rangle\left\langle x_{j^{\prime}} y_{\beta} \mid x_{j} y_{\gamma}\right\rangle=\sum_{i^{\prime}, \alpha} \rho_{i^{\prime} \alpha, j \gamma}\left|x_{i}^{\prime} y_{\alpha}\right\rangle
2004.05631v1_EQ0089_p048	\sum_{i^{\prime}, \alpha} \rho_{i^{\prime} \alpha, j \gamma}\left\langle x_{i} B_{\gamma} \mid x_{i^{\prime}} y_{\alpha}\right\rangle=\sum_{i^{\prime}} \rho_{i^{\prime} \gamma, j \gamma}\left\langle x_{i} \mid x_{i^{\prime}}\right\rangle=\rho_{i \gamma, i \gamma \prime}
2004.05631v1_EQ0090_p048	\sum_{\alpha} B_{\alpha}^{\dagger} B_{\alpha}\left(\left|x_{j}\right\rangle \otimes\left|y_{\beta}\right\rangle\right)=B_{\beta}^{\dagger}\left|x_{j}\right\rangle=\left|x_{j}\right\rangle \otimes\left|y_{\beta}\right\rangle
2004.05631v1_EQ0091_p048(2.12)	V \otimes W \cong V^{*} \otimes W \cong \operatorname{hom}(V, W)
2004.05631v1_EQ0092_p049	|\psi\rangle=\sum_{x, y} \psi_{x y}|x\rangle \otimes|y\rangle,
2004.05631v1_EQ0093_p049	\left[\begin{array}{l} 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ 6 \end{array}\right] \quad \rightsquigarrow \quad\left[\begin{array}{ll} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{array}\right]
2004.05631v1_EQ0094_p049(2.13)	M=V \Sigma U^{\dagger} .
2004.05631v1_EQ0095_p049(2.14)	M=V D U_{0}^{\dagger}
2004.05631v1_EQ0096_p050
2004.05631v1_EQ0097_p050	\begin{aligned} & \left.\left[\begin{array}{lll} * & * & * \\ * & * & * \end{array}\right]=\left[\begin{array}{ll} * & * \\ * & * \end{array}\right]\left[\begin{array}{ll} * & \\ & * \end{array}\right] \begin{array}{l} 0 \\ 0 \end{array}\right]\left[\begin{array}{lll} * & * & * \\ * & * & * \end{array}\right] \\ & M \quad=\quad V \quad D \quad U_{0}^{\dagger} \end{aligned}
2004.05631v1_EQ0098_p050(2.15)	|\psi\rangle=\sum_{i=1}^{m} \sigma_{i}\left|f_{i}\right\rangle \otimes\left|e_{i}\right\rangle
2004.05631v1_EQ0099_p050	|\psi\rangle=\sum_{i, \alpha} \psi_{i \alpha}\left|x_{i}\right\rangle \otimes\left|y_{\alpha}\right\rangle .
2004.05631v1_EQ0100_p051	\begin{aligned} \rho & =|\psi\rangle\langle\psi| \\ & =\left(\sum_{i=1}^{m} \sigma_{i}\left|e_{i}\right\rangle \otimes\left|f_{i}\right\rangle\right)\left(\sum_{j=1}^{m} \sigma_{j}\left\langle f_{j}\right| \otimes\left\langle e_{j}\right|\right) \\ & =\sum_{i, j=1}^{m} \sigma_{i} \sigma_{j}\left|e_{i}\right\rangle\left\langle e_{j}\right| \otimes\left|f_{i}\right\rangle\left\langle f_{j}\right| \end{aligned}
2004.05631v1_EQ0101_p051(2.16)	\begin{aligned} \operatorname{tr}_{W}\left(\sum_{i, j=1}^{m} \sigma_{i} \sigma_{j}\left|e_{i}\right\rangle\left\langle e_{j}\right| \otimes\left|f_{i}\right\rangle\left\langle f_{j}\right|\right) & =\sum_{i, j=1}^{m} \sigma_{i} \sigma_{j}\left|e_{i}\right\rangle\left\langle e_{j}\right| \cdot \operatorname{tr}\left(\left|f_{i}\right\rangle\left\langle f_{j}\right|\right) \\ & =\sum_{i, j=1}^{m} \sigma_{i} \sigma_{j}\left|e_{i}\right\rangle\left\langle e_{j}\right|\left\langle f_{i} \mid f_{j}\right\rangle \\ & =\sum_{i=1}^{m} \sigma_{i}^{2}\left|e_{i}\right\rangle\left\langle e_{i}\right| \end{aligned}
2004.05631v1_EQ0102_p052(2.17)	\begin{aligned} \operatorname{tr}_{V}\left(\sum_{i, j=1}^{m} \sigma_{i} \sigma_{j}\left|e_{i}\right\rangle\left\langle e_{j}\right| \otimes\left|f_{i}\right\rangle\left\langle f_{j}\right|\right) & =\sum_{i, j=1}^{m} \sigma_{i} \sigma_{j} \operatorname{tr}\left(\left|e_{i}\right\rangle\left\langle e_{j}\right|\right) \cdot\left|f_{i}\right\rangle\left\langle f_{j}\right| \\ & =\sum_{i, j=1}^{m} \sigma_{i} \sigma_{j}\left\langle e_{i} \mid e_{j}\right\rangle\left|f_{i}\right\rangle\left\langle f_{j}\right| \\ & =\sum_{i=1}^{m} \sigma_{i}^{2}\left|f_{i}\right\rangle\left\langle f_{i}\right| \end{aligned}
2004.05631v1_EQ0103_p052(2.17)	\rho_{V}\left|e_{i}\right\rangle=\lambda_{i}\left|e_{i}\right\rangle \quad \text { and } \quad \rho_{W}\left|f_{i}\right\rangle=\lambda_{i}\left|f_{i}\right\rangle
2004.05631v1_EQ0104_p052	\left|e_{i}\right\rangle \stackrel{\lambda_{i}}{\longleftrightarrow}\left|f_{i}\right\rangle
2004.05631v1_EQ0105_p052	M\left|e_{i}\right\rangle=\lambda_{i}\left|f_{i}\right\rangle \quad M^{\dagger}\left|f_{i}\right\rangle=\lambda_{i}\left|e_{i}\right\rangle .
2004.05631v1_EQ0106_p052(2.18)	\rho_{V}=U D^{2} U^{\dagger} \quad \rho_{W}=V D^{2} V^{\dagger}
2004.05631v1_EQ0107_p052(2.19)	\rho_{V}=M^{\dagger} M \quad \rho_{W}=M M^{\dagger}
2004.05631v1_EQ0108_p052	\begin{aligned} M^{\dagger} M & =\left(V D U^{\dagger}\right)^{\dagger}\left(V D U^{\dagger}\right) \\ & =U D^{\dagger} V^{\dagger} V D U^{\dagger} \\ & =U D^{2} U^{\dagger} \\ & =\rho_{V} \end{aligned}
2004.05631v1_EQ0109_p053	\begin{aligned} M M^{\dagger} & =\left(V D U^{\dagger}\right)\left(V D U^{\dagger}\right)^{\dagger} \\ & =V D^{\dagger} U^{\dagger} U D V^{\dagger} \\ & =V D^{2} V^{\dagger} \\ & =\rho_{W} \end{aligned}
2004.05631v1_EQ0110_p056	\pi(x, y)=\pi_{X}(x) \pi_{Y}(y)
2004.05631v1_EQ0111_p056	\pi_{\rho}(x, y)=\pi_{\rho_{X}}(x) \pi_{\rho_{Y}}(y)
2004.05631v1_EQ0112_p056	\begin{aligned} \pi_{\rho}(x, y) & =\langle x y \mid a b\rangle\langle a b \mid x y\rangle \\ & =\langle x \mid a\rangle\langle y \mid b\rangle\langle a \mid x\rangle\langle b \mid y\rangle \\ & =|\langle a \mid x\rangle|^{2}|\langle b \mid y\rangle|^{2} \\ & =\pi_{\rho_{X}}(x) \pi_{\rho_{Y}}(y) . \end{aligned}
2004.05631v1_EQ0113_p056	\pi_{\rho}(x, y)=\left(\pi_{\rho}\right)_{X}(x)\left(\pi_{\rho}\right)_{Y}(y) .
2004.05631v1_EQ0114_p062(\3.1\)	|\psi\rangle=\sum_{i, \alpha} \sqrt{\pi\left(x_{i}, y_{\alpha}\right)}\left|x_{i}\right\rangle \otimes\left|y_{\alpha}\right\rangle
2004.05631v1_EQ0115_p063	\begin{aligned} \rho & =|\psi\rangle\langle\psi| \\ & =\left(\sum_{i, \alpha} \sqrt{\pi\left(x_{i}, y_{\alpha}\right)}\left|x_{i}\right\rangle \otimes\left|y_{\beta}\right\rangle\right)\left(\sum_{j, \beta} \sqrt{\pi\left(x_{j}, y_{\beta}\right)}\left\langle x_{j}\right| \otimes\left\langle y_{\beta}\right|\right) \\ & =\sum_{\substack{i, \alpha \\ j, \beta}} \sqrt{\pi\left(x_{i}, y_{\alpha}\right) \pi\left(x_{j}, y_{\beta}\right)}\left|x_{i}\right\rangle\left\langle x_{j}\right| \otimes\left|y_{\alpha}\right\rangle\left\langle y_{\beta}\right| \end{aligned}
2004.05631v1_EQ0116_p063	\rho_{X}=M^{\dagger} M \quad \rho_{Y}=M M^{\dagger}
2004.05631v1_EQ0117_p064	M_{\alpha i}:=\sqrt{\pi\left(x_{i}, y_{\alpha}\right)}
2004.05631v1_EQ0118_p064	\begin{aligned} \rho_{X} & =\operatorname{tr}_{Y}|\psi\rangle\langle\psi| \\ & =\sum_{\substack{i, \alpha \\ j, \beta}} \sqrt{\pi\left(x_{i}, y_{\alpha}\right) \pi\left(x_{j}, y_{\beta}\right)} \operatorname{tr}_{Y}\left(\left|x_{i}\right\rangle\left\langle x_{j}\right| \otimes\left|y_{\alpha}\right\rangle\left\langle y_{\beta}\right|\right) \\ & =\sum_{\substack{i, \alpha \\ j, \beta}} \sqrt{\pi\left(x_{i}, y_{\alpha}\right) \pi\left(x_{j}, y_{\beta}\right)}\left|x_{i}\right\rangle\left\langle x_{j}\right| \cdot \operatorname{tr}\left|y_{\alpha}\right\rangle\left\langle y_{\beta}\right| \\ & =\sum_{\substack{i, j \\ \alpha}} \sqrt{\pi\left(x_{i}, y_{\alpha}\right) \pi\left(x_{j}, y_{\alpha}\right)}\left|x_{i}\right\rangle\left\langle x_{j}\right| \end{aligned}
2004.05631v1_EQ0119_p064	\rho_{X}=\left[\begin{array}{cccc}\pi_{X}\left(x_{1}\right) & & & * \\ & \pi_{X}\left(x_{2}\right) & & \\ * & & \ddots & \\ * & & & \pi_{X}\left(x_{n}\right)\end{array}\right]
2004.05631v1_EQ0120_p064(\3.2\)	\left(\rho_{X}\right)_{i j}=\sum_{\alpha} \sqrt{\pi\left(x_{i}, y_{\alpha}\right) \pi\left(x_{j}, y_{\alpha}\right)}
2004.05631v1_EQ0121_p064(\3 \cdot 3\)	\rho_{Y}=\sum_{\substack{\alpha, \beta \\ i}} \sqrt{\pi\left(x_{i}, y_{\alpha}\right) \pi\left(x_{i}, y_{\beta}\right)}\left|y_{\alpha}\right\rangle\left\langle y_{\beta}\right|
2004.05631v1_EQ0122_p065(\3 \cdot 4\)	\left(\rho_{Y}\right)_{\alpha \beta}=\sum_{i} \sqrt{\pi\left(x_{i}, y_{\alpha}\right) \pi\left(x_{i}, y_{\beta}\right)}
2004.05631v1_EQ0123_p065(\3 \cdot 4\)	\rho_{Y}=\left[\begin{array}{cccc} \pi_{Y}\left(y_{1}\right) & & & * \\ & \pi_{Y}\left(y_{2}\right) & & \\ * & & \ddots & \\ * & & & \pi_{Y}\left(y_{m}\right) \end{array}\right]
2004.05631v1_EQ0124_p066	\rho_{X}=\sum_{i=1}^{m} \lambda_{i}\left|e_{i}\right\rangle\left\langle e_{i}\right| \quad \rho_{Y}=\sum_{i=1}^{m} \lambda_{i}\left|f_{i}\right\rangle\left\langle f_{i}\right|
2004.05631v1_EQ0125_p066(\3 \cdot 5\)	\pi(x, y)=\pi(x \mid y) \pi_{Y}(y)
2004.05631v1_EQ0126_p067	\begin{aligned} \pi(\text { orange, fruit }) & =1 / 3 \\ \pi(\text { green, fruit }) & =1 / 3 \\ \pi(\text { purple, vegetable }) & =1 / 3 \end{aligned}
2004.05631v1_EQ0127_p067	M=V D U^{\dagger}
2004.05631v1_EQ0128_p068	\begin{array}{r} \left.\left.\mid \text { orange }\rangle \left.=\left[\begin{array}{l} 1 \\ 0 \\ 0 \end{array}\right] \quad \right\rvert\, \text { green }\right\rangle \left.=\left[\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right] \quad \right\rvert\, \text { purple }\right\rangle=\left[\begin{array}{l} 0 \\ 0 \\ 1 \end{array}\right] \\ \left.\mid \text { fruit }\rangle \left.=\left[\begin{array}{l} 1 \\ 0 \end{array}\right] \quad \right\rvert\, \text { vegetable }\right\rangle=\left[\begin{array}{l} 0 \\ 1 \end{array}\right] \end{array}
2004.05631v1_EQ0129_p068	\mid \text { orange }\rangle \otimes \mid \text { fruit }\rangle=\left[\begin{array}{l} 1 \\ 0 \\ 0 \end{array}\right]\left[\begin{array}{ll} 1 & 0 \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\right]
2004.05631v1_EQ0130_p068	\begin{aligned} \mid \text { orange }\rangle \otimes \mid \text { fruit }\rangle & =\left[\begin{array}{llllll} 1 & 0 & 0 & 0 & 0 & 0 \end{array}\right]^{\top} \\ \mid \text { green }\rangle \otimes \mid \text { fruit }\rangle & =\left[\begin{array}{llllll} 0 & 1 & 0 & 0 & 0 & 0 \end{array}\right]^{\top} \\ \mid \text { purple }\rangle \otimes \mid \text { vegetable }\rangle & =\left[\begin{array}{llllll} 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right]^{\top} \end{aligned}
2004.05631v1_EQ0131_p068	|\psi\rangle=\left[\begin{array}{llllll} \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & 0 & 0 & 0 & \frac{1}{\sqrt{3}} \end{array}\right]^{\top} \text {. }
2004.05631v1_EQ0132_p069	\left.\left.\left.\left.\left.|\psi\rangle\langle\psi|=\frac{1}{3}(\mid \text { orange }\rangle\langle\text { orange }| \otimes \right\rvert\, \text { fruit }\right\rangle\langle\text { fruit }|+\mid \text { orange }\right\rangle\langle\text { green }| \otimes \mid \text { fruit }\right\rangle\langle\text { vegetable }|+\cdots\right)
2004.05631v1_EQ0133_p069(\3.6\)	o=\text { orange } \quad g=\text { green } \quad p=\text { purple } \quad F=\text { fruit } \quad V=\text { vegetable }
2004.05631v1_EQ0134_p069(\3.6\)	|\psi\rangle\langle\psi|=\frac{1}{3}\left(\begin{array}{rl|l|} |0\rangle\langle 0| \otimes|F\rangle\langle F| & +|0\rangle\langle g| \otimes|F\rangle\langle F| & +|0\rangle\langle p| \otimes|F\rangle\langle V| \\ |g\rangle\langle 0| \otimes|F\rangle\langle F| & +|g\rangle\langle g| \otimes|F\rangle\langle F| & +|g\rangle\langle p| \otimes|F\rangle\langle V| \\ |p\rangle\langle 0| \otimes|V\rangle\langle F| & +|p\rangle\langle g| \otimes|V\rangle\langle F| & +|p\rangle\langle p| \otimes|V\rangle\langle V| \end{array}\right)
2004.05631v1_EQ0135_p069(\3.6\)	|\psi\rangle\langle\psi|=\left[\begin{array}{cccccc} \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 & \frac{1}{3} \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 & \frac{1}{3} \end{array}\right]
2004.05631v1_EQ0136_p069	\rho_{X}=\operatorname{tr}_{Y}|\psi\rangle\langle\psi| \quad \rho_{Y}=\operatorname{tr}_{X}|\psi\rangle\langle\psi|
2004.05631v1_EQ0137_p070	\begin{aligned} \rho_{X}=\operatorname{tr}_{Y}|\psi\rangle\langle\psi| & =\frac{1}{3}\left(\begin{array}{ll} |0\rangle\langle 0| \otimes\langle F \mid F\rangle & +|0\rangle\langle g| \otimes\langle F \mid F\rangle \\ |g\rangle\langle 0| \otimes\langle F \mid F\rangle & +|0\rangle\langle p| \otimes\langle F \mid V\rangle \\ |p\rangle\langle 0| \otimes\langle V \mid E\rangle & +|p\rangle\langle g| \otimes\langle V \mid F\rangle+|g\rangle\langle p| \otimes\langle F \mid V\rangle \\ |g| & +|p\rangle\langle p| \otimes\langle V \mid V\rangle \end{array}\right) \\ & =\frac{1}{3}(|0\rangle\langle 0|+|0\rangle\langle g|+|g\rangle\langle 0|+|g\rangle\langle g|+|p\rangle\langle p|) \end{aligned}
2004.05631v1_EQ0138_p070(\3 \cdot 7\)	\begin{array}{ll} |0\rangle\langle 0|=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] & |p\rangle\langle p|=\left[\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right] \\ |g\rangle\langle g|=\left[\begin{array}{lll} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right] & |0\rangle\langle g|=\left[\begin{array}{lll} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \end{array}
2004.05631v1_EQ0139_p070(\3 \cdot 7\)	\rho_{X}=\frac{g}{p}\left[\begin{array}{ccc} o & g & p \\ 1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \frac{1}{3}
2004.05631v1_EQ0140_p071	\begin{aligned} & \rho_{Y}=\operatorname{tr}_{X}|\psi\rangle\langle\psi|=\frac{1}{3}\left(\begin{array}{lll} \langle 0 \mid 0\rangle \otimes|F\rangle\langle F| & +\langle\mid g\rangle \otimes|F\rangle\langle F| & +\langle\mid p\rangle \otimes|F\rangle\langle V| \\ \langle g \mid 0\rangle \otimes|F\rangle\langle F| & +\langle g \mid g\rangle \otimes|F\rangle\langle F| & +\langle g \mid p\rangle \otimes|F\rangle\langle V| \\ \langle p \mid 0\rangle \otimes|V\rangle\langle F| & +\langle p \mid g\rangle \otimes|V\rangle\langle F| & +\langle p \mid p\rangle \otimes|V\rangle\langle V| \end{array}\right) \\ &= \frac{1}{3}(|F\rangle\langle F|+|F\rangle\langle F|+|V\rangle\langle V|) \end{aligned}
2004.05631v1_EQ0141_p071(\3.8\)	|F\rangle\langle F|=\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right] \quad|V\rangle\langle V|=\left[\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right]
2004.05631v1_EQ0142_p071(\3.8\)	\rho_{Y}=\begin{gathered} \\ F \\ V \end{gathered} \begin{array}{cc} F & V \\ {\left[\begin{array}{cc} 2 & 0 \\ 0 & 1 \end{array}\right] \frac{1}{3}} \end{array}
2004.05631v1_EQ0143_p071	M=\left[\begin{array}{ccc} \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & 0 \\ 0 & 0 & \frac{1}{\sqrt{3}} \end{array}\right] \leftrightarrow|\psi\rangle
2004.05631v1_EQ0144_p071	\begin{gathered} M^{\dagger} M=\left[\begin{array}{ccc} \frac{1}{3} & \frac{1}{3} & 0 \\ \frac{1}{3} & \frac{1}{3} & 0 \\ 0 & 0 & \frac{1}{3} \end{array}\right]=\rho_{X} \\ M M^{\dagger}=\left[\begin{array}{cc} \frac{2}{3} & 0 \\ 0 & \frac{1}{3} \end{array}\right]=\rho_{Y} \end{gathered}
2004.05631v1_EQ0145_p073	\hat{\pi}(x, y)= \begin{cases}\frac{1}{|T|} & \text { if }(x, y) \in T \\ 0 & \text { otherwise }\end{cases}
2004.05631v1_EQ0146_p073(\3 \cdot 9\)	|\psi\rangle=\frac{1}{\sqrt{|T|}} \sum_{\left(x_{i}, y_{\alpha}\right) \in T}\left|x_{i}\right\rangle \otimes\left|y_{\alpha}\right\rangle
2004.05631v1_EQ0147_p073	\begin{aligned} d\left(x_{i}, x_{j}\right) & :=\mid\left\{y \in Y \mid\left(x_{i}, y\right) \in T \text { and }\left(x_{j}, y\right) \in T\right\} \mid \\ d\left(y_{\alpha}, y_{\beta}\right) & :=\mid\left\{x \in X \mid\left(x, y_{\alpha}\right) \in T \text { and }\left(x, y_{\beta}\right) \in T\right\} \mid \end{aligned}
2004.05631v1_EQ0148_p074(\3.10\)	\begin{aligned} \left(\rho_{X}\right)_{i j} & =\frac{d\left(x_{i}, x_{j}\right)}{|T|} \\ \left(\rho_{Y}\right)_{\alpha \beta} & =\frac{d\left(y_{\alpha}, y_{\beta}\right)}{|T|} \end{aligned}
2004.05631v1_EQ0149_p075	\rho_{X}=x_{2}\left[\begin{array}{ccc} x_{1} & x_{2} & x_{3} \\ x_{1} \end{array}\left[\begin{array}{cc} 1 & 1 \\ 1 & 2 \\ 1 & 2 \\ 2 \end{array}\right] \frac{1}{5} \quad \rho_{Y}=\begin{array}{c} y_{1} \\ y_{2} \end{array}\left[\begin{array}{cc} y_{1} & y_{2} \\ 3 & 2 \\ 2 & 2 \end{array}\right] \frac{1}{5}\right.
2004.05631v1_EQ0150_p075	\left[\begin{array}{ll} 3 & 2 \\ 2 & 2 \end{array}\right]
2004.05631v1_EQ0151_p075	\left[\begin{array}{lll} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 2 \end{array}\right]
2004.05631v1_EQ0152_p075	\left[\begin{array}{lll} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 2 \end{array}\right]
2004.05631v1_EQ0153_p076	\left[\begin{array}{lll} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 2 \end{array}\right]
2004.05631v1_EQ0154_p076	T=\{(\text { orange, fruit }),(\text { green, fruit }),(\text { purple, vegetable })\}
2004.05631v1_EQ0155_p076	\rho_{X}=\frac{o}{g}\left[\begin{array}{ccc} o & g & p \\ p & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \frac{1}{3}
2004.05631v1_EQ0156_p076	\left.\rho_{Y}=\begin{array}{c} \\ F \\ V \end{array} \begin{array}{cc} F & V \\ {\left[\begin{array}{l} 2 \\ 0 \end{array}\right.} & 0 \\ 0 & 1 \end{array}\right] \frac{1}{3}
2004.05631v1_EQ0157_p077	|\psi\rangle=\sum_{(x, y) \in X \times Y} \sqrt{\pi(x, y)}|x\rangle \otimes|y\rangle .
2004.05631v1_EQ0158_p077	\left(\rho_{X}\right)_{i i}=\sum_{\alpha} \pi\left(x_{i}, y_{\alpha}\right)=\pi_{X}\left(x_{i}\right)
2004.05631v1_EQ0159_p077	\left(\rho_{Y}\right)_{\alpha \alpha}=\sum_{i} \pi\left(x_{i}, y_{\alpha}\right)=\pi_{Y}\left(y_{\alpha}\right)
2004.05631v1_EQ0160_p077	\begin{aligned} \left(\rho_{X}\right)_{i j} & =\sum_{\alpha} \sqrt{\pi\left(x_{i}, y_{\alpha}\right) \pi\left(x_{j}, y_{\alpha}\right)} \\ \left(\rho_{Y}\right)_{\alpha \beta} & =\sum_{i} \sqrt{\pi\left(x_{i}, y_{\alpha}\right) \pi\left(x_{i}, y_{\beta}\right)} \end{aligned}
2004.05631v1_EQ0161_p080	\begin{array}{ll} \left.\left.\left.\left\{\left|e_{1}\right\rangle,\left|f_{1}\right\rangle\right\}=\left\{\frac{1}{\sqrt{2}}(\mid \text { orange }\rangle+\mid \text { green }\right\rangle\right), \mid \text { fruit }\right\rangle\right\} & \lambda_{1}=\frac{2}{3} \\ \left.\left.\left\{\left|e_{2}\right\rangle,\left|f_{2}\right\rangle\right\}=\{\mid \text { purple }\rangle, \mid \text { vegetable }\right\rangle\right\} & \lambda_{2}=\frac{1}{3} \end{array}
2004.05631v1_EQ0162_p080	\begin{aligned} & X \longrightarrow 2^{Y} \\ & x \longmapsto a(x):=\{y \in Y \mid R(x, y)=1\} \\ & Y \longrightarrow 2^{X} \longrightarrow X(y):=\{x \in X \mid R(x, y)=1\} \\ & y \longmapsto 2^{X} \end{aligned}
2004.05631v1_EQ0163_p080	\begin{aligned} & X \xrightarrow{\alpha} \mathbb{C}^{Y} \\ & x_{i} \longmapsto A_{i}|\psi\rangle \end{aligned}
2004.05631v1_EQ0164_p080	\begin{gathered} Y \xrightarrow{\beta} \mathbb{C}^{X} \\ y_{\alpha} \longmapsto B_{\alpha}|\psi\rangle \end{gathered}
2004.05631v1_EQ0165_p081	\begin{aligned} & (\{\text { orange, green }\},\{\text { fruit }\}) \\ & (\{\text { green, purple }\},\{\text { vegetable }\}) \\ & (\{\text { green }\},\{\text { fruit, vegetable }\}) \end{aligned}
2004.05631v1_EQ0166_p081(6)	\begin{array}{ll} \left.\left.\left.\left.\left.\left\{\frac{1}{\sqrt{6}}(\mid \text { orange }\rangle+\mid \text { green }\right\rangle+\mid \text { purple }\right\rangle\right), \frac{1}{\sqrt{2}}(\mid \text { fruit }\rangle+\mid \text { vegetable }\right\rangle\right)\right\} & \lambda_{1}=\frac{3}{4} \\ \left.\left.\left.\left.\left\{\frac{1}{\sqrt{2}}(\mid \text { orange }\rangle-\mid \text { purple }\right\rangle\right), \frac{1}{\sqrt{2}}(\mid \text { fruit }\rangle-\mid \text { vegetable }\right\rangle\right)\right\} & \lambda_{2}=\frac{1}{4} \end{array}
2004.05631v1_EQ0167_p084(\3.12\)	\rho_{\text {orange }} \geq \pi(\text { small ripe orange } \mid \text { orange }) \rho_{\text {small ripe orange }}
2004.05631v1_EQ0168_p085	\begin{array}{ll} \text { small ripe orange fruit } & \text { large rotten green vegetable } \\ \text { large ripe orange vegetable } & \text { small ripe orange vegetable } \\ \text { small rotten orange fruit } & \end{array}
2004.05631v1_EQ0169_p085	\begin{array}{ll} A:=\{\text { small, large }\} & C:=\{\text { orange, green }\} \\ B:=\{\text { ripe, rotten }\} & Y:=\{\text { fruit, vegetable }\} \end{array}
2004.05631v1_EQ0170_p085	\hat{\pi}(x, y)= \begin{cases}\frac{1}{5} & \text { if }(x, y) \in T \\ 0 & \text { if }(x, y) \notin T .\end{cases}
2004.05631v1_EQ0171_p085	\begin{aligned} |\psi\rangle & =\sum_{x, y} \sqrt{\hat{\pi}(x, y)}|x\rangle \otimes|y\rangle \\ & =\sum_{a, b, c, y} \sqrt{\hat{\pi}(a, b, c, y)}|a\rangle \otimes|b\rangle \otimes|c\rangle \otimes|y\rangle \\ & =\frac{1}{\sqrt{5}}\left(\begin{array}{ccccccc} \mid \text { small }\rangle & \otimes & \mid \text { ripe }\rangle & \otimes & \mid \text { orange }\rangle & \otimes & \mid \text { fruit } \\ \mid \text { small }\rangle & \otimes & \mid \text { rotten }\rangle & \otimes & \mid \text { orange }\rangle & \otimes & +\mid \text { fruit }\rangle \\ \mid \text { large }\rangle & \otimes & \mid \text { ripe }\rangle & \otimes & \mid \text { orange }\rangle & \otimes & \mid \text { vegetable }\rangle \\ \mid \text { large }\rangle & \otimes & \mid \text { rotten }\rangle & \otimes & \mid \text { green }\rangle & \otimes & \mid \text { vegetable }\rangle \\ \mid \text { small }\rangle & \otimes & \mid \text { ripe }\rangle & \otimes & \mid \text { orange }\rangle & \otimes & \mid \text { vegetable }\rangle \end{array}\right) \end{aligned}
2004.05631v1_EQ0172_p085	|\psi\rangle=\sqrt{\|}
2004.05631v1_EQ0173_p086(\3.13\)	\rho_{Y}=\sum_{x \in X} A_{x}|\psi\rangle\langle\psi| A_{x}^{\dagger}
2004.05631v1_EQ0174_p086(3.14)	\begin{aligned} A_{x}|\psi\rangle & =\sum_{y} \sqrt{\hat{\pi}(x, y)}|y\rangle \\ & =\sqrt{\hat{\pi}(x, \text { fruit })} \mid \text { fruit }\rangle+\sqrt{\hat{\pi}(x, \text { vegetable })} \mid \text { vegetable }\rangle . \end{aligned}
2004.05631v1_EQ0175_p086	\begin{array}{ll} \mathbf{x}_{\mathbf{1}}=(\text { small, ripe, orange }) & x_{5}=(\text { small, ripe, green }) \\ \mathbf{x}_{\mathbf{2}}=(\text { large, ripe, orange }) & x_{6}=(\text { large, ripe, green }) \\ \mathbf{x}_{\mathbf{3}}=(\text { small, rotten, orange }) & x_{7}=(\text { small, rotten, green }) \\ x_{4}=(\text { large, rotten, orange }) & \mathbf{x}_{\mathbf{8}}=(\text { large, rotten, green }) \end{array}
2004.05631v1_EQ0176_p086	M=\frac{1}{\sqrt{5}}\left[\begin{array}{llllllll} 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right]
2004.05631v1_EQ0177_p086	\rho_{Y}=\sum_{(a, b, c) \in A \times B \times C} M|a b c\rangle\langle a b c| M^{\dagger}
2004.05631v1_EQ0178_p087	M=\begin{gathered} y_{1} \\ y_{2} \end{gathered}\left[\begin{array}{cccccccc} x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right] \begin{gathered} 1 \\ \frac{1}{\sqrt{5}} \end{gathered}
2004.05631v1_EQ0179_p087	M=\begin{gathered} \\ \text { fruit } \\ \text { vegetable } \end{gathered}
2004.05631v1_EQ0180_p087	\begin{aligned} & \left.\left.A_{1}|\psi\rangle=\frac{1}{\sqrt{5}}(\mid \text { fruit }\rangle+\mid \text { vegetable }\right\rangle\right)=\frac{1}{\sqrt{5}}\left[\begin{array}{l} 1 \\ 1 \end{array}\right] \\ & \left.\left.A_{2}|\psi\rangle=\frac{1}{\sqrt{5}} \right\rvert\, \text { vegetable }\right\rangle=\frac{1}{\sqrt{5}}\left[\begin{array}{l} 0 \\ 1 \end{array}\right] \\ & \left.\left.A_{3}|\psi\rangle=\frac{1}{\sqrt{5}} \right\rvert\, \text { fruit }\right\rangle=\frac{1}{\sqrt{5}}\left[\begin{array}{l} 1 \\ 0 \end{array}\right] \\ & \left.\left.A_{8}|\psi\rangle=\frac{1}{\sqrt{5}} \right\rvert\, \text { vegetable }\right\rangle=\frac{1}{\sqrt{5}}\left[\begin{array}{l} 0 \\ 1 \end{array}\right] \end{aligned}
2004.05631v1_EQ0181_p088	\begin{aligned} \rho_{Y} & =A_{1}|\psi\rangle\langle\psi| A_{1}^{\dagger}+A_{2}|\psi\rangle\langle\psi| A_{2}^{\dagger}+A_{3}|\psi\rangle\langle\psi| A_{3}^{\dagger}+A_{8}|\psi\rangle\langle\psi| A_{8}^{\dagger} \\ & =\frac{1}{5}\left(\left[\begin{array}{l} 1 \\ 1 \end{array}\right]\left[\begin{array}{ll} 1 & 1 \end{array}\right]+\left[\begin{array}{l} 0 \\ 1 \end{array}\right]\left[\begin{array}{ll} 0 & 1 \end{array}\right]+\left[\begin{array}{l} 1 \\ 0 \end{array}\right]\left[\begin{array}{ll} 1 & 0 \end{array}\right]+\left[\begin{array}{l} 0 \\ 1 \end{array}\right]\left[\begin{array}{ll} 0 & 1 \end{array}\right]\right) \\ & =\frac{1}{5}\left(\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right]+\left[\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right]+\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right]+\left[\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right]\right) \\ & =\frac{1}{5}\left[\begin{array}{ll} 2 & 1 \\ 1 & 3 \end{array}\right] \end{aligned}
2004.05631v1_EQ0182_p088	\hat{\rho}_{x_{i}}=A_{i}|\psi\rangle\langle\psi| A_{i}^{\dagger}
2004.05631v1_EQ0183_p088	\rho_{x_{i}}:=\frac{1}{\hat{\pi}_{X}\left(x_{i}\right)} \hat{\rho}_{x_{i}}
2004.05631v1_EQ0184_p088	\begin{aligned} & \rho_{x_{3}}=\rho_{\text {small rotten orange }}=\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right] \\ & \rho_{x_{8}}=\rho_{\text {large rotten green }}=\left[\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right] \end{aligned}
2004.05631v1_EQ0185_p088	\begin{aligned} & \hat{\pi}_{X}\left(x_{1}\right)=2 / 5 \\ & \hat{\pi}_{X}\left(x_{2}\right)=1 / 5 \\ & \hat{\pi}_{X}\left(x_{3}\right)=1 / 5 \\ & \hat{\pi}_{X}\left(x_{8}\right)=1 / 5 \end{aligned}
2004.05631v1_EQ0186_p089	\rho_{Y}=\overbrace{A_{1}|\psi\rangle\langle\psi| A_{1}^{+}+A_{2}|\psi\rangle\langle\psi| A_{2}^{+}+A_{3}|\psi\rangle\langle\psi| A_{3}^{+}}^{\rho_{\text {orange }}}+\overbrace{A_{8}|\psi\rangle\langle\psi| A_{8}^{+}}^{\rho_{\text {green }}}
2004.05631v1_EQ0187_p089	\begin{aligned} \rho_{\text {orange }} & :=\frac{1}{\hat{\pi}_{C}(\text { orange })} \hat{\rho}_{\text {orange }} \\ & =\frac{1}{\hat{\pi}_{C}(\text { orange })}\left(\hat{\rho}_{x_{1}}+\hat{\rho}_{x_{2}}+\hat{\rho}_{x_{3}}\right) \\ & =\frac{1}{4}\left[\begin{array}{ll} 2 & 1 \\ 1 & 2 \end{array}\right] \end{aligned}
2004.05631v1_EQ0188_p089	\begin{aligned} \rho_{\text {orange }} & =\frac{1}{\pi_{C}(\text { orange })} \sum_{i=1,2,3} \hat{\rho}_{x_{i}} \\ & =\sum_{i=1,2,3} \frac{\pi_{X}\left(x_{i}\right)}{\pi_{C}(\text { orange })} \frac{\hat{\rho}_{x_{i}}}{\pi_{X}\left(x_{i}\right)} \\ & =\sum_{i=1,2,3} \pi\left(x_{i} \mid \text { orange }\right) \rho_{x_{i}} . \end{aligned}
2004.05631v1_EQ0189_p089	\rho_{\text {orange }} \geq \pi\left(x_{i} \mid \text { orange }\right) \rho_{x_{i}}
2004.05631v1_EQ0190_p089	\rho_{\text {orange }} \geq \pi(\text { small ripe orange } \mid \text { orange }) \rho_{\text {small ripe orange }} .
2004.05631v1_EQ0191_p090	\hat{\rho}_{\text {ripe orange }}:=A_{1}|\psi\rangle\langle\psi| A_{1}^{\dagger}+A_{2}|\psi\rangle\langle\psi| A_{2}^{\dagger}=\frac{1}{5}\left[\begin{array}{ll} 1 & 1 \\ 1 & 2 \end{array}\right] .
2004.05631v1_EQ0192_p090	\hat{\rho}_{x_{i}}:=M\left|x_{i}\right\rangle\left\langle x_{i}\right| M^{\dagger}
2004.05631v1_EQ0193_p090	\rho_{x_{i}}:=\frac{1}{\hat{\pi}_{X}\left(x_{i}\right)} \hat{\rho}_{x_{i}}
2004.05631v1_EQ0194_p091	\rho_{Y}=\sum_{i} \hat{\rho}_{x_{i}}
2004.05631v1_EQ0195_p091	\hat{\rho}_{a}=\sum_{x \in X(a)} \hat{\rho}_{x}
2004.05631v1_EQ0196_p091	\begin{aligned} \rho_{a} & :=\frac{1}{\hat{\pi}(a)} \hat{\rho}_{a} \\ & =\sum_{x \in X(a)} \frac{1}{\hat{\pi}(a)} \hat{\rho}_{x} \\ & =\sum_{x \in X(a)} \frac{\hat{\pi}(x)}{\hat{\pi}(a)} \frac{1}{\hat{\pi}(x)} \hat{\rho}_{x} \\ & =\sum_{x \in X(a)} \pi(x \mid a) \rho_{x} . \end{aligned}
2004.05631v1_EQ0197_p096	\hat{\pi}(s)= \begin{cases}\frac{1}{N_{T}} & \text { if } s \in T \\ 0 & \text { otherwise }\end{cases}
2004.05631v1_EQ0198_p096(\4 \cdot 1\)	|\psi\rangle=\frac{1}{\sqrt{N}_{T}} \sum_{s \in T}|s\rangle .
2004.05631v1_EQ0199_p096(\4 \cdot 1\)	|\psi\rangle=\|_{\||\cdots\rangle}=\frac{1}{\sqrt{N}_{T}} \sum \mathrm{p} \mathrm{p}, \cdots \mathrm{p}
2004.05631v1_EQ0200_p102	\rho_{2}=\begin{gathered} \\ 00 \\ 11 \\ 01 \\ 10 \end{gathered}\left[\begin{array}{cccc} 00 & 11 & 01 & 10 \\ \operatorname{deg} 00 & s_{e} & 0 & 0 \\ s_{e} & \operatorname{deg} 11 & 0 & 0 \\ 0 & 0 & \operatorname{deg} 01 & s_{o} \\ 0 & 0 & s_{o} & \operatorname{deg} 10 \end{array}\right] \frac{1}{2^{4}}=\frac{1}{2^{4}}\left[\begin{array}{cccc} 4 & 4 & 0 & 0 \\ 4 & 4 & 0 & 0 \\ 0 & 0 & 4 & 4 \\ 0 & 0 & 4 & 4 \end{array}\right]
2004.05631v1_EQ0201_p102	\left|E_{2}\right\rangle=\frac{|00\rangle+|11\rangle}{\sqrt{2}}=\left[\begin{array}{llll} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 & 0 \end{array}\right]^{\top}
2004.05631v1_EQ0202_p102	\left|O_{2}\right\rangle=\frac{|01\rangle+|01\rangle}{\sqrt{2}}=\left[\begin{array}{llll} 0 & 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{array}\right]^{\top}
2004.05631v1_EQ0203_p103	\rho_{2}=\left[\begin{array}{cc} \frac{1}{\sqrt{2}} & 0 \\ \frac{1}{\sqrt{2}} & 0 \\ 0 & \frac{1}{\sqrt{2}} \\ 0 & \frac{1}{\sqrt{2}} \end{array}\right]\left[\begin{array}{cc} \frac{1}{2} & 0 \\ 0 & \frac{1}{2} \end{array}\right]\left[\begin{array}{cccc} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 & 0 \\ 0 & 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{array}\right]
2004.05631v1_EQ0204_p103	\begin{array}{lll} U^{\dagger}|00\rangle=\frac{1}{\sqrt{2}}|0\rangle & \text { (even) } & U^{\dagger}|01\rangle=\frac{1}{\sqrt{2}}|1\rangle \\ U^{\dagger}|11\rangle=\frac{1}{\sqrt{2}}|0\rangle & \text { (even) } & U^{\dagger}|10\rangle=\frac{1}{\sqrt{2}}|1\rangle \\ \text { (odd) } \end{array}
2004.05631v1_EQ0205_p104	\rho_{2}=\frac{1}{N_{T}}\left[\begin{array}{cccc} d_{1} & s_{e} & 0 & 0 \\ s_{3} & d_{2} & 0 & 0 \\ 0 & 0 & d_{3} & s_{o} \\ 0 & 0 & s_{o} & d_{4} \end{array}\right]
2004.05631v1_EQ0206_p104	\rho_{2}=\operatorname{tr}_{V^{\otimes 3}}|\psi\rangle\langle\psi|=\frac{1}{7}\left[\begin{array}{llll} 2 & 1 & 0 & 0 \\ 1 & 2 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 2 \end{array}\right]
2004.05631v1_EQ0207_p104	\theta=\arctan \left(\frac{2 s_{e}}{\sqrt{G_{e}^{2}+4 s_{e}^{2}}+G_{e}}\right)
2004.05631v1_EQ0208_p104	\left.\begin{array}{ll} 0 & 0 \end{array}\right]^{\top} \quad \phi=\arctan \left(\frac{2 s_{o}}{\sqrt{G_{o}^{2}+4 s_{o}^{2}}+G_{o}}\right)
2004.05631v1_EQ0209_p105	\left|E_{N}\right\rangle=\frac{1}{\sqrt{2^{N-1}}} \sum_{s \text { even }}|s\rangle
2004.05631v1_EQ0210_p109	\mathbb{C}^{X} \quad 2^{X} \quad \mathrm{Set}^{\mathrm{C}^{\text {op }}}
2004.05631v1_EQ0211_p110	\mathrm{C}(F c, d) \cong \mathrm{D}(c, G d)
2004.05631v1_EQ0212_p111	U: \text { Vect } \rightleftarrows \text { Set: } F \text {. }
2004.05631v1_EQ0213_p111(\5.1\)	|v\rangle:=\sum_{x} v(x) x
2004.05631v1_EQ0214_p111(\5.2\)	\hat{f}|v\rangle=\sum_{x} v(x) f(x) .
2004.05631v1_EQ0215_p112(\5 \cdot 3\)	U F X=U C^{X},
2004.05631v1_EQ0216_p112(\5 \cdot 4\)	\mathrm{C}(x \times y, z) \cong \mathrm{C}\left(y, z^{x}\right)
2004.05631v1_EQ0217_p112(\5 \cdot 5\)	\langle M v, w\rangle=\left\langle v, M^{\dagger} w\right\rangle \quad \text { for all } v \in F X \text { and } w \in F Y .
2004.05631v1_EQ0218_p114	U: \text { CocompCAT } \rightleftarrows \text { CAT : } F \text {. }
2004.05631v1_EQ0219_p114	F C=\mathrm{Set}^{\mathrm{C}^{\mathrm{Op}}} .
2004.05631v1_EQ0220_p115(\5.6\)	V \cong \int^{c \in \mathrm{C}} \mathrm{C}(-, c) \cdot V c
2004.05631v1_EQ0221_p116(\5 \cdot 7\)	\hat{f} V:=\int^{c \in \mathrm{C}} f_{c} \cdot V c
2004.05631v1_EQ0222_p116	|v\rangle=\sum_{x \in X} v(x)|x\rangle \quad \mid \quad V \cong \int^{c \in \mathrm{C}} \mathrm{C}(-, c) \cdot V c
2004.05631v1_EQ0223_p116	\hat{f}|v\rangle=\sum_{x \in X} v(x) f(x) \quad \mid \hat{f} V=\int^{c \in \mathrm{C}} f c \cdot V c
2004.05631v1_EQ0224_p116(\5.8\)	U F C=U S_{\mathrm{et}} \mathrm{C}^{\mathrm{Cop}},
2004.05631v1_EQ0225_p117	U: \text { CompCAT ⇄ CAT }: \bar{F} .
2004.05631v1_EQ0226_p117	U \overline{F C}=\left(U S_{\mathrm{et}}{ }^{\mathrm{C}}\right)^{\mathrm{op}},
2004.05631v1_EQ0227_p118(5.9)	\operatorname{Set}^{\mathrm{D}}\left(W, M^{*} V\right) \cong \operatorname{Set}^{\mathrm{Cop}}\left(V, M_{*} W\right) .
2004.05631v1_EQ0228_p119	\operatorname{Fix}\left(M_{*} M^{*}\right) \cong \operatorname{Fix}\left(M^{*} M_{*}\right)
2004.05631v1_EQ0229_p119(5.10)	\operatorname{Set}^{\mathrm{D}}\left(W, M^{*} V\right) \cong \operatorname{Set}^{\mathrm{C}^{\mathrm{op}}}\left(V, M_{*} W\right)
2004.05631v1_EQ0230_p119	\langle M v, w\rangle=\left\langle v, M^{\dagger} w\right\rangle
2004.05631v1_EQ0231_p121	U: \text { Join } \rightleftarrows \text { Set: } F \text {. }
2004.05631v1_EQ0232_p122(5.11)	\hat{f} A:=\bigvee_{x \in A} f x .
2004.05631v1_EQ0233_p122	2:=\{0 \leq 1\} .
2004.05631v1_EQ0234_p122	0 \vee 0=0 \quad 0 \vee 1=1 \vee 0=1 \vee 1=1 .
2004.05631v1_EQ0235_p122	0 \wedge 0=0 \wedge 1=1 \wedge 0=0 \quad 1 \wedge 1=1 .
2004.05631v1_EQ0236_p123(\5.12\)	2(A, B)= \begin{cases}1 & \text { if } A \subseteq B \\ 0 & \text { otherwise }\end{cases}
2004.05631v1_EQ0237_p123(5.13)	2(\bar{A}, \bar{B})= \begin{cases}1 & \text { if } \bar{A} x \leq \bar{B} x \text { for all } x \in X \\ 0 & \text { otherwise }\end{cases}
2004.05631v1_EQ0238_p124	\mathbb{C}^{X} \quad \operatorname{Set}^{\mathrm{C}^{\mathrm{Cop}}} \quad \mathcal{V}^{\mathrm{C}^{\mathrm{op}}} \quad 2^{X}
2004.05631v1_EQ0239_p124(5.14)	U F X=U 2^{X},
2004.05631v1_EQ0240_p124	U: \text { Meet } \rightleftarrows \text { Set }: \bar{F}
2004.05631v1_EQ0241_p125(\5.15\)	\hat{g} B:=\bigwedge_{x \in B} g x .
2004.05631v1_EQ0242_p125(\5.16\)	\bar{F} X=(F X)^{\mathrm{op}}
2004.05631v1_EQ0243_p125	U \bar{F} Y=U 2^{Y}=U F Y .
2004.05631v1_EQ0244_p126(\5.17\)	a x(y)= \begin{cases}1 & \text { if } R(x, y)=1 \\ 0 & \text { if } R(x, y)=0,\end{cases}
2004.05631v1_EQ0245_p126(5.18)	f A=\bigcap_{x \in A} a x \quad \text { and } \quad g B=\bigcap_{y \in B} b y
2004.05631v1_EQ0246_p126	F Y(B, f A) \cong F X(A, g B)
2004.05631v1_EQ0247_p126	B \subseteq f A \quad \text { if and only if } \quad A \subseteq g B .
2004.05631v1_EQ0248_p128	\text { eval: } U E^{X} \times X \rightarrow U E
2004.05631v1_EQ0249_p128(5.19)	\mathrm{D}(X \times Y, U E) \cong \mathrm{D}\left(Y, U E^{X}\right) .
2004.05631v1_EQ0250_p128(\5.20\)	U F X \cong U E^{X} \quad \text { for every } X \in \mathrm{D} .
2004.05631v1_EQ0251_p128(\5.20\)	\mathrm{D}\left(I, U E^{X}\right) \cong \mathrm{D}(X \times I, U E) \cong \mathrm{D}(X, U E)
2004.05631v1_EQ0252_p128(\5.21\)	\mathrm{C}(F X, F Y) \cong \mathrm{D}\left(X, U E^{Y}\right) .
2004.05631v1_EQ0253_p129(\5.22\)	\mathrm{C}(F Y, F X) \cong \mathrm{D}\left(Y, U E^{X}\right)
2004.05631v1_EQ0254_p129	\begin{aligned} \mathrm{C}(F X, F Y) & \stackrel{\text { by }}{\stackrel{(5.21)}{\cong}} \mathrm{D}\left(X, U E^{Y}\right) \\ & \stackrel{\text { by }(5.19)}{\cong} \mathrm{D}(X \times Y, U E) \\ & \text { by }\left(\stackrel{5.19)}{\cong} \mathrm{D}\left(Y, U E^{X}\right)\right. \\ & \stackrel{\text { by }(5.22)}{\cong} \mathrm{C}(F Y, F X) \end{aligned}