Expand knowledge base, add /sources/

9 files changed, 866 insertions, 15 deletions

diff --git a/.gitignore b/.gitignore
index 6a2fe92..eebeee5 100644
--- a/.gitignore
+++ b/.gitignore
@@ -1 +1,4 @@
 /public/*
+/sources/**/*.aux
+/sources/**/*.log
+/sources/**/*.pdf
diff --git a/content/know/concept/cauchy-strain-tensor/index.pdc b/content/know/concept/cauchy-strain-tensor/index.pdc
index f150723..cb48377 100644
--- a/content/know/concept/cauchy-strain-tensor/index.pdc
+++ b/content/know/concept/cauchy-strain-tensor/index.pdc
@@ -82,7 +82,7 @@ we expand the middle term to first order in $\va{a}$:
 $$\begin{aligned}
     \va{u}(\va{x} + \va{a})
     \approx \va{u}(\va{x}) + a_x \pdv{\va{u}}{x} + a_y \pdv{\va{u}}{y} + a_z \pdv{\va{u}}{z}
-    = \va{u}(\va{x}) + \va{a} \cdot \nabla \va{u}(\va{x})
+    = \va{u}(\va{x}) + (\va{a} \cdot \nabla) \va{u}(\va{x})
 \end{aligned}$$
 
 With this, we can now define the "shift" $\delta\va{a}$
@@ -91,7 +91,7 @@ as the difference between $\va{a}$ and $\va{A}$ like so:
 $$\begin{aligned}
     \delta{\va{a}}
     \equiv \va{a} - \va{A}
-    = \va{a} \cdot \nabla \va{u}(\va{x})
+    = (\va{a} \cdot \nabla) \va{u}(\va{x})
 \end{aligned}$$
 
 In index notation, we write this expression as follows,
@@ -245,8 +245,8 @@ is easy to express using the displacement field $\va{u}$:
 $$\begin{aligned}
     \boxed{
         \delta(\dd{\va{l}})
-        = \dd{\va{l}} \cdot \nabla \va{u}
-        = (\nabla \vec{u})^\top \cdot \dd{\va{l}}
+        = (\dd{\va{l}} \cdot \nabla) \va{u}
+        %= (\nabla \vec{u})^\top \cdot \dd{\va{l}}
     }
 \end{aligned}$$
 
@@ -259,9 +259,9 @@ $$\begin{aligned}
     = \delta(\va{a} \cross \va{b} \cdot \va{c})
     &= \delta\va{a} \cross \va{b} \cdot \va{c} + \va{a} \cross \delta\va{b} \cdot \va{c} + \va{a} \cross \va{b} \cdot \delta\va{c}
     \\
-    &= (\va{a} \cdot \nabla\va{u}) \cross \va{b} \cdot \va{c}
-    + \va{a} \cross (\va{b} \cdot \nabla\va{u}) \cdot \va{c}
-    + \va{a} \cross \va{b} \cdot (\va{c} \cdot \nabla\va{u})
+    &= (\va{a} \cdot \nabla) \va{u} \cross \va{b} \cdot \va{c}
+    + \va{a} \cross (\va{b} \cdot \nabla )\va{u} \cdot \va{c}
+    + \va{a} \cross \va{b} \cdot (\va{c} \cdot \nabla) \va{u}
 \end{aligned}$$
 
 We can reorder the factors like so
@@ -303,7 +303,7 @@ $$\begin{aligned}
     \delta(\dd{V})
     = \delta(\va{c} \cdot \dd{\va{S}})
     = \delta\va{c} \cdot \dd{\va{S}} + \va{c} \cdot \delta(\dd{\va{S}})
-    = (\va{c} \cdot \nabla\va{u}) \cdot \dd{\va{S}} + \va{c} \cdot \delta(\dd{\va{S}})
+    = (\va{c} \cdot \nabla) \va{u} \cdot \dd{\va{S}} + \va{c} \cdot \delta(\dd{\va{S}})
 \end{aligned}$$
 
 By comparing this to the previous result for $\delta(\dd{V})$,
@@ -311,7 +311,7 @@ we arrive at the following equation:
 
 $$\begin{aligned}
     \nabla \cdot \va{u} (\va{c} \cdot \dd{\va{S}})
-    = (\va{c} \cdot \nabla\va{u}) \cdot \dd{\va{S}} + \va{c} \cdot \delta(\dd{\va{S}})
+    = (\va{c} \cdot \nabla) \va{u} \cdot \dd{\va{S}} + \va{c} \cdot \delta(\dd{\va{S}})
 \end{aligned}$$
 
 Since $\va{c}$ is dot-multiplied at the front of each term,
diff --git a/content/know/concept/deutsch-jozsa-algorithm/deutsch-circuit.png b/content/know/concept/deutsch-jozsa-algorithm/deutsch-circuit.png
new file mode 100644
index 0000000..04f81fe
--- /dev/null
+++ b/content/know/concept/deutsch-jozsa-algorithm/deutsch-circuit.png
diff --git a/content/know/concept/deutsch-jozsa-algorithm/deutsch-jozsa-circuit.png b/content/know/concept/deutsch-jozsa-algorithm/deutsch-jozsa-circuit.png
new file mode 100644
index 0000000..a43ae6a
--- /dev/null
+++ b/content/know/concept/deutsch-jozsa-algorithm/deutsch-jozsa-circuit.png
diff --git a/content/know/concept/deutsch-jozsa-algorithm/index.pdc b/content/know/concept/deutsch-jozsa-algorithm/index.pdc
new file mode 100644
index 0000000..b2b3d98
--- /dev/null
+++ b/content/know/concept/deutsch-jozsa-algorithm/index.pdc
@@ -0,0 +1,234 @@
+---
+title: "Deutsch-Jozsa algorithm"
+firstLetter: "D"
+publishDate: 2021-04-08
+categories:
+- Quantum information
+
+date: 2021-04-08T10:31:45+02:00
+draft: false
+markup: pandoc
+---
+
+# Deutsch-Jozsa algorithm
+
+The **Deutsch algorithm** and its extension, the **Deutsch-Jozsa algorithm**,
+were first to prove that quantum computers can
+solve certain problems more efficiently
+than any classical system.
+
+Given an unknown "black box" binary function $f(x)$ of one or more bits $x$,
+the goal is determine whether $f$ is
+**constant** (i.e. $f(x)$ is the same for all $x$)
+or **balanced** (i.e. exactly 50\% of all $x$-values yield $f(x) = 0$,
+and the other 50\% yield $f(x) = 1$).
+We can query $f$ as many times as we want with inputs of our choice,
+but we want to solve the problem using as few queries as possible.
+
+The problem is extremely artificial and of no practical use,
+but quantum computers can solve it with a single query,
+while classical computers need up to $2^{N - 1} + 1$ queries
+for an $N$-bit $x$.
+
+
+## Deutsch algorithm
+
+The Deutsch algorithm handles the simplest case,
+where $x$ is only a single bit.
+Only four $f$ exist:
+
++ **Constant**: $(f(0) = f(1) = 0)$ or $(f(0) = f(1) = 1)$.
++ **Balanced**: $(f(0) = 0, f(1) = 1)$, or $(f(0) = 1, f(1) = 0)$.
+
+In other words, we only need to determine if $f(0) = f(1)$ or $f(0) \neq f(1)$.
+To do this, we use the following quantum circuit,
+where $U_f$ is the oracle we query:
+
+<a href="deutsch-circuit.png">
+<img src="deutsch-circuit.png" style="width:50%;display:block;margin:auto;">
+</a>
+
+Due to unitarity constraints,
+the action of $U_f$ is defined to be as follows,
+with $\oplus$ meaning XOR:
+
+$$\begin{aligned}
+    \ket{x} \ket{y}
+    \quad \to \boxed{U_f} \to \quad
+    \ket{x} \ket{y \oplus f(x)}
+\end{aligned}$$
+
+Starting on the left from two qubits $\ket{0}$ and $\ket{1}$,
+we apply the Hadamard gate $H$ to both:
+
+$$\begin{aligned}
+    \ket{0} \ket{1}
+    \quad \to \boxed{H^{\otimes 2}} \to \quad
+    \ket{+} \ket{-}
+    = \frac{1}{2} \Big( \ket{0} + \ket{1} \Big) \Big( \ket{0} - \ket{1} \Big)
+\end{aligned}$$
+
+Feeding this result into the oracle $U_f$ then leads us to:
+
+$$\begin{aligned}
+    \to \boxed{U_f} \to \quad
+    \frac{1}{2} \ket{0} \Big( \ket{0 \oplus f(0)} - \ket{1 \oplus f(0)} \Big)
+    + \frac{1}{2} \ket{1} \Big( \ket{0 \oplus f(1)} - \ket{1 \oplus f(1)} \Big)
+\end{aligned}$$
+
+The parenthesized superpositions can be reduced.
+Assuming that $f(b) = 0$, we notice:
+
+$$\begin{aligned}
+    \ket{0 \oplus f(b)} - \ket{1 \oplus f(b)}
+    = \ket{0 \oplus 0} - \ket{1 \oplus 0}
+    = \ket{0} - \ket{1}
+\end{aligned}$$
+
+On the other hand, if we assume that $f(b) = 1$,
+we get the opposite result:
+
+$$\begin{aligned}
+    \ket{0 \oplus f(b)} - \ket{1 \oplus f(b)}
+    = \ket{0 \oplus 1} - \ket{1 \oplus 1}
+    = - \big(\ket{0} - \ket{1}\big)
+\end{aligned}$$
+
+We can thus combine both cases, $f(b) = 0$ or $f(b) = 1$,
+into the following single expression:
+
+$$\begin{aligned}
+    \ket{0 \oplus f(b)} - \ket{1 \oplus f(b)}
+    = (-1)^{f(b)} \big(\ket{0} - \ket{1}\big)
+\end{aligned}$$
+
+Using this, we rewrite the intermediate state of the quantum circuit like so:
+
+$$\begin{aligned}
+    \ket{0} \ket{1}
+    \quad \to \boxed{H^{\otimes 2}} \to \boxed{U_f} \to \quad
+    \frac{1}{2} \Big( (-1)^{f(0)} \ket{0} + (-1)^{f(1)} \ket{1} \Big) \Big( \ket{0} - \ket{1} \Big)
+\end{aligned}$$
+
+The second qubit in state $\ket{-}$ is garbage; it is no longer of interest.
+The first qubit is given by:
+
+$$\begin{aligned}
+    \frac{1}{\sqrt{2}} \Big( (-1)^{f(0)} \ket{0} + (-1)^{f(1)} \ket{1} \Big)
+    = \frac{(-1)^{f(0)}}{\sqrt{2}} \Big( \ket{0} + (-1)^{f(0) \oplus f(1)} \ket{1} \Big)
+\end{aligned}$$
+
+If $f$ is constant, then $f(0) \oplus f(1) = 0$,
+meaning this state is $(-1)^{f(0)} \ket{+}$.
+On the other hand, if $f$ is balanced, then $f(0) \oplus f(1) = 1$,
+meaning this state is $(-1)^{f(0)} \ket{-}$.
+Taking the Hadamard transform of this qubit therefore yields:
+
+$$\begin{aligned}
+    \to \boxed{H} \to \quad
+    (-1)^{f(0)} \ket{f(0) \oplus f(1)}
+\end{aligned}$$
+
+Depending on whether $f$ is constant or balanced,
+the mearurement outcome of this state will be $\ket{0}$ or $\ket{1}$
+with 100\% probability. We have solved the problem!
+
+Note that we only consulted the oracle (i.e. applied $U_f$) once.
+A classical computer would need to query it twice,
+once with input $x = 0$, and again with $x = 1$.
+
+
+## Full Deutsch-Jozsa algorithm
+
+The Deutsch-Jozsa algorithm generalizes the above to $N$-bit inputs $x$.
+We are promised that $f(x)$ is either constant or balanced;
+other possibilities are assumed to be impossible.
+This algorithm is then implemented by the following quantum circuit:
+
+<a href="deutsch-jozsa-circuit.png">
+<img src="deutsch-jozsa-circuit.png" style="width:50%;display:block;margin:auto;">
+</a>
+
+There are $N$ qubits in initial state $\ket{0}$, and one in $\ket{1}$.
+For clarity, the oracle $U_f$ works like so:
+
+$$\begin{aligned}
+    \ket{x_1} \ket{x_2} \cdots \ket{x_N} \ket{y}
+    \quad \to \boxed{U_f} \to \quad
+    \ket{x_1} \cdots \ket{x_N} \ket{y \oplus f(x_1, ..., x_N)}
+\end{aligned}$$
+
+Applying the $N + 1$ Hadamard gates to the initial state
+yields the following superposition:
+
+$$\begin{aligned}
+    \ket{0}^{\otimes N} \ket{1}
+    \quad \to \boxed{H^{\otimes N + 1}} \to \quad
+    \ket{+}^{\otimes N} \ket{-}
+    = \frac{1}{\sqrt{2^N}} \sum_{x = 0}^{2^N - 1} \ket{x} \ket{-}
+\end{aligned}$$
+
+Where $\ket{x} = \ket{x_1} \cdots \ket{x_N}$ denotes a classical binary state.
+For example, if $x = 5 = 2^0 + 2^2$ in the summation,
+then $\ket{x} = \ket{1} \ket{0} \ket{1} \ket{0}^{\otimes N-3}$
+(from least to most significant).
+
+We give this state to the oracle,
+and, by the same logic as for the Deutsch algorithm,
+get back:
+
+$$\begin{aligned}
+    \to \boxed{U_f} \to \quad
+    \frac{1}{\sqrt{2^N}} \sum_{x = 0}^{2^N - 1} (-1)^{f(x)} \ket{x} \ket{-}
+\end{aligned}$$
+
+The last qubit $\ket{-}$ is garbage.
+Next, applying the Hadamard transform to the other $N$ gives:
+
+$$\begin{aligned}
+    \to \boxed{H^{\otimes N}} \to \quad
+    \frac{1}{\sqrt{2^N}} \sum_{x = 0}^{2^N - 1} (-1)^{f(x)}
+    \bigg( \frac{1}{\sqrt{2^N}} \sum_{y = 0}^{2^N - 1} (-1)^{x \cdot y} \ket{y} \bigg)
+\end{aligned}$$
+
+Where $x \cdot y$ is the bitwise dot product of the binary representations of $x$ and $y$,
+so, for example, if $N = 2$, then $x \cdot y = x_1 y_1 + x_2 y_2$.
+Note that the above expression has not been reduced at all;
+it follows from the definition of the Hadamard transform.
+We can rewrite it like so:
+
+$$\begin{aligned}
+    \frac{1}{2^N} \sum_{x = 0}^{2^N - 1} \sum_{y = 0}^{2^N - 1} (-1)^{f(x) + x \cdot y} \ket{y}
+    = \sum_{y = 0}^{2^N - 1} \bigg( \frac{1}{2^N} \sum_{x = 0}^{2^N - 1} (-1)^{f(x) + x \cdot y} \bigg) \ket{y}
+    = \sum_{y = 0}^{2^N - 1} c_y \ket{y}
+\end{aligned}$$
+
+The parenthesized expression can be interpreted as the coefficients
+of a superposition of several $y$-values.
+Therefore, the probability that a measurement yields $y = 0$,
+i.e. $\ket{y} = \ket{0}^{\otimes N}$, is:
+
+$$\begin{aligned}
+    |c_0|^2
+    = \bigg| \frac{1}{2^N} \sum_{x = 0}^{2^N - 1} (-1)^{f(x)} \bigg|^2
+\end{aligned}$$
+
+The summation always contains an even number of terms, for all values of $N$.
+Consequently, if $f$ is constant, then $|c_0|^2 = |\!\pm\! 2^N / 2^N|^2 = 1$.
+Otherwise, if $f$ is balanced, all the terms cancel out, so we are left with $|c_0|^2 = 0$.
+In other words, we reach the same result as the Deutsch algorithm:
+we only need to measure the $N$ qubits once;
+$f$ is constant if and only if all are zero.
+
+The Deutsch-Jozsa algorithm needs only one oracle query to give an error-free result,
+whereas a classical computer needs $2^{N-1} + 1$ queries in the worst case;
+a revolutionary discovery.
+
+
+## References
+1.  J.S. Neergaard-Nielsen,
+    *Quantum information: lectures notes*,
+    2021, unpublished.
+2.  S. Aaronson,
+    *Introduction to quantum information science: lecture notes*,
+    2018, unpublished.
diff --git a/content/know/concept/euler-equations/index.pdc b/content/know/concept/euler-equations/index.pdc
index 37d2fea..cedfd93 100644
--- a/content/know/concept/euler-equations/index.pdc
+++ b/content/know/concept/euler-equations/index.pdc
@@ -149,7 +149,7 @@ to which we apply a vector identity:
 $$\begin{aligned}
     0
     = \dv{\rho}{t} + \nabla \cdot (\rho \va{v})
-    = \dv{\rho}{t} + \va{v} \cdot \nabla \rho + \rho (\nabla \cdot \va{v})
+    = \dv{\rho}{t} + (\va{v} \cdot \nabla) \rho + \rho (\nabla \cdot \va{v})
 \end{aligned}$$
 
 Thanks to incompressibility, the last term disappears,
@@ -159,7 +159,7 @@ $$\begin{aligned}
     \boxed{
         0
         = \frac{\mathrm{D} \rho}{\mathrm{D} t}
-        = \dv{\rho}{t} + \va{v} \cdot \nabla \rho
+        = \dv{\rho}{t} + (\va{v} \cdot \nabla) \rho
     }
 \end{aligned}$$
 
diff --git a/content/know/concept/material-derivative/index.pdc b/content/know/concept/material-derivative/index.pdc
index af65ca0..1c6bfdc 100644
--- a/content/know/concept/material-derivative/index.pdc
+++ b/content/know/concept/material-derivative/index.pdc
@@ -57,7 +57,7 @@ then we can rewrite this expression like so:
 
 $$\begin{aligned}
     \dv{t} f\big(x(t), y(t), z(t), t\big)
-    &= \pdv{f}{t} + \va{v} \cdot \nabla f
+    &= \pdv{f}{t} + (\va{v} \cdot \nabla) f
 \end{aligned}$$
 
 Note that $\va{v} = \va{v}(\va{r}, t)$,
@@ -73,7 +73,7 @@ and is known as the **material derivative** or **comoving derivative**:
 $$\begin{aligned}
     \boxed{
         \frac{\mathrm{D}f}{\mathrm{D}t}
-        \equiv \pdv{f}{t} + \va{v} \cdot \nabla f
+        \equiv \pdv{f}{t} + (\va{v} \cdot \nabla) f
     }
 \end{aligned}$$
 
@@ -89,14 +89,14 @@ but in fact the definition also works for vector fields $\va{U}(\va{r}, t)$:
 $$\begin{aligned}
     \boxed{
         \frac{\mathrm{D} \va{U}}{\mathrm{D}t}
-        \equiv \pdv{f}{t} + \va{v} \cdot \nabla \va{U}
+        \equiv \pdv{\va{U}}{t} + (\va{v} \cdot \nabla) \va{U}
     }
 \end{aligned}$$
 
 Where the advective term is to be evaluated in the following way in Cartesian coordinates:
 
 $$\begin{aligned}
-    \va{v} \cdot \nabla \va{U}
+    (\va{v} \cdot \nabla) \va{U}
     =
     \begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix}
     \cdot
diff --git a/sources/know/concept/bloch-sphere/bloch.svg b/sources/know/concept/bloch-sphere/bloch.svg
new file mode 100644
index 0000000..be67f6f
--- /dev/null
+++ b/sources/know/concept/bloch-sphere/bloch.svg
@@ -0,0 +1,561 @@
+<?xml version="1.0" encoding="UTF-8" standalone="no"?>
+<svg
+   xmlns:dc="http://purl.org/dc/elements/1.1/"
+   xmlns:cc="http://creativecommons.org/ns#"
+   xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
+   xmlns:svg="http://www.w3.org/2000/svg"
+   xmlns="http://www.w3.org/2000/svg"
+   xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
+   xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
+   width="150mm"
+   height="150mm"
+   viewBox="0 0 150 150"
+   version="1.1"
+   id="svg8"
+   inkscape:version="1.0.2 (e86c870879, 2021-01-15)"
+   sodipodi:docname="bloch.svg"
+   inkscape:export-filename="/home/user/bloch.png"
+   inkscape:export-xdpi="256"
+   inkscape:export-ydpi="256">
+  <defs
+     id="defs2">
+    <marker
+       style="overflow:visible;"
+       id="marker1988"
+       refX="0.0"
+       refY="0.0"
+       orient="auto"
+       inkscape:stockid="Arrow1Mend"
+       inkscape:isstock="true">
+      <path
+         transform="scale(0.4) rotate(180) translate(10,0)"
+         style="fill-rule:evenodd;stroke:#222222;stroke-width:1pt;stroke-opacity:1;fill:#222222;fill-opacity:1"
+         d="M 0.0,0.0 L 5.0,-5.0 L -12.5,0.0 L 5.0,5.0 L 0.0,0.0 z "
+         id="path1986" />
+    </marker>
+    <marker
+       style="overflow:visible"
+       id="Arrow1Mstart"
+       refX="0.0"
+       refY="0.0"
+       orient="auto"
+       inkscape:stockid="Arrow1Mstart"
+       inkscape:isstock="true">
+      <path
+         transform="scale(0.4) translate(10,0)"
+         style="fill-rule:evenodd;stroke:#222222;stroke-width:1pt;stroke-opacity:1;fill:#222222;fill-opacity:1"
+         d="M 0.0,0.0 L 5.0,-5.0 L -12.5,0.0 L 5.0,5.0 L 0.0,0.0 z "
+         id="path972" />
+    </marker>
+    <marker
+       style="overflow:visible;"
+       id="Arrow1Mend"
+       refX="0.0"
+       refY="0.0"
+       orient="auto"
+       inkscape:stockid="Arrow1Mend"
+       inkscape:isstock="true">
+      <path
+         transform="scale(0.4) rotate(180) translate(10,0)"
+         style="fill-rule:evenodd;stroke:#222222;stroke-width:1pt;stroke-opacity:1;fill:#222222;fill-opacity:1"
+         d="M 0.0,0.0 L 5.0,-5.0 L -12.5,0.0 L 5.0,5.0 L 0.0,0.0 z "
+         id="path975" />
+    </marker>
+    <marker
+       style="overflow:visible;"
+       id="marker1395"
+       refX="0.0"
+       refY="0.0"
+       orient="auto"
+       inkscape:stockid="Arrow1Lend"
+       inkscape:isstock="true">
+      <path
+         transform="scale(0.8) rotate(180) translate(12.5,0)"
+         style="fill-rule:evenodd;stroke:#ff0000;stroke-width:1pt;stroke-opacity:1;fill:#ff0000;fill-opacity:1"
+         d="M 0.0,0.0 L 5.0,-5.0 L -12.5,0.0 L 5.0,5.0 L 0.0,0.0 z "
+         id="path1393" />
+    </marker>
+    <marker
+       style="overflow:visible;"
+       id="Arrow2Mend"
+       refX="0.0"
+       refY="0.0"
+       orient="auto"
+       inkscape:stockid="Arrow2Mend"
+       inkscape:isstock="true">
+      <path
+         transform="scale(0.6) rotate(180) translate(0,0)"
+         d="M 8.7185878,4.0337352 L -2.2072895,0.016013256 L 8.7185884,-4.0017078 C 6.9730900,-1.6296469 6.9831476,1.6157441 8.7185878,4.0337352 z "
+         style="fill-rule:evenodd;stroke-width:0.625;stroke-linejoin:round;stroke:#ff0000;stroke-opacity:1;fill:#ff0000;fill-opacity:1"
+         id="path993" />
+    </marker>
+    <marker
+       style="overflow:visible;"
+       id="Arrow2Lend"
+       refX="0.0"
+       refY="0.0"
+       orient="auto"
+       inkscape:stockid="Arrow2Lend"
+       inkscape:isstock="true">
+      <path
+         transform="scale(1.1) rotate(180) translate(1,0)"
+         d="M 8.7185878,4.0337352 L -2.2072895,0.016013256 L 8.7185884,-4.0017078 C 6.9730900,-1.6296469 6.9831476,1.6157441 8.7185878,4.0337352 z "
+         style="fill-rule:evenodd;stroke-width:0.625;stroke-linejoin:round;stroke:#ff0000;stroke-opacity:1;fill:#ff0000;fill-opacity:1"
+         id="path987" />
+    </marker>
+    <marker
+       style="overflow:visible;"
+       id="marker1229"
+       refX="0.0"
+       refY="0.0"
+       orient="auto"
+       inkscape:stockid="Arrow1Lend"
+       inkscape:isstock="true">
+      <path
+         transform="scale(0.8) rotate(180) translate(12.5,0)"
+         style="fill-rule:evenodd;stroke:#ff0000;stroke-width:1pt;stroke-opacity:1;fill:#ff0000;fill-opacity:1"
+         d="M 0.0,0.0 L 5.0,-5.0 L -12.5,0.0 L 5.0,5.0 L 0.0,0.0 z "
+         id="path969" />
+    </marker>
+    <marker
+       style="overflow:visible"
+       id="Arrow1Lstart"
+       refX="0.0"
+       refY="0.0"
+       orient="auto"
+       inkscape:stockid="Arrow1Lstart"
+       inkscape:isstock="true">
+      <path
+         transform="scale(0.8) translate(12.5,0)"
+         style="fill-rule:evenodd;stroke:#000000;stroke-width:1pt;stroke-opacity:1;fill:#000000;fill-opacity:1"
+         d="M 0.0,0.0 L 5.0,-5.0 L -12.5,0.0 L 5.0,5.0 L 0.0,0.0 z "
+         id="path962" />
+    </marker>
+    <clipPath
+       clipPathUnits="userSpaceOnUse"
+       id="clipPath932">
+      <rect
+         style="fill:#ff00ff;stroke:#000000;stroke-width:0.264999;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0"
+         id="rect934"
+         width="150"
+         height="20"
+         x="29.999998"
+         y="65" />
+    </clipPath>
+    <clipPath
+       clipPathUnits="userSpaceOnUse"
+       id="clipPath956">
+      <rect
+         style="fill:#ff00ff;stroke:#000000;stroke-width:0.264999;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0"
+         id="rect958"
+         width="150"
+         height="20"
+         x="-39"
+         y="-100"
+         transform="rotate(90)" />
+    </clipPath>
+  </defs>
+  <sodipodi:namedview
+     id="base"
+     pagecolor="#ffffff"
+     bordercolor="#666666"
+     borderopacity="1.0"
+     inkscape:pageopacity="1"
+     inkscape:pageshadow="2"
+     inkscape:zoom="1.7197115"
+     inkscape:cx="329.86489"
+     inkscape:cy="274.90496"
+     inkscape:document-units="mm"
+     inkscape:current-layer="layer1"
+     inkscape:document-rotation="0"
+     showgrid="true"
+     inkscape:window-width="2560"
+     inkscape:window-height="1440"
+     inkscape:window-x="0"
+     inkscape:window-y="0"
+     inkscape:window-maximized="1"
+     inkscape:pagecheckerboard="false"
+     inkscape:snap-bbox="true"
+     inkscape:snap-bbox-midpoints="true"
+     inkscape:snap-object-midpoints="true"
+     inkscape:snap-grids="true"
+     inkscape:snap-page="false"
+     inkscape:snap-bbox-edge-midpoints="false"
+     inkscape:snap-intersection-paths="true"
+     inkscape:object-paths="true"
+     inkscape:snap-smooth-nodes="true"
+     inkscape:snap-nodes="true"
+     inkscape:snap-others="true"
+     inkscape:bbox-nodes="true">
+    <inkscape:grid
+       units="mm"
+       spacingx="0.99999999"
+       spacingy="0.99999999"
+       type="xygrid"
+       id="grid44" />
+  </sodipodi:namedview>
+  <metadata
+     id="metadata5">
+    <rdf:RDF>
+      <cc:Work
+         rdf:about="">
+        <dc:format>image/svg+xml</dc:format>
+        <dc:type
+           rdf:resource="http://purl.org/dc/dcmitype/StillImage" />
+        <dc:title />
+      </cc:Work>
+    </rdf:RDF>
+  </metadata>
+  <g
+     inkscape:label="Layer 1"
+     inkscape:groupmode="layer"
+     id="layer1">
+    <ellipse
+       style="fill:none;stroke:#222222;stroke-width:0.57154763;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
+       id="path59-7"
+       cx="100"
+       cy="38"
+       rx="10"
+       ry="60"
+       clip-path="url(#clipPath956)"
+       transform="matrix(1.4999999,0,0,1,-74.999992,36.999999)" />
+    <ellipse
+       style="fill:none;stroke:#222222;stroke-width:0.57154761;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1"
+       id="path57-3"
+       cx="105"
+       cy="65"
+       rx="60"
+       ry="10"
+       clip-path="url(#clipPath932)"
+       transform="matrix(1,0,0,1.5,-30.000001,-22.500001)" />
+    <circle
+       style="fill:none;stroke:#222222;stroke-width:0.69999998;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1"
+       id="path55"
+       cx="75"
+       cy="75"
+       r="60" />
+    <ellipse
+       style="fill:none;stroke:#222222;stroke-width:0.7;stroke-miterlimit:4;stroke-dasharray:2.1, 2.1;stroke-dashoffset:0;stroke-opacity:1"
+       id="path57"
+       cx="75"
+       cy="75"
+       rx="60"
+       ry="14.999998" />
+    <ellipse
+       style="fill:none;stroke:#222222;stroke-width:0.7;stroke-miterlimit:4;stroke-dasharray:2.1, 2.1;stroke-dashoffset:0;stroke-opacity:1"
+       id="path59"
+       cx="74.971001"
+       cy="75"
+       rx="14.971004"
+       ry="59.970505" />
+    <path
+       style="fill:none;stroke:#222222;stroke-width:0.69999998;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;marker-start:url(#Arrow1Mstart);marker-end:url(#Arrow1Mend);stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0"
+       d="M 74.999997,5 V 74.999997 L 145,74.999999"
+       id="path1304" />
+    <path
+       style="fill:#222222;stroke:#222222;stroke-width:0.7;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;marker-end:url(#marker1988);stroke-miterlimit:4;stroke-dasharray:none;fill-opacity:1"
+       d="M 74.999997,74.999997 49.999999,99.999999"
+       id="path1326" />
+    <circle
+       style="fill:#0000ff;stroke:#0000ff;stroke-width:0.7;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0"
+       id="path1454"
+       cx="89.507233"
+       cy="60.468727"
+       r="0.6963979" />
+    <circle
+       style="fill:#0000ff;stroke:#0000ff;stroke-width:0.7;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0"
+       id="path1454-5"
+       cx="60.472923"
+       cy="89.555412"
+       r="0.6963979" />
+    <circle
+       style="fill:#0000ff;stroke:#0000ff;stroke-width:0.7;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0"
+       id="path1454-3"
+       cx="15.000178"
+       cy="74.994789"
+       r="0.6963979" />
+    <circle
+       style="fill:#0000ff;stroke:#0000ff;stroke-width:0.7;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0"
+       id="path1454-3-5"
+       cx="135.00212"
+       cy="74.997704"
+       r="0.6963979" />
+    <circle
+       style="fill:#0000ff;stroke:#0000ff;stroke-width:0.7;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0"
+       id="path1454-3-6"
+       cx="75.009789"
+       cy="14.994744"
+       r="0.6963979" />
+    <circle
+       style="fill:#0000ff;stroke:#0000ff;stroke-width:0.7;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0"
+       id="path1454-3-2"
+       cx="75.000175"
+       cy="134.9948"
+       r="0.6963979" />
+    <path
+       style="fill:none;stroke:#ff0000;stroke-width:0.69902;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow2Mend)"
+       d="M 75.020493,75.032989 104.11688,54.677175"
+       id="path964" />
+    <circle
+       style="fill:#ff0000;stroke:#ff0000;stroke-width:0.7;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0"
+       id="path1454-3-5-1"
+       cx="107"
+       cy="53"
+       r="0.6963979" />
+    <path
+       style="fill:none;stroke:#ff0000;stroke-width:0.7;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:1.4, 1.4;stroke-dashoffset:0;stroke-opacity:1"
+       d="m 107,54.499999 v 27"
+       id="path1463" />
+    <path
+       style="fill:none;stroke:#ff0000;stroke-width:0.758195;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:1.51639, 1.51639;stroke-dashoffset:0;stroke-opacity:1"
+       d="M 106.92497,81.62366 74.995731,74.989042"
+       id="path1477" />
+    <path
+       style="fill:none;stroke:#ff0000;stroke-width:0.7;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0"
+       id="path1544"
+       sodipodi:type="arc"
+       sodipodi:cx="75.038399"
+       sodipodi:cy="74.929611"
+       sodipodi:rx="7.9961185"
+       sodipodi:ry="8.1049089"
+       sodipodi:start="4.7097345"
+       sodipodi:end="5.6633059"
+       sodipodi:open="true"
+       sodipodi:arc-type="arc"
+       d="m 75.017173,66.824731 a 7.9961185,8.1049089 0 0 1 6.529655,3.396439" />
+    <path
+       id="path1548"
+       style="fill:none;stroke:#ff0000;stroke-width:0.7;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0"
+       d="M 84.261781,76.907541 A 12,3 0 0 1 72.042408,77.907454" />
+    <g
+       aria-label="0〉"
+       id="text1009"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:6.21118px;line-height:1.25;font-family:sans-serif;-inkscape-font-specification:'sans-serif, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-feature-settings:normal;text-align:start;letter-spacing:0px;word-spacing:0px;writing-mode:lr-tb;text-anchor:start;fill:#0000ff;fill-opacity:1;stroke:none;stroke-width:0.388199">
+      <path
+         d="m 82.305355,10.30709 c 0,-0.4968948 -0.03106,-0.9937893 -0.248448,-1.4534166 -0.285714,-0.5962733 -0.795031,-0.6956522 -1.0559,-0.6956522 -0.372671,0 -0.826087,0.1614907 -1.080746,0.7391305 -0.198757,0.4285714 -0.229813,0.9130435 -0.229813,1.4099383 0,0.465838 0.02484,1.024844 0.279503,1.496894 0.267081,0.503106 0.720497,0.627329 1.024845,0.627329 0.335403,0 0.807453,-0.130435 1.080745,-0.720497 0.198758,-0.428571 0.229814,-0.913043 0.229814,-1.403726 z m -0.515528,-0.07454 c 0,0.465839 0,0.888199 -0.06832,1.285715 -0.09317,0.590062 -0.447205,0.776397 -0.726708,0.776397 -0.242236,0 -0.608696,-0.155279 -0.720497,-0.751553 -0.06832,-0.37267 -0.06832,-0.944099 -0.06832,-1.310559 0,-0.3975151 0,-0.807453 0.04969,-1.1428567 0.118013,-0.7391305 0.583851,-0.7950311 0.739131,-0.7950311 0.204968,0 0.614906,0.1118012 0.732919,0.7267081 0.06211,0.3478261 0.06211,0.8198758 0.06211,1.2111797 z"
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:6.21118px;font-family:'Latin Modern Math';-inkscape-font-specification:'Latin Modern Math, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;fill:#0000ff;stroke-width:0.388199"
+         id="path1373" />
+      <path
+         d="m 82.876771,12.648705 0.391305,0.180124 1.84472,-2.8944102 -1.84472,-2.89441 -0.391305,0.1801242 1.739131,2.7142858 z"
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:6.21118px;font-family:'Latin Modern Math';-inkscape-font-specification:'Latin Modern Math, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;fill:#0000ff;stroke-width:0.388199"
+         id="path1375" />
+    </g>
+    <g
+       aria-label="−i〉"
+       transform="scale(0.99957339,1.0004268)"
+       id="text1009-8"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:6.20872px;line-height:1.25;font-family:C059;-inkscape-font-specification:'C059, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;text-align:start;letter-spacing:0px;word-spacing:0px;writing-mode:lr-tb;text-anchor:start;fill:#0000ff;fill-opacity:1;stroke:none;stroke-width:0.388045">
+      <path
+         d="m 4.8721422,76.037993 h 3.129195 v -0.496697 h -3.129195 z"
+         style="font-style:italic;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:6.20872px;font-family:C059;-inkscape-font-specification:'C059, Italic';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;stroke-width:0.388045"
+         id="path1405" />
+      <path
+         d="m 9.8142772,74.442352 -1.1920743,0.09313 -0.024835,0.223514 h 0.1490093 c 0.2545575,0 0.3538971,0.05588 0.3538971,0.19247 0,0.06209 -0.012417,0.130383 -0.055879,0.273184 l -0.3601058,1.2107 c -0.1179657,0.397359 -0.1365918,0.484281 -0.1365918,0.627081 0,0.217305 0.1862616,0.384941 0.4221929,0.384941 0.2731837,0 0.5277413,-0.142801 0.7760901,-0.428402 0.1117569,-0.124174 0.2110965,-0.260766 0.3849407,-0.527741 L 9.9384516,76.385682 c -0.3228535,0.490489 -0.5836197,0.751255 -0.7512552,0.751255 -0.062087,0 -0.1055482,-0.05588 -0.1055482,-0.130383 0,-0.06209 0.024835,-0.173844 0.074505,-0.347689 z m -0.111757,-1.514927 c -0.1862616,0 -0.3352709,0.149009 -0.3352709,0.33527 0,0.180053 0.1490093,0.335271 0.3290622,0.335271 0.1924703,0 0.3414795,-0.149009 0.3414795,-0.335271 0,-0.186261 -0.1490092,-0.33527 -0.3352708,-0.33527 z"
+         style="font-style:italic;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:6.20872px;font-family:C059;-inkscape-font-specification:'C059, Italic';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;stroke-width:0.388045"
+         id="path1407" />
+      <path
+         d="m 10.770409,77.714348 0.39115,0.180053 1.84399,-2.893264 -1.84399,-2.893264 -0.39115,0.180053 1.738442,2.713211 z"
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:6.20872px;font-family:C059;-inkscape-font-specification:'C059, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;fill:#0000ff;stroke-width:0.388045"
+         id="path1409" />
+    </g>
+    <g
+       aria-label="1〉"
+       id="text1009-4"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:6.18129px;line-height:1.25;font-family:sans-serif;-inkscape-font-specification:'sans-serif, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-feature-settings:normal;text-align:start;letter-spacing:0px;word-spacing:0px;writing-mode:lr-tb;text-anchor:start;fill:#0000ff;fill-opacity:1;stroke:none;stroke-width:0.38633">
+      <path
+         d="m 76.496233,142.34889 v -0.19162 h -0.197801 c -0.556316,0 -0.57486,-0.068 -0.57486,-0.2967 v -3.4677 c 0,-0.14835 0,-0.16072 -0.142169,-0.16072 -0.38324,0.39561 -0.927194,0.39561 -1.124995,0.39561 v 0.19162 c 0.123626,0 0.488322,0 0.809749,-0.16072 v 3.20191 c 0,0.22253 -0.01854,0.2967 -0.57486,0.2967 h -0.197802 v 0.19162 c 0.216346,-0.0185 0.754118,-0.0185 1.001369,-0.0185 0.247252,0 0.785024,0 1.001369,0.0185 z"
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:6.18129px;font-family:'Latin Modern Math';-inkscape-font-specification:'Latin Modern Math, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;fill:#0000ff;stroke-width:0.38633"
+         id="path1415" />
+      <path
+         d="m 77.318333,142.70123 0.389422,0.17925 L 79.543598,140 77.707755,137.11952 77.318333,137.29878 79.049095,140 Z"
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:6.18129px;font-family:'Latin Modern Math';-inkscape-font-specification:'Latin Modern Math, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;fill:#0000ff;stroke-width:0.38633"
+         id="path1417" />
+    </g>
+    <g
+       aria-label="+i〉"
+       id="text1009-8-3"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:6.21118px;line-height:1.25;font-family:C059;-inkscape-font-specification:'C059, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;text-align:start;letter-spacing:0px;word-spacing:0px;writing-mode:lr-tb;text-anchor:start;fill:#0000ff;fill-opacity:1;stroke:none;stroke-width:0.388198">
+      <path
+         d="m 138.49015,70.580278 h -1.31677 v 0.496895 h 1.31677 v 1.322981 h 0.4969 v -1.322981 h 1.31677 v -0.496895 h -1.31677 v -1.31677 h -0.4969 z"
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:6.21118px;font-family:C059;-inkscape-font-specification:'C059, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;fill:#0000ff;stroke-width:0.388198"
+         id="path1423" />
+      <path
+         d="m 142.05536,69.480899 -1.19254,0.09317 -0.0248,0.223603 h 0.14907 c 0.25466,0 0.35404,0.0559 0.35404,0.192546 0,0.06211 -0.0124,0.130435 -0.0559,0.273292 l -0.36025,1.21118 c -0.11801,0.397516 -0.13665,0.484472 -0.13665,0.627329 0,0.217392 0.18634,0.385094 0.42236,0.385094 0.2733,0 0.52795,-0.142858 0.7764,-0.428572 0.1118,-0.124223 0.21118,-0.260869 0.38509,-0.52795 l -0.19254,-0.10559 c -0.32298,0.490683 -0.58385,0.751553 -0.75155,0.751553 -0.0621,0 -0.10559,-0.0559 -0.10559,-0.130435 0,-0.06211 0.0248,-0.173913 0.0745,-0.347826 z m -0.1118,-1.515528 c -0.18633,0 -0.3354,0.149069 -0.3354,0.335404 0,0.180124 0.14907,0.335404 0.32919,0.335404 0.19255,0 0.34162,-0.149068 0.34162,-0.335404 0,-0.186335 -0.14907,-0.335404 -0.33541,-0.335404 z"
+         style="font-style:italic;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:6.21118px;font-family:C059;-inkscape-font-specification:'C059, Italic';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;stroke-width:0.388198"
+         id="path1425" />
+      <path
+         d="m 143.01188,72.754191 0.3913,0.180125 1.84472,-2.89441 -1.84472,-2.89441 -0.3913,0.180124 1.73913,2.714286 z"
+         style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:6.21118px;font-family:C059;-inkscape-font-specification:'C059, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;fill:#0000ff;stroke-width:0.388198"
+         id="path1427" />
+    </g>
+    <text