diff --git a/doc/pub/lecture2/html/lecture2-bs.html b/doc/pub/lecture2/html/lecture2-bs.html
index 4fe30b4..f4a2cf0 100644
--- a/doc/pub/lecture2/html/lecture2-bs.html
+++ b/doc/pub/lecture2/html/lecture2-bs.html
@@ -330,7 +330,7 @@ <h2 id="measurements" class="anchor">Measurements </h2>
 \sum_{i=0}^{n-1}\boldsymbol{P}_i=\boldsymbol{I}.
 $$
 
-<p>As an example, consider the basis of two qubits \( \vert 0\rangle \) and \( \vert 1\rangle \) with the correspong sum</p>
+<p>As an example, consider the basis of two qubits \( \vert 0\rangle \) and \( \vert 1\rangle \) with the corresponding sum</p>
 $$
 \sum_{i=0}^{1}\boldsymbol{P}_i=\begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}.
 $$
diff --git a/doc/pub/lecture2/html/lecture2-reveal.html b/doc/pub/lecture2/html/lecture2-reveal.html
index ba714a0..159b93d 100644
--- a/doc/pub/lecture2/html/lecture2-reveal.html
+++ b/doc/pub/lecture2/html/lecture2-reveal.html
@@ -416,7 +416,7 @@ <h2 id="measurements">Measurements </h2>
 $$
 <p>&nbsp;<br>
 
-<p>As an example, consider the basis of two qubits \( \vert 0\rangle \) and \( \vert 1\rangle \) with the correspong sum</p>
+<p>As an example, consider the basis of two qubits \( \vert 0\rangle \) and \( \vert 1\rangle \) with the corresponding sum</p>
 <p>&nbsp;<br>
 $$
 \sum_{i=0}^{1}\boldsymbol{P}_i=\begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}.
diff --git a/doc/pub/lecture2/html/lecture2-solarized.html b/doc/pub/lecture2/html/lecture2-solarized.html
index d59c3b2..89b7109 100644
--- a/doc/pub/lecture2/html/lecture2-solarized.html
+++ b/doc/pub/lecture2/html/lecture2-solarized.html
@@ -316,7 +316,7 @@ <h2 id="measurements">Measurements </h2>
 \sum_{i=0}^{n-1}\boldsymbol{P}_i=\boldsymbol{I}.
 $$
 
-<p>As an example, consider the basis of two qubits \( \vert 0\rangle \) and \( \vert 1\rangle \) with the correspong sum</p>
+<p>As an example, consider the basis of two qubits \( \vert 0\rangle \) and \( \vert 1\rangle \) with the corresponding sum</p>
 $$
 \sum_{i=0}^{1}\boldsymbol{P}_i=\begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}.
 $$
diff --git a/doc/pub/lecture2/html/lecture2.html b/doc/pub/lecture2/html/lecture2.html
index bac1c10..9bf846f 100644
--- a/doc/pub/lecture2/html/lecture2.html
+++ b/doc/pub/lecture2/html/lecture2.html
@@ -393,7 +393,7 @@ <h2 id="measurements">Measurements </h2>
 \sum_{i=0}^{n-1}\boldsymbol{P}_i=\boldsymbol{I}.
 $$
 
-<p>As an example, consider the basis of two qubits \( \vert 0\rangle \) and \( \vert 1\rangle \) with the correspong sum</p>
+<p>As an example, consider the basis of two qubits \( \vert 0\rangle \) and \( \vert 1\rangle \) with the corresponding sum</p>
 $$
 \sum_{i=0}^{1}\boldsymbol{P}_i=\begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}.
 $$
diff --git a/doc/pub/lecture2/ipynb/ipynb-lecture2-src.tar.gz b/doc/pub/lecture2/ipynb/ipynb-lecture2-src.tar.gz
index 4008fc7..8c1715e 100644
Binary files a/doc/pub/lecture2/ipynb/ipynb-lecture2-src.tar.gz and b/doc/pub/lecture2/ipynb/ipynb-lecture2-src.tar.gz differ
diff --git a/doc/pub/lecture2/ipynb/lecture2.ipynb b/doc/pub/lecture2/ipynb/lecture2.ipynb
index 0e5a41c..f7fdaa8 100644
--- a/doc/pub/lecture2/ipynb/lecture2.ipynb
+++ b/doc/pub/lecture2/ipynb/lecture2.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "3b366bf5",
+   "id": "401d7115",
    "metadata": {
     "editable": true
    },
@@ -14,7 +14,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "955256a2",
+   "id": "735c5bc3",
    "metadata": {
     "editable": true
    },
@@ -27,7 +27,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4198917d",
+   "id": "a9262957",
    "metadata": {
     "editable": true
    },
@@ -41,7 +41,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b5b9705d",
+   "id": "a5752fb1",
    "metadata": {
     "editable": true
    },
@@ -61,7 +61,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e2591179",
+   "id": "2956966d",
    "metadata": {
     "editable": true
    },
@@ -80,7 +80,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1ab98ab5",
+   "id": "f327140e",
    "metadata": {
     "editable": true
    },
@@ -92,7 +92,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "60eff931",
+   "id": "c6526abc",
    "metadata": {
     "editable": true
    },
@@ -104,7 +104,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a229ec26",
+   "id": "7da7bffc",
    "metadata": {
     "editable": true
    },
@@ -116,7 +116,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ae79913d",
+   "id": "083fc583",
    "metadata": {
     "editable": true
    },
@@ -126,7 +126,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "011102ac",
+   "id": "853c5770",
    "metadata": {
     "editable": true
    },
@@ -138,7 +138,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "81306a14",
+   "id": "aace289a",
    "metadata": {
     "editable": true
    },
@@ -155,7 +155,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1978c773",
+   "id": "cca58037",
    "metadata": {
     "editable": true
    },
@@ -167,7 +167,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "23acc823",
+   "id": "1bc192eb",
    "metadata": {
     "editable": true
    },
@@ -178,7 +178,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0827fe74",
+   "id": "2ca582b5",
    "metadata": {
     "editable": true
    },
@@ -190,7 +190,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a57340df",
+   "id": "f75ac9be",
    "metadata": {
     "editable": true
    },
@@ -207,7 +207,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c7bc8190",
+   "id": "2506fad4",
    "metadata": {
     "editable": true
    },
@@ -219,7 +219,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "652d16d5",
+   "id": "0393b6b3",
    "metadata": {
     "editable": true
    },
@@ -230,7 +230,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d46fccc0",
+   "id": "e0a6d969",
    "metadata": {
     "editable": true
    },
@@ -242,7 +242,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1b14d396",
+   "id": "ea608059",
    "metadata": {
     "editable": true
    },
@@ -252,7 +252,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "31f5a0ad",
+   "id": "0c757d23",
    "metadata": {
     "editable": true
    },
@@ -264,7 +264,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f6d28939",
+   "id": "6e610b04",
    "metadata": {
     "editable": true
    },
@@ -274,7 +274,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "21ed9064",
+   "id": "dd68cf41",
    "metadata": {
     "editable": true
    },
@@ -286,7 +286,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "70a430a3",
+   "id": "22c596fb",
    "metadata": {
     "editable": true
    },
@@ -296,7 +296,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "23ef4080",
+   "id": "03b9f7e3",
    "metadata": {
     "editable": true
    },
@@ -308,7 +308,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1da098cc",
+   "id": "f6384bf2",
    "metadata": {
     "editable": true
    },
@@ -318,7 +318,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f416af51",
+   "id": "f500874b",
    "metadata": {
     "editable": true
    },
@@ -330,7 +330,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e321f2f8",
+   "id": "d0b04341",
    "metadata": {
     "editable": true
    },
@@ -341,7 +341,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b2bba701",
+   "id": "32e7fd92",
    "metadata": {
     "editable": true
    },
@@ -353,7 +353,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f7a2f09b",
+   "id": "14766f71",
    "metadata": {
     "editable": true
    },
@@ -363,7 +363,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ae54590a",
+   "id": "074df91f",
    "metadata": {
     "editable": true
    },
@@ -375,7 +375,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "298ec8de",
+   "id": "f71d86ae",
    "metadata": {
     "editable": true
    },
@@ -392,7 +392,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2e7b88c8",
+   "id": "5b9ed301",
    "metadata": {
     "editable": true
    },
@@ -404,7 +404,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f0f62b63",
+   "id": "a1b48682",
    "metadata": {
     "editable": true
    },
@@ -414,7 +414,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "53b5bac2",
+   "id": "66922046",
    "metadata": {
     "editable": true
    },
@@ -426,7 +426,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "58f239b4",
+   "id": "5cd74ca7",
    "metadata": {
     "editable": true
    },
@@ -436,7 +436,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a18606cb",
+   "id": "4e7b51b6",
    "metadata": {
     "editable": true
    },
@@ -449,7 +449,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "37044286",
+   "id": "6ab1acab",
    "metadata": {
     "editable": true
    },
@@ -459,7 +459,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "492d9cde",
+   "id": "24b0ac10",
    "metadata": {
     "editable": true
    },
@@ -472,7 +472,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6e992378",
+   "id": "cfa9ee17",
    "metadata": {
     "editable": true
    },
@@ -482,7 +482,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "78a1988d",
+   "id": "a2a3e8e8",
    "metadata": {
     "editable": true
    },
@@ -494,7 +494,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "08bb3e7e",
+   "id": "b201f53d",
    "metadata": {
     "editable": true
    },
@@ -504,7 +504,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9e1fc06a",
+   "id": "78eac811",
    "metadata": {
     "editable": true
    },
@@ -518,7 +518,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8d3d6e5a",
+   "id": "a179594e",
    "metadata": {
     "editable": true
    },
@@ -535,7 +535,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "72b604be",
+   "id": "e50479bc",
    "metadata": {
     "editable": true
    },
@@ -547,17 +547,17 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8c80c1f8",
+   "id": "afa654d6",
    "metadata": {
     "editable": true
    },
    "source": [
-    "As an example, consider the basis of two qubits $\\vert 0\\rangle$ and $\\vert 1\\rangle$ with the correspong sum"
+    "As an example, consider the basis of two qubits $\\vert 0\\rangle$ and $\\vert 1\\rangle$ with the corresponding sum"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "67cc621b",
+   "id": "9e06b8fb",
    "metadata": {
     "editable": true
    },
@@ -569,7 +569,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "107e8749",
+   "id": "ba19fa07",
    "metadata": {
     "editable": true
    },
@@ -579,7 +579,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d4cbf87f",
+   "id": "fb29fa6d",
    "metadata": {
     "editable": true
    },
@@ -591,7 +591,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9de7cf7e",
+   "id": "95daa787",
    "metadata": {
     "editable": true
    },
@@ -602,7 +602,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "76f8ce95",
+   "id": "bc12d69d",
    "metadata": {
     "editable": true
    },
@@ -614,7 +614,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a5f79a88",
+   "id": "86476a17",
    "metadata": {
     "editable": true
    },
@@ -624,7 +624,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b39b3f0a",
+   "id": "2d0a1aa2",
    "metadata": {
     "editable": true
    },
@@ -636,7 +636,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "43587d2b",
+   "id": "99ec5e57",
    "metadata": {
     "editable": true
    },
@@ -646,7 +646,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "25540f61",
+   "id": "2c1b3122",
    "metadata": {
     "editable": true
    },
@@ -658,7 +658,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "37fb4c27",
+   "id": "9d5d0ce9",
    "metadata": {
     "editable": true
    },
@@ -668,7 +668,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1d4e5da1",
+   "id": "174bb39d",
    "metadata": {
     "editable": true
    },
@@ -680,7 +680,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "685cf0b5",
+   "id": "d26eb6a9",
    "metadata": {
     "editable": true
    },
@@ -690,7 +690,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5d3edddc",
+   "id": "8f824d05",
    "metadata": {
     "editable": true
    },
@@ -702,7 +702,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f5fc286a",
+   "id": "fa6dd044",
    "metadata": {
     "editable": true
    },
@@ -712,7 +712,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e9ae7264",
+   "id": "93dfc8e6",
    "metadata": {
     "editable": true
    },
@@ -724,7 +724,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6ac43ea0",
+   "id": "de56b214",
    "metadata": {
     "editable": true
    },
@@ -736,7 +736,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d675039a",
+   "id": "a7baf79d",
    "metadata": {
     "editable": true
    },
@@ -746,7 +746,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "530d948a",
+   "id": "962b15eb",
    "metadata": {
     "editable": true
    },
@@ -758,7 +758,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7b836d4e",
+   "id": "a3c3d3cf",
    "metadata": {
     "editable": true
    },
@@ -768,7 +768,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8c518739",
+   "id": "f41abf20",
    "metadata": {
     "editable": true
    },
@@ -780,7 +780,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8fd360ab",
+   "id": "be4dc3b6",
    "metadata": {
     "editable": true
    },
@@ -790,7 +790,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "007f5cd2",
+   "id": "d61d2f85",
    "metadata": {
     "editable": true
    },
@@ -802,7 +802,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6dbebdc6",
+   "id": "21b48457",
    "metadata": {
     "editable": true
    },
@@ -812,7 +812,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3a2750a6",
+   "id": "a779332e",
    "metadata": {
     "editable": true
    },
@@ -824,7 +824,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "203b1ae4",
+   "id": "f6ce324b",
    "metadata": {
     "editable": true
    },
@@ -834,7 +834,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a01a5d7a",
+   "id": "a688dce4",
    "metadata": {
     "editable": true
    },
@@ -846,7 +846,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b0e7d4ab",
+   "id": "71589cb6",
    "metadata": {
     "editable": true
    },
@@ -856,7 +856,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e2882b2b",
+   "id": "2053c5f1",
    "metadata": {
     "editable": true
    },
@@ -868,7 +868,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2b1bcdc5",
+   "id": "c5bb8090",
    "metadata": {
     "editable": true
    },
@@ -878,7 +878,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0378325f",
+   "id": "d287de4c",
    "metadata": {
     "editable": true
    },
@@ -890,7 +890,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5ba582fb",
+   "id": "8562e9ab",
    "metadata": {
     "editable": true
    },
@@ -900,7 +900,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f3fd9f64",
+   "id": "60d59868",
    "metadata": {
     "editable": true
    },
@@ -912,7 +912,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bc511148",
+   "id": "cdad1601",
    "metadata": {
     "editable": true
    },
@@ -922,7 +922,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2e2132de",
+   "id": "91078ea6",
    "metadata": {
     "editable": true
    },
@@ -934,7 +934,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8278bbbe",
+   "id": "486c11f3",
    "metadata": {
     "editable": true
    },
@@ -944,7 +944,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "23f6765b",
+   "id": "859566f5",
    "metadata": {
     "editable": true
    },
@@ -956,7 +956,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4eb9987e",
+   "id": "7d2b6c02",
    "metadata": {
     "editable": true
    },
@@ -966,7 +966,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cced4714",
+   "id": "738f2157",
    "metadata": {
     "editable": true
    },
@@ -978,7 +978,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f5d22fc0",
+   "id": "68631a76",
    "metadata": {
     "editable": true
    },
@@ -988,7 +988,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d5799e04",
+   "id": "3f5c3ad3",
    "metadata": {
     "editable": true
    },
@@ -1000,7 +1000,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "565925db",
+   "id": "daf99a64",
    "metadata": {
     "editable": true
    },
@@ -1010,7 +1010,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3282c60a",
+   "id": "6d8ea234",
    "metadata": {
     "editable": true
    },
@@ -1022,7 +1022,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f3255761",
+   "id": "9c144c0d",
    "metadata": {
     "editable": true
    },
@@ -1032,7 +1032,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5ee9d8da",
+   "id": "8b558d7c",
    "metadata": {
     "editable": true
    },
@@ -1044,7 +1044,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f5576d38",
+   "id": "c8a80fa2",
    "metadata": {
     "editable": true
    },
@@ -1055,7 +1055,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "db16c5d2",
+   "id": "774bdd8b",
    "metadata": {
     "editable": true
    },
@@ -1067,7 +1067,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f5e1c661",
+   "id": "47e5ccc4",
    "metadata": {
     "editable": true
    },
@@ -1077,7 +1077,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "197aa1a3",
+   "id": "331b9e45",
    "metadata": {
     "editable": true
    },
@@ -1089,7 +1089,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e233a764",
+   "id": "50834438",
    "metadata": {
     "editable": true
    },
@@ -1099,7 +1099,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bbfc685d",
+   "id": "695022a7",
    "metadata": {
     "editable": true
    },
@@ -1111,7 +1111,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "86bac51b",
+   "id": "2f97dbfd",
    "metadata": {
     "editable": true
    },
@@ -1121,7 +1121,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5f844217",
+   "id": "65ccca4c",
    "metadata": {
     "editable": true
    },
@@ -1133,7 +1133,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0a11931c",
+   "id": "1697ffc1",
    "metadata": {
     "editable": true
    },
@@ -1145,7 +1145,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bd2e76c5",
+   "id": "fd217010",
    "metadata": {
     "editable": true
    },
@@ -1157,7 +1157,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d7f70dc7",
+   "id": "1f40c07d",
    "metadata": {
     "editable": true
    },
@@ -1169,7 +1169,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f8a7748c",
+   "id": "eeec3b48",
    "metadata": {
     "editable": true
    },
@@ -1179,7 +1179,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "76e6074a",
+   "id": "64c80f10",
    "metadata": {
     "editable": true
    },
@@ -1191,7 +1191,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f5d17d1f",
+   "id": "38608c04",
    "metadata": {
     "editable": true
    },
@@ -1201,7 +1201,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "41a9a8e1",
+   "id": "baf9e76d",
    "metadata": {
     "editable": true
    },
@@ -1213,7 +1213,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "547155b5",
+   "id": "21ecfa66",
    "metadata": {
     "editable": true
    },
@@ -1223,7 +1223,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c4d121b9",
+   "id": "e7815868",
    "metadata": {
     "editable": true
    },
@@ -1235,7 +1235,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "06e9933f",
+   "id": "57b6096d",
    "metadata": {
     "editable": true
    },
@@ -1254,7 +1254,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dcc2df6a",
+   "id": "7888dfea",
    "metadata": {
     "editable": true
    },
@@ -1266,7 +1266,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "01205d80",
+   "id": "bf1bb3ad",
    "metadata": {
     "editable": true
    },
@@ -1287,7 +1287,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "980c4209",
+   "id": "9452dead",
    "metadata": {
     "editable": true
    },
@@ -1297,7 +1297,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bc5f1cc6",
+   "id": "1dc6be2c",
    "metadata": {
     "editable": true
    },
@@ -1310,7 +1310,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ba4da992",
+   "id": "decf26dc",
    "metadata": {
     "editable": true
    },
@@ -1322,7 +1322,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8e9a4afb",
+   "id": "ae04e57a",
    "metadata": {
     "editable": true
    },
@@ -1333,7 +1333,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4cb86693",
+   "id": "4dd5b9c3",
    "metadata": {
     "editable": true
    },
@@ -1345,7 +1345,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2185b018",
+   "id": "c9f17cd3",
    "metadata": {
     "editable": true
    },
@@ -1355,7 +1355,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6005b9a4",
+   "id": "4d8a6665",
    "metadata": {
     "editable": true
    },
@@ -1367,7 +1367,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "30e3590f",
+   "id": "251aae55",
    "metadata": {
     "editable": true
    },
@@ -1377,7 +1377,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bd99ef66",
+   "id": "4cf80c4a",
    "metadata": {
     "editable": true
    },
@@ -1389,7 +1389,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "67560c43",
+   "id": "c687f727",
    "metadata": {
     "editable": true
    },
@@ -1399,7 +1399,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f4170998",
+   "id": "b5dd98e5",
    "metadata": {
     "editable": true
    },
@@ -1411,7 +1411,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "68bcba68",
+   "id": "2a8ac3d8",
    "metadata": {
     "editable": true
    },
@@ -1425,7 +1425,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bf2a4fea",
+   "id": "b9fb0e45",
    "metadata": {
     "editable": true
    },
@@ -1437,7 +1437,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "891e4351",
+   "id": "cd23bf0d",
    "metadata": {
     "editable": true
    },
@@ -1449,7 +1449,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0aa4855c",
+   "id": "8932305a",
    "metadata": {
     "editable": true
    },
@@ -1461,7 +1461,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "498b2743",
+   "id": "9c9eba2b",
    "metadata": {
     "editable": true
    },
@@ -1471,7 +1471,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0bad099e",
+   "id": "26f0eec9",
    "metadata": {
     "editable": true
    },
@@ -1483,7 +1483,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "635bdd87",
+   "id": "6db102a1",
    "metadata": {
     "editable": true
    },
@@ -1497,7 +1497,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "77eda1cd",
+   "id": "037aca78",
    "metadata": {
     "editable": true
    },
@@ -1509,7 +1509,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c0bf32f3",
+   "id": "26a7b52b",
    "metadata": {
     "editable": true
    },
@@ -1519,7 +1519,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1cb0263a",
+   "id": "16f61c06",
    "metadata": {
     "editable": true
    },
@@ -1531,7 +1531,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3321ae9e",
+   "id": "0d0f6d2a",
    "metadata": {
     "editable": true
    },
@@ -1544,7 +1544,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "beb52f21",
+   "id": "2c45f7af",
    "metadata": {
     "editable": true
    },
@@ -1556,7 +1556,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "16d24de3",
+   "id": "996c28a3",
    "metadata": {
     "editable": true
    },
@@ -1566,7 +1566,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "92720b51",
+   "id": "54de8741",
    "metadata": {
     "editable": true
    },
@@ -1578,7 +1578,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2741ecd3",
+   "id": "0809430a",
    "metadata": {
     "editable": true
    },
@@ -1588,7 +1588,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ec3c5432",
+   "id": "449cc360",
    "metadata": {
     "editable": true
    },
@@ -1600,7 +1600,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5044a64c",
+   "id": "b3f04c09",
    "metadata": {
     "editable": true
    },
@@ -1610,7 +1610,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "452086e8",
+   "id": "e938be4d",
    "metadata": {
     "editable": true
    },
@@ -1622,7 +1622,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5977d3b9",
+   "id": "dd7f31d5",
    "metadata": {
     "editable": true
    },
@@ -1636,7 +1636,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c10b21d9",
+   "id": "f2c30b41",
    "metadata": {
     "editable": true
    },
@@ -1648,7 +1648,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "20d9bef2",
+   "id": "43eb1482",
    "metadata": {
     "editable": true
    },
@@ -1664,7 +1664,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6b6dd87e",
+   "id": "2ddfb1cf",
    "metadata": {
     "editable": true
    },
@@ -1676,7 +1676,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7867ae29",
+   "id": "98aace60",
    "metadata": {
     "editable": true
    },
@@ -1703,7 +1703,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f884bcc2",
+   "id": "b7e9e0a8",
    "metadata": {
     "editable": true
    },
@@ -1715,7 +1715,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3c651cee",
+   "id": "514de72d",
    "metadata": {
     "editable": true
    },
@@ -1726,7 +1726,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "be92f65f",
+   "id": "3f4f4818",
    "metadata": {
     "editable": true
    },
@@ -1738,7 +1738,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7d7d7834",
+   "id": "504c023b",
    "metadata": {
     "editable": true
    },
@@ -1750,7 +1750,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a7a0b152",
+   "id": "1ac8f869",
    "metadata": {
     "editable": true
    },
@@ -1762,7 +1762,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "605f8cf0",
+   "id": "1810ade8",
    "metadata": {
     "editable": true
    },
@@ -1777,7 +1777,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "74e1065d",
+   "id": "36754434",
    "metadata": {
     "editable": true
    },
@@ -1789,7 +1789,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6938b21f",
+   "id": "3129ce98",
    "metadata": {
     "editable": true
    },
@@ -1805,7 +1805,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4a389e6e",
+   "id": "384bf22f",
    "metadata": {
     "editable": true
    },
@@ -1817,7 +1817,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "26c9f496",
+   "id": "3412ab35",
    "metadata": {
     "editable": true
    },
@@ -1836,7 +1836,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4022b58b",
+   "id": "9e74198b",
    "metadata": {
     "editable": true
    },
@@ -1848,7 +1848,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "535d0b5b",
+   "id": "0a186e03",
    "metadata": {
     "editable": true
    },
@@ -1858,7 +1858,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "81a5e571",
+   "id": "22ef02d3",
    "metadata": {
     "editable": true
    },
@@ -1870,7 +1870,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bfa158e8",
+   "id": "50c9be30",
    "metadata": {
     "editable": true
    },
@@ -1880,7 +1880,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f29dc92f",
+   "id": "b1cf9201",
    "metadata": {
     "editable": true
    },
@@ -1892,7 +1892,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3312b124",
+   "id": "bb2455e3",
    "metadata": {
     "editable": true
    },
@@ -1902,7 +1902,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3be1581e",
+   "id": "0ba904c5",
    "metadata": {
     "editable": true
    },
@@ -1914,7 +1914,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2458269a",
+   "id": "c3dfb791",
    "metadata": {
     "editable": true
    },
@@ -1924,7 +1924,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3459c097",
+   "id": "6a43cad7",
    "metadata": {
     "editable": true
    },
@@ -1936,7 +1936,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "55272445",
+   "id": "f5e8b42a",
    "metadata": {
     "editable": true
    },
@@ -1946,7 +1946,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "11173956",
+   "id": "1cb8d3b7",
    "metadata": {
     "editable": true
    },
@@ -1958,7 +1958,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3ccf4843",
+   "id": "33c2257d",
    "metadata": {
     "editable": true
    },
@@ -1970,7 +1970,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d1e47c16",
+   "id": "33a10817",
    "metadata": {
     "editable": true
    },
@@ -1983,7 +1983,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b17b21e2",
+   "id": "167459e8",
    "metadata": {
     "editable": true
    },
@@ -1995,7 +1995,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1a4f0f36",
+   "id": "aad28722",
    "metadata": {
     "editable": true
    },
@@ -2007,7 +2007,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c16169d3",
+   "id": "cfa156dc",
    "metadata": {
     "editable": true
    },
@@ -2019,7 +2019,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "649d9796",
+   "id": "96403141",
    "metadata": {
     "editable": true
    },
@@ -2029,7 +2029,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7368491b",
+   "id": "5dffd2e5",
    "metadata": {
     "editable": true
    },
@@ -2041,7 +2041,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "564c67ae",
+   "id": "3f00240e",
    "metadata": {
     "editable": true
    },
@@ -2051,7 +2051,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1921599f",
+   "id": "1f5118ff",
    "metadata": {
     "editable": true
    },
@@ -2063,7 +2063,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f5866591",
+   "id": "c679ee66",
    "metadata": {
     "editable": true
    },
@@ -2073,7 +2073,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "11fa39eb",
+   "id": "04b4b2da",
    "metadata": {
     "editable": true
    },
@@ -2088,7 +2088,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fb43d542",
+   "id": "046b6f93",
    "metadata": {
     "editable": true
    },
@@ -2100,7 +2100,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0580c408",
+   "id": "c3e23411",
    "metadata": {
     "editable": true
    },
@@ -2112,7 +2112,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d5da774b",
+   "id": "1cfafc7f",
    "metadata": {
     "editable": true
    },
@@ -2126,7 +2126,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "46fe3ed3",
+   "id": "0faac02e",
    "metadata": {
     "editable": true
    },
@@ -2138,7 +2138,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e4a6cbcf",
+   "id": "24c5f080",
    "metadata": {
     "editable": true
    },
@@ -2148,7 +2148,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4e0bbcc3",
+   "id": "eceff8dc",
    "metadata": {
     "editable": true
    },
@@ -2160,7 +2160,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6553a1ed",
+   "id": "15b16854",
    "metadata": {
     "editable": true
    },
@@ -2177,7 +2177,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e571f65f",
+   "id": "f4707be4",
    "metadata": {
     "editable": true
    },
@@ -2189,7 +2189,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "67d5193b",
+   "id": "89fcca0d",
    "metadata": {
     "editable": true
    },
@@ -2199,7 +2199,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2cb330d4",
+   "id": "1ea8072b",
    "metadata": {
     "editable": true
    },
@@ -2211,7 +2211,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "88baa1f1",
+   "id": "37096156",
    "metadata": {
     "editable": true
    },
@@ -2221,7 +2221,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "04652256",
+   "id": "85658d59",
    "metadata": {
     "editable": true
    },
@@ -2235,7 +2235,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4b78107c",
+   "id": "cee7703f",
    "metadata": {
     "editable": true
    },
@@ -2247,7 +2247,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7dc44944",
+   "id": "418bb064",
    "metadata": {
     "editable": true
    },
@@ -2269,7 +2269,7 @@
   {
    "cell_type": "code",
    "execution_count": 1,
-   "id": "df0bcd9a",
+   "id": "8d0ed303",
    "metadata": {
     "collapsed": false,
     "editable": true
diff --git a/doc/pub/lecture2/pdf/lecture2.pdf b/doc/pub/lecture2/pdf/lecture2.pdf
index f0efd4f..2e69ee0 100644
Binary files a/doc/pub/lecture2/pdf/lecture2.pdf and b/doc/pub/lecture2/pdf/lecture2.pdf differ
diff --git a/doc/src/LecturesQTECNOS/lecture2.do.txt b/doc/src/LecturesQTECNOS/lecture2.do.txt
index 6af9065..2adb43f 100644
--- a/doc/src/LecturesQTECNOS/lecture2.do.txt
+++ b/doc/src/LecturesQTECNOS/lecture2.do.txt
@@ -204,7 +204,7 @@ We assume now we can contruct a series of such orthogonal operators based on $\v
 \sum_{i=0}^{n-1}\bm{P}_i=\bm{I}.
 \]
 !et
-As an example, consider the basis of two qubits $\vert 0\rangle$ and $\vert 1\rangle$ with the correspong sum
+As an example, consider the basis of two qubits $\vert 0\rangle$ and $\vert 1\rangle$ with the corresponding sum
 !bt
 \[
 \sum_{i=0}^{1}\bm{P}_i=\begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}.