From b8bd6ecc75c97404df8927be6739b1adf7363ad5 Mon Sep 17 00:00:00 2001
From: Jue-Xu <Jue-Xu@users.noreply.github.com>
Date: Tue, 30 Jul 2024 07:34:48 +0000
Subject: [PATCH] deploy: cd44b08313eaf7aefec32e41b344700a3f6e9da9

---
 ...1c9a2a36e0bcc40fc2048f6bd2200c75142567.png | Bin 0 -> 34923 bytes
 ...0e8cc07282c0e15f1564663f9a1b4ada4dbbb4.png | Bin 0 -> 54743 bytes
 _sources/channel.ipynb                        | 418 +++++++++++++++++-
 _sources/operator.ipynb                       |   4 +-
 channel.html                                  | 295 +++++++++++-
 operator.html                                 |  16 +-
 searchindex.js                                |   2 +-
 7 files changed, 697 insertions(+), 38 deletions(-)
 create mode 100644 _images/49a6274a9b9a0e1d2821e40efe1c9a2a36e0bcc40fc2048f6bd2200c75142567.png
 create mode 100644 _images/8da3ff51fd54ba7622dc810fbe0e8cc07282c0e15f1564663f9a1b4ada4dbbb4.png

diff --git a/_images/49a6274a9b9a0e1d2821e40efe1c9a2a36e0bcc40fc2048f6bd2200c75142567.png b/_images/49a6274a9b9a0e1d2821e40efe1c9a2a36e0bcc40fc2048f6bd2200c75142567.png
new file mode 100644
index 0000000000000000000000000000000000000000..87a12c98bd1868a0f8bbed1be17ae8126e17b00f
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BTY3Nh

literal 0
HcmV?d00001

diff --git a/_sources/channel.ipynb b/_sources/channel.ipynb
index bd7212a..02e81b5 100644
--- a/_sources/channel.ipynb
+++ b/_sources/channel.ipynb
@@ -17,7 +17,7 @@
    "source": [
     "from qiskit import QuantumCircuit\n",
     "from qiskit_aer import AerSimulator\n",
-    "from qiskit.quantum_info import Kraus, SuperOp\n",
+    "from qiskit.quantum_info import Kraus, SuperOp, random_quantum_channel, PTM, Choi, Chi\n",
     "from qiskit.visualization import plot_histogram\n",
     "from qiskit.transpiler.preset_passmanagers import generate_preset_pass_manager\n",
     "\n",
@@ -32,6 +32,411 @@
     ")"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "id": "ddc7b1fd",
+   "metadata": {},
+   "source": [
+    "## Representation"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "70fbdf81",
+   "metadata": {},
+   "source": [
+    "### Stinespring"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 130,
+   "id": "a4b2376e",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "Stinespring([[ 0.03849769+0.02036647j, -0.698461  -0.09797363j],\n",
+       "             [-0.0819251 +0.11373122j, -0.05175353+0.46665556j],\n",
+       "             [-0.15717588+0.11410124j,  0.07142001+0.09756838j],\n",
+       "             [ 0.00692667+0.03544208j,  0.00834859+0.31888786j],\n",
+       "             [ 0.14203522-0.13634234j, -0.09643058+0.12861438j],\n",
+       "             [-0.09117948-0.34925634j,  0.02034073+0.10721822j],\n",
+       "             [-0.56400128+0.6295895j , -0.09551063-0.09133277j],\n",
+       "             [-0.0836914 -0.22110541j, -0.33102433-0.030609j  ]],\n",
+       "            input_dims=(2,), output_dims=(2,))"
+      ]
+     },
+     "execution_count": 130,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "## random quantum channel in Stinespring form\n",
+    "from qiskit.quantum_info import random_quantum_channel\n",
+    "random_quantum_channel(2, 2)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "398ba931",
+   "metadata": {},
+   "source": [
+    "### Pauli Transfer Matrix (PTM) \n",
+    "https://docs.quantum.ibm.com/api/qiskit/qiskit.quantum_info.PTM\n",
+    "\n",
+    "Pauli Transfer Matrix (PTM) representation of a Quantum Channel.\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 96,
+   "id": "9eadfea1",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "num_qubits:  1\n"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "PTM([[ 1.00000000e+00-9.52787305e-18j, -2.08166817e-17+0.00000000e+00j,\n",
+       "       2.77555756e-17+0.00000000e+00j, -3.33066907e-16-6.88837700e-18j],\n",
+       "     [-9.09489194e-02+0.00000000e+00j, -3.72195758e-01+0.00000000e+00j,\n",
+       "       2.01356134e-02+0.00000000e+00j, -1.88193128e-01+0.00000000e+00j],\n",
+       "     [ 1.07371749e-01+0.00000000e+00j,  3.69139869e-01+0.00000000e+00j,\n",
+       "      -1.20645755e-01+0.00000000e+00j, -4.81331497e-01+0.00000000e+00j],\n",
+       "     [-1.56748013e-01+8.54415943e-18j,  1.14009043e-01+0.00000000e+00j,\n",
+       "      -4.85784633e-01+0.00000000e+00j, -1.72685021e-01+4.62303219e-19j]],\n",
+       "    input_dims=(2,), output_dims=(2,))"
+      ]
+     },
+     "execution_count": 96,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "rqc = PTM(random_quantum_channel(2, 2))\n",
+    "print('num_qubits: ', rqc.num_qubits)\n",
+    "rqc"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 60,
+   "id": "d5e0cce4",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "PTM([[ 1.00000000e+00-2.40259894e-18j, -3.65286159e-17+1.06400890e-17j,\n",
+       "      -3.25381917e-17-3.48337848e-19j,  8.10475729e-17-2.95275781e-18j],\n",
+       "     [-5.03863959e-01-3.97212202e-18j,  8.72077238e-02+7.41407448e-18j,\n",
+       "      -4.36095725e-02-9.82187287e-18j, -2.73736239e-02+4.42821287e-18j],\n",
+       "     [ 3.66501244e-01+1.11459993e-17j, -5.33738689e-02-5.89698216e-18j,\n",
+       "       8.10998407e-02-1.25503009e-17j, -9.32187871e-02+2.08189613e-18j],\n",
+       "     [-2.18558871e-01-7.45593673e-18j,  5.86923694e-02-9.37442687e-18j,\n",
+       "       2.00888015e-01-1.19893761e-17j, -7.11955275e-03-7.38513938e-19j]],\n",
+       "    input_dims=(2,), output_dims=(2,))"
+      ]
+     },
+     "execution_count": 60,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "rqc.compose(rqc)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 56,
+   "id": "a656e232",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "True"
+      ]
+     },
+     "execution_count": 56,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "rqc.power(4)\n",
+    "rqc.adjoint()\n",
+    "rqc.is_unitary()\n",
+    "rqc.is_cptp() # cp, tp, unitary"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f2d144cc",
+   "metadata": {},
+   "source": [
+    "### Kraus"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 132,
+   "id": "a8692063",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "Kraus([[[ 0.40797192-0.21907435j, -0.04302789-0.46760563j],\n",
+       "        [-0.02762658-0.30734793j, -0.02968974+0.05646335j]],\n",
+       "\n",
+       "       [[-0.31594324-0.24582982j,  0.22465833-0.25842112j],\n",
+       "        [-0.29698492+0.15016866j,  0.33448024-0.03782379j]],\n",
+       "\n",
+       "       [[ 0.29609313-0.1089395j ,  0.17119268-0.49213072j],\n",
+       "        [-0.19383795-0.15392175j,  0.30024441-0.14315255j]],\n",
+       "\n",
+       "       [[ 0.15164906-0.06327016j,  0.22433035+0.00823283j],\n",
+       "        [ 0.47619079-0.06909437j, -0.07334134+0.32703776j]]],\n",
+       "      input_dims=(2,), output_dims=(2,))"
+      ]
+     },
+     "execution_count": 132,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from qiskit.quantum_info import Kraus\n",
+    "rqc_kraus = Kraus(random_quantum_channel(2, 2))\n",
+    "rqc_kraus"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7b3c8a3c",
+   "metadata": {},
+   "source": [
+    "### SuperOp"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 139,
+   "id": "e31b9be2",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "SuperOp([[ 0.35072481-2.86610956e-18j, -0.14007719-7.01581293e-02j,\n",
+       "          -0.14007719+7.01581293e-02j,  0.31865109-3.53173471e-18j],\n",
+       "         [ 0.30439855+2.40310794e-01j, -0.10928561+8.78976417e-03j,\n",
+       "          -0.03296743-5.41481067e-03j,  0.20376672+2.19722547e-01j],\n",
+       "         [ 0.30439855-2.40310794e-01j, -0.03296743+5.41481067e-03j,\n",
+       "          -0.10928561-8.78976417e-03j,  0.20376672-2.19722547e-01j],\n",
+       "         [ 0.64927519+1.59760684e-19j,  0.14007719+7.01581293e-02j,\n",
+       "           0.14007719-7.01581293e-02j,  0.68134891-2.87830609e-18j]],\n",
+       "        input_dims=(2,), output_dims=(2,))"
+      ]
+     },
+     "execution_count": 139,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from qiskit.quantum_info import SuperOp\n",
+    "rqc_superop = SuperOp(random_quantum_channel(2, 2))\n",
+    "rqc_superop"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b73a9dce",
+   "metadata": {},
+   "source": [
+    "### Choi "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 115,
+   "id": "5cb1724d",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "Choi([[ 5.79887972e-01-1.35042435e-17j, -3.83764342e-01+8.14399434e-02j,\n",
+       "       -3.70265417e-04+8.17702963e-02j,  2.46990613e-01+4.54470675e-02j],\n",
+       "      [-3.83764342e-01-8.14399434e-02j,  4.20112028e-01+1.10093054e-17j,\n",
+       "       -9.87365214e-02-2.68883634e-01j,  3.70265417e-04-8.17702963e-02j],\n",
+       "      [-3.70265417e-04-8.17702963e-02j, -9.87365214e-02+2.68883634e-01j,\n",
+       "        4.72549646e-01+1.03334622e-18j, -1.67075303e-01+1.19816900e-01j],\n",
+       "      [ 2.46990613e-01-4.54470675e-02j,  3.70265417e-04+8.17702963e-02j,\n",
+       "       -1.67075303e-01-1.19816900e-01j,  5.27450354e-01-4.91109101e-18j]],\n",
+       "     input_dims=(2,), output_dims=(2,))"
+      ]
+     },
+     "execution_count": 115,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from qiskit.quantum_info import Choi\n",
+    "\n",
+    "rqc_choi = Choi(random_quantum_channel(2, 2))\n",
+    "rqc_choi"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "dfcdd0e1",
+   "metadata": {},
+   "source": [
+    "### Chi"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 128,
+   "id": "88c545fd",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "Chi([[ 0.11737619-3.46944695e-18j, -0.06467212+2.71818885e-02j,\n",
+       "      -0.00854459+2.51498699e-01j, -0.02205206+4.17213701e-02j],\n",
+       "     [-0.06467212-2.71818885e-02j,  0.39124272+6.93889390e-18j,\n",
+       "       0.09044332+2.20520604e-02j,  0.05584356-8.54458617e-03j],\n",
+       "     [-0.00854459-2.51498699e-01j,  0.09044332-2.20520604e-02j,\n",
+       "       1.3812895 +6.93889390e-18j,  0.13840778+6.46721240e-02j],\n",
+       "     [-0.02205206-4.17213701e-02j,  0.05584356+8.54458617e-03j,\n",
+       "       0.13840778-6.46721240e-02j,  0.11009159-3.46944695e-18j]],\n",
+       "    input_dims=(2,), output_dims=(2,))"
+      ]
+     },
+     "execution_count": 128,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from qiskit.quantum_info import Chi\n",
+    "rqc_chi = Chi(random_quantum_channel(2, 2))\n",
+    "rqc_chi"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fe557d72",
+   "metadata": {},
+   "source": [
+    "## Measures"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 160,
+   "id": "1e31f7fc",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "0.5734653416497364"
+      ]
+     },
+     "execution_count": 160,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from qiskit.quantum_info import average_gate_fidelity\n",
+    "average_gate_fidelity(random_quantum_channel(2, 2))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 171,
+   "id": "c19ce54b",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "0.5232190682672958"
+      ]
+     },
+     "execution_count": 171,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from qiskit.quantum_info import process_fidelity\n",
+    "process_fidelity(random_quantum_channel(2, 2), random_quantum_channel(2, 2))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 180,
+   "id": "27913856",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "0.4836283339885725"
+      ]
+     },
+     "execution_count": 180,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from qiskit.quantum_info import gate_error\n",
+    "gate_error(random_quantum_channel(2, 2))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 204,
+   "id": "a3ae5f49",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "1.180153665892265"
+      ]
+     },
+     "execution_count": 204,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from qiskit.quantum_info import diamond_norm\n",
+    "diamond_norm(random_quantum_channel(2, 2)-random_quantum_channel(2, 2))"
+   ]
+  },
   {
    "cell_type": "markdown",
    "id": "a299aec9",
@@ -452,17 +857,6 @@
     "plot_histogram(counts_bit_flip)"
    ]
   },
-  {
-   "cell_type": "markdown",
-   "id": "162148b7",
-   "metadata": {},
-   "source": [
-    "## Pauli Transfer Matrix (PTM) \n",
-    "https://docs.quantum.ibm.com/api/qiskit/qiskit.quantum_info.PTM\n",
-    "\n",
-    "Pauli Transfer Matrix (PTM) representation of a Quantum Channel.\n"
-   ]
-  },
   {
    "cell_type": "markdown",
    "id": "e10f3644",
diff --git a/_sources/operator.ipynb b/_sources/operator.ipynb
index 542fa74..e4c4cb8 100644
--- a/_sources/operator.ipynb
+++ b/_sources/operator.ipynb
@@ -606,7 +606,7 @@
    "id": "c445f6d2",
    "metadata": {},
    "source": [
-    "## Different norms"
+    "## Norms"
    ]
   },
   {
@@ -679,7 +679,7 @@
    "id": "bcba2d92",
    "metadata": {},
    "source": [
-    "## Entenglement measures"
+    "## Entenglement "
    ]
   },
   {
diff --git a/channel.html b/channel.html
index 4f7f78e..f1451f3 100644
--- a/channel.html
+++ b/channel.html
@@ -394,6 +394,16 @@ <h2> Contents </h2>
             </div>
             <nav aria-label="Page">
                 <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#representation">Representation</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#stinespring">Stinespring</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#pauli-transfer-matrix-ptm">Pauli Transfer Matrix (PTM)</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#kraus">Kraus</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#superop">SuperOp</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#choi">Choi</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#chi">Chi</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#measures">Measures</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#noise-channel">Noise channel</a><ul class="nav section-nav flex-column">
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#converting-to-and-from-quantumchannel-operators">Converting to and from QuantumChannel operators</a></li>
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#readout-error">Readout error</a></li>
@@ -403,10 +413,9 @@ <h2> Contents </h2>
 </li>
 </ul>
 </li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#pauli-transfer-matrix-ptm">Pauli Transfer Matrix (PTM)</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#measurement">Measurement</a><ul class="nav section-nav flex-column">
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#povm">POVM</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#kraus">Kraus</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Kraus</a></li>
 </ul>
 </li>
 </ul>
@@ -426,7 +435,7 @@ <h1>Channels<a class="headerlink" href="#channels" title="Permalink to this head
 <div class="cell_input docutils container">
 <div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">qiskit</span> <span class="kn">import</span> <span class="n">QuantumCircuit</span>
 <span class="kn">from</span> <span class="nn">qiskit_aer</span> <span class="kn">import</span> <span class="n">AerSimulator</span>
-<span class="kn">from</span> <span class="nn">qiskit.quantum_info</span> <span class="kn">import</span> <span class="n">Kraus</span><span class="p">,</span> <span class="n">SuperOp</span>
+<span class="kn">from</span> <span class="nn">qiskit.quantum_info</span> <span class="kn">import</span> <span class="n">Kraus</span><span class="p">,</span> <span class="n">SuperOp</span><span class="p">,</span> <span class="n">random_quantum_channel</span><span class="p">,</span> <span class="n">PTM</span><span class="p">,</span> <span class="n">Choi</span><span class="p">,</span> <span class="n">Chi</span>
 <span class="kn">from</span> <span class="nn">qiskit.visualization</span> <span class="kn">import</span> <span class="n">plot_histogram</span>
 <span class="kn">from</span> <span class="nn">qiskit.transpiler.preset_passmanagers</span> <span class="kn">import</span> <span class="n">generate_preset_pass_manager</span>
 
@@ -443,6 +452,258 @@ <h1>Channels<a class="headerlink" href="#channels" title="Permalink to this head
 </div>
 </div>
 </div>
+<section id="representation">
+<h2>Representation<a class="headerlink" href="#representation" title="Permalink to this heading">#</a></h2>
+<section id="stinespring">
+<h3>Stinespring<a class="headerlink" href="#stinespring" title="Permalink to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">## random quantum channel in Stinespring form</span>
+<span class="kn">from</span> <span class="nn">qiskit.quantum_info</span> <span class="kn">import</span> <span class="n">random_quantum_channel</span>
+<span class="n">random_quantum_channel</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Stinespring([[ 0.06885476+0.3910152j ,  0.20438933+0.05910322j],
+             [ 0.31245599+0.02311884j,  0.16757821-0.21559717j],
+             [-0.05263711-0.00905652j, -0.1945255 +0.53793483j],
+             [ 0.02102957-0.49189433j,  0.16111173-0.21351109j],
+             [ 0.29642876+0.22348274j, -0.10087805-0.28069541j],
+             [-0.34187756-0.00845965j, -0.20053018-0.31480756j],
+             [ 0.03999834+0.42614222j, -0.13431833-0.1325844j ],
+             [-0.24527358+0.02872948j,  0.46028553-0.07512615j]],
+            input_dims=(2,), output_dims=(2,))
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="pauli-transfer-matrix-ptm">
+<h3>Pauli Transfer Matrix (PTM)<a class="headerlink" href="#pauli-transfer-matrix-ptm" title="Permalink to this heading">#</a></h3>
+<p><a class="reference external" href="https://docs.quantum.ibm.com/api/qiskit/qiskit.quantum_info.PTM">https://docs.quantum.ibm.com/api/qiskit/qiskit.quantum_info.PTM</a></p>
+<p>Pauli Transfer Matrix (PTM) representation of a Quantum Channel.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">rqc</span> <span class="o">=</span> <span class="n">PTM</span><span class="p">(</span><span class="n">random_quantum_channel</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;num_qubits: &#39;</span><span class="p">,</span> <span class="n">rqc</span><span class="o">.</span><span class="n">num_qubits</span><span class="p">)</span>
+<span class="n">rqc</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>num_qubits:  1
+</pre></div>
+</div>
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>PTM([[ 1.00000000e+00+8.74076374e-18j,  5.89805982e-17-6.93889390e-18j,
+       3.46944695e-17+0.00000000e+00j, -1.11022302e-16-7.82562180e-18j],
+     [-2.86593783e-01+8.67361738e-19j,  1.47855987e-01+0.00000000e+00j,
+      -1.62431989e-01+0.00000000e+00j, -3.17758765e-01+8.67361738e-19j],
+     [ 1.47424935e-01+0.00000000e+00j, -1.62548064e-01+0.00000000e+00j,
+       1.31079590e-01+0.00000000e+00j, -1.30989106e-01+0.00000000e+00j],
+     [-6.63227365e-01-1.03564694e-17j,  2.69536560e-02+6.93889390e-18j,
+       1.06805607e-01+0.00000000e+00j, -4.01931013e-02+2.94093626e-18j]],
+    input_dims=(2,), output_dims=(2,))
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">rqc</span><span class="o">.</span><span class="n">compose</span><span class="p">(</span><span class="n">rqc</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>PTM([[ 1.00000000e+00+2.46603379e-17j,  5.90692569e-17-8.17578003e-18j,
+       1.78040661e-17+2.91278054e-19j, -1.29846171e-16-5.30619143e-18j],
+     [-1.42168607e-01+1.20615869e-18j,  3.96996378e-02-1.92871932e-19j,
+      -7.92464783e-02+9.26390967e-20j, -1.29340044e-02+1.40164895e-18j],
+     [ 3.00210160e-01+2.50420322e-18j, -4.88710735e-02-1.93188549e-18j,
+       2.95944934e-02+0.00000000e+00j,  3.97459322e-02-1.67991037e-18j],
+     [-6.28549141e-01-1.96530991e-17j, -1.44591363e-02+5.42839465e-18j,
+       5.32905065e-03-8.12989857e-19j, -2.09396460e-02+2.77224004e-18j]],
+    input_dims=(2,), output_dims=(2,))
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">rqc</span><span class="o">.</span><span class="n">power</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
+<span class="n">rqc</span><span class="o">.</span><span class="n">adjoint</span><span class="p">()</span>
+<span class="n">rqc</span><span class="o">.</span><span class="n">is_unitary</span><span class="p">()</span>
+<span class="n">rqc</span><span class="o">.</span><span class="n">is_cptp</span><span class="p">()</span> <span class="c1"># cp, tp, unitary</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>True
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="kraus">
+<h3>Kraus<a class="headerlink" href="#kraus" title="Permalink to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">qiskit.quantum_info</span> <span class="kn">import</span> <span class="n">Kraus</span>
+<span class="n">rqc_kraus</span> <span class="o">=</span> <span class="n">Kraus</span><span class="p">(</span><span class="n">random_quantum_channel</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">rqc_kraus</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Kraus([[[-0.43707631-0.14516188j, -0.034397  +0.27381402j],
+        [ 0.24785798-0.06810455j, -0.08515486-0.18925227j]],
+
+       [[ 0.0379263 -0.00665024j, -0.16190967+0.02613112j],
+        [-0.25128186+0.00193994j,  0.19069565+0.07950482j]],
+
+       [[ 0.13031368+0.247972j  ,  0.68426905+0.12155623j],
+        [ 0.10067869+0.53356301j, -0.28467422+0.14111131j]],
+
+       [[ 0.51606571+0.00561311j, -0.15976635+0.26480532j],
+        [ 0.06668372+0.11442053j, -0.30541943+0.19573325j]]],
+      input_dims=(2,), output_dims=(2,))
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="superop">
+<h3>SuperOp<a class="headerlink" href="#superop" title="Permalink to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">qiskit.quantum_info</span> <span class="kn">import</span> <span class="n">SuperOp</span>
+<span class="n">rqc_superop</span> <span class="o">=</span> <span class="n">SuperOp</span><span class="p">(</span><span class="n">random_quantum_channel</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">rqc_superop</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>SuperOp([[ 0.52026977-9.66067536e-18j,  0.10063323-1.76811433e-01j,
+           0.10063323+1.76811433e-01j,  0.32567927+7.52895145e-19j],
+         [ 0.03864309-4.79708656e-02j,  0.31098596-5.40737904e-02j,
+           0.08899708-1.02966869e-01j,  0.34797041+2.02380933e-01j],
+         [ 0.03864309+4.79708656e-02j,  0.08899708+1.02966869e-01j,
+           0.31098596+5.40737904e-02j,  0.34797041-2.02380933e-01j],
+         [ 0.47973023+1.13154920e-21j, -0.10063323+1.76811433e-01j,
+          -0.10063323-1.76811433e-01j,  0.67432073-1.06008950e-17j]],
+        input_dims=(2,), output_dims=(2,))
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="choi">
+<h3>Choi<a class="headerlink" href="#choi" title="Permalink to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">qiskit.quantum_info</span> <span class="kn">import</span> <span class="n">Choi</span>
+
+<span class="n">rqc_choi</span> <span class="o">=</span> <span class="n">Choi</span><span class="p">(</span><span class="n">random_quantum_channel</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">rqc_choi</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Choi([[ 0.62624433+1.58476117e-17j,  0.08534607-4.01810435e-02j,
+        0.23678685+3.01178015e-02j,  0.12303162-5.61603539e-02j],
+      [ 0.08534607+4.01810435e-02j,  0.37375567-6.10074507e-18j,
+        0.16896483-3.21122870e-02j, -0.23678685-3.01178015e-02j],
+      [ 0.23678685-3.01178015e-02j,  0.16896483+3.21122870e-02j,
+        0.54821592+2.32410142e-18j,  0.15737015-1.64499647e-01j],
+      [ 0.12303162+5.61603539e-02j, -0.23678685+3.01178015e-02j,
+        0.15737015+1.64499647e-01j,  0.45178408-3.47516213e-18j]],
+     input_dims=(2,), output_dims=(2,))
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="chi">
+<h3>Chi<a class="headerlink" href="#chi" title="Permalink to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">qiskit.quantum_info</span> <span class="kn">import</span> <span class="n">Chi</span>
+<span class="n">rqc_chi</span> <span class="o">=</span> <span class="n">Chi</span><span class="p">(</span><span class="n">random_quantum_channel</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">rqc_chi</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Chi([[ 0.79836415+6.93889390e-18j, -0.03488378-7.09711601e-02j,
+      -0.0518835 +4.87656411e-01j,  0.02820931-1.12935664e-01j],
+     [-0.03488378+7.09711601e-02j,  0.41181799-6.93889390e-18j,
+      -0.05601561-2.82093077e-02j, -0.01635159-5.18835015e-02j],
+     [-0.0518835 -4.87656411e-01j, -0.05601561+2.82093077e-02j,
+       0.5439036 +6.93889390e-18j,  0.09223869+3.48837838e-02j],
+     [ 0.02820931+1.12935664e-01j, -0.01635159+5.18835015e-02j,
+       0.09223869-3.48837838e-02j,  0.24591426-2.08166817e-17j]],
+    input_dims=(2,), output_dims=(2,))
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+</section>
+<section id="measures">
+<h2>Measures<a class="headerlink" href="#measures" title="Permalink to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">qiskit.quantum_info</span> <span class="kn">import</span> <span class="n">average_gate_fidelity</span>
+<span class="n">average_gate_fidelity</span><span class="p">(</span><span class="n">random_quantum_channel</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.5675002373830127
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">qiskit.quantum_info</span> <span class="kn">import</span> <span class="n">process_fidelity</span>
+<span class="n">process_fidelity</span><span class="p">(</span><span class="n">random_quantum_channel</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">random_quantum_channel</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.40233711255718513
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">qiskit.quantum_info</span> <span class="kn">import</span> <span class="n">gate_error</span>
+<span class="n">gate_error</span><span class="p">(</span><span class="n">random_quantum_channel</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.5587452460410387
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">qiskit.quantum_info</span> <span class="kn">import</span> <span class="n">diamond_norm</span>
+<span class="n">diamond_norm</span><span class="p">(</span><span class="n">random_quantum_channel</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">-</span><span class="n">random_quantum_channel</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>1.203391603312065
+</pre></div>
+</div>
+</div>
+</div>
+</section>
 <section id="noise-channel">
 <h2>Noise channel<a class="headerlink" href="#noise-channel" title="Permalink to this heading">#</a></h2>
 <p><a class="reference external" href="https://docs.quantum.ibm.com/guides/build-noise-models#build-noise-models">https://docs.quantum.ibm.com/guides/build-noise-models#build-noise-models</a></p>
@@ -675,7 +936,7 @@ <h3>Examples<a class="headerlink" href="#examples" title="Permalink to this head
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="_images/ae310a47b70948ff788d194b8aee7c25c940bf25b20c54f59de0fc5a902f4a06.png" src="_images/ae310a47b70948ff788d194b8aee7c25c940bf25b20c54f59de0fc5a902f4a06.png" />
+<img alt="_images/49a6274a9b9a0e1d2821e40efe1c9a2a36e0bcc40fc2048f6bd2200c75142567.png" src="_images/49a6274a9b9a0e1d2821e40efe1c9a2a36e0bcc40fc2048f6bd2200c75142567.png" />
 </div>
 </div>
 <section id="from-qiskit-aer-import-aersimulator">
@@ -707,7 +968,7 @@ <h4>from qiskit_aer import AerSimulator<a class="headerlink" href="#from-qiskit-
 <div class="cell_output docutils container">
 <div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>NoiseModel:
   Basis gates: [&#39;cx&#39;, &#39;id&#39;, &#39;rz&#39;, &#39;sx&#39;, &#39;u1&#39;, &#39;u2&#39;, &#39;u3&#39;]
-  Instructions with noise: [&#39;reset&#39;, &#39;u3&#39;, &#39;u1&#39;, &#39;cx&#39;, &#39;measure&#39;, &#39;u2&#39;]
+  Instructions with noise: [&#39;measure&#39;, &#39;reset&#39;, &#39;u3&#39;, &#39;cx&#39;, &#39;u1&#39;, &#39;u2&#39;]
   All-qubits errors: [&#39;reset&#39;, &#39;measure&#39;, &#39;u1&#39;, &#39;u2&#39;, &#39;u3&#39;, &#39;cx&#39;]
 </pre></div>
 </div>
@@ -738,24 +999,19 @@ <h4>from qiskit_aer import AerSimulator<a class="headerlink" href="#from-qiskit-
   r = _umath_linalg.det(a, signature=signature)
 </pre></div>
 </div>
-<img alt="_images/59844ab3115aca3e6210653113827faec8bf38a4fe731016ea2b60b7eefb4f10.png" src="_images/59844ab3115aca3e6210653113827faec8bf38a4fe731016ea2b60b7eefb4f10.png" />
+<img alt="_images/8da3ff51fd54ba7622dc810fbe0e8cc07282c0e15f1564663f9a1b4ada4dbbb4.png" src="_images/8da3ff51fd54ba7622dc810fbe0e8cc07282c0e15f1564663f9a1b4ada4dbbb4.png" />
 </div>
 </div>
 </section>
 </section>
 </section>
-<section id="pauli-transfer-matrix-ptm">
-<h2>Pauli Transfer Matrix (PTM)<a class="headerlink" href="#pauli-transfer-matrix-ptm" title="Permalink to this heading">#</a></h2>
-<p><a class="reference external" href="https://docs.quantum.ibm.com/api/qiskit/qiskit.quantum_info.PTM">https://docs.quantum.ibm.com/api/qiskit/qiskit.quantum_info.PTM</a></p>
-<p>Pauli Transfer Matrix (PTM) representation of a Quantum Channel.</p>
-</section>
 <section id="measurement">
 <h2>Measurement<a class="headerlink" href="#measurement" title="Permalink to this heading">#</a></h2>
 <section id="povm">
 <h3>POVM<a class="headerlink" href="#povm" title="Permalink to this heading">#</a></h3>
 </section>
-<section id="kraus">
-<h3>Kraus<a class="headerlink" href="#kraus" title="Permalink to this heading">#</a></h3>
+<section id="id1">
+<h3>Kraus<a class="headerlink" href="#id1" title="Permalink to this heading">#</a></h3>
 </section>
 </section>
 </section>
@@ -823,6 +1079,16 @@ <h3>Kraus<a class="headerlink" href="#kraus" title="Permalink to this heading">#
   </div>
   <nav class="bd-toc-nav page-toc">
     <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#representation">Representation</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#stinespring">Stinespring</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#pauli-transfer-matrix-ptm">Pauli Transfer Matrix (PTM)</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#kraus">Kraus</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#superop">SuperOp</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#choi">Choi</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#chi">Chi</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#measures">Measures</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#noise-channel">Noise channel</a><ul class="nav section-nav flex-column">
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#converting-to-and-from-quantumchannel-operators">Converting to and from QuantumChannel operators</a></li>
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#readout-error">Readout error</a></li>
@@ -832,10 +1098,9 @@ <h3>Kraus<a class="headerlink" href="#kraus" title="Permalink to this heading">#
 </li>
 </ul>
 </li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#pauli-transfer-matrix-ptm">Pauli Transfer Matrix (PTM)</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#measurement">Measurement</a><ul class="nav section-nav flex-column">
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#povm">POVM</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#kraus">Kraus</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Kraus</a></li>
 </ul>
 </li>
 </ul>
diff --git a/operator.html b/operator.html
index dee8077..a466e02 100644
--- a/operator.html
+++ b/operator.html
@@ -419,12 +419,12 @@ <h2> Contents </h2>
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#anti-commutator">(Anti) commutator</a></li>
 </ul>
 </li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#different-norms">Different norms</a><ul class="nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#norms">Norms</a><ul class="nav section-nav flex-column">
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#vector-induced-norm">Vector induced norm</a></li>
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#schatten-p-norm">Schatten p norm</a></li>
 </ul>
 </li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#entenglement-measures">Entenglement measures</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#entenglement">Entenglement</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#quantum-circuit">Quantum circuit</a></li>
 </ul>
             </nav>
@@ -815,8 +815,8 @@ <h2>Measures<a class="headerlink" href="#measures" title="Permalink to this head
 <h3>(Anti) commutator<a class="headerlink" href="#anti-commutator" title="Permalink to this heading">#</a></h3>
 </section>
 </section>
-<section id="different-norms">
-<h2>Different norms<a class="headerlink" href="#different-norms" title="Permalink to this heading">#</a></h2>
+<section id="norms">
+<h2>Norms<a class="headerlink" href="#norms" title="Permalink to this heading">#</a></h2>
 <section id="vector-induced-norm">
 <h3>Vector induced norm<a class="headerlink" href="#vector-induced-norm" title="Permalink to this heading">#</a></h3>
 <div class="cell docutils container">
@@ -859,8 +859,8 @@ <h3>Schatten p norm<a class="headerlink" href="#schatten-p-norm" title="Permalin
 </div>
 </section>
 </section>
-<section id="entenglement-measures">
-<h2>Entenglement measures<a class="headerlink" href="#entenglement-measures" title="Permalink to this heading">#</a></h2>
+<section id="entenglement">
+<h2>Entenglement<a class="headerlink" href="#entenglement" title="Permalink to this heading">#</a></h2>
 <div class="cell docutils container">
 <div class="cell_input docutils container">
 <div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">### purity</span>
@@ -1033,12 +1033,12 @@ <h2>Quantum circuit<a class="headerlink" href="#quantum-circuit" title="Permalin
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#anti-commutator">(Anti) commutator</a></li>
 </ul>
 </li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#different-norms">Different norms</a><ul class="nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#norms">Norms</a><ul class="nav section-nav flex-column">
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#vector-induced-norm">Vector induced norm</a></li>
 <li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#schatten-p-norm">Schatten p norm</a></li>
 </ul>
 </li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#entenglement-measures">Entenglement measures</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#entenglement">Entenglement</a></li>
 <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#quantum-circuit">Quantum circuit</a></li>
 </ul>
   </nav></div>
diff --git a/searchindex.js b/searchindex.js
index 9f077bd..3df1b3a 100644
--- a/searchindex.js
+++ b/searchindex.js
@@ -1 +1 @@
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