fixed formulas for purity RB

Author: ShellyGarion

ignis/RB_advanced.ipynb

@@ -431,39 +431,39 @@
     "\n",
     "### Definition of the purity\n",
     "The purity is defined as \n",
-    "\\begin{equation}\n",
+    "$$\n",
     "\\mathcal{P}  = {Tr}(\\rho^2)\n",
-    "\\end{equation}\n",
+    "$$\n",
     "where $\\rho$ can be expressed as a sum of Pauli matrices\n",
-    "\\begin{equation}\n",
+    "$$\n",
     "\\rho = \\sum_{i} \\alpha_i P_i\n",
-    "\\end{equation}\n",
+    "$$\n",
     "where $i$ is a sum over $4^n$ Paulis. Therefore,\n",
-    "\\begin{equation}\n",
+    "$$\n",
     "\\rho^2 = \\sum_{ij} \\alpha_i \\alpha_j P_i \\cdot P_j\n",
-    "\\end{equation}\n",
+    "$$\n",
     "and\n",
-    "\\begin{equation}\n",
+    "$$\n",
     "Tr(\\rho^2) = \\sum_{ij} \\alpha_i \\alpha_j Tr(P_i P_j)\n",
-    "\\end{equation}\n",
+    "$$\n",
     "If $i\\ne j$ then $P_i P_j = P_k$ (where $P_k \\ne \\mathbf{I}$ and if $i==j$ then $P_i P_j = \\mathcal{I}$). Since $Tr(P_k)=0$, then\n",
-    "\\begin{equation}\n",
+    "$$\n",
     "Tr(\\rho^2) = \\sum_i \\alpha_i^2 d\n",
-    "\\end{equation}\n",
+    "$$\n",
     "Now we can calculate any expectation value as,\n",
-    "\\begin{eqnarray}\n",
+    "$$\n",
     "\\langle \\hat{A} \\rangle = Tr(\\hat{A} \\rho)\n",
-    "\\end{eqnarray}\n",
+    "$$\n",
     "so the Pauli expectation value is\n",
-    "\\begin{eqnarray}\n",
-    "\\langle P_k \\rangle & = & Tr(P_k \\rho) \\\\\n",
-    "& = & \\sum_i \\alpha_i Tr(P_k P_i)\\\\\n",
-    "& = & \\alpha_k d\n",
-    "\\end{eqnarray}\n",
+    "$$\n",
+    "\\langle P_k \\rangle  =  Tr(P_k \\rho) \n",
+    " =  \\sum_i \\alpha_i Tr(P_k P_i)\n",
+    " =  \\alpha_k d\n",
+    "$$\n",
     "so\n",
-    "\\begin{eqnarray}\n",
+    "$$\n",
     "Tr(\\rho^2) = \\sum_k \\langle P_k \\rangle^2 /d\n",
-    "\\end{eqnarray}"
+    "$$"
    ]
   },
   {
@@ -473,11 +473,11 @@
     "### Step 1: Generating the purity RB sequences \n",
     "\n",
     "To calculate all $Z$ correlators we only need the on diagonal elements of $\\rho$ (i.e. what is measured in experiment). If we apply a $\\pi/2$ rotation to the density matrix and then measure the $Z$ correlator,\n",
-    "\\begin{eqnarray}\n",
-    "\\rho^{'} & = & e^{i P_j \\pi/4} \\rho e^{-i P_j \\pi/4} \\\\\n",
-    "Tr(Z \\rho^{'}) & = & Tr(Z e^{i P_j \\pi/4} \\rho e^{-i P_j \\pi/4}) \\\\\n",
-    "& = & Tr(e^{-i P_j \\pi/4}Z e^{i P_j \\pi/4} \\rho)\n",
-    "\\end{eqnarray}\n",
+    "$$\n",
+    "\\rho^{'}  =  e^{i P_j \\pi/4} \\rho e^{-i P_j \\pi/4} \\\\\n",
+    "Tr(Z \\rho^{'})  =  Tr(Z e^{i P_j \\pi/4} \\rho e^{-i P_j \\pi/4}) \n",
+    " =  Tr(e^{-i P_j \\pi/4}Z e^{i P_j \\pi/4} \\rho)\n",
+    "$$\n",
     "which looks like calculating the expectation value of the rotated Pauli. \n",
     "\n",
     "Therefore, in order to generate each of the $3^n$ circuits, we need to do (per each of the $n$ qubits) either:\n",