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8 changes: 4 additions & 4 deletions doc/pub/week1/ipynb/week1.ipynb
Original file line number Diff line number Diff line change
Expand Up @@ -659,7 +659,7 @@
"id": "f60e43ec",
"metadata": {},
"source": [
"For two arbitrary vectors $\\vert x\\rangle$ and $\\vert y\\rangle$ with the same lentgh, we have the\n",
"For two arbitrary vectors $\\vert x\\rangle$ and $\\vert y\\rangle$ with the same length, we have the\n",
"general expression"
]
},
Expand Down Expand Up @@ -1414,7 +1414,7 @@
"metadata": {},
"source": [
"## Examples of tensor products\n",
"If we now go back to our original one-qubit basis states, we can form teh following tensor products"
"If we now go back to our original one-qubit basis states, we can form the following tensor products"
]
},
{
Expand Down Expand Up @@ -1981,7 +1981,7 @@
"\n",
"Since our original basis $\\vert \\psi\\rangle$ is orthogonal and normalized with $\\vert\\alpha\\vert^2+\\vert\\beta\\vert^2=1$, the new basis is also orthogonal and normalized, as we can see below here.\n",
"\n",
"Since the inverse of a hermitian matrix is equal to its hermitian\n",
"Since the inverse of a unitary matrix is equal to its hermitian\n",
"conjugate/adjoint), unitary transformations are always reversible.\n",
"\n",
"Why are only unitary transformations allowed? The key lies in the way the inner product tranforms.\n",
Expand All @@ -2004,7 +2004,7 @@
"id": "1d1c60c1",
"metadata": {},
"source": [
"or in terms of a matrix-vector notatio we have"
"or in terms of a matrix-vector notation we have"
]
},
{
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30 changes: 15 additions & 15 deletions doc/pub/week2/ipynb/week2.ipynb
Original file line number Diff line number Diff line change
Expand Up @@ -500,7 +500,7 @@
"source": [
"$$\n",
"\\boldsymbol{P}_a=\\vert \\psi_a\\rangle \\langle \\psi_a\\vert = \\begin{bmatrix} \\vert \\alpha_0\\vert^2 &\\alpha_0\\alpha_1^* \\\\\n",
" \\alpha_1\\alpha_0^* & \\vert \\alpha_1\\vert^* \\end{bmatrix},\n",
" \\alpha_1\\alpha_0^* & \\vert \\alpha_1\\vert^2 \\end{bmatrix},\n",
"$$"
]
},
Expand All @@ -519,7 +519,7 @@
"source": [
"$$\n",
"\\boldsymbol{P}_b=\\vert \\psi_b\\rangle \\langle \\psi_b\\vert = \\begin{bmatrix} \\vert \\beta_0\\vert^2 &\\beta_0\\beta_1^* \\\\\n",
" \\beta_1\\beta_0^* & \\vert \\beta_1\\vert^* \\end{bmatrix}.\n",
" \\beta_1\\beta_0^* & \\vert \\beta_1\\vert^2 \\end{bmatrix}.\n",
"$$"
]
},
Expand Down Expand Up @@ -558,8 +558,8 @@
"source": [
"$$\n",
"\\boldsymbol{A}=\\lambda_a\\begin{bmatrix} \\vert \\alpha_0\\vert^2 &\\alpha_0\\alpha_1^* \\\\\n",
" \\alpha_1\\alpha_0^* & \\vert \\alpha_1\\vert^* \\end{bmatrix} +\\lambda_b\\begin{bmatrix} \\vert \\beta_0\\vert^2 &\\beta_0\\beta_1^* \\\\\n",
" \\beta_1\\beta_0^* & \\vert \\beta_1\\vert^* \\end{bmatrix}.\n",
" \\alpha_1\\alpha_0^* & \\vert \\alpha_1\\vert^2 \\end{bmatrix} +\\lambda_b\\begin{bmatrix} \\vert \\beta_0\\vert^2 &\\beta_0\\beta_1^* \\\\\n",
" \\beta_1\\beta_0^* & \\vert \\beta_1\\vert^2 \\end{bmatrix}.\n",
"$$"
]
},
Expand Down Expand Up @@ -742,7 +742,7 @@
"metadata": {},
"source": [
"$$\n",
"\\sum_{i=0}^1\\boldsymbol{P}_i^{\\dagger}\\boldsymbol{P}_1=\\boldsymbol{I},\n",
"\\sum_{i=0}^1\\boldsymbol{P}_i^{\\dagger}\\boldsymbol{P}_i=\\boldsymbol{I},\n",
"$$"
]
},
Expand Down Expand Up @@ -1500,14 +1500,14 @@
"source": [
"## Simple Hamiltonian models\n",
"\n",
"In order to study get started with coding, we will study two simple Hamiltonian systems, one which we can use for a single qubit systems and one which has as basis functions a two-qubit system. These two simple Hamiltonians exhibit also something which is called level crossing, a feature which we will use in later studies of entanglement.\n",
"In order to get started with coding, we will study two simple Hamiltonian systems, one which we can use for a single qubit systems and one which has as basis functions a two-qubit system. These two simple Hamiltonians exhibit also something which is called level crossing, a feature which we will use in later studies of entanglement.\n",
"\n",
"We study first a simple two-level system. Thereafter we\n",
"extend our model to a four-level system which can be\n",
"interpreted as composed of two separate (not necesseraly identical)\n",
"subsystems.\n",
"\n",
"We let our hamiltonian depend linearly on a strength parameter $z$"
"We let our hamiltonian depend linearly on a strength parameter $\\lambda$"
]
},
{
Expand Down Expand Up @@ -1668,7 +1668,7 @@
"is $\\vert 0 \\rangle$. At $\\lambda=1$ the $\\vert 0 \\rangle$ mixing of\n",
"the lowest eigenvalue is $1\\%$ while for $\\lambda\\leq 2/3$ we have a\n",
"$\\vert 0 \\rangle$ component of more than $90\\%$. The character of the\n",
"eigenvectors has therefore been interchanged when passing $z=2/3$. The\n",
"eigenvectors has therefore been interchanged when passing $\\lambda=2/3$. The\n",
"value of the parameter $X$ represents the strength of the coupling\n",
"between the model space and the excluded space. The following code\n",
"computes and plots the eigenvalues."
Expand Down Expand Up @@ -1785,7 +1785,7 @@
"metadata": {},
"source": [
"$$\n",
"\\vert 10\\rangle = \\vert 1\\rangle_{\\mathrm{A}}\\otimes \\vert 0\\rangle_{\\mathrm{B}}=\\begin{bmatrix} 0 & 1 & 0 &0\\end{bmatrix}^T,\n",
"\\vert 01\\rangle = \\vert 0\\rangle_{\\mathrm{A}}\\otimes \\vert 1\\rangle_{\\mathrm{B}}=\\begin{bmatrix} 0 & 1 & 0 &0\\end{bmatrix}^T,\n",
"$$"
]
},
Expand All @@ -1803,7 +1803,7 @@
"metadata": {},
"source": [
"$$\n",
"\\vert 01\\rangle = \\vert 0\\rangle_{\\mathrm{A}}\\otimes \\vert 1\\rangle_{\\mathrm{B}}=\\begin{bmatrix} 0 & 0 & 1 &0\\end{bmatrix}^T,\n",
"\\vert 10\\rangle = \\vert 1\\rangle_{\\mathrm{A}}\\otimes \\vert 0\\rangle_{\\mathrm{B}}=\\begin{bmatrix} 0 & 0 & 1 &0\\end{bmatrix}^T,\n",
"$$"
]
},
Expand Down Expand Up @@ -1849,7 +1849,7 @@
"metadata": {},
"source": [
"$$\n",
"H_0\\vert 10 \\rangle = \\epsilon_{10}\\vert 10 \\rangle,\n",
"H_0\\vert 01 \\rangle = \\epsilon_{01}\\vert 01 \\rangle,\n",
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These changes are strictly speaking not necessary, but I guess it makes sense to change the ordering, because of the correction above in the definition of |01> and |10>.

"$$"
]
},
Expand All @@ -1859,7 +1859,7 @@
"metadata": {},
"source": [
"$$\n",
"H_0\\vert 01 \\rangle = \\epsilon_{01}\\vert 01 \\rangle,\n",
"H_0\\vert 10 \\rangle = \\epsilon_{10}\\vert 10 \\rangle,\n",
"$$"
]
},
Expand Down Expand Up @@ -1914,8 +1914,8 @@
"source": [
"$$\n",
"\\boldsymbol{H}=\\begin{bmatrix} \\epsilon_{00}+H_z & 0 & 0 & H_x \\\\\n",
" 0 & \\epsilon_{10}-H_z & H_x & 0 \\\\\n",
"\t\t 0 & H_x & \\epsilon_{01}-H_z & 0 \\\\\n",
" 0 & \\epsilon_{01}-H_z & H_x & 0 \\\\\n",
"\t\t 0 & H_x & \\epsilon_{10}-H_z & 0 \\\\\n",
"\t\t H_x & 0 & 0 & \\epsilon_{11} +H_z \\end{bmatrix}.\n",
"$$"
]
Expand All @@ -1936,7 +1936,7 @@
"metadata": {},
"source": [
"$$\n",
"\\rho_0=\\left(\\alpha_{00}\\vert 00 \\rangle\\langle 00\\vert+\\alpha_{10}\\vert 10 \\rangle\\langle 10\\vert+\\alpha_{01}\\vert 01 \\rangle\\langle 01\\vert+\\alpha_{11}\\vert 11 \\rangle\\langle 11\\vert\\right),\n",
"\\rho_0=\\left(\\alpha_{00}\\vert 00 \\rangle\\langle 00\\vert+\\alpha_{01}\\vert 01 \\rangle\\langle 01\\vert+\\alpha_{10}\\vert 10 \\rangle\\langle 10\\vert+\\alpha_{11}\\vert 11 \\rangle\\langle 11\\vert\\right),\n",
"$$"
]
},
Expand Down