Update week3.do.txt

Author: mhjensen

doc/src/week3/week3.do.txt

@@ -8,31 +8,14 @@ DATE: February 5, 2025
 
 !bblock 
 o Reminder and review of  density matrices and measurements from last week
-o Schmidt decomposition and entanglement
+# Postpone thiso Schmidt decomposition and entanglement
 o Discussion of entropies, classical information entropy (Shannon entropy) and von Neumann entropy
-o Single and two-qubit gates
+o Single-qubit and two-qubit gates and codes
 o "Video of lecture to be added":"https://youtu.be/"
 #Chapters 3 and 4 of Scherer's text contains useful discussions of several of these topics. More reading suggestions will be added.
 !eblock
 
 
-!split
-===== Motivation =====
-
-In order to study entanglement and why it is so important for quantum
-computing, we need to introduce some basic measures and useful
-quantities.  For these endeavors, we  will use our two-qubit system from
-the second lecture in order to introduce and repeat, through examples, density
-matrices and entropy. These two quantities, together with
-technicalities like the Schmidt decomposition define important quantities in analyzing quantum computing examples.
-
-The Schmidt decomposition is again a
-linear decomposition which allows us to express a vector in terms of
-tensor product of two inner product spaces. In quantum information
-theory and quantum computing it is widely used as away to define and
-describe entanglement.
-
-
 
 !split
 ===== Reminder on density matrices and traces =====
@@ -322,118 +305,6 @@ $\rho_B$, that is we have for a given probability distribution $p_i$
 \]
 !et
 
-!split
-===== Schmidt decomposition =====
-If we cannot write the density matrix in this form, we say the system
-$AB$ is entangled. In order to see this, we can use the so-called
-Schmidt decomposition, which is essentially an application of the
-singular-value decomposition.
-
-
-!split
-===== Pure states and Schmidt decomposition =====
-
-The Schmidt decomposition allows us to define a pure state in a
-bipartite Hilbert space composed of systems $A$ and $B$ as
-
-!bt
-\[
-\vert\psi\rangle=\sum_{i=0}^{d-1}\sigma_i\vert i\rangle_A\vert i\rangle_B,
-\]
-!et
-where the amplitudes $\sigma_i$ are real and positive and their
-squared values sum up to one, $\sum_i\sigma_i^2=1$. The states $\vert
-i\rangle_A$ and $\vert i\rangle_B$ form orthornormal bases for systems
-$A$ and $B$ respectively, the amplitudes $\lambda_i$ are the so-called
-Schmidt coefficients and the Schmidt rank $d$ is equal to the number
-of Schmidt coefficients and is smaller or equal to the minimum
-dimensionality of system $A$ and system $B$, that is $d\leq
-\mathrm{min}(\mathrm{dim}(A), \mathrm{dim}(B))$.
-
-
-!split
-===== Proof of Schmidt decomposition =====
-
-The proof for the above decomposition is based on the singular-value
-decomposition. To see this, assume that we have two orthonormal bases
-sets for systems $A$ and $B$, respectively. That is we have two ONBs
-$\vert i\rangle_A$ and $\vert j\rangle_B$. We can always construct a
-product state (a pure state) as
-
-!bt
-\[ \vert\psi \rangle = \sum_{ij} c_{ij}\vert i\rangle_A\vert
-j\rangle_B,
-\]
-!et
-where the coefficients $c_{ij}$ are the overlap coefficients which
-belong to a matrix $\bm{C}$.
-
-
-
-!split
-===== Further parts of proof =====
-
-If we now assume that the
-dimensionalities of the two subsystems $A$ and $B$ are the same $d$,
-we can always rewrite the matrix $\bm{C}$ in terms of a singular-value
-decomposition with unitary/orthogonal matrices $\bm{U}$ and $\bm{V}$
-of dimension $d\times d$ and a matrix $\bm{\Sigma}$ which contains the
-(diagonal) singular values $\sigma_0\leq \sigma_1 \leq \dots 0$ as
-
-!bt
-\[
-\bm{C}=\bm{U}\bm{\Sigma}\bm{V}^{\dagger}.
-\]
-!et
-
-!split
-===== SVD parts in proof =====
-
-This means we can rewrite the coefficients $c_{ij}$ in terms of the singular-value decomposition
-!bt
-\[
-c_{ij}=\sum_k u_{ik}\sigma_kv_{kj},
-\]
-!et
-and inserting this in the definition of the pure state $\vert \psi\rangle$ we have
-
-!bt
-\[
-\vert\psi \rangle = \sum_{ij} \left(\sum_k u_{ik}\sigma_kv_{kj} \right)\vert i\rangle_A\vert j\rangle_B.
-\]
-!et
-
-!split
-===== Slight rewrite =====
-We rewrite the last equation as
-
-!bt
-\[
-\vert\psi \rangle = \sum_{k}\sigma_k \left(\sum_i u_{ik}\vert i\rangle_A\right)\otimes\left(\sum_jv_{kj}\vert j\rangle_B\right),
-\]
-!et
-which we identify simply as, since the matrices $\bm{U}$ and $\bm{V}$ represent unitary transformations,
-!bt
-\[
-\vert\psi \rangle = \sum_{k}\sigma_k \vert k\rangle_A\vert k\rangle_B.
-\]
-!et
-
-
-!split
-===== Different dimensionalities =====
-
-It is straight forward to prove this relation in case systems $A$ and
-$B$ have different dimensionalities.  Once we know the Schmidt
-decomposition of a state, we can immmediately say whether it is
-entangled or not. If a state $\psi$ has is entangled, then its Schmidt
-decomposition has more than one term. Stated differently, the state is
-entangled if the so-called Schmidt rank is is greater than one.  There
-is another important property of the Schmidt decomposition which is
-related to the properties of the density matrices and their trace
-operations and the entropies. In order to introduce these concepts let
-us look at the two-qubit Hamiltonian described here.
-
 
 !split
 ===== Entropies and density matrices =====
@@ -471,8 +342,6 @@ S=-\mathrm{Tr}[\rho\log_2{\rho}].
 
 _This part is best seen using the jupyter-notebook_.
 
-The system we discuss here is a continuation of the two qubit example from week 2.
-
 
 This system can be thought of as composed of two subsystems
 $A$ and $B$. Each subsystem has computational basis states
@@ -669,29 +538,3 @@ plt.show
 !ec
 
 
-
-
-!split
-===== Exercise: Two-qubit Hamiltonian =====
-
-Use the Hamiltonian for the two-qubit example to find the eigenpairs
-as functions of the interaction strength $\lambda$ and study the final
-eigenvectors as functions of the admixture of the original basis
-states.  Discuss the results as functions of the parameter $\lambda$ and compute the von Neumann
-entropy and discuss the results. You will need to calculate the entropy of the subsystems $A$ or $B$.
-
-
-
-
-!split
-===== The next lecture  =====
-
-In our next lecture, we will discuss
-o Reminder and review of  entropy and entanglement
-o Gates and circuits and how to perform operations on states
-
-"Reading: Chapters 2.1-2.11 of Hundt's text":"https://github.com/CompPhysics/QuantumComputingMachineLearning/blob/gh-pages/doc/Textbooks/Programming/chapter2.pdf"
-
-
-
-