Update week5.do.txt

Author: mhjensen

doc/src/week5/week5.do.txt

@@ -97,7 +97,7 @@ and with the unitary transformation it is easy to show that thet trace of the tr
 
 
 !split
-===== First entanglement encounter, two qubit system =====
+===== Revisiting our Bell states and ur  entanglement encounter, two qubit system =====
 
 We define a system that can be thought of as composed of two subsystems
 $A$ and $B$. Each subsystem has computational basis states
@@ -273,36 +273,6 @@ state $\vert\psi\rangle_{A}\otimes\vert\phi\rangle_B$ for any choice
 of the states $\vert\psi\rangle_{A}$ and $\vert\phi\rangle_B$. Otherwise we say the state is separable.
 
 
-!split
-===== Examples of entanglement =====
-
-As an example, considere an ansatz for the ground state of the helium
-atom with two electrons in the lowest $1s$ state (hydrogen-like
-orbits) and with spin $s=1/2$ and spin projections $m_s=-1/2$ and
-$m_s=1/2$.  The two single-particle states are given by the tensor
-products of their spatial $1s$ single-particle states
-$\vert\phi_{1s}\rangle$ and and their spin up or spin down spinors
-$\vert\xi_{sm_s}\rangle$. The ansatz for the ground state is given by a Slater
-determinant with total orbital momentum $L=l_1+l_2=0$ and totalt spin
-$S=s_1+s_2=0$, normally labeled as a spin-singlet state.
-
-!split
-===== Ground state of helium =====
-This ansatz
-for the ground state is then written as, using the compact notations
-!bt
-\[
-\vert \psi_{i}\rangle = \vert\phi_{1s}\rangle_i\otimes \vert \xi\rangle_{s_im_{s_i}}=\vert 1s,s,m_s\rangle_i,  \]
-!et
-with $i$ being electron $1$ or $2$, and the tensor product of the two single-electron states as
-$\vert 1s,s,m_s\rangle_1\vert 1s,s,m_s\rangle_2=\vert 1s,s,m_s\rangle_1\otimes \vert 1s,s,m_s\rangle_2$, we arrive at
-!bt
-\[
-\Psi(\bm{r}_1,\bm{r}_2;s_1,s_2)=\frac{1}{\sqrt{2}}\left[\vert 1s,1/2,1/2\rangle_1\vert 1s,1/2,-1/2\rangle_2-\vert 1s,1/2,-1/2\rangle_1\vert 1s,1/2,1/2\rangle_2\right].
-\]
-!et
-This is also an example of a state which cannot be written out as a pure state. We call this for an entangled state as well.
-
 
 !split
 ===== Maximally entangled =====
@@ -327,6 +297,38 @@ $\rho_B$, that is we have for a given probability distribution $p_i$
 \]
 !et
 
+!split
+===== Entropies and density matrices =====
+
+
+
+!split
+===== Shannon information entropy =====
+
+We start our discussions with the classical information entropy, or
+just Shannon entropy, before we move over to a quantum mechanical way
+to define the entropy based on the density matrices discussed earlier.
+
+We define a set of random variables $X=\{x_0,x_1,\dots,x_{n-1}\}$ with probability for an outcome $x\in X$ given by $p_X(x)$, the
+information entropy is defined as
+!bt
+\[
+S=-\sum_{x\in X}p_X(x)\log_2{p_X(x)}.
+\]
+!et
+
+!split
+===== Von Neumann entropy =====
+
+!bt
+\[
+S=-\mathrm{Tr}[\rho\log_2{\rho}].
+\]
+!et
+
+
+
+
 !split
 ===== Schmidt decomposition =====
 If we cannot write the density matrix in this form, we say the system
@@ -440,36 +442,6 @@ 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 =====
-
-_Note: more details on whiteboard. This material will be added later_
-
-!split
-===== Shannon information entropy =====
-
-We start our discussions with the classical information entropy, or
-just Shannon entropy, before we move over to a quantum mechanical way
-to define the entropy based on the density matrices discussed earlier.
-
-We define a set of random variables $X=\{x_0,x_1,\dots,x_{n-1}\}$ with probability for an outcome $x\in X$ given by $p_X(x)$, the
-information entropy is defined as
-!bt
-\[
-S=-\sum_{x\in X}p_X(x)\log_2{p_X(x)}.
-\]
-!et
-
-!split
-===== Von Neumann entropy =====
-
-!bt
-\[
-S=-\mathrm{Tr}[\rho\log_2{\rho}].
-\]
-!et
-
-
 
 !split
 ===== Two-qubit system and calculation of density matrices and exercise =====