update test

chapters/chap0.tex

@@ -518,9 +518,9 @@ $$
 % 	Gleason's theorem (Theorem 1.1.15 in~\cite{parthasarathy2005mathematical})
 
 %     Let $\mathscr{H}$ be a Hilbert space over $\mathbb{C}$ or $\mathbb{R}$ of dimension $n\geq 3$. Let $\mu$ be a state on the space $\mathscr{P}$ of projections on $\mathscr{H}$. Then there exists a unique density operator $\rho$ such that
-%     \[
+%     $$
 %     \mu(P)=\operatorname{Tr}(\rho P)
-%     \]
+%     $$
 %     for all $P\in\mathscr{P}$. $\mathscr{P}$ is the space of all orthogonal projections on $\mathscr{H}$.
 % \end{theorem}
 
@@ -532,7 +532,7 @@ $$
 
 % This theorem is a very important theorem in non-commutative probability theory; it states that any state on the space of all orthogonal projections on $\mathscr{H}$ can be represented by a density operator.
 
-The counterpart of the random variable in the non-commutative probability theory is called an observable, which is a Hermitian operator on $\mathscr{H}$ (for all $\psi,\phi$ in the domain of $A$, we have $\langle A\psi,\phi\rangle=\langle\psi,A\phi\rangle$. This kind of operator ensures that our outcome interpreted as probability is a real number).
+The counterpart of the random variable in the non-commutative probability theory is called an observable, which is a Hermitian operator on $\mathscr{H}$ (for all $\psi,\phi$ in the domain of $A$, we have $\langle A\psi,\phi\rangle=\langle\psi,A\phi\rangle$. This kind of operator ensures that our outcome interpreted as probability is a real number). 
 
 \begin{defn}
     \label{defn:observable}
@@ -540,30 +540,210 @@ The counterpart of the random variable in the non-commutative probability theory
 
     Let $\mathscr{B}(\mathbb{R})$ be the set of all Borel sets on $\mathbb{R}$.
 
-    A random variable on the Hilbert space $\mathscr{H}$ is a projection-valued map (measure) $P:\mathscr{B}(\mathbb{R})\to\mathscr{P}$.
+    An (real-valued) observable (random variable) on the Hilbert space $\mathscr{H}$, denoted by $A$, is a projection-valued map (measure) $P_A:\mathscr{B}(\mathbb{R})\to\mathscr{P}(\mathscr{H})$.
 
-    With the following properties:
+    Satisfies the following properties:
     \begin{itemize}
-        \item $P(\emptyset)=O$ (the zero projection)
-        \item $P(\mathbb{R})=I$ (the identity projection)
+        \item $P_A(\emptyset)=O$ (the zero projection)
+        \item $P_A(\mathbb{R})=I$ (the identity projection)
         \item For any sequence $A_1,A_2,\cdots,A_n\in \mathscr{B}(\mathbb{R})$, the following holds:
               \begin{itemize}
-                  \item $P(\bigcup_{i=1}^n A_i)=\bigvee_{i=1}^n P(A_i)$
-                  \item $P(\bigcap_{i=1}^n A_i)=\bigwedge_{i=1}^n P(A_i)$
-                  \item $P(A^c)=I-P(A)$
-                  \item If $A_j$ are mutually disjoint (that is $P(A_i)P(A_j)=P(A_j)P(A_i)=O$ for $i\neq j$), then $P(\bigcup_{j=1}^n A_j)=\sum_{j=1}^n P(A_j)$
+                  \item $P_A(\bigcup_{i=1}^n A_i)=\bigvee_{i=1}^n P_A(A_i)$
+                  \item $P_A(\bigcap_{i=1}^n A_i)=\bigwedge_{i=1}^n P_A(A_i)$
+                  \item $P_A(A^c)=I-P_A(A),\forall A\in\mathscr{B}(\mathbb{R})$
               \end{itemize}
     \end{itemize}
 \end{defn}
 
+If $A$ is an observable determined by the map $P_A:\mathcal{B}(\mathbb{R})\to\mathcal{P}(\mathscr{H})$, $P_A$ is a spectral measure (a complete additive orthogonal projection valued measure on $\mathcal{B}(\mathbb{R})$). And every spectral measure can be represented by an observable. \cite{parthasarathy2005mathematical}
+
+\begin{prop}
+    If $A_j$ are mutually disjoint (that is $P_A(A_i)P_A(A_j)=P_A(A_j)P_A(A_i)=O$ for $i\neq j$), then $P_A(\bigcup_{j=1}^n A_j)=\sum_{j=1}^n P_A(A_j)$
+\end{prop}
+
 \begin{defn}
     \label{defn:probability_of_random_variable}
     Probability of a random variable:
 
-    For a system prepared in state $\rho$, the probability that the random variable given by the projection-valued measure $P$ is in the Borel set $A$ is $\operatorname{Tr}(\rho P(A))$.
+    Let $A$ be a real-valued observable on a Hilbert space $\mathscr{H}$. $\rho$ be a state. The probability of observing the outcome $E\in \mathcal{B}(\mathbb{R})$ is given by:
+
+    $$
+    \mu(E)=\operatorname{Tr}(\rho P_A(E))
+    $$
 \end{defn}
 
-When operators commute, we recover classical probability measures.
+Restriction of a quantum state to a commutative subalgebra defines an ordinary probability measure.
+
+\begin{examples}
+Let
+$$
+Z=\begin{pmatrix}
+1 & 0\\
+0 & -1
+\end{pmatrix}.
+$$
+
+The eigenvalues of $Z$ are $+1$ and $-1$, with corresponding normalized eigenvectors
+
+$$
+\ket{0}=\begin{pmatrix}1\\0\end{pmatrix},
+\qquad
+\ket{1}=\begin{pmatrix}0\\1\end{pmatrix}.
+$$
+
+The spectral projections are
+$$
+P_Z(\{1\}) = \ket{0}\bra{0}
+=
+\begin{pmatrix}
+1 & 0\\
+0 & 0
+\end{pmatrix},
+\qquad
+P_Z(\{-1\}) =  \ket{1}\bra{1}
+=
+\begin{pmatrix}
+0 & 0\\
+0 & 1
+\end{pmatrix}.
+$$
+
+The associated projection-valued measure $P_Z$ satisfies
+$$
+P_Z(\{1,-1\}) = I,
+\qquad
+P_Z(\emptyset)=0.
+$$
+
+%==============================
+% 4. Example: X measurement and its PVM
+%==============================
+
+Let
+$$
+X=\begin{pmatrix}
+0 & 1\\
+1 & 0
+\end{pmatrix}.
+$$
+
+The normalized eigenvectors of $X$ are
+$$
+\ket{+}=\frac{1}{\sqrt{2}}\left(\ket{0}+\ket{1}\right),
+\qquad
+\ket{-}=\frac{1}{\sqrt{2}}\left(\ket{0}-\ket{1}\right),
+$$
+with eigenvalues $+1$ and $-1$, respectively.
+
+The corresponding spectral projections are
+$$
+P_X(\{1\}) = \ket{+}\bra{+}
+=
+\frac{1}{2}
+\begin{pmatrix}
+1 & 1\\
+1 & 1
+\end{pmatrix},
+$$
+$$
+P_X(\{-1\}) = \ket{-}\bra{-}
+=
+\frac{1}{2}
+\begin{pmatrix}
+1 & -1\\
+-1 & 1
+\end{pmatrix}.
+$$
+
+%==============================
+% 5. Noncommutativity of the projections
+%==============================
+
+Compute
+$$
+P_Z(\{1\})P_X(\{1\})
+=
+\begin{pmatrix}
+1 & 0\\
+0 & 0
+\end{pmatrix}
+\cdot
+\frac{1}{2}
+\begin{pmatrix}
+1 & 1\\
+1 & 1
+\end{pmatrix}
+=
+\frac{1}{2}
+\begin{pmatrix}
+1 & 1\\
+0 & 0
+\end{pmatrix}.
+$$
+
+On the other hand,
+$$
+P_X(\{1\})P_Z(\{1\})
+=
+\frac{1}{2}
+\begin{pmatrix}
+1 & 1\\
+1 & 1
+\end{pmatrix}
+\cdot
+\begin{pmatrix}
+1 & 0\\
+0 & 0
+\end{pmatrix}
+=
+\frac{1}{2}
+\begin{pmatrix}
+1 & 0\\
+1 & 0
+\end{pmatrix}.
+$$
+
+Since
+$$
+P_Z(\{1\})P_X(\{1\}) \neq P_X(\{1\})P_Z(\{1\}),
+$$
+the projections do not commute.
+
+Let $\rho$ be a density operator on $\mathbb C^2$, i.e.
+$$
+\rho \ge 0,
+\qquad
+\operatorname{Tr}(\rho)=1.
+$$
+
+For a pure state $\ket{\psi}$, one has
+$$
+\rho = \ket{\psi}\bra{\psi}.
+$$
+
+The probability that a measurement associated with a PVM $P$ yields an outcome in a Borel set $A$ is
+$$
+\mathbb P(A) = \operatorname{Tr}(\rho\, P(A)).
+$$
+
+For example, let
+$$
+\rho = \ket{0}\langle 0|
+=
+\begin{pmatrix}
+1 & 0\\
+0 & 0
+\end{pmatrix}.
+$$
+
+Then
+$$
+\operatorname{Tr}\bigl(\rho\, P_Z(\{1\})\bigr) = 1,
+\qquad
+\operatorname{Tr}\bigl(\rho\, P_X(\{1\})\bigr) = \frac{1}{2}.
+$$
+
+\end{examples}
 
 \begin{defn}
     \label{defn:measurement}
@@ -572,14 +752,14 @@ When operators commute, we recover classical probability measures.
     A measurement (observation) of a system prepared in a given state produces an outcome $x$, $x$ is a physical event that is a subset of the set of all possible outcomes. For each $x$, we associate a measurement operator $M_x$ on $\mathscr{H}$.
 
     Given the initial state (pure state, unit vector) $u$, the probability of measurement outcome $x$ is given by:
-    \[
+    $$
         p(x)=\|M_xu\|^2
-    \]
+    $$
 
     Note that to make sense of this definition, the collection of measurement operators $\{M_x\}$ must satisfy the completeness requirement:
-    \[
+    $$
         1=\sum_{x\in X} p(x)=\sum_{x\in X}\|M_xu\|^2=\sum_{x\in X}\langle M_xu,M_xu\rangle=\langle u,(\sum_{x\in X}M_x^*M_x)u\rangle
-    \]
+    $$
     So $\sum_{x\in X}M_x^*M_x=I$.
 
 \end{defn}

chapters/chap1.tex

@@ -505,10 +505,7 @@ Then we have bound for Lipschitz constant $\eta$ of the map $S(\varphi_A): \math
 \end{lemma}
 
 \begin{proof}
-    The proof use lagrange multiplier method to find the maximum of the gradient of $S(\varphi_A)$.
-    %
-    TODO: use lagrange multiplier method to find the maximum of the gradient of $S(\varphi_A)$.
-    %
+    Consider the Lipschitz constant of the function $g:A\otimes B\to \R$ defined as $g(\varphi)=H(M(\varphi_A))$, where $M:A\otimes B\to \mathcal{P}(A)$ is the complete von Neumann measurement and $H: \mathcal{P}(A)\otimes \mathcal{P}(B)\to \R$ is the Shannon entropy.
 \end{proof}
 
 From Levy's lemma, we have