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proper multiplication Let $V = \mathbb{C}^2, W = \mathbb{C}^3$, choose bases $\{\ket{0}, \ket{1}\} \subset V, \{\ket{0}, \ket{1}, \ket{2}\} \subset W$. $$ - v=\begin{pmatrix} + v=\begin{pmatrix} v_1 \\ v_2 \end{pmatrix}=v_1\ket{0}+v_2\ket{1}\in V,w=\begin{pmatrix} @@ -194,72 +194,72 @@ This enables basis-free construction of vector spaces with proper multiplication $$ v\otimes w=\begin{pmatrix} - v_1 w_1 &v_1 w_2 &v_1 w_3 \\ - v_2 w_1 &v_2 w_2 &v_2 w_3 + v_1 w_1 & v_1 w_2 & v_1 w_3 \\ + v_2 w_1 & v_2 w_2 & v_2 w_3 \end{pmatrix}\in \mathbb{C}^6 $$ \end{examples} \begin{examples}[Examples of tensor product for vector spaces] -Let $V = \mathbb{C}^2, W = \mathbb{C}^3$, choose bases $\{\ket{0}, \ket{1}\} \subset V, \{\ket{0}, \ket{1}, \ket{2}\} \subset W.$ + Let $V = \mathbb{C}^2, W = \mathbb{C}^3$, choose bases $\{\ket{0}, \ket{1}\} \subset V, \{\ket{0}, \ket{1}, \ket{2}\} \subset W.$ -Then a basis of the tensor product is -$$ -\{ -\ket{00}, \ket{01}, \ket{02}, -\ket{10}, \ket{11}, \ket{12} -\}, -$$ -where $\ket{ij} := \ket{i}\otimes\ket{j}$. + Then a basis of the tensor product is + $$ + \{ + \ket{00}, \ket{01}, \ket{02}, + \ket{10}, \ket{11}, \ket{12} + \}, + $$ + where $\ket{ij} := \ket{i}\otimes\ket{j}$. -An example element of $V \otimes W$ is -$$ -\ket{\psi} -= -2\,\ket{0}\otimes\ket{1} -+ -(1+i)\,\ket{1}\otimes\ket{0} -- -i\,\ket{1}\otimes\ket{2}. -$$ + An example element of $V \otimes W$ is + $$ + \ket{\psi} + = + 2\,\ket{0}\otimes\ket{1} + + + (1+i)\,\ket{1}\otimes\ket{0} + - + i\,\ket{1}\otimes\ket{2}. + $$ -With respect to the ordered basis -$$ -(\ket{00}, \ket{01}, \ket{02}, \ket{10}, \ket{11}, \ket{12}), -$$ -this tensor corresponds to the coordinate vector -$$ -\ket{\psi} -\;\longleftrightarrow\; -\begin{pmatrix} -0\\ -2\\ -0\\ -1+i\\ -0\\ --i -\end{pmatrix} -\in \mathbb{C}^6. -$$ + With respect to the ordered basis + $$ + (\ket{00}, \ket{01}, \ket{02}, \ket{10}, \ket{11}, \ket{12}), + $$ + this tensor corresponds to the coordinate vector + $$ + \ket{\psi} + \;\longleftrightarrow\; + \begin{pmatrix} + 0 \\ + 2 \\ + 0 \\ + 1+i \\ + 0 \\ + -i + \end{pmatrix} + \in \mathbb{C}^6. + $$ -Using the canonical identification -$$ -\mathbb{C}^2 \otimes \mathbb{C}^3 \cong \mathbb{C}^{2\times 3}, -$$ -where -$$ -\ket{i}\otimes\ket{j} \longmapsto E_{ij}, -$$ -the same tensor is represented by the matrix -$$ -\ket{\psi} -\;\longleftrightarrow\; -\begin{pmatrix} -0 & 2 & 0\\ -1+i & 0 & -i -\end{pmatrix}. -$$ + Using the canonical identification + $$ + \mathbb{C}^2 \otimes \mathbb{C}^3 \cong \mathbb{C}^{2\times 3}, + $$ + where + $$ + \ket{i}\otimes\ket{j} \longmapsto E_{ij}, + $$ + the same tensor is represented by the matrix + $$ + \ket{\psi} + \;\longleftrightarrow\; + \begin{pmatrix} + 0 & 2 & 0 \\ + 1+i & 0 & -i + \end{pmatrix}. + $$ \end{examples} @@ -278,13 +278,13 @@ Partial trace \begin{defn} -\label{defn:trace} + \label{defn:trace} -Let $T$ be a linear operator on $\mathscr{H}$, $(e_1,e_2,\cdots,e_n)$ be a basis of $\mathscr{H}$ and $(\epsilon_1,\epsilon_2,\cdots,\epsilon_n)$ be a basis of dual space $\mathscr{H}^*$. Then the trace of $T$ is defined by + Let $T$ be a linear operator on $\mathscr{H}$, $(e_1,e_2,\cdots,e_n)$ be a basis of $\mathscr{H}$ and $(\epsilon_1,\epsilon_2,\cdots,\epsilon_n)$ be a basis of dual space $\mathscr{H}^*$. Then the trace of $T$ is defined by -$$ -\operatorname{Tr}(T)=\sum_{i=1}^n \epsilon_i(T(e_i))=\sum_{i=1}^n \langle e_i,T(e_i)\rangle -$$ + $$ + \operatorname{Tr}(T)=\sum_{i=1}^n \epsilon_i(T(e_i))=\sum_{i=1}^n \langle e_i,T(e_i)\rangle + $$ \end{defn} @@ -293,27 +293,27 @@ This is equivalent to the sum of the diagonal elements of $T$. \begin{defn} \label{defn:partial_trace} -Let $T$ be a linear operator on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$, where $\mathscr{A}$ and $\mathscr{B}$ are finite-dimensional Hilbert spaces. + Let $T$ be a linear operator on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$, where $\mathscr{A}$ and $\mathscr{B}$ are finite-dimensional Hilbert spaces. -An operator $T$ on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$ can be written as + An operator $T$ on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$ can be written as -$$ -T=\sum_{i=1}^n a_i A_i\otimes B_i -$$ + $$ + T=\sum_{i=1}^n a_i A_i\otimes B_i + $$ -where $A_i$ is a linear operator on $\mathscr{A}$ and $B_i$ is a linear operator on $\mathscr{B}$. + where $A_i$ is a linear operator on $\mathscr{A}$ and $B_i$ is a linear operator on $\mathscr{B}$. -The $\mathscr{B}$-partial trace of $T$ ($\operatorname{Tr}_{\mathscr{B}}(T):\mathcal{L}(\mathscr{A}\otimes \mathscr{B})\to \mathcal{L}(\mathscr{A})$) is the linear operator on $\mathscr{A}$ defined by + The $\mathscr{B}$-partial trace of $T$ ($\operatorname{Tr}_{\mathscr{B}}(T):\mathcal{L}(\mathscr{A}\otimes \mathscr{B})\to \mathcal{L}(\mathscr{A})$) is the linear operator on $\mathscr{A}$ defined by -$$ -\operatorname{Tr}_{\mathscr{B}}(T)=\sum_{i=1}^n a_i \operatorname{Tr}(B_i) A_i -$$ + $$ + \operatorname{Tr}_{\mathscr{B}}(T)=\sum_{i=1}^n a_i \operatorname{Tr}(B_i) A_i + $$ \end{defn} Or we can define the map $L_v: \mathscr{A}\to \mathscr{A}\otimes \mathscr{B}$ by $$ -L_v(u)=u\otimes v + L_v(u)=u\otimes v $$ Note that $\langle u,L_v^*(u')\otimes v'\rangle=\langle u,u'\rangle \langle v,v'\rangle=\langle u\otimes v,u'\otimes v'\rangle=\langle L_v(u),u'\otimes v'\rangle$. @@ -325,21 +325,21 @@ Then the partial trace of $T$ can also be defined by Let $\{v_j\}$ be a set of orthonormal basis of $\mathscr{B}$. $$ -\operatorname{Tr}_{\mathscr{B}}(T)=\sum_{j} L^*_{v_j}(T)L_{v_j} + \operatorname{Tr}_{\mathscr{B}}(T)=\sum_{j} L^*_{v_j}(T)L_{v_j} $$ \begin{defn} \label{defn:partial_trace_with_respect_to_state} -Let $T$ be a linear operator on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$, where $\mathscr{A}$ and $\mathscr{B}$ are finite-dimensional Hilbert spaces. + Let $T$ be a linear operator on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$, where $\mathscr{A}$ and $\mathscr{B}$ are finite-dimensional Hilbert spaces. -Let $\rho$ be a state on $\mathscr{B}$ consisting of orthonormal basis $\{v_j\}$ and eigenvalue $\{\lambda_j\}$. + Let $\rho$ be a state on $\mathscr{B}$ consisting of orthonormal basis $\{v_j\}$ and eigenvalue $\{\lambda_j\}$. -The partial trace of $T$ with respect to $\rho$ is the linear operator on $\mathscr{A}$ defined by + The partial trace of $T$ with respect to $\rho$ is the linear operator on $\mathscr{A}$ defined by -$$ -\operatorname{Tr}_{\mathscr{A}}(T)=\sum_{j} \lambda_j L^*_{v_j}(T)L_{v_j} -$$ + $$ + \operatorname{Tr}_{\mathscr{A}}(T)=\sum_{j} \lambda_j L^*_{v_j}(T)L_{v_j} + $$ \end{defn} @@ -362,12 +362,12 @@ First, we define the Hilbert space in case one did not make the step from the li That is, a vector space equipped with an inner product, with the induced metric defined by the norm of the inner product, we have a metric space, which is complete. Reminds that complete mean that every Cauchy sequence, the sequence such that for any $\epsilon>0$, there exists an $N$ such that for all $m,n\geq N$, we have $|x_m-x_n|<\epsilon$, converges to a limit. -As a side note we will use later, we also defined the Borel measure on a space, here we use the following definition specialized for the space (manifolds) we are interested in. +As a side note we will use later, we also defined the Borel measure on a space, here we use the following definition specialized for the space (manifolds) we are interested in. \begin{defn} \label{defn:Borel_measure} Borel measure: - + Let $X$ be a topological space, then a Borel measure $\mu:\mathscr{B}(X)\to [0,\infty]$ on $X$ is a measure on the Borel $\sigma$-algebra of $X$ $\mathscr{B}(X)$ satisfying the following properties: \begin{enumerate} @@ -381,46 +381,46 @@ In later sections, we will use Lebesgue measure, and Haar measure for various ci \begin{examples} -To introduce an example of Hilbert space we use when studying quantum mechanics, we need to introduce a common inner product used in $\mathbb{C}^n$. + To introduce an example of Hilbert space we use when studying quantum mechanics, we need to introduce a common inner product used in $\mathbb{C}^n$. -\begin{prop} - \label{prop:Hermitian_inner_product_with_complex_vectorspace} - The Hermitian inner product on the complex vector space $\C^n$ makes it a Hilbert space. -\end{prop} + \begin{prop} + \label{prop:Hermitian_inner_product_with_complex_vectorspace} + The Hermitian inner product on the complex vector space $\C^n$ makes it a Hilbert space. + \end{prop} -\begin{proof} - We first verify that the Hermitian inner product - $$ - \langle u,v\rangle = \sum_{i=1}^n \overline{u_i} v_i - $$ - on $\C^n$ satisfies the axioms of an inner product: - \begin{enumerate} - \item \textbf{Conjugate symmetry:} For all $u,v\in\C^n$, - $$ - \langle u,v\rangle =\sum_{i=1}^n \overline{u_i} v_i=\overline{\sum_{i=1}^n \overline{v_i} u_i}=\overline{\langle v,u\rangle}. - $$ - \item \textbf{Linearity:} For any $u,v,w\in\C^n$ and scalars $a,b\in\C$, we have - $$ - \langle u, av + bw\rangle = \sum_{i=1}^n \overline{u_i} (av_i + bw_i)=a\langle u,v\rangle + b\langle u,w\rangle. - $$ - \item \textbf{Positive definiteness:} For every $u=(u_1,u_2,\cdots,u_n)\in\C^n$, let $u_j=a_j+b_ji$, where $a_j,b_j\in\mathbb{R}$. - $$ - \langle u,u\rangle = \sum_{j=1}^n \overline{u_j} u_j=\sum_{i=1}^n (a_i^2+b_i^2)\geq 0, - $$ - with equality if and only if $u=0$. + \begin{proof} + We first verify that the Hermitian inner product + $$ + \langle u,v\rangle = \sum_{i=1}^n \overline{u_i} v_i + $$ + on $\C^n$ satisfies the axioms of an inner product: + \begin{enumerate} + \item \textbf{Conjugate symmetry:} For all $u,v\in\C^n$, + $$ + \langle u,v\rangle =\sum_{i=1}^n \overline{u_i} v_i=\overline{\sum_{i=1}^n \overline{v_i} u_i}=\overline{\langle v,u\rangle}. + $$ + \item \textbf{Linearity:} For any $u,v,w\in\C^n$ and scalars $a,b\in\C$, we have + $$ + \langle u, av + bw\rangle = \sum_{i=1}^n \overline{u_i} (av_i + bw_i)=a\langle u,v\rangle + b\langle u,w\rangle. + $$ + \item \textbf{Positive definiteness:} For every $u=(u_1,u_2,\cdots,u_n)\in\C^n$, let $u_j=a_j+b_ji$, where $a_j,b_j\in\mathbb{R}$. + $$ + \langle u,u\rangle = \sum_{j=1}^n \overline{u_j} u_j=\sum_{i=1}^n (a_i^2+b_i^2)\geq 0, + $$ + with equality if and only if $u=0$. - Therefore, the Hermitian inner product is an inner product. - \end{enumerate} + Therefore, the Hermitian inner product is an inner product. + \end{enumerate} - Next, we show that $\C^n$ is complete with respect to the norm induced by this inner product: - $$ - \|u\| = \sqrt{\langle u,u\rangle}. - $$ - Since $\C^n$ is finite-dimensional, every Cauchy sequence (with respect to any norm) converges in $\C^n$. This is a standard result in finite-dimensional normed spaces, which implies that $\C^n$ is indeed complete. + Next, we show that $\C^n$ is complete with respect to the norm induced by this inner product: + $$ + \|u\| = \sqrt{\langle u,u\rangle}. + $$ + Since $\C^n$ is finite-dimensional, every Cauchy sequence (with respect to any norm) converges in $\C^n$. This is a standard result in finite-dimensional normed spaces, which implies that $\C^n$ is indeed complete. - Therefore, since the Hermitian inner product fulfills the inner product axioms and $\C^n$ is complete, the complex vector space $\C^n$ with the Hermitian inner product is a Hilbert space. -\end{proof} + Therefore, since the Hermitian inner product fulfills the inner product axioms and $\C^n$ is complete, the complex vector space $\C^n$ with the Hermitian inner product is a Hilbert space. + \end{proof} \end{examples} @@ -440,44 +440,44 @@ Another classical example of Hilbert space is $L^2(\Omega, \mathscr{F}, P)$, whe \end{enumerate} \begin{examples} - -\begin{prop} - \label{prop:L2_space_is_a_Hilbert_space} - $L^2(\Omega, \mathscr{F}, P)$ is a Hilbert space. -\end{prop} -\begin{proof} - We check the two conditions of the Hilbert space: - \begin{itemize} - \item Completeness: - Let $(f_n)$ be a Cauchy sequence in $L^2(\Omega, \mathscr{F}, P)$. Then for any $\epsilon>0$, there exists an $N$ such that for all $m,n\geq N$, we have - $$ - \int_\Omega |f_m(\omega)-f_n(\omega)|^2 dP(\omega)<\epsilon^2 - $$ - This means that $(f_n)$ is a Cauchy sequence in the norm of $L^2(\Omega, \mathscr{F}, P)$. - \item Inner product: - The inner product is defined by - $$ - \langle f,g\rangle=\int_\Omega \overline{f(\omega)}g(\omega)dP(\omega) - $$ - This is a well-defined inner product on $L^2(\Omega, \mathscr{F}, P)$. We can check the properties of the inner product: - \begin{itemize} - \item Linearity: - $$ - \langle af+bg,h\rangle=a\langle f,h\rangle+b\langle g,h\rangle - $$ - \item Conjugate symmetry: - $$ - \langle f,g\rangle=\overline{\langle g,f\rangle} - $$ - \item Positive definiteness: - $$ - \langle f,f\rangle\geq 0 - $$ - \end{itemize} - \end{itemize} -\end{proof} + \begin{prop} + \label{prop:L2_space_is_a_Hilbert_space} + $L^2(\Omega, \mathscr{F}, P)$ is a Hilbert space. + \end{prop} + + \begin{proof} + We check the two conditions of the Hilbert space: + \begin{itemize} + \item Completeness: + Let $(f_n)$ be a Cauchy sequence in $L^2(\Omega, \mathscr{F}, P)$. Then for any $\epsilon>0$, there exists an $N$ such that for all $m,n\geq N$, we have + $$ + \int_\Omega |f_m(\omega)-f_n(\omega)|^2 dP(\omega)<\epsilon^2 + $$ + This means that $(f_n)$ is a Cauchy sequence in the norm of $L^2(\Omega, \mathscr{F}, P)$. + \item Inner product: + The inner product is defined by + $$ + \langle f,g\rangle=\int_\Omega \overline{f(\omega)}g(\omega)dP(\omega) + $$ + This is a well-defined inner product on $L^2(\Omega, \mathscr{F}, P)$. We can check the properties of the inner product: + \begin{itemize} + \item Linearity: + $$ + \langle af+bg,h\rangle=a\langle f,h\rangle+b\langle g,h\rangle + $$ + \item Conjugate symmetry: + $$ + \langle f,g\rangle=\overline{\langle g,f\rangle} + $$ + \item Positive definiteness: + $$ + \langle f,f\rangle\geq 0 + $$ + \end{itemize} + \end{itemize} + \end{proof} \end{examples} @@ -508,8 +508,8 @@ Recall from classical probability theory, we call the initial probability distri Given a non-commutative probability space $(\mathscr{B}(\mathscr{H}),\mathscr{P})$, - A state is a unit vector $\ket{\psi}$ in the Hilbert space $\mathscr{H}$, such that $\bra{\psi}\ket{\psi}=1$. - + A state is a unit vector $\ket{\psi}$ in the Hilbert space $\mathscr{H}$, such that $\bra{\psi}\ket{\psi}=1$. + Every state uniquely defines a map $\rho:\mathscr{P}\to[0,1]$, $\rho(P)=\bra{\psi}P\ket{\psi}$ (commonly named as density operator) such that: \begin{itemize} \item $\rho(O)=0$, where $O$ is the zero projection, and $\rho(I)=1$, where $I$ is the identity projection. @@ -548,7 +548,7 @@ $$ % 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} @@ -584,180 +584,180 @@ If $A$ is an observable determined by the map $P_A:\mathcal{B}(\mathbb{R})\to\ma 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: $$ - \mu(E)=\operatorname{Tr}(\rho P_A(E)) + \mu(E)=\operatorname{Tr}(\rho P_A(E)) $$ \end{defn} Restriction of a quantum state to a commutative subalgebra defines an ordinary probability measure. \begin{examples} -Let -$$ -Z=\begin{pmatrix} -1 & 0\\ -0 & -1 -\end{pmatrix}. -$$ + Let + $$ + Z=\begin{pmatrix} + 1 & 0 \\ + 0 & -1 + \end{pmatrix}. + $$ -The eigenvalues of $Z$ are $+1$ and $-1$, with corresponding normalized eigenvectors + The eigenvalues of $Z$ are $+1$ and $-1$, with corresponding normalized eigenvectors -$$ -\ket{0}=\begin{pmatrix}1\\0\end{pmatrix}, -\qquad -\ket{1}=\begin{pmatrix}0\\1\end{pmatrix}. -$$ + $$ + \ket{0}=\begin{pmatrix}1\\0\end{pmatrix}, + \qquad + \ket{1}=\begin{pmatrix}0\\1\end{pmatrix}. + $$ -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 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 -$$ -P_Z(\{1,-1\}) = I, -\qquad -P_Z(\emptyset)=0. -$$ + 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 -%============================== + %============================== + % 4. Example: X measurement and its PVM + %============================== -Let -$$ -X=\begin{pmatrix} -0 & 1\\ -1 & 0 -\end{pmatrix}. -$$ + 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 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 -$$ -P_X(\{1\}) = \ket{+}\bra{+} -= -\frac{1}{2} -\begin{pmatrix} -1 & 1\\ -1 & 1 -\end{pmatrix}, -$$ -$$ -P_X(\{-1\}) = \ket{-}\bra{-} -= -\frac{1}{2} -\begin{pmatrix} -1 & -1\\ --1 & 1 -\end{pmatrix}. -$$ + 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 -%============================== + %============================== + % 5. Noncommutativity of the projections + %============================== -Compute -$$ -P_Z(\{1\})P_X(\{1\}) -= -\begin{pmatrix} -1 & 0\\ -0 & 0 -\end{pmatrix} -\cdot -\frac{1}{2} -\begin{pmatrix} -1 & 1\\ -1 & 1 -\end{pmatrix} -= -\frac{1}{2} -\begin{pmatrix} -1 & 1\\ -0 & 0 -\end{pmatrix}. -$$ + 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, -$$ -P_X(\{1\})P_Z(\{1\}) -= -\frac{1}{2} -\begin{pmatrix} -1 & 1\\ -1 & 1 -\end{pmatrix} -\cdot -\begin{pmatrix} -1 & 0\\ -0 & 0 -\end{pmatrix} -= -\frac{1}{2} -\begin{pmatrix} -1 & 0\\ -1 & 0 -\end{pmatrix}. -$$ + 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. + 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. -$$ -\rho \ge 0, -\qquad -\operatorname{Tr}(\rho)=1. -$$ + 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 -$$ -\rho = \ket{\psi}\bra{\psi}. -$$ + 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 $A\in \mathcal{B}$ is -$$ -\mathbb P(A) = \operatorname{Tr}(\rho\, P(A)). -$$ + The probability that a measurement associated with a PVM $P$ yields an outcome in a Borel set $A\in \mathcal{B}$ is + $$ + \mathbb P(A) = \operatorname{Tr}(\rho\, P(A)). + $$ -For example, let -$$ -\rho = \ket{0}\langle 0| -= -\begin{pmatrix} -1 & 0\\ -0 & 0 -\end{pmatrix}. -$$ + For example, let + $$ + \rho = \ket{0}\langle 0| + = + \begin{pmatrix} + 1 & 0 \\ + 0 & 0 + \end{pmatrix}. + $$ -Then -$$ -\operatorname{Tr}\bigl(\rho\, P_Z(\{1\})\bigr) = 1, -\qquad -\operatorname{Tr}\bigl(\rho\, P_X(\{1\})\bigr) = \frac{1}{2}. -$$ + Then + $$ + \operatorname{Tr}\bigl(\rho\, P_Z(\{1\})\bigr) = 1, + \qquad + \operatorname{Tr}\bigl(\rho\, P_X(\{1\})\bigr) = \frac{1}{2}. + $$ \end{examples} @@ -841,13 +841,13 @@ In this section, we will introduce some basic definitions and theorems used in m \begin{defn} \label{defn:m-manifold} -An $m$-manifold is a Topological space $X$ that is + An $m$-manifold is a Topological space $X$ that is -\begin{enumerate} - \item Hausdroff: every distinct two points in $X$ can be separated by two disjoint open sets. - \item Second countable: $X$ has countable basis. - \item Every point $p$ has an open neighborhood $p\in U$ that is homeomorphic to an open subset of $\mathbb{R}^m$. -\end{enumerate} + \begin{enumerate} + \item Hausdroff: every distinct two points in $X$ can be separated by two disjoint open sets. + \item Second countable: $X$ has countable basis. + \item Every point $p$ has an open neighborhood $p\in U$ that is homeomorphic to an open subset of $\mathbb{R}^m$. + \end{enumerate} \end{defn} @@ -864,15 +864,102 @@ An $m$-manifold is a Topological space $X$ that is \begin{examples} \label{example:manifold} -1-manifold is a curve and 2-manifold is a surface. + 1-manifold is a curve and 2-manifold is a surface. \end{examples} \begin{theorem} \label{Theorem of imbedded space} + + Whithney's Embedding Theorem: + If $X$ is a compact $m$-manifold, then $X$ can be imbedded in $\mathbb{R}^n$ for some $n$. \end{theorem} -This theorem might save you from imagining abstract structures back to real dimension. Good news, at least you stay in some real numbers. +This proof is from topology course, and use additional one lemma: + +\begin{lemma} + \label{lemma:finite_partition_of_unity} + + Let $\{U_i\}_{i=1}^n$ be a finite open cover of a normal space $X$ (Every pair of closed sets in $X$ can be separated by two open sets in $X$). + +Then there exists a partition of unity dominated by $\{U_i\}_{i=1}^n$. +\end{lemma} + +\begin{proof} + +Since $X$ is a $m$ compact manifold, $\forall x\in X$, there is an open neighborhood $U_x$ of $x$ such that $U_x$ is homeomorphic to $\mathbb{R}^m$. That means there exists $\varphi_i:U_x\to \varphi(U_x)\subseteq \mathbb{R}^m$. + +Where $\{U_x\}_{x\in X}$ is an open cover of $X$. Since $X$ is compact, there is a finite subcover $\bigcup_{i=1}^k U_{x_i}=X$. + +Apply the existence of a finite partition of unity, we can find a partition of unity dominated by $\{U_{x_i}\}_{i=1}^k$. With family of functions $\phi_i:\mathbb{R}^d\to[0,1]$. + +Define $h_i:X\to \mathbb{R}^m$ by + +$$ +h_i(x)=\begin{cases} +\phi_i(x)\varphi_i(x) & \text{if }x=x_i\\ +0 & \text{otherwise} +\end{cases} +$$ + +We claim that $h_i$ is continuous using pasting lemma. + +On $U_i$, $h_i=\phi_i\varphi_i$ is product of two continuous functions therefore continuous. + +On $X-\operatorname{supp}(\phi_i)$, $h_i=0$ is continuous. + +By pasting lemma, $h_i$ is continuous. + +Define + +$$ +F: X\to (\mathbb{R}^m\times \mathbb{R})^n +$$ + +where $x\mapsto (h_1(x),\varphi_1(x),h_2(x),\varphi_2(x),\dots,h_n(x),\varphi_n(x))$ + +We want to show that $F$ is imbedding map. + +\begin{enumerate} + + \item $F$ is continuous + + +since it is a product of continuous functions. + +\item $F$ is injective + +that is, if $F(x_1)=F(x_2)$, then $x_1=x_2$. + +By partition of unity, we have, + +$h_1(x_1)=h_1(x_2), h_2(x_1)=h_2(x_2), \dots, h_n(x_1)=h_n(x_2)$. + +And $\varphi_1(x_1)=\varphi_1(x_2), \varphi_2(x_1)=\varphi_2(x_2), \dots, \varphi_n(x_1)=\varphi_n(x_2)$. + +Because $\sum_{i=1}^n \varphi_i(x_1)=1$, therefore the exists $\varphi_i(x_1)=\varphi_i(x_2)>0$. + +Therefore $x1,x_2\in \operatorname{supp}(\phi_i)\subseteq U_i$. + +By definition of $h$, $h_i(x_1)=h_i(x_2)$, $\varphi_i(x_1)\phi_i(x_1)=\varphi_i(x_2)\phi_i(x_2)$. + +Using cancellation, $\phi_i(x_1)=\phi_i(x_2)$. + +Therefore $x_1=x_2$ since $\phi_i(x_1)=\phi_i(x_2)$ is a homeomorphism. + +\textit{In this proof, $\varphi$ ensures the imbedding is properly defined on the open sets} + +\item $F$ is a homeomorphism. + +Note that if $f:X\to Y$ is continuous and $X$ is compact, $Y$ is Hausdorff, then $f$ is a closed map. + +$F:X\to F(X)$ is a bijective map from a compact space to a Hausdorff space, therefore $F$ is a closed map. + +Since $F$ is continuous, then $F^{-1}(C)$ where $C$ is a closed set in $F(X)$, $F^{-1}(C)$ is closed in $X$. + +Therefore $F$ is a homeomorphism. +\end{enumerate} +\end{proof} \subsection{Smooth manifolds and Lie groups} @@ -886,11 +973,11 @@ This section is adopted from \cite{lee_introduction_2012} For any $a=(a_1,\cdots,a_n)\in U$, $j\in \{1,\cdots,n\}$, the $j$-th partial derivative of $F$ at $a$ is defined as $$ - \begin{aligned} - \frac{\partial f}{\partial x_j}(a)&=\lim_{h\to 0}\frac{f(a_1,\cdots,a_j+h,\cdots,a_n)-f(a_1,\cdots,a_j,\cdots,a_n)}{h} \\ - &=\lim_{h\to 0}\frac{f(a+he_j)-f(a)}{h} - \end{aligned} -$$ + \begin{aligned} + \frac{\partial f}{\partial x_j}(a) & =\lim_{h\to 0}\frac{f(a_1,\cdots,a_j+h,\cdots,a_n)-f(a_1,\cdots,a_j,\cdots,a_n)}{h} \\ + & =\lim_{h\to 0}\frac{f(a+he_j)-f(a)}{h} + \end{aligned} + $$ \end{defn} @@ -901,31 +988,31 @@ $$ If for any $j\in \{1,\cdots,n\}$, the $j$-th partial derivative of $f$ is continuous at $a$, then $f$ is continuously differentiable at $a$. If $\forall a\in U$, $\frac{\partial f}{\partial x_j}$ exists and is continuous at $a$, then $f$ is continuously differentiable on $U$. or $C^1$ map. (Note that $C^0$ map is just a continuous map.) -\end{defn} +\end{defn} \begin{defn} \label{defn:smooth_map} - A function $f:U\to \mathbb{R}^n$ is smooth if it is of class $C^k$ for every $k\geq 0$ on $U$. Such function is called a diffeomorphism if it is also a \textbf{\texttt{bijection}} and its \textbf{\texttt{inverse is also smooth}}. + A function $f:U\to \mathbb{R}^n$ is smooth if it is of class $C^k$ for every $k\geq 0$ on $U$. Such function is called a diffeomorphism if it is also a \textbf{bijection} and its \textbf{inverse is also smooth}. \end{defn} \begin{defn} \label{defn:chart} - Let $M$ be a smooth manifold. A \textbf{\texttt{chart}} is a pair $(U,\varphi)$ where $U\subseteq M$ is an open subset and $\varphi:U\to \hat{U}\subseteq \mathbb{R}^n$ is a homeomorphism (a continuous bijection map and its inverse is also continuous). + Let $M$ be a smooth manifold. A \textbf{chart} is a pair $(U,\varphi)$ where $U\subseteq M$ is an open subset and $\varphi:U\to \hat{U}\subseteq \mathbb{R}^n$ is a homeomorphism (a continuous bijection map and its inverse is also continuous). If $p\in U$ and $\varphi(p)=0$, then we say that $p$ is the origin of the chart $(U,\varphi)$. - For $p\in U$, we note that the continuous function $\varphi(p)=(x_1(p),\cdots,x_n(p))$ gives a vector in $\mathbb{R}^n$. The $(x_1(p),\cdots,x_n(p))$ is called the \textbf{\texttt{local coordinates}} of $p$ in the chart $(U,\varphi)$. + For $p\in U$, we note that the continuous function $\varphi(p)=(x_1(p),\cdots,x_n(p))$ gives a vector in $\mathbb{R}^n$. The $(x_1(p),\cdots,x_n(p))$ is called the \textbf{local coordinates} of $p$ in the chart $(U,\varphi)$. \end{defn} \begin{defn} \label{defn:atlas} - Let $M$ be a smooth manifold. An \textbf{\texttt{atlas}} is a collection of charts $\mathcal{A}=\{(U_\alpha,\phi_\alpha)\}_{\alpha\in I}$ such that $M=\bigcup_{\alpha\in I} U_\alpha$. + Let $M$ be a smooth manifold. An \textbf{atlas} is a collection of charts $\mathcal{A}=\{(U_\alpha,\phi_\alpha)\}_{\alpha\in I}$ such that $M=\bigcup_{\alpha\in I} U_\alpha$. - An atlas is said to be \textbf{\texttt{smooth}} if the transition maps $\phi_\alpha\circ \phi_\beta^{-1}:\phi_\beta(U_\alpha\cap U_\beta)\to \phi_\alpha(U_\alpha\cap U_\beta)$ are smooth for all $\alpha, \beta\in I$. + An atlas is said to be \textbf{smooth} if the transition maps $\phi_\alpha\circ \phi_\beta^{-1}:\phi_\beta(U_\alpha\cap U_\beta)\to \phi_\alpha(U_\alpha\cap U_\beta)$ are smooth for all $\alpha, \beta\in I$. \end{defn} @@ -935,13 +1022,18 @@ $$ \end{defn} \begin{defn} - \label{defn:differential} - + \label{defn:differential} + Let $M$ and $N$ be smooth manifolds, and $f:M\to N$ be a smooth map. For each $p\in M$, the \textbf{differential} of $f$ at $p$ is the linear map + $$ + df_p:T_pM\to T_{f(p)}N + $$ \end{defn} \begin{defn} - \label{defn:smooth-submersion} - + \label{defn:smooth-submersion} + A smooth map $f:M\to N$ is a \textbf{smooth submersion} if for each $p\in M$, the differential $F:M\to N$ is surjective. + + Or equivalently $\operatorname{rank}(F)=\dim N$ for each $p\in M$. \end{defn} Here are some additional propositions that will be helpful for our study in later sections: @@ -951,153 +1043,205 @@ This one is from \cite{lee_introduction_2012} Theorem 4.26 \begin{theorem} \label{theorem:local_section_theorem} - Let $M$ and $N$ be smooth manifolds and $\pi:M\to N$ is a smooth map. Then $\pi$ is a smooth submersion if and only if every point of $M$ is in the image of a smooth local section of $\pi$ (a local section of $\pi$ is a map $\sigma:U\to M$ defined on some open subset $U\subseteq N$ with $\pi\circ \sigma=Id_U$). + Let $M$ and $N$ be smooth manifolds and $\pi:M\to N$ is a smooth map. Then $\pi$ is a smooth submersion if and only if every point of $M$ is in the image of a smooth local section of $\pi$ (a local section of $\pi$ is a map $\sigma:U\to M$ defined on some open subset $U\subseteq N$ with $\pi\circ \sigma=Id_U$). \end{theorem} \subsection{Riemannian manifolds} \begin{defn} - \label{defn:riemannian-metric} + \label{defn:riemannian-metric} - Let $M$ be a smooth manifold. A \textit{\textbf{Riemannian metric}} on $M$ is a smooth covariant tensor field $g\in \mathcal{T}^2(M)$ such that for each $p\in M$, $g_p$ is an inner product on $T_pM$. + Let $M$ be a smooth manifold. A \textit{\textbf{Riemannian metric}} on $M$ is a smooth covariant tensor field $g\in \mathcal{T}^2(M)$ such that for each $p\in M$, $g_p$ is an inner product on $T_pM$. - $g_p(v,v)\geq 0$ for each $p\in M$ and each $v\in T_pM$. equality holds if and only if $v=0$. + $g_p(v,v)\geq 0$ for each $p\in M$ and each $v\in T_pM$. equality holds if and only if $v=0$. \end{defn} \begin{defn} - \label{defn:riemannian-submersion} - Suppose $(\tilde{M},\tilde{g})$ and $(M,g)$ are smooth Riemannian manifolds, and $\pi:\tilde{M}\to M$ is a smooth submersion. Then $\pi$ is said to be a \textit{\textbf{Riemannian submersion}} if for each $x\in \tilde{M}$, the differential $d\pi_x:\tilde{g}_x\to g_{\pi(x)}$ restricts to a linear isometry from $H_x$ onto $T_{\pi(x)}M$. + \label{defn:riemannian-submersion} + Suppose $(\tilde{M},\tilde{g})$ and $(M,g)$ are smooth Riemannian manifolds, and $\pi:\tilde{M}\to M$ is a smooth submersion. Then $\pi$ is said to be a \textit{\textbf{Riemannian submersion}} if for each $x\in \tilde{M}$, the differential $d\pi_x:\tilde{g}_x\to g_{\pi(x)}$ restricts to a linear isometry from $H_x$ onto $T_{\pi(x)}M$. - In other words, $\tilde{g}_x(v,w)=g_{\pi(x)}(d\pi_x(v),d\pi_x(w))$ whenever $v,w\in H_x$. + In other words, $\tilde{g}_x(v,w)=g_{\pi(x)}(d\pi_x(v),d\pi_x(w))$ whenever $v,w\in H_x$. \end{defn} \begin{theorem} - \label{theorem:riemannian-submersion} + \label{theorem:riemannian-submersion} - Let $(\tilde{M},\tilde{g})$ be a Riemannian manifold, let $\pi:\tilde{M}\to M$ be a surjective smooth submersion, and let $G$ be a group acting on $\tilde{M}$. If the \textbf{action} is - \begin{enumerate} - \item isometric: the map $x\mapsto \varphi\cdot x$ is an isometry for each $\varphi\in G$. - \item vertical: every element $\varphi\in G$ takes each fiber to itself, that is $\pi(\varphi\cdot p)=\pi(p)$ for all $p\in \tilde{M}$. - \item transitive on fibers: for each $p,q\in \tilde{M}$ such that $\pi(p)=\pi(q)$, there exists $\varphi\in G$ such that $\varphi\cdot p = q$. - \end{enumerate} - Then there is a unique Riemannian metric on $M$ such that $\pi$ is a Riemannian submersion. + Let $(\tilde{M},\tilde{g})$ be a Riemannian manifold, let $\pi:\tilde{M}\to M$ be a surjective smooth submersion, and let $G$ be a group acting on $\tilde{M}$. If the \textbf{action} is + \begin{enumerate} + \item isometric: the map $x\mapsto \varphi\cdot x$ is an isometry for each $\varphi\in G$. + \item vertical: every element $\varphi\in G$ takes each fiber to itself, that is $\pi(\varphi\cdot p)=\pi(p)$ for all $p\in \tilde{M}$. + \item transitive on fibers: for each $p,q\in \tilde{M}$ such that $\pi(p)=\pi(q)$, there exists $\varphi\in G$ such that $\varphi\cdot p = q$. + \end{enumerate} + Then there is a unique Riemannian metric on $M$ such that $\pi$ is a Riemannian submersion. \end{theorem} \begin{proof} -For each $p\in \tilde{M}$, let -$$ -V_p:=\ker(d\pi_p)\subseteq T_p\tilde{M} -$$ -be the vertical space, and let -$$ -H_p:=V_p^{\perp_{\tilde g}} -$$ -be its $\tilde g$-orthogonal complement. Since $\pi$ is a surjective smooth submersion, each $d\pi_p:T_p\tilde M\to T_{\pi(p)}M$ is surjective, so -$$ -T_p\tilde M = V_p\oplus H_p, -$$ -and therefore the restriction -$$ -d\pi_p|_{H_p}:H_p\to T_{\pi(p)}M -$$ -is a linear isomorphism. + For each $p\in \tilde{M}$, let + $$ + V_p:=\ker(d\pi_p)\subseteq T_p\tilde{M} + $$ + be the vertical space, and let + $$ + H_p:=V_p^{\perp_{\tilde g}} + $$ + be its $\tilde g$-orthogonal complement. Since $\pi$ is a surjective smooth submersion, each $d\pi_p:T_p\tilde M\to T_{\pi(p)}M$ is surjective, so + $$ + T_p\tilde M = V_p\oplus H_p, + $$ + and therefore the restriction + $$ + d\pi_p|_{H_p}:H_p\to T_{\pi(p)}M + $$ + is a linear isomorphism. -We first show that the group action preserves the horizontal distribution. Fix $\varphi\in G$. Since the action is vertical, we have -$$ -\pi(\varphi\cdot x)=\pi(x)\qquad\text{for all }x\in \tilde M. -$$ -Differentiating at $p$ gives -$$ -d\pi_{\varphi\cdot p}\circ d\varphi_p = d\pi_p. -$$ -Hence if $v\in V_p=\ker(d\pi_p)$, then -$$ -d\pi_{\varphi\cdot p}(d\varphi_p v)=d\pi_p(v)=0, -$$ -so $d\varphi_p(V_p)\subseteq V_{\varphi\cdot p}$. Since $\varphi$ acts isometrically, $d\varphi_p$ is a linear isometry, and thus it preserves orthogonal complements. Therefore -$$ -d\varphi_p(H_p)=H_{\varphi\cdot p}. -$$ + We first show that the group action preserves the horizontal distribution. Fix $\varphi\in G$. Since the action is vertical, we have + $$ + \pi(\varphi\cdot x)=\pi(x)\qquad\text{for all }x\in \tilde M. + $$ + Differentiating at $p$ gives + $$ + d\pi_{\varphi\cdot p}\circ d\varphi_p = d\pi_p. + $$ + Hence if $v\in V_p=\ker(d\pi_p)$, then + $$ + d\pi_{\varphi\cdot p}(d\varphi_p v)=d\pi_p(v)=0, + $$ + so $d\varphi_p(V_p)\subseteq V_{\varphi\cdot p}$. Since $\varphi$ acts isometrically, $d\varphi_p$ is a linear isometry, and thus it preserves orthogonal complements. Therefore + $$ + d\varphi_p(H_p)=H_{\varphi\cdot p}. + $$ -We now define a metric on $M$. Let $m\in M$, and choose any $p\in \pi^{-1}(m)$. For $u,v\in T_mM$, let $\tilde u,\tilde v\in H_p$ be the unique horizontal lifts satisfying -$$ -d\pi_p(\tilde u)=u,\qquad d\pi_p(\tilde v)=v. -$$ -Define -$$ -g_m(u,v):=\tilde g_p(\tilde u,\tilde v). -$$ -This is a symmetric bilinear form on $T_mM$, and it is positive definite because $\tilde g_p$ is positive definite on $H_p$ and $d\pi_p|_{H_p}$ is an isomorphism. + We now define a metric on $M$. Let $m\in M$, and choose any $p\in \pi^{-1}(m)$. For $u,v\in T_mM$, let $\tilde u,\tilde v\in H_p$ be the unique horizontal lifts satisfying + $$ + d\pi_p(\tilde u)=u,\qquad d\pi_p(\tilde v)=v. + $$ + Define + $$ + g_m(u,v):=\tilde g_p(\tilde u,\tilde v). + $$ + This is a symmetric bilinear form on $T_mM$, and it is positive definite because $\tilde g_p$ is positive definite on $H_p$ and $d\pi_p|_{H_p}$ is an isomorphism. -It remains to show that this definition is independent of the choice of $p$ in the fiber. Suppose $p,q\in \pi^{-1}(m)$. By transitivity of the action on fibers, there exists $\varphi\in G$ such that $\varphi\cdot p=q$. Let $\tilde u_p,\tilde v_p\in H_p$ be the horizontal lifts of $u,v$ at $p$, and define -$$ -\tilde u_q:=d\varphi_p(\tilde u_p),\qquad \tilde v_q:=d\varphi_p(\tilde v_p). -$$ -By the previous paragraph, $\tilde u_q,\tilde v_q\in H_q$. Moreover, -$$ -d\pi_q(\tilde u_q) -= -d\pi_q(d\varphi_p\tilde u_p) -= -d\pi_p(\tilde u_p) -= -u, -$$ -and similarly $d\pi_q(\tilde v_q)=v$. Thus $\tilde u_q,\tilde v_q$ are exactly the horizontal lifts of $u,v$ at $q$. Since $\varphi$ is an isometry, -$$ -\tilde g_q(\tilde u_q,\tilde v_q) -= -\tilde g_q(d\varphi_p\tilde u_p,d\varphi_p\tilde v_p) -= -\tilde g_p(\tilde u_p,\tilde v_p). -$$ -Therefore $g_m(u,v)$ is independent of the chosen point $p\in \pi^{-1}(m)$, so $g$ is well defined on $M$. + It remains to show that this definition is independent of the choice of $p$ in the fiber. Suppose $p,q\in \pi^{-1}(m)$. By transitivity of the action on fibers, there exists $\varphi\in G$ such that $\varphi\cdot p=q$. Let $\tilde u_p,\tilde v_p\in H_p$ be the horizontal lifts of $u,v$ at $p$, and define + $$ + \tilde u_q:=d\varphi_p(\tilde u_p),\qquad \tilde v_q:=d\varphi_p(\tilde v_p). + $$ + By the previous paragraph, $\tilde u_q,\tilde v_q\in H_q$. Moreover, + $$ + d\pi_q(\tilde u_q) + = + d\pi_q(d\varphi_p\tilde u_p) + = + d\pi_p(\tilde u_p) + = + u, + $$ + and similarly $d\pi_q(\tilde v_q)=v$. Thus $\tilde u_q,\tilde v_q$ are exactly the horizontal lifts of $u,v$ at $q$. Since $\varphi$ is an isometry, + $$ + \tilde g_q(\tilde u_q,\tilde v_q) + = + \tilde g_q(d\varphi_p\tilde u_p,d\varphi_p\tilde v_p) + = + \tilde g_p(\tilde u_p,\tilde v_p). + $$ + Therefore $g_m(u,v)$ is independent of the chosen point $p\in \pi^{-1}(m)$, so $g$ is well defined on $M$. -Next we prove that $g$ is smooth. Let $m_0\in M$. Since $\pi$ is a smooth submersion, there exists an open neighborhood $U\subseteq M$ of $m_0$ and a smooth local section -$$ -s:U\to \tilde M -\qquad\text{such that}\qquad -\pi\circ s=\mathrm{id}_U. -$$ -Over $s(U)$, the vertical bundle $V=\ker d\pi$ is a smooth subbundle of $T\tilde M$, and hence so is its orthogonal complement $H=V^\perp$. For each $x\in U$, the restriction -$$ -d\pi_{s(x)}|_{H_{s(x)}}:H_{s(x)}\to T_xM -$$ -is a linear isomorphism, and these isomorphisms depend smoothly on $x$. Thus they define a smooth vector bundle isomorphism -$$ -d\pi|_H:H|_{s(U)}\to TU, -$$ -whose inverse is also smooth. + Next we prove that $g$ is smooth. Let $m_0\in M$. Since $\pi$ is a smooth submersion, there exists an open neighborhood $U\subseteq M$ of $m_0$ and a smooth local section + $$ + s:U\to \tilde M + \qquad\text{such that}\qquad + \pi\circ s=\mathrm{id}_U. + $$ + Over $s(U)$, the vertical bundle $V=\ker d\pi$ is a smooth subbundle of $T\tilde M$, and hence so is its orthogonal complement $H=V^\perp$. For each $x\in U$, the restriction + $$ + d\pi_{s(x)}|_{H_{s(x)}}:H_{s(x)}\to T_xM + $$ + is a linear isomorphism, and these isomorphisms depend smoothly on $x$. Thus they define a smooth vector bundle isomorphism + $$ + d\pi|_H:H|_{s(U)}\to TU, + $$ + whose inverse is also smooth. -If $X,Y$ are smooth vector fields on $U$, define their horizontal lifts along $s$ by -$$ -X_x^H:=\bigl(d\pi_{s(x)}|_{H_{s(x)}}\bigr)^{-1}(X_x), -\qquad -Y_x^H:=\bigl(d\pi_{s(x)}|_{H_{s(x)}}\bigr)^{-1}(Y_x). -$$ -Then $X^H$ and $Y^H$ are smooth vector fields along $s(U)$, and by construction, -$$ -g(X,Y)(x)=\tilde g_{s(x)}(X_x^H,Y_x^H). -$$ -Since the right-hand side depends smoothly on $x$, it follows that $g$ is a smooth Riemannian metric on $M$. + If $X,Y$ are smooth vector fields on $U$, define their horizontal lifts along $s$ by + $$ + X_x^H:=\bigl(d\pi_{s(x)}|_{H_{s(x)}}\bigr)^{-1}(X_x), + \qquad + Y_x^H:=\bigl(d\pi_{s(x)}|_{H_{s(x)}}\bigr)^{-1}(Y_x). + $$ + Then $X^H$ and $Y^H$ are smooth vector fields along $s(U)$, and by construction, + $$ + g(X,Y)(x)=\tilde g_{s(x)}(X_x^H,Y_x^H). + $$ + Since the right-hand side depends smoothly on $x$, it follows that $g$ is a smooth Riemannian metric on $M$. -By construction, for every $p\in \tilde M$ and every $\tilde u,\tilde v\in H_p$, -$$ -g_{\pi(p)}(d\pi_p\tilde u,d\pi_p\tilde v)=\tilde g_p(\tilde u,\tilde v). -$$ -Thus $d\pi_p:H_p\to T_{\pi(p)}M$ is an isometry for every $p$, so $\pi:(\tilde M,\tilde g)\to (M,g)$ is a Riemannian submersion. + By construction, for every $p\in \tilde M$ and every $\tilde u,\tilde v\in H_p$, + $$ + g_{\pi(p)}(d\pi_p\tilde u,d\pi_p\tilde v)=\tilde g_p(\tilde u,\tilde v). + $$ + Thus $d\pi_p:H_p\to T_{\pi(p)}M$ is an isometry for every $p$, so $\pi:(\tilde M,\tilde g)\to (M,g)$ is a Riemannian submersion. -Finally, uniqueness is immediate. Indeed, if $g'$ is another Riemannian metric on $M$ such that $\pi:(\tilde M,\tilde g)\to (M,g')$ is a Riemannian submersion, then for any $m\in M$, any $p\in \pi^{-1}(m)$, and any $u,v\in T_mM$, letting $\tilde u,\tilde v\in H_p$ denote the horizontal lifts of $u,v$, we must have -$$ -g'_m(u,v)=\tilde g_p(\tilde u,\tilde v)=g_m(u,v). -$$ -Hence $g'=g$. + Finally, uniqueness is immediate. Indeed, if $g'$ is another Riemannian metric on $M$ such that $\pi:(\tilde M,\tilde g)\to (M,g')$ is a Riemannian submersion, then for any $m\in M$, any $p\in \pi^{-1}(m)$, and any $u,v\in T_mM$, letting $\tilde u,\tilde v\in H_p$ denote the horizontal lifts of $u,v$, we must have + $$ + g'_m(u,v)=\tilde g_p(\tilde u,\tilde v)=g_m(u,v). + $$ + Hence $g'=g$. -Therefore there exists a unique Riemannian metric on $M$ such that $\pi$ is a Riemannian submersion. + Therefore there exists a unique Riemannian metric on $M$ such that $\pi$ is a Riemannian submersion. \end{proof} +\subsection{Hopf fibration} + +There are some remaining steps for showing how the metric on Sphere induces the metric on complex projective space, now we will just drop the conclusion here so that we can continue our discussion: + +\begin{itemize} + \item Every nonzero vector can be normalized, so each pure state has a representative on the unit sphere + $$ + S^{2n+1} \subset \mathbb C^{n+1}. + $$ + \item Two unit vectors represent the same pure state exactly when they differ by a phase: + $$ + z \sim e^{i\theta} z. + $$ + \item Therefore + $$ + \mathbb C P^n = S^{2n+1}/S^1. + $$ +\end{itemize} + +\vspace{0.4em} +The quotient map +$$ + p:S^{2n+1}\to \mathbb C P^n, \qquad p(z)=[z]=\{\lambda z : \lambda \in \mathbb C^\times\}, +$$ +is the \textbf{Hopf fibration}. + + +The geometric picture is +$$ + S^{2n+1} + \xrightarrow{\text{Hopf fibration}} + \mathbb C P^n, + \qquad + \text{round metric} + \rightsquigarrow + \text{Fubini--Study metric}. +$$ + + +The sphere $S^{2n+1}\subset \mathbb C^{n+1}$ has the \textbf{round metric} +$$ + g_{\mathrm{round}}=\sum_{j=0}^n (dx_j^2+dy_j^2)\big|_{S^{2n+1}}, +$$ +induced from the Euclidean metric on $\mathbb R^{2n+2}$. + +In homogeneous coordinates $[z]\in\mathbb C P^n$, the \textbf{Fubini--Study metric} is +$$ + g_{FS} + = + \frac{\langle dz,dz\rangle \langle z,z\rangle-|\langle z,dz\rangle|^2}{\langle z,z\rangle^2}, +$$ + \section{Quantum physics and terminologies} In this section, we will introduce some terminologies and theorems used in quantum physics that are relevant to our study. Assuming no prior knowledge of quantum physics, we will provide brief definitions and explanations for each term. @@ -1105,21 +1249,21 @@ In this section, we will introduce some terminologies and theorems used in quant One might ask, what is the fundamental difference between a quantum system and a classical system, and why can we not directly apply those theorems in classical computers to a quantum computer? It turns out that quantum error-correcting codes are hard due to the following definitions and features for quantum computing. \begin{defn} - All quantum operations can be constructed by composing four kinds of transformations: (adapted from Chapter 10 of \cite{Bengtsson_Zyczkowski_2017}) + All quantum operations can be constructed by composing four kinds of transformations: (adapted from Chapter 10 of \cite{Bengtsson_Zyczkowski_2017}) - \begin{enumerate} - \item Unitary operations. $U(\cdot)$ for any quantum state. It is possible to apply a non-unitary operation for an open quantum system, but that is usually not the focus for quantum computing and usually leads to non-recoverable loss of information that we wish to obtain. - \item Extend the system. Given a quantum state $\rho\in\mathcal{H}^N$, we can extend it to a larger quantum system by "entangle" (For this report, you don't need to worry for how quantum entanglement works) it with some new states $\sigma\in \mathcal{H}^K$ (The space where the new state dwells is usually called ancilla system) and get $\rho'=\rho\otimes\sigma\in \mathcal{H}^N\otimes \mathcal{K}$. - \item Partial trace. Given a quantum state $\rho\in\mathcal{H}^N$ and some reference state $\sigma\in\mathcal {H}^K$, we can trace out some subsystems and get a new state $\rho'\in\mathcal{H}^{N-K}$. - \item Selective measurement. Given a quantum state, we measure it and get a classical bit; unlike the classical case, the measurement is a probabilistic operation. (More specifically, this is some projection to a reference state corresponding to a classical bit output. For this report, you don't need to worry about how such a result is obtained and how the reference state is constructed.) - \end{enumerate} + \begin{enumerate} + \item Unitary operations. $U(\cdot)$ for any quantum state. It is possible to apply a non-unitary operation for an open quantum system, but that is usually not the focus for quantum computing and usually leads to non-recoverable loss of information that we wish to obtain. + \item Extend the system. Given a quantum state $\rho\in\mathcal{H}^N$, we can extend it to a larger quantum system by "entangle" (For this report, you don't need to worry for how quantum entanglement works) it with some new states $\sigma\in \mathcal{H}^K$ (The space where the new state dwells is usually called ancilla system) and get $\rho'=\rho\otimes\sigma\in \mathcal{H}^N\otimes \mathcal{K}$. + \item Partial trace. Given a quantum state $\rho\in\mathcal{H}^N$ and some reference state $\sigma\in\mathcal {H}^K$, we can trace out some subsystems and get a new state $\rho'\in\mathcal{H}^{N-K}$. + \item Selective measurement. Given a quantum state, we measure it and get a classical bit; unlike the classical case, the measurement is a probabilistic operation. (More specifically, this is some projection to a reference state corresponding to a classical bit output. For this report, you don't need to worry about how such a result is obtained and how the reference state is constructed.) + \end{enumerate} \end{defn} -$U(n)$ is the group of all $n\times n$ \textbf{unitary matrices} over $\mathbb{C}$, +$U(n)$ is the group of all $n\times n$ \textbf{unitary matrices} over $\mathbb{C}$, $$ -U(n)=\{A\in \mathbb{C}^{n\times n}: A^*A=AA^*=I_n\} + U(n)=\{A\in \mathbb{C}^{n\times n}: A^*A=AA^*=I_n\} $$ The uniqueness of such measurement came from the lemma below~\cite{Elizabeth_book} @@ -1130,28 +1274,57 @@ The uniqueness of such measurement came from the lemma below~\cite{Elizabeth_boo Let $(U(n), \| \cdot \|, \mu)$ be a metric measure space where $\| \cdot \|$ is the Hilbert-Schmidt norm and $\mu$ is the measure function. The Haar measure on $U(n)$ is the unique probability measure that is invariant under the action of $U(n)$ on itself. - + That is, fixing $B\in U(n)$, $\forall A\in U(n)$, $\mu(A\cdot B)=\mu(B\cdot A)=\mu(B)$. - + The Haar measure is the unique probability measure that is invariant under the action of $U(n)$ on itself. \end{lemma} -\begin{defn} - \label{defn:pure_state} - Pure state: +A finite-dimensional quantum system is modeled by a complex Hilbert space (a complete inner product space) +$$ + \mathcal H \cong \mathbb C^{n+1}. +$$ -A random pure state $\varphi$ is any random variable distributed according to the unitarily invariant probability measure on the pure states $\mathcal{P}(A)$ of the system $A$, denoted by $\varphi\in_R\mathcal{P}(A)$. -\end{defn} +A \textbf{pure state} is represented by a unit vector +$$ + \psi \in \mathcal H, \qquad \|\psi\|=1. +$$ -It is trivial that for the space of pure state, we can easily apply the Haar measure as the unitarily invariant probability measure since the space of pure state is $S^n$ for some $n$. However, for the case of mixed states, that is a bit complicated and we need to use partial tracing to defined the rank-$s$ random states. +A \textbf{mixed state} is represented by a density matrix +$$ + \rho=\sum_{j=1}^n p_j|\psi_j\rangle\langle\psi_j|, \qquad \sum_{j=1}^n p_j=1, \qquad p_j\geq 0. +$$ -\begin{defn} - \label{defn:rank_s_random_state} - Rank-$s$ random state. +Some key comparisons between pure states and mixed states: + +Pure states describe maximal information; mixed states describe probabilistic mixtures or partial information. + +Pure states form a curved geometric space; mixed states form a convex set inside the space of matrices. + +Pure states live in the complex projective space. + +\begin{itemize} + \item Two nonzero vectors that differ by a nonzero complex scalar represent the same physical state: + $$ + \psi \sim \lambda \psi, \qquad \lambda \in \mathbb C^\times. + $$ + \item In particular, multiplying by a phase $e^{i\theta}$ does not change any physical predictions. + \item Therefore the physical pure state is not a single vector, but the \emph{complex line} spanned by that vector. +\end{itemize} + +Hence the space of pure states (denoted by $\mathcal{P}(\mathcal H)$) is +$$ + \mathcal{P}(\mathcal H) + = + (\mathcal H \setminus \{0\})/\mathbb C^\times. +$$ + +After choosing a basis $\mathcal H \cong \mathbb C^{n+1}$, this becomes +$$ + \mathcal{P}(\mathcal H) \cong \mathbb C P^n. +$$ - For a system $A$ and an integer $s\geq 1$, consider the distribution onn the mixed states $\mathcal{S}(A)$ of A induced by the partial trace over the second factor form the uniform distribution on pure states of $A\otimes\mathbb{C}^s$. Any random variable $\rho$ distributed as such will be called a rank-$s$ random states; denoted as $\rho\in_R \mathcal{S}_s(A)$. And $\mathcal{P}(A)=\mathcal{S}_1(A)$. -\end{defn} \begin{prop} @@ -1180,6 +1353,26 @@ Intuitively, if the two states are not orthogonal, then for any measurement (pro First, we need to define what is a random state in a bipartite system. + + +\begin{defn} + \label{defn:random_pure_state} + Pure state: + + A random pure state $\varphi$ is any random variable distributed according to the unitarily invariant probability measure on the pure states $\mathcal{P}(A)$ of the system $A$, denoted by $\varphi\in_R\mathcal{P}(A)$. +\end{defn} + + +It is trivial that for the space of pure state, we can easily apply the Haar measure as the unitarily invariant probability measure since the space of pure state is $S^n$ for some $n$. However, for the case of mixed states, that is a bit complicated and we need to use partial tracing to defined the rank-$s$ random states. + +\begin{defn} + \label{defn:rank_s_random_state} + Rank-$s$ random state. + + For a system $A$ and an integer $s\geq 1$, consider the distribution onn the mixed states $\mathcal{S}(A)$ of A induced by the partial trace over the second factor form the uniform distribution on pure states of $A\otimes\mathbb{C}^s$. Any random variable $\rho$ distributed as such will be called a rank-$s$ random states; denoted as $\rho\in_R \mathcal{S}_s(A)$. And $\mathcal{P}(A)=\mathcal{S}_1(A)$. +\end{defn} + + % When compiled standalone, print this chapter's references at the end. \ifSubfilesClassLoaded{ \printbibliography[title={References}] diff --git a/latex/chapters/chap1.pdf b/latex/chapters/chap1.pdf index a7c1a37482d5f4089ec717b6392641e7e7fd1a08..92a76a1895a87c68c8344e2bc8db5fd84ee8c287 100644 GIT binary patch delta 41802 zcmV(|K+(VY=^Kmc8?amn0Wy;z6DfbaT1#`>HWI$;S8z{DRe2cVMF6hdgS{0yNySyk zMmNWm;>=JQZf3{{$&sx6_1ldGpTbC^s$?$~2W&R_@uBfG;5536(&*z=y7>G4=IWad z!bUtzvW%@5J=L&cS(b=QfQ9gv zDN=E>0}o`H#vAgMn~As9I&XjR^5 zyRN3eZs)bM-D&$sV=ksJ1R7T^j$ka=${>T%lC7Kia6ki39T<3zPoRlw9Uq%(7T4>1 zUGB^7n(O!x$1|L&ZXU0Vj1!oLXYu>IEjRgo@5bJsv4-qHX`H-k%KLvZ-_t<7Zj0vr zT4k{dh?n1#U0L0c#BEa_=;>oquX+SbZ9dd!=l$O?Kt{(8_^WPy$6s0enNT>-bjA{+ zH0W{PfBIM->V1uV zKbFUKgQo4u?!P2+d#Zmn4hCeRoPm*>WHj+MWEMOs#(R;sAuT#dx#=Z-@e&G#txmmw zU?viYRX&)#F|(O$k_iCHMC@-Qrm~qq5+1XDx_^gkhR7-KOD$11T^KH?%*0iZH}sU_ zA04+Pu4x(~%h?2o1d+;6m?eDg-UAJ7J;9ixUj*eUsIKaw8(@@u?Wpl4Ht5vhK0>V 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