optimize entropy
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@@ -28,7 +28,6 @@
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\chapter{Concentration of Measure And Quantum Entanglement}
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As the future version of me might forgot everything we have over the summer, as I did for now, I will make a review again from the simple definition to recall the necessary information to tell you why we are here and how we are going to proceed.
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First, we will build the mathematical model describing the behavior of quantum system and why they makes sense for physicists and meaningful for general publics.
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@@ -44,7 +43,7 @@ The light intensity decreases with $\alpha$ (the angle between the two filters).
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However, for a system of 3 polarizing filters $F_1,F_2,F_3$, having directions $\alpha_1,\alpha_2,\alpha_3$, if we put them on the optical bench in pairs, then we will have three random variables $P_1,P_2,P_3$.
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\begin{figure}[H]
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\begin{figure}[h]
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\centering
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\includegraphics[width=0.7\textwidth]{Filter_figure.png}
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\caption{The light polarization experiment, image from \cite{kummer1998elements}}
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@@ -122,481 +121,60 @@ Probability of a photon passing through the filter $P_\alpha$ is given by $\lang
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Since the probability of a photon passing through the three filters is not commutative, it is impossible to discuss $\operatorname{Prob}(P_1=1,P_3=0)$ in the classical setting.
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The main vector space we are interested in is $\mathbb{C}^n$; therefore, all the linear operators we defined are from $\mathbb{C}^n$ to $\mathbb{C}^n$.
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We denote a vector in vector space as $\ket{\psi}=(z_1,\ldots,z_n)$ (might also be infinite dimensional, and $z_i\in\mathbb{C}$).
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A natural inner product space defined on $\mathbb{C}^n$ is given by the Hermitian inner product:
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We now show how the experimentally observed probability
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$$
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\langle\psi|\varphi\rangle=\sum_{i=1}^n z_iz_i^*
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\frac{1}{2}\sin^2(\alpha_i-\alpha_j)
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$$
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arises from the operator model.
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Assume the incoming light is \emph{unpolarized}. It is therefore described by
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the density matrix
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$$
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\rho=\frac{1}{2} I .
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$$
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This satisfies the following properties:
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Let $P_{\alpha_i}$ and $P_{\alpha_j}$ be the orthogonal projections corresponding
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to the two polarization filters with angles $\alpha_i$ and $\alpha_j$.
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\begin{enumerate}
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\item $\bra{\psi}\sum_i \lambda_i\ket{\varphi}=\sum_i \lambda_i \langle\psi|\varphi\rangle$ (linear on the second argument. Note that in physics \cite{Nielsen_Chuang_2010} we use linear on the second argument and conjugate linear on the first argument. But in math, we use linear on the first argument and conjugate linear on the second argument \cite{Axler_2024}. As promised in the beginning, we will use the physics convention in this report.)
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\item $\langle\varphi|\psi\rangle=(\langle\psi|\varphi\rangle)^*$
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\item $\langle\psi|\psi\rangle\geq 0$ with equality if and only if $\ket{\psi}=0$
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\end{enumerate}
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Here $\psi$ is just a label for the vector, and you don't need to worry about it too much. This is also called the ket, where the counterpart:
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\begin{itemize}
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\item $\langle\psi\rangle$ is called the bra, used to denote the vector dual to $\psi$; such an element is a linear functional if you really want to know what that is.
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\item $\langle\psi|\varphi\rangle$ is the inner product between two vectors, and $\bra{\psi} A\ket{\varphi}$ is the inner product between $A\ket{\varphi}$ and $\bra{\psi}$, or equivalently $A^\dagger \bra{\psi}$ and $\ket{\varphi}$.
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\item Given a complex matrix $A=\mathbb{C}^{n\times n}$,
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\begin{enumerate}
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\item $A^*$ is the complex conjugate of $A$.
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i.e.,
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$$
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A=\begin{bmatrix}
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1+i & 2+i & 3+i\\
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4+i & 5+i & 6+i\\
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7+i & 8+i & 9+i\end{bmatrix},
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A^*=\begin{bmatrix}
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1-i & 2-i & 3-i\\
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4-i & 5-i & 6-i\\
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7-i & 8-i & 9-i
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\end{bmatrix}
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$$
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\item $A^\top$ denotes the transpose of $A$.
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i.e.,
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$$
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A=\begin{bmatrix}
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1+i & 2+i & 3+i\\
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4+i & 5+i & 6+i\\
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7+i & 8+i & 9+i
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\end{bmatrix},
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A^\top=\begin{bmatrix}
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1+i & 4+i & 7+i\\
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2+i & 5+i & 8+i\\
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3+i & 6+i & 9+i
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\end{bmatrix}
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$$
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\item $A^\dagger=(A^*)^\top$ denotes the complex conjugate transpose, referred to as the adjoint, or Hermitian conjugate of $A$.
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i.e.,
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$$
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A=\begin{bmatrix}
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1+i & 2+i & 3+i\\
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4+i & 5+i & 6+i\\
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7+i & 8+i & 9+i
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\end{bmatrix},
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A^\dagger=\begin{bmatrix}
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1-i & 4-i & 7-i\\
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2-i & 5-i & 8-i\\
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3-i & 6-i & 9-i
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\end{bmatrix}
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$$
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\item $A$ is unitary if $A^\dagger A=AA^\dagger=I$.
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\item $A$ is hermitian (self-adjoint in mathematics literature) if $A^\dagger=A$.
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\end{enumerate}
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\end{itemize}
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\subsubsection{Motivation of Tensor product}
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Recall from the traditional notation of product space of two vector spaces $V$ and $W$, that is, $V\times W$, is the set of all ordered pairs $(\ket{v},\ket{w})$ where $\ket{v}\in V$ and $\ket{w}\in W$.
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The space has dimension $\dim V+\dim W$.
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We want to define a vector space with the notation of multiplication of two vectors from different vector spaces.
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That is
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The probability that a photon passes the first filter $P_{\alpha_i}$ is given by the Born rule:
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$$
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(\ket{v_1}+\ket{v_2})\otimes \ket{w}=(\ket{v_1}\otimes \ket{w})+(\ket{v_2}\otimes \ket{w})
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$$
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$$
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\ket{v}\otimes (\ket{w_1}+\ket{w_2})=(\ket{v}\otimes \ket{w_1})+(\ket{v}\otimes \ket{w_2})
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\operatorname{Prob}(P_i=1)
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=\operatorname{tr}(\rho P_{\alpha_i})
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=\frac{1}{2} \operatorname{tr}(P_{\alpha_i})
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=\frac{1}{2}
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$$
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and enables scalar multiplication by
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If the photon passes the first filter, the post-measurement state is given by the L\"uders rule:
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$$
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\lambda (\ket{v}\otimes \ket{w})=(\lambda \ket{v})\otimes \ket{w}=\ket{v}\otimes (\lambda \ket{w})
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\rho \longmapsto
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\rho_i
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=\frac{P_{\alpha_i}\rho P_{\alpha_i}}{\operatorname{tr}(\rho P_{\alpha_i})}
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= P_{\alpha_i}.
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$$
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And we wish to build a way to associate the basis of $V$ and $W$ with the basis of $V\otimes W$. That makes the tensor product a vector space with dimension $\dim V\times \dim W$.
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\begin{defn}
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Definition of linear functional
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A linear functional is a linear map from $V$ to $\mathbb{F}$.
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\end{defn}
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Note the difference between a linear functional and a linear map.
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A generalized linear map is a function $f: V\to W$ satisfying the condition.
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\begin{itemize}
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\item $f(\ket{u}+\ket{v})=f(\ket{u})+f(\ket{v})$
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\item $f(\lambda \ket{v})=\lambda f(\ket{v})$
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\end{itemize}
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\begin{defn}
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A bilinear functional is a bilinear function $\beta:V\times W\to \mathbb{F}$ satisfying the condition that $\ket{v}\to \beta(\ket{v},\ket{w})$ is a linear functional for all $\ket{w}\in W$ and $\ket{w}\to \beta(\ket{v},\ket{w})$ is a linear functional for all $\ket{v}\in V$.
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\end{defn}
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The vector space of all bilinear functionals is denoted by $\mathcal{B}(V, W)$.
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\begin{defn}
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Let $V, W$ be two vector spaces.
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Let $V'$ and $W'$ be the dual spaces of $V$ and $W$, respectively, that is $V'=\{\psi:V\to \mathbb{F}\}$ and $W'=\{\phi:W\to \mathbb{F}\}$, $\psi, \phi$ are linear functionals.
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The tensor product of vectors $v\in V$ and $w\in W$ is the bilinear functional defined by $\forall (\psi,\phi)\in V'\times W'$ given by the notation
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The probability that the photon then passes the second filter is
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$$
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(v\otimes w)(\psi,\phi)\coloneqq\psi(v)\phi(w)
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\operatorname{Prob}(P_j=1 \mid P_i=1)
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=\operatorname{tr}(P_{\alpha_i} P_{\alpha_j})
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=\cos^2(\alpha_i-\alpha_j).
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$$
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The tensor product of two vector spaces $V$ and $W$ is the vector space $\mathcal{B}(V',W')$
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Notice that the basis of such vector space is the linear combination of the basis of $V'$ and $W'$, that is, if $\{e_i\}$ is the basis of $V'$ and $\{f_j\}$ is the basis of $W'$, then $\{e_i\otimes f_j\}$ is the basis of $\mathcal{B}(V', W')$.
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That is, every element of $\mathcal{B}(V', W')$ can be written as a linear combination of the basis.
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Since $\{e_i\}$ and $\{f_j\}$ are bases of $V'$ and $W'$, respectively, then we can always find a set of linear functionals $\{\phi_i\}$ and $\{\psi_j\}$ such that $\phi_i(e_j)=\delta_{ij}$ and $\psi_j(f_i)=\delta_{ij}$.
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Here $\delta_{ij}=\begin{cases}
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1 & \text{if } i=j \\
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0 & \text{otherwise}
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\end{cases}$ is the Kronecker delta.
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\end{defn}
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Hence, the probability that the photon passes $P_{\alpha_i}$ and is then blocked by $P_{\alpha_j}$ is
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$$
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V\otimes W=\left\{\sum_{i=1}^n \sum_{j=1}^m a_{ij} \phi_i(v)\psi_j(w): \phi_i\in V', \psi_j\in W'\right\}
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\begin{aligned}
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\operatorname{Prob}(P_i=1, P_j=0)
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&= \operatorname{Prob}(P_i=1)
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- \operatorname{Prob}(P_i=1, P_j=1) \\
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&= \frac12 - \frac12 \cos^2(\alpha_i-\alpha_j) \\
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&= \frac12 \sin^2(\alpha_i-\alpha_j).
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\end{aligned}
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$$
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Note that $\sum_{i=1}^n \sum_{j=1}^m a_{ij} \phi_i(v)\psi_j(w)$ is a bilinear functional that maps $V'\times W'$ to $\mathbb{F}$.
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This enables basis-free construction of vector spaces with proper multiplication and scalar multiplication.
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This vector space is equipped with the unique inner product $\langle v\otimes w, u\otimes x\rangle_{V\otimes W}$ defined by
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$$
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\langle v\otimes w, u\otimes x\rangle=\langle v,u\rangle_V\langle w,x\rangle_W
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$$
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In practice, we ignore the subscript of the vector space and just write $\langle v\otimes w, u\otimes x\rangle=\langle v,u\rangle\langle w,x\rangle$.
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This introduces a new model in mathematics explaining quantum mechanics: the non-commutative probability theory.
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\section{Non-commutative probability theory}
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The non-commutative probability theory is a branch of generalized probability theory that studies the probability of events in non-commutative algebras.
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There are several main components of the generalized probability theory; let's see how we can formulate them, comparing with the classical probability theory.
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First, we define the Hilbert space in case one did not make the step from the linear algebra courses like me.
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\begin{defn}
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\label{defn:Hilbert_space}
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Hilbert space:
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A Hilbert space is a complete inner product space.
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\end{defn}
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That is, a vector space equipped with an inner product that is complete (every Cauchy sequence converges to a limit).
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To introduce an example of Hilbert space we use when studying quantum mechanics, we need to introduce a common inner product used in $\mathbb{C}^n$.
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\begin{defn}
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\label{defn:Hermitian_inner_product}
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Hermitian inner product:
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On $\mathbb{C}^n$, the Hermitian inner product is defined by
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$$
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\langle u,v\rangle=\sum_{i=1}^n \overline{u_i}v_i
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$$
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\end{defn}
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\begin{prop}
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\label{prop:Hermitian_inner_product_with_complex_vectorspace}
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The Hermitian inner product on the complex vector space $\C^n$ makes it a Hilbert space.
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\end{prop}
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\begin{proof}
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We first verify that the Hermitian inner product
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$$
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\langle u,v\rangle = \sum_{i=1}^n \overline{u_i} v_i
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$$
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on $\C^n$ satisfies the axioms of an inner product:
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\begin{enumerate}
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\item \textbf{Conjugate symmetry:} For all $u,v\in\C^n$,
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$$
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\langle u,v\rangle =\sum_{i=1}^n \overline{u_i} v_i=\overline{\sum_{i=1}^n \overline{v_i} u_i}=\overline{\langle v,u\rangle}.
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$$
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\item \textbf{Linearity:} For any $u,v,w\in\C^n$ and scalars $a,b\in\C$, we have
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$$
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\langle u, av + bw\rangle = \sum_{i=1}^n \overline{u_i} (av_i + bw_i)=a\langle u,v\rangle + b\langle u,w\rangle.
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$$
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\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 $a_j,b_j\in\mathbb{R}$.
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$$
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\langle u,u\rangle = \sum_{j=1}^n \overline{u_j} u_j=\sum_{i=1}^n (a_i^2+b_i^2)\geq 0,
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$$
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with equality if and only if $u=0$.
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Therefore, the Hermitian inner product is an inner product.
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\end{enumerate}
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Next, we show that $\C^n$ is complete with respect to the norm induced by this inner product:
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$$
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\|u\| = \sqrt{\langle u,u\rangle}.
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$$
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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.
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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.
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\end{proof}
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Another classical example of Hilbert space is $L^2(\Omega, \mathscr{F}, P)$, where $(\Omega, \mathscr{F}, P)$ is a measure space ($\Omega$ is a set, $\mathscr{F}$ is a $\sigma$-algebra on $\Omega$, and $P$ is a measure on $\mathscr{F}$). The $L^2$ space is the space of all function on $\Omega$ that is
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\begin{enumerate}
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\item \textbf{square integrable}: square integrable functions are the functions $f:\Omega\to \mathbb{C}$ such that
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$$
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\int_\Omega |f(\omega)|^2 dP(\omega)<\infty
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$$
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with inner product defined by
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$$
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\langle f,g\rangle=\int_\Omega \overline{f(\omega)}g(\omega)dP(\omega)
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$$
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\item \textbf{complex-valued}: functions are complex-valued measurable. $f=u+v i$ is complex-valued if $u$ and $v$ are real-valued measurable.
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\end{enumerate}
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\begin{prop}
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\label{prop:L2_space_is_a_Hilbert_space}
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$L^2(\Omega, \mathscr{F}, P)$ is a Hilbert space.
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\end{prop}
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\begin{proof}
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We check the two conditions of the Hilbert space:
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\begin{itemize}
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\item Completeness:
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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
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$$
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\int_\Omega |f_m(\omega)-f_n(\omega)|^2 dP(\omega)<\epsilon^2
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$$
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This means that $(f_n)$ is a Cauchy sequence in the norm of $L^2(\Omega, \mathscr{F}, P)$.
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\item Inner product:
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The inner product is defined by
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$$
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\langle f,g\rangle=\int_\Omega \overline{f(\omega)}g(\omega)dP(\omega)
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$$
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This is a well-defined inner product on $L^2(\Omega, \mathscr{F}, P)$. We can check the properties of the inner product:
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\begin{itemize}
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\item Linearity:
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$$
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\langle af+bg,h\rangle=a\langle f,h\rangle+b\langle g,h\rangle
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$$
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\item Conjugate symmetry:
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$$
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\langle f,g\rangle=\overline{\langle g,f\rangle}
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$$
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\item Positive definiteness:
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$$
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\langle f,f\rangle\geq 0
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$$
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\end{itemize}
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\end{itemize}
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\end{proof}
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Let $\mathscr{H}$ be a Hilbert space. $\mathscr{H}$ consists of complex-valued functions on a finite set $\Omega=\{1,2,\cdots,n\}$, and the functions $(e_1,e_2,\cdots,e_n)$ form an orthonormal basis of $\mathscr{H}$. (We use Dirac notation $|k\rangle$ to denote the basis vector $e_k$~\cite{parthasarathy1992quantum}.)
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As an analog to the classical probability space $(\Omega,\mathscr{F},\mu)$, which consists of a sample space $\Omega$ and a probability measure $\mu$ on the state space $\mathscr{F}$, the non-commutative probability space $(\mathscr{H},\mathscr{P},\rho)$ consists of a Hilbert space $\mathscr{H}$ and a state $\rho$ on the space of all orthogonal projections $\mathscr{P}$.
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The detailed definition of the non-commutative probability space is given below:
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\begin{defn}
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\label{defn:non-commutative_probability_space}
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Non-commutative probability space:
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A non-commutative probability space is a pair $(\mathscr{B}(\mathscr{H}),\mathscr{P})$, where $\mathscr{B}(\mathscr{H})$ is the set of all \textbf{bounded} linear operators on $\mathscr{H}$.
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A linear operator on $\mathscr{H}$ is \textbf{bounded} if for all $u$ such that $\|u\|\leq 1$, we have $\|Au\|\leq M$ for some $M>0$.
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$\mathscr{P}$ is the set of all orthogonal projections on $\mathscr{B}(\mathscr{H})$.
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The set $\mathscr{P}=\{P\in\mathscr{B}(\mathscr{H}):P^*=P=P^2\}$ is the set of all orthogonal projections on $\mathscr{B}(\mathscr{H})$.
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\end{defn}
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Recall from classical probability theory, we call the initial probability distribution for possible outcomes in the classical probability theory as our \textit{state}, simillarly, we need to define the \textit{state} in the non-commutative probability theory.
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\begin{defn}
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\label{defn:state}
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Non-commutative probability state:
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A state on $(\mathscr{B}(\mathscr{H}),\mathscr{P})$ is a map $\rho:\mathscr{P}\to[0,1]$, (commonly named as density operator) such that:
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\begin{itemize}
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\item $\rho(O)=0$, where $O$ is the zero projection, and $\rho(I)=1$, where $I$ is the identity projection.
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\item If $P_1,P_2,\ldots,P_n$ are pairwise disjoint orthogonal projections, then $\rho(P_1 + P_2 + \cdots + P_n) = \sum_{i=1}^n \rho(P_i)$.
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\end{itemize}
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\end{defn}
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An example of a density operator can be given as follows:
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|
||||
If $(|\psi_1\rangle,|\psi_2\rangle,\cdots,|\psi_n\rangle)$ is an orthonormal basis of $\mathscr{H}$ consisting of eigenvectors of $\rho$, for the eigenvalues $p_1,p_2,\cdots,p_n$, then $p_j\geq 0$ and $\sum_{j=1}^n p_j=1$.
|
||||
|
||||
We can write $\rho$ as
|
||||
\[
|
||||
\rho=\sum_{j=1}^n p_j|\psi_j\rangle\langle\psi_j|
|
||||
\]
|
||||
(Under basis $|\psi_j\rangle$, it is a diagonal matrix with $p_j$ on the diagonal.)
|
||||
|
||||
% Then we need to introduce a theorem that ensures that every state on the space of all orthogonal projections on $\mathscr{H}$ can be represented by a density operator.
|
||||
|
||||
% \begin{theorem}
|
||||
% \label{theorem:Gleason's_theorem}
|
||||
% 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}
|
||||
|
||||
% This proof came from~\cite{parthasarathy2005mathematical}.
|
||||
|
||||
% \begin{proof}
|
||||
% % TODO: FILL IN THE PROOF
|
||||
% \end{proof}
|
||||
|
||||
% 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).
|
||||
|
||||
\begin{defn}
|
||||
\label{defn:observable}
|
||||
Observable:
|
||||
|
||||
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) $P:\mathscr{B}(\mathbb{R})\to\mathscr{P}$.
|
||||
|
||||
With the following properties:
|
||||
\begin{itemize}
|
||||
\item $P(\emptyset)=O$ (the zero projection)
|
||||
\item $P(\mathbb{R})=I$ (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 $P(\bigcup_{j=1}^n A_j)=\sum_{j=1}^n P(A_j)$
|
||||
\end{itemize}
|
||||
\end{itemize}
|
||||
\end{defn}
|
||||
|
||||
\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 $\operatorname{Tr}(\rho P(A))$.
|
||||
\end{defn}
|
||||
|
||||
When operators commute, we recover classical probability measures.
|
||||
|
||||
\begin{defn}
|
||||
\label{defn:measurement}
|
||||
Definition of measurement:
|
||||
|
||||
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}
|
||||
|
||||
\begin{prop}
|
||||
\label{prop:indistinguishability}
|
||||
Proposition of indistinguishability:
|
||||
|
||||
Suppose that we have two systems $u_1,u_2\in \mathscr{H}_1$, the two states are distinguishable if and only if they are orthogonal.
|
||||
\end{prop}
|
||||
|
||||
\begin{proof}
|
||||
|
||||
Ways to distinguish the two states:
|
||||
\begin{enumerate}
|
||||
\item Set $X=\{0,1,2\}$ and $M_i=|u_i\rangle\langle u_i|$, $M_0=I-M_1-M_2$
|
||||
\item Then $\{M_0,M_1,M_2\}$ is a complete collection of measurement operators on $\mathscr{H}$.
|
||||
\item Suppose the prepared state is $u_1$, then $p(1)=\|M_1u_1\|^2=\|u_1\|^2=1$, $p(2)=\|M_2u_1\|^2=0$, $p(0)=\|M_0u_1\|^2=0$.
|
||||
\end{enumerate}
|
||||
|
||||
If they are not orthogonal, then there is no choice of measurement operators to perfectly distinguish the two states.
|
||||
|
||||
\end{proof}
|
||||
|
||||
\textit{Intuitively, if the two states are not orthogonal, then for any measurement (projection) there exists non-zero probability of getting the same outcome for both states.}
|
||||
|
||||
Here is Table~\ref{tab:analog_of_classical_probability_theory_and_non_commutative_probability_theory} summarizing the analog of classical probability theory and non-commutative (\textit{quantum}) probability theory~\cite{Feres}:
|
||||
|
||||
\begin{table}
|
||||
\centering
|
||||
\renewcommand{\arraystretch}{1.5}
|
||||
\caption{Analog of classical probability theory and non-commutative (\textit{quantum}) probability theory}
|
||||
\label{tab:analog_of_classical_probability_theory_and_non_commutative_probability_theory}
|
||||
{\tiny
|
||||
\begin{tabular}{|p{0.5\linewidth}|p{0.5\linewidth}|}
|
||||
\hline
|
||||
\textbf{Classical probability} & \textbf{Non-commutative probability} \\
|
||||
\hline
|
||||
Sample space $\Omega$, cardinality $\vert\Omega\vert=n$, example: $\Omega=\{0,1\}$ & Complex Hilbert space $\mathscr{H}$, dimension $\dim\mathscr{H}=n$, example: $\mathscr{H}=\mathbb{C}^2$ \\
|
||||
\hline
|
||||
Common algebra of $\mathbb{C}$ valued functions & Algebra of bounded operators $\mathscr{B}(\mathscr{H})$ \\
|
||||
\hline
|
||||
$f\mapsto \bar{f}$ complex conjugation & $P\mapsto P^*$ adjoint \\
|
||||
\hline
|
||||
Events: indicator functions of sets & Projections: space of orthogonal projections $\mathscr{P}\subseteq\mathscr{B}(\mathscr{H})$ \\
|
||||
\hline
|
||||
functions $f$ such that $f^2=f=\overline{f}$ & orthogonal projections $P$ such that $P^*=P=P^2$ \\
|
||||
\hline
|
||||
$\mathbb{R}$-valued functions $f=\overline{f}$ & self-adjoint operators $A=A^*$ \\
|
||||
\hline
|
||||
$\mathbb{I}_{f^{-1}(\{\lambda\})}$ is the indicator function of the set $f^{-1}(\{\lambda\})$ & $P(\lambda)$ is the orthogonal projection to eigenspace \\
|
||||
\hline
|
||||
$f=\sum_{\lambda\in \operatorname{Range}(f)}\lambda \mathbb{I}_{f^{-1}(\{\lambda\})}$ & $A=\sum_{\lambda\in \operatorname{sp}(A)}\lambda P(\lambda)$ \\
|
||||
\hline
|
||||
Probability measure $\mu$ on $\Omega$ & Density operator $\rho$ on $\mathscr{H}$ \\
|
||||
\hline
|
||||
Delta measure $\delta_\omega$ & Pure state $\rho=\vert\psi\rangle\langle\psi\vert$ \\
|
||||
\hline
|
||||
$\mu$ is non-negative measure and $\sum_{i=1}^n\mu(\{i\})=1$ & $\rho$ is positive semi-definite and $\operatorname{Tr}(\rho)=1$ \\
|
||||
\hline
|
||||
Expected value of random variable $f$ is $\mathbb{E}_{\mu}(f)=\sum_{i=1}^n f(i)\mu(\{i\})$ & Expected value of operator $A$ is $\mathbb{E}_\rho(A)=\operatorname{Tr}(\rho A)$ \\
|
||||
\hline
|
||||
Variance of random variable $f$ is $\operatorname{Var}_\mu(f)=\sum_{i=1}^n (f(i)-\mathbb{E}_\mu(f))^2\mu(\{i\})$ & Variance of operator $A$ is $\operatorname{Var}_\rho(A)=\operatorname{Tr}(\rho A^2)-\operatorname{Tr}(\rho A)^2$ \\
|
||||
\hline
|
||||
Covariance of random variables $f$ and $g$ is $\operatorname{Cov}_\mu(f,g)=\sum_{i=1}^n (f(i)-\mathbb{E}_\mu(f))(g(i)-\mathbb{E}_\mu(g))\mu(\{i\})$ & Covariance of operators $A$ and $B$ is $\operatorname{Cov}_\rho(A,B)=\operatorname{Tr}(\rho A\circ B)-\operatorname{Tr}(\rho A)\operatorname{Tr}(\rho B)$ \\
|
||||
\hline
|
||||
Composite system is given by Cartesian product of the sample spaces $\Omega_1\times\Omega_2$ & Composite system is given by tensor product of the Hilbert spaces $\mathscr{H}_1\otimes\mathscr{H}_2$ \\
|
||||
\hline
|
||||
Product measure $\mu_1\times\mu_2$ on $\Omega_1\times\Omega_2$ & Tensor product of space $\rho_1\otimes\rho_2$ on $\mathscr{H}_1\otimes\mathscr{H}_2$ \\
|
||||
\hline
|
||||
Marginal distribution $\pi_*v$ & Partial trace $\operatorname{Tr}_2(\rho)$ \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
}
|
||||
\vspace{0.5cm}
|
||||
\end{table}
|
||||
This agrees with the experimentally observed transmission probabilities, but it should be emphasized that this quantity corresponds to a \emph{sequential measurement} rather than a joint probability in the classical sense.
|
||||
|
||||
\section{Concentration of measure phenomenon}
|
||||
|
||||
@@ -612,6 +190,8 @@ Here is Table~\ref{tab:analog_of_classical_probability_theory_and_non_commutativ
|
||||
|
||||
That basically means that the function $f$ should not change the distance between any two pairs of points in $X$ by more than a factor of $L$.
|
||||
|
||||
This is a stronger condition than continuity, every Lipschitz function is continuous, but not every continuous function is Lipschitz.
|
||||
|
||||
\begin{lemma}
|
||||
\label{lemma:isoperimetric_inequality_on_sphere}
|
||||
Isoperimetric inequality on the sphere:
|
||||
@@ -681,7 +261,7 @@ If $X\sim \operatorname{Unif}(S^n(\sqrt{n}))$, then for any fixed unit vector $x
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.8\textwidth]{./images/maxwell.png}
|
||||
\includegraphics[width=0.8\textwidth]{../images/maxwell.png}
|
||||
\caption{Maxwell-Boltzmann distribution law, image from \cite{romanvershyni}}
|
||||
\label{fig:Maxwell-Boltzmann_distribution_law}
|
||||
\end{figure}
|
||||
@@ -968,16 +548,13 @@ Experimentally, we can have the following result:
|
||||
|
||||
As the dimension of the Hilbert space increases, the chance of getting an almost maximally entangled state increases (see Figure~\ref{fig:entropy_vs_dA}).
|
||||
|
||||
\begin{figure}
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.8\textwidth]{entropy_vs_dA.png}
|
||||
\caption{Entropy vs $d_A$}
|
||||
\label{fig:entropy_vs_dA}
|
||||
\end{figure}
|
||||
|
||||
In Hayden's work, the result is also extended to the multiparty case~\cite{Hayden}, and the result is still under research and I will show the result in the final report if I have enough time.
|
||||
|
||||
|
||||
% When compiled standalone, print this chapter's references at the end.
|
||||
\ifSubfilesClassLoaded{
|
||||
\printbibliography[title={References for Chapter 1}]
|
||||
|
||||
Reference in New Issue
Block a user