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@@ -335,19 +335,21 @@ $$
Let $\rho$ be a state on $\mathscr{B}$ consisting of orthonormal basis $\{v_j\}$ and eigenvalue $\{\lambda_j\}$.
The partial trace of $T$ with respect to $\rho$ is the linear operator on $\mathscr{A}$ defined by
$$
\operatorname{Tr}_{\mathscr{A}}(T)=\sum_{j} \lambda_j L^*_{v_j}(T)L_{v_j}
$$
\end{defn}
The partial trace of $T$ with respect to $\rho$ is the linear operator on $\mathscr{A}$ defined by
$$
\operatorname{Tr}_{\mathscr{B},\rho}(T)=\sum_{j} \lambda_j L^*_{v_j}(T)L_{v_j}
$$
\end{defn}
This introduces a new model in mathematics explaining quantum mechanics: the non-commutative probability theory.
\section{Non-commutative probability theory}
The non-commutative probability theory is a branch of generalized probability theory that studies the probability of events in non-commutative algebras.
\section{Non-commutative probability theory}
The constructions above explain why tensor products and traces appear before probability is mentioned again: they are the algebraic devices that let composite quantum systems behave like probabilistic systems with marginals and expectations. The next section packages these operations into the operator-theoretic language of states, observables, and expectation values, which is the setting used later for random quantum states and entropy.
The non-commutative probability theory is a branch of generalized probability theory that studies the probability of events in non-commutative algebras.
There are several main components of the generalized probability theory; let's see how we can formulate them, comparing with the classical probability theory.
@@ -368,14 +370,14 @@ As a side note we will use later, we also defined the Borel measure on a space,
\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}
\item $X \in \mathscr{B}$.
\item Close under complement: If $A\subseteq X$, then $\mu(A^c)=\mu(X)-\mu(A)$
\item Close under countable unions; If $E_1,E_2,\cdots$ are disjoint sets, then $\mu(\bigcup_{i=1}^\infty E_i)=\sum_{i=1}^\infty \mu(E_i)$
\end{enumerate}
\end{defn}
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}
\item $X \in \mathscr{B}(X)$.
\item Close under complement: If $A\in \mathscr{B}(X)$, then $A^c\in \mathscr{B}(X)$.
\item Close under countable unions: If $E_1,E_2,\cdots$ are disjoint Borel sets, then $\mu(\bigcup_{i=1}^\infty E_i)=\sum_{i=1}^\infty \mu(E_i)$.
\end{enumerate}
\end{defn}
In later sections, we will use Lebesgue measure, and Haar measure for various circumstances, their detailed definition may be introduced in later sections.
@@ -832,23 +834,25 @@ Here is Table~\ref{tab:analog_of_classical_probability_theory_and_non_commutativ
\vspace{0.5cm}
\end{table}
\section{Manifolds}
In this section, we will introduce some basic definitions and theorems used in manifold theory that are relevant to our study. Assuming no prior knowledge of manifold theory but basic topology understanding. We will provide brief definitions and explanations for each term. From the most abstract Manifold definition to the Riemannian manifolds and related theorems.
\section{Manifolds}
Up to this point the emphasis has been algebraic and probabilistic. The concentration results used later, however, live naturally on curved spaces equipped with metrics and measures. For that reason the discussion now shifts from operator theory to manifold theory, starting with topological manifolds and then adding smooth and Riemannian structure until we can describe complex projective space as a genuine geometric state space.
In this section, we will introduce some basic definitions and theorems used in manifold theory that are relevant to our study. Assuming no prior knowledge of manifold theory but basic topology understanding. We will provide brief definitions and explanations for each term. From the most abstract Manifold definition to the Riemannian manifolds and related theorems.
\subsection{Manifolds}
\begin{defn}
\label{defn:m-manifold}
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}
\end{defn}
An $m$-manifold is a topological space $X$ that is
\begin{enumerate}
\item Hausdorff: every distinct two points in $X$ can be separated by two disjoint open sets.
\item Second countable: $X$ has a 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}
\begin{examples}
@@ -937,7 +941,7 @@ $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$.
Because $\sum_{i=1}^n \varphi_i(x_1)=1$, therefore there exists $\varphi_i(x_1)=\varphi_i(x_2)>0$.
Therefore $x1,x_2\in \operatorname{supp}(\phi_i)\subseteq U_i$.
@@ -961,9 +965,11 @@ Therefore $F$ is a homeomorphism.
\end{enumerate}
\end{proof}
\subsection{Smooth manifolds and Lie groups}
This section is adopted from \cite{lee_introduction_2012}
\subsection{Smooth manifolds and Lie groups}
This section is adopted from \cite{lee_introduction_2012}
The topological definition of a manifold tells us what the space looks like locally, but not how to differentiate on it. The next step is therefore to add charts with smooth transition maps. Once this smooth structure is available, notions such as differentials, submersions, and group actions can be stated precisely, and these are exactly the tools needed later for the Hopf fibration.
\begin{defn}
\label{defn:partial_derivative}
@@ -987,7 +993,7 @@ This section is adopted from \cite{lee_introduction_2012}
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.)
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 a $C^1$ map. (Note that $C^0$ map is just a continuous map.)
\end{defn}
@@ -1029,12 +1035,12 @@ This section is adopted from \cite{lee_introduction_2012}
$$
\end{defn}
\begin{defn}
\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}
\begin{defn}
\label{defn:smooth-submersion}
A smooth map $f:M\to N$ is a \textbf{smooth submersion} if for each $p\in M$, the differential $df_p:T_pM\to T_{f(p)}N$ is surjective.
Or equivalently $\operatorname{rank}(df_p)=\dim N$ for each $p\in M$.
\end{defn}
Here are some additional propositions that will be helpful for our study in later sections:
@@ -1046,10 +1052,12 @@ This one is from \cite{lee_introduction_2012} Theorem 4.26
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}
\subsection{Riemannian manifolds}
Smooth manifolds still do not measure lengths, angles, or volumes. To connect the manifold side of the thesis with concentration of measure, we need a metric structure that turns local smooth data into global geometric data. Riemannian metrics provide exactly that extra layer, and Riemannian submersions will later transfer this geometry from spheres to complex projective spaces.
\begin{defn}
\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 $T_pM$.
@@ -1057,9 +1065,9 @@ This one is from \cite{lee_introduction_2012} Theorem 4.26
\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 $T_{\pi(x)}M$.
\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 $d\pi_x:T_x\tilde{M}\to T_{\pi(x)}M$ 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 $v,w\in H_x$.
\end{defn}
@@ -1190,9 +1198,11 @@ This one is from \cite{lee_introduction_2012} Theorem 4.26
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:
\subsection{Hopf fibration}
The previous subsection gives the abstract mechanism for pushing a metric through a quotient map. The Hopf fibration is the concrete instance needed in this thesis: it explains why the geometry of the sphere descends to complex projective space, and therefore why concentration on spheres is relevant to the geometry of pure quantum states.
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
@@ -1235,16 +1245,18 @@ $$
$$
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.
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}
The geometric discussion above identifies the right state space, but the thesis ultimately studies physical observables on that space. We now return to quantum terminology and translate the geometric objects into the language of states, measurements, Haar sampling, and reduced density matrices. This is the point where the manifold picture and the operator picture meet.
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.
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.
@@ -1252,12 +1264,12 @@ One might ask, what is the fundamental difference between a quantum system and a
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}
\end{defn}
\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 reduced state on the remaining subsystem.
\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 $\mathbb{C}$,
@@ -1271,14 +1283,14 @@ The uniqueness of such measurement came from the lemma below~\cite{Elizabeth_boo
\begin{lemma}
\label{lemma:haar_measure}
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}
Let $(U(n), \| \cdot \|, \mu)$ be a metric measure space where $\| \cdot \|$ is the Hilbert-Schmidt norm and $\mu$ is a Borel probability measure.
The Haar measure on $U(n)$ is the unique probability measure that is invariant under the action of $U(n)$ on itself.
That is, for every Borel set $E\subseteq U(n)$ and every $A\in U(n)$, $\mu(AE)=\mu(EA)=\mu(E)$.
The Haar measure is the unique probability measure that is invariant under the action of $U(n)$ on itself.
\end{lemma}
A finite-dimensional quantum system is modeled by a complex Hilbert space (a complete inner product space)
@@ -1348,9 +1360,11 @@ $$
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.
\subsection{Random quantum states}
First, we need to define what is a random state in a bipartite system.
\subsection{Random quantum states}
The preceding material identifies the spaces and symmetries of quantum states. The next step is probabilistic: once Haar invariance is available, we can speak about random pure states and random mixed states in a way that matches the geometric viewpoint developed earlier. These definitions are the starting point for the concentration statements proved in the next chapter.
First, we need to define what is a random state in a bipartite system.
@@ -1363,14 +1377,14 @@ First, we need to define what is a random state in a bipartite system.
\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.
It is trivial that for the space of pure state, we can easily apply the Haar measure as the unitarily invariant probability measure by sampling unit vectors on $S^n$ for some $n$. However, for the case of mixed states, that is a bit complicated and we need to use partial tracing to define 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}
For a system $A$ and an integer $s\geq 1$, consider the distribution on the mixed states $\mathcal{S}(A)$ of $A$ induced by the partial trace over the second factor from 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 state, denoted by $\rho\in_R \mathcal{S}_s(A)$. And $\mathcal{P}(A)=\mathcal{S}_1(A)$.
\end{defn}
% When compiled standalone, print this chapter's references at the end.