From d956004179f6591c59d0388c27766b1df595b261 Mon Sep 17 00:00:00 2001 From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com> Date: Fri, 20 Jun 2025 17:01:08 -0500 Subject: [PATCH] Update Math401_T4.md --- pages/Math401/Math401_T4.md | 125 +++++++++++++++++++++++++++++++----- 1 file changed, 110 insertions(+), 15 deletions(-) diff --git a/pages/Math401/Math401_T4.md b/pages/Math401/Math401_T4.md index 555ee26..d656154 100644 --- a/pages/Math401/Math401_T4.md +++ b/pages/Math401/Math401_T4.md @@ -2,14 +2,14 @@ > In mathematics, on often speaks of non-commutative instead of quantum constructions. +**Note, in this section, we will see a lot of mixed used terms used in physics and mathematics. I will use _italic_ to denote the terminology used in physics. It is safe to ignore them if you just care about the mathematics.** + ## Section 1: Generalities about classical versus quantum systems -In classical physics, we assume that a systema's properties have well-defined values regardless of how we choose to measure them. +In classical physics, we assume that a system's properties have well-defined values regardless of how we choose to measure them. ### Basic terminology -#### Observables - #### Set of states The preparation of a system builds a convex set of states as our initial condition for the system. @@ -24,17 +24,17 @@ The set of effects is the set of all possible outcomes of a measurement. $\Omega A pair of state and effect determines a probability $E_i(P)=p(\omega_i|P)$. By the law of large numbers, this probability shall converge to $N(\omega_i)/N$ as $N$ increases. -**Quantum states, observables, and effects can be represented mathematically by linear operators on a Hilbert space.** +**Quantum states, _observables_ (random variables), and effects can be represented mathematically by linear operators on a Hilbert space.** ## Section 2: Examples of physical experiment in language of mathematics ### Sten-Gernach experiment -**State preparation:** Silver tams are emitted from a thermal source and collimated to form a beam. +_**State preparation:**_ Silver tams are emitted from a thermal source and collimated to form a beam. -**Measurement:** Silver atoms interact with the field produced by the magnet and impinges on the class plate. +_**Measurement:**_ Silver atoms interact with the field produced by the magnet and impinges on the class plate. -**Registration:** The impression left on the glass pace by the condensed silver atoms. +_**Registration:**_ The impression left on the glass pace by the condensed silver atoms. ## Section 3: Finite probability spaces in the language of Hilbert space and operators @@ -44,9 +44,13 @@ A quantum coin can be represented mathematically by linear combination of $|0\ra > For the rest of the material, we shall take the $\mathscr{H}$ to be vector space over $\mathbb{C}$. -### Rewrite the language of probability +### Definitions in classical probability under generalized probability theory -To each event $A\in \Omega$, we associate the operator on $\mathscr{H}$ of multiplication by the indicator function $M_{\mathbb{I}_A}:f\mapsto \mathbb{I}_A f$ that projects onto the subspace of $\mathscr{H}$ corresponding to the event $A$. +#### Definition of states (classical probability) + +[definition of states continue here.] + +To each event $A\in \Omega$, we associate the operator on $\mathscr{H}$ of multiplication by the indicator function $P_A\coloneqq M_{\mathbb{I}_A}:f\mapsto \mathbb{I}_A f$ that projects onto the subspace of $\mathscr{H}$ corresponding to the event $A$. $$ P_A=\sum_{k=1}^n a_k|k\rangle\langle k| @@ -54,7 +58,7 @@ $$ where $a_k\in\{0,1\}$, and $a_k=1$ if and only if $k\in A$. Note that $P_A^*=P_A$ and $P_A^2=P_A$. -#### Density operator +#### Definition of density operator (classical probability) Let $(p_1,p_2,\cdots,p_n)$ be a probability distribution on $X$, where $p_k\geq 0$ and $\sum_{k=1}^n p_k=1$. The density operator $\rho$ is defined by @@ -65,10 +69,10 @@ $$ The probability of event $A$ relative to the probability distribution $(p_1,p_2,\cdots,p_n)$ becomes the trace of the product of $\rho$ and $P_A$. $$ -\operatorname{Prob}_\rho(A)=\text{Tr}(\rho P_A) +\operatorname{Prob}_\rho(A)\coloneqq\text{Tr}(\rho P_A) $$ -#### Random variables +#### Definition of random variables (classical probability) A random variable is a function $f:X\to\mathbb{R}$. It can also be written in operator form: @@ -82,19 +86,110 @@ $$ \mathbb{E}_\rho(f)=\sum_{k=1}^n p_k f(k)=\operatorname{Tr}(\rho F) $$ -Note, by our definition of the operator $F,P_A,\rho$ (all diagonal operators) commute among themselves, which does not hold in general quantum theory. +Note, by our definition of the operator $F,P_A,\rho$ (all diagonal operators) commute among themselves, which does not hold in general, in non-commutative (_quantum_) theory. ## Section 4: Why we need generalized probability theory to study quantum systems -Story of light polarization. +Story of light polarization and violation of Bell's inequality. ### Classical picture of light polarization and Bell's inequality > An interesting story will be presented here. -## Section 5: The quantum probability theory +## Section 5: The non-commutative (_quantum_) probability theory Let $\mathscr{H}$ be a Hilbert space. $\mathscr{H}$ consists of complex-valued functions on a finite set $\Omega=\{1,2,\cdots,n\}$. and that 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$. In classical settings, multiplication operators is now be the full space of bounded linear operators on $\mathscr{H}$. (Denoted by $\mathscr{B}(\mathscr{H})$) +Let $A,B\in\mathscr{F}$ be the set of all events in the classical probability settings. $X$ denotes the set of all possible outcomes. + +> A orthogonal projection on a Hilbert space is a projection operator satisfying $P^*=P$ and $P^2=P$. We denote the set of all orthogonal projections on $\mathscr{H}$ by $\mathscr{P}$. +> +> This can be found in linear algebra. [Orthogonal projection](https://notenextra.trance-0.com/Math429/Math429_L28#definition-655) + +Let $P,Q\in\mathscr{P}$ be the event in probability space. $R(\cdot)$ is the range of the operator. $P^\perp$ is the orthogonal complement of $P$. + +| Classical | Classical interpretation | Non-commutative (_Quantum_) | Non-commutative (_Quantum_) interpretation | +| --------- | ------- | -------- | -------- | +| $A\subset B$| Event $A$ is a subset of event $B$ | $P\leq Q$| $R(P)\subseteq R(Q)$ Range of event $P$ is a subset of range of event $Q$ | +| $A\cap B$| Both event $A$ and $B$ happened | $P\land Q$| projection to $R(P)\cap R(Q)$ Range of event $P$ and event $Q$ happened | +| $A\cup B$| Any of the event $A$ or $B$ happened | $P\lor Q$| projection to $R(P)\cup R(Q)$ Range of event $P$ or event $Q$ happened | +| $X\subset A$ or $A^c$| Event $A$ did not happen | $P^\perp$| projection$R(P)^\perp$ Range of event $P$ is the orthogonal complement of range of event $P$ | + +In such setting, some rules of classical probability theory are not valid in quantum probability theory. + +In classical probability theory, $A\cap(B\cup C)=(A\cap B)\cup(A\cap C)$. + +In quantum probability theory, $P\land(Q\lor R)\neq(P\land Q)\lor(P\land R)$ in general. + +### Definitions of non-commutative (_quantum_) probability theory under generalized probability theory + +#### Definition of states (non-commutative (_quantum_) probability theory) + +A state on $(\mathscr{H},\mathscr{P})$ is a map $\mu:\mathscr{P}\to[0,1]$ such that: + +1. $\mu(O)=0$, where $O$ is the zero projection. +2. If $P_1,P_2,\cdots,P_n$ are pairwise disjoint orthogonal projections, then $\mu(P_1\lor P_2\lor\cdots\lor P_n)=\sum_{i=1}^n\mu(P_i)$. + +Where projections are disjoint if $P_iP_j=P_jP_i=O$. + +#### Definition of density operator (non-commutative (_quantum_) probability theory) + +A density operator $\rho$ on the finite-dimensional Hilbert space $\mathscr{H}$ is: + +1. self-adjoint ($A^*=A$, that is $\langle Ax,y\rangle=\langle x,Ay\rangle$ for all $x,y\in\mathscr{H}$) +2. positive semi-definite (all eigenvalues are non-negative) +3. $\operatorname{Tr}(\rho)=1$. + +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 eigenvalue $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| +$$ + +(under basis $|\psi_j\rangle$, it is a diagonal matrix with $p_j$ on the diagonal) + +#### Theorem: Born's rule + +Let $\rho$ be a density operator on $\mathscr{H}$. then + +$$ +\mu(P)\coloneqq\operatorname{Tr}(\rho P)=\sum_{j=1}^n p_j\langle\psi_j|P|\psi_j\rangle +$$ + +Defines a probability measure on the space $\mathscr{P}$. + +[Proof ignored here] + +#### Theorem: Gleason's theorem + +Let $\mathscr{H}$ be a Hilbert space over $\mathbb{C}$ or $\mathbb{R}$ of dimension $n\geq 3$. Let $\mu$ be a state on the space $\mathscr{P}$ of projections on $\mathscr{H}$. Then there exists a unique density operator $\rho$ such that + +$$ +\mu(P)=\operatorname{Tr}(\rho P) +$$ + +for all $P\in\mathscr{P}$. $\mathscr{P}$ is the space of all orthogonal projections on $\mathscr{H}$. + +[Proof ignored here] + +#### Definition of random variable _or Observables_ (non-commutative (_quantum_) probability theory) + +_It is the physical measurement of a system that we are interested in. (kinetic energy, position, momentum, etc.)_ + +$\mathscr{B}(\mathbb{R})$ is the set of all Borel sets on $\mathbb{R}$. + +An random variable on the Hilbert space $\mathscr{H}$ is a projection valued map $P:\mathscr{B}(\mathbb{R})\to\mathscr{P}$. + +With the following properties: + +1. $P(\emptyset)=O$ (the zero projection) +2. $P(\mathbb{R})=I$ (the identity projection) +3. For any sequence $A_1,A_2,\cdots,A_n\in \mathscr{B}(\mathbb{R})$. the following holds: + (a) $P(\bigcup_{i=1}^n A_i)=\bigvee_{i=1}^n P(A_i)$ + (b) $P(\bigcap_{i=1}^n A_i)=\bigwedge_{i=1}^n P(A_i)$ + (c) $P(A^c)=I-P(A)$ + (d) 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 $P(\bigcup_{j=1}^n A_j)=\sum_{j=1}^n P(A_j)$