196 lines
9.0 KiB
Markdown
196 lines
9.0 KiB
Markdown
# Topic 4: The quantum version of probabilistic concepts
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> In mathematics, on often speaks of non-commutative instead of quantum constructions.
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**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.**
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## Section 1: Generalities about classical versus quantum systems
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In classical physics, we assume that a system's properties have well-defined values regardless of how we choose to measure them.
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### Basic terminology
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#### Set of states
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The preparation of a system builds a convex set of states as our initial condition for the system.
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For a collection of $N$ system. Let procedure $N_1=\lambda P_1$ be a preparation procedure for state $P_1$, and $N_2=(1-\lambda) P_2$ be a preparation procedure for state $P_2$. The state of the collection is $N=\lambda N_1+(1-\lambda) N_2$.
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#### Set of effects
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The set of effects is the set of all possible outcomes of a measurement. $\Omega=\{\omega_1, \omega_2, \ldots, \omega_n\}$. Where each $\omega_i$ is an associated effect, or some query problems regarding the system. (For example, is outcome $\omega_i$ observed?)
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#### Registration of outcomes
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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.
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**Quantum states, _observables_ (random variables), and effects can be represented mathematically by linear operators on a Hilbert space.**
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## Section 2: Examples of physical experiment in language of mathematics
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### Sten-Gernach experiment
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_**State preparation:**_ Silver tams are emitted from a thermal source and collimated to form a beam.
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_**Measurement:**_ Silver atoms interact with the field produced by the magnet and impinges on the class plate.
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_**Registration:**_ The impression left on the glass pace by the condensed silver atoms.
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## Section 3: Finite probability spaces in the language of Hilbert space and operators
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> Superposition is a linear combination of two or more states.
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A quantum coin can be represented mathematically by linear combination of $|0\rangle$ and $|1\rangle$.$\alpha|0\rangle+\beta|1\rangle\in\mathscr{H}\cong\mathbb{C}^2$.
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> For the rest of the material, we shall take the $\mathscr{H}$ to be vector space over $\mathbb{C}$.
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### Definitions in classical probability under generalized probability theory
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#### Definition of states (classical probability)
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[definition of states continue here.]
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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 $A$.
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$$
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P_A=\sum_{k=1}^n a_k|k\rangle\langle k|
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$$
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where $a_k\in\{0,1\}$, and $a_k=1$ if and only if $k\in A$. Note that $P_A^*=P_A$ and $P_A^2=P_A$.
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#### Definition of density operator (classical probability)
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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
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$$
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\rho=\sum_{k=1}^n p_k|k\rangle\langle k|
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$$
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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 $P_A$.
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$$
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\operatorname{Prob}_\rho(A)\coloneqq\text{Tr}(\rho P_A)
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$$
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#### Definition of random variables (classical probability)
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A random variable is a function $f:X\to\mathbb{R}$. It can also be written in operator form:
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$$
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F=\sum_{k=1}^n f(k)P_{\{k\}}
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$$
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The expectation of $f$ relative to the probability distribution $(p_1,p_2,\cdots,p_n)$ is given by
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$$
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\mathbb{E}_\rho(f)=\sum_{k=1}^n p_k f(k)=\operatorname{Tr}(\rho F)
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$$
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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.
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## Section 4: Why we need generalized probability theory to study quantum systems
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Story of light polarization and violation of Bell's inequality.
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### Classical picture of light polarization and Bell's inequality
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> An interesting story will be presented here.
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## Section 5: The non-commutative (_quantum_) probability theory
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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 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 $e_k$.
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In classical settings, multiplication operators is now be the full space of bounded linear operators on $\mathscr{H}$. (Denoted by $\mathscr{B}(\mathscr{H})$)
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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.
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> 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 $\mathscr{P}$.
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>
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> This can be found in linear algebra. [Orthogonal projection](https://notenextra.trance-0.com/Math429/Math429_L28#definition-655)
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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 $P$.
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| Classical | Classical interpretation | Non-commutative (_Quantum_) | Non-commutative (_Quantum_) interpretation |
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| --------- | ------- | -------- | -------- |
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| $A\subset B$| 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 $Q$ |
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| $A\cap B$| Both event $A$ and $B$ happened | $P\land Q$| projection to $R(P)\cap R(Q)$ Range of event $P$ and event $Q$ happened |
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| $A\cup B$| 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 |
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| $X\subset A$ 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 $P$ |
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In such setting, some rules of classical probability theory are not valid in quantum probability theory.
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In classical probability theory, $A\cap(B\cup C)=(A\cap B)\cup(A\cap C)$.
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In quantum probability theory, $P\land(Q\lor R)\neq(P\land Q)\lor(P\land R)$ in general.
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### Definitions of non-commutative (_quantum_) probability theory under generalized probability theory
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#### Definition of states (non-commutative (_quantum_) probability theory)
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A state on $(\mathscr{H},\mathscr{P})$ is a map $\mu:\mathscr{P}\to[0,1]$ such that:
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1. $\mu(O)=0$, where $O$ is the zero projection.
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2. If $P_1,P_2,\cdots,P_n$ are pairwise disjoint orthogonal projections, then $\mu(P_1\lor P_2\lor\cdots\lor P_n)=\sum_{i=1}^n\mu(P_i)$.
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Where projections are disjoint if $P_iP_j=P_jP_i=O$.
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#### Definition of density operator (non-commutative (_quantum_) probability theory)
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A density operator $\rho$ on the finite-dimensional Hilbert space $\mathscr{H}$ is:
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1. self-adjoint ($A^*=A$, that is $\langle Ax,y\rangle=\langle x,Ay\rangle$ for all $x,y\in\mathscr{H}$)
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2. positive semi-definite (all eigenvalues are non-negative)
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3. $\operatorname{Tr}(\rho)=1$.
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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 eigenvalue $p_1,p_2,\cdots,p_n$, then $p_j\geq 0$ and $\sum_{j=1}^n p_j=1$.
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We can write $\rho$ as
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$$
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\rho=\sum_{j=1}^n p_j|\psi_j\rangle\langle\psi_j|
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$$
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(under basis $|\psi_j\rangle$, it is a diagonal matrix with $p_j$ on the diagonal)
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#### Theorem: Born's rule
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Let $\rho$ be a density operator on $\mathscr{H}$. then
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$$
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\mu(P)\coloneqq\operatorname{Tr}(\rho P)=\sum_{j=1}^n p_j\langle\psi_j|P|\psi_j\rangle
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$$
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Defines a probability measure on the space $\mathscr{P}$.
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[Proof ignored here]
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#### Theorem: Gleason's theorem
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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
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$$
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\mu(P)=\operatorname{Tr}(\rho P)
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$$
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for all $P\in\mathscr{P}$. $\mathscr{P}$ is the space of all orthogonal projections on $\mathscr{H}$.
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[Proof ignored here]
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#### Definition of random variable _or Observables_ (non-commutative (_quantum_) probability theory)
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_It is the physical measurement of a system that we are interested in. (kinetic energy, position, momentum, etc.)_
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$\mathscr{B}(\mathbb{R})$ is the set of all Borel sets on $\mathbb{R}$.
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An random variable on the Hilbert space $\mathscr{H}$ is a projection valued map $P:\mathscr{B}(\mathbb{R})\to\mathscr{P}$.
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With the following properties:
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1. $P(\emptyset)=O$ (the zero projection)
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2. $P(\mathbb{R})=I$ (the identity projection)
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3. For any sequence $A_1,A_2,\cdots,A_n\in \mathscr{B}(\mathbb{R})$. the following holds:
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(a) $P(\bigcup_{i=1}^n A_i)=\bigvee_{i=1}^n P(A_i)$
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(b) $P(\bigcap_{i=1}^n A_i)=\bigwedge_{i=1}^n P(A_i)$
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(c) $P(A^c)=I-P(A)$
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(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 $P(\bigcup_{j=1}^n A_j)=\sum_{j=1}^n P(A_j)$
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