4.1 KiB
Topic 4: The quantum version of probabilistic concepts
In mathematics, on often speaks of non-commutative instead of quantum constructions.
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.
Basic terminology
Observables
Set of states
The preparation of a system builds a convex set of states as our initial condition for the system.
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.
Set of effects
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?)
Registration of outcomes
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.
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.
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.
Section 3: Finite probability spaces in the language of Hilbert space and operators
Superposition is a linear combination of two or more states.
A quantum coin can be represented mathematically by linear combination of |0\rangle and |1\rangle.$\alpha|0\rangle+\beta|1\rangle\in\mathscr{H}\cong\mathbb{C}^2$.
For the rest of the material, we shall take the
\mathscr{H}to be vector space over\mathbb{C}.
Rewrite the language of probability
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.
P_A=\sum_{k=1}^n a_k|k\rangle\langle k|
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
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
\rho=\sum_{k=1}^n p_k|k\rangle\langle k|
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)
Random variables
A random variable is a function f:X\to\mathbb{R}. It can also be written in operator form:
F=\sum_{k=1}^n f(k)P_{\{k\}}
The expectation of f relative to the probability distribution (p_1,p_2,\cdots,p_n) is given by
\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.
Section 4: Why we need generalized probability theory to study quantum systems
Story of light polarization.
Classical picture of light polarization and Bell's inequality
An interesting story will be presented here.
Section 5: The 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}))