2.1 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.
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}.