Lecture 2 of 4
Quantum Techniques for Stochastic Mechanics, by Jacob Biamonte, QIC 890/891 Selected Advanced Topics in Quantum Information, The University of Waterloo, Waterloo Ontario, Canada, (Spring term 2012).
Given Aug 9th, 2012 in Waterloo Canada
Review of lecture 1
Probabilities vs amplitudes (chapter 4)
Operators (creation and annihilation)
Amoeba field theory
Preamble to stochastic mechanics vs quantum mechanics
Note: the lecture content here is to be read along with the youtube video of the course. It should contain what was written on the board. Without the video, it might not make sense.
A stochastic petri net has
we want to know
one for each $1\leq i \leq k$.
For each transition we end up with a term like this
We will use index free notation.
is the concentration vector and
input vector: $m = (m_1, \ldots, m_k) \in N^k$
output vector: $n = (n_1, \ldots, n_k) \in N^k$
notation: $x^m = x_1^{m_1}\cdots x_k^{m_k}$
then
Example
In birth, one rabbit comes in and two go out. This is a caricature of reality: these bunnies reproduce asexually, splitting in two like amoebas.
In predation, one wolf and one rabbit come in and two wolves go out. This is a caricature of how predators need to eat prey to reproduce.
In death, one wolf comes in and nothing goes out. Note that we are pretending rabbits don’t die unless they’re eaten by wolves.
$R(t)$ rabbits
$W(t)$ wolves
The rate equations of motion are
Let $\psi_{n_1, \ldots, n_k}$ be the probability that we have $n_1$ of the first thing, $n_2$ of the second, etc.
The master equation says how these “things” change with time
It contains all possible “histories” of the possible interactions
Let $n = (n_1, \ldots, n_k) \in N^k$ and $\psi_n = \psi_{n_1}, \ldots, \psi_{n_k}$ and then write a monomial
and express any stochastic state as
A state has
Quantum Techniques for Stochastic Mechanics, John Baez and Jacob Biamonte, (2012).