📖 Definition:
- State space: a set $S$
- Initial state: fixed elements $s_I\in S$
- Current state: a variable element $s\in S$
- Action: a possible operation or event that can cause a transition between states in the system. It belongs to a set of all possible actions, denoted as Act. Each action is associated with at least one transition in the system, determining how states change.
- Transition: a triple $(s,act,s')$ where $s,s'$ are states an $act$ is an action
- Transition system: $S$ on $(S,Act)$ is just a set of transitions built from $(S,Act)$
💃 Example: a diagram of a state space
💃 Example: equations of motion of an object describe the behaviour from $(\vec x(t),\vec v (t))$ $\to$ $(\vec x(t'),\vec v (t'))$ where $t'>t$ this is an action
📖 Definitions:
Nondeterministic: if for some state $s$ and an action $a$ there are at least 2 transitions $(s,a,s')$ and $(s,a,s'')$ where $s'\ne s''$
Stochastic: each transition $(s,s')$ has an associates probability $p(s,s')$ (nondeterministic)
The probabilities must add up to one as from one state there must be some outcome so
$$ \sum_{s'} p(s,s')=1 $$
Quantum TS: each transition $(s,s')$ has an associated probability amplitude $\psi(s,s')$
Here to get our total probability we have to square the probability amplitude. thus $\psi(s,s')$ can be complex or negative
$$ \sum_{s'}|\psi(s,s')|^2=1 $$
💼 Case: lets consider a sequential composition with $s,s',s''$ as states visited in order
We get
$$ \begin{aligned} \text{Stochastic:} \quad &p(s,s',s'')=p(s,s')\times p(s',s'') \\ \text{Quantum:} \quad &\psi(s,s',s'')=\psi(s,s')\times \psi(s',s'')
\end{aligned} $$
💎 Conclusion: we see that we get the same behaviour for both since $p_q=\psi^2(s,s')\times\psi^2(s',s'')$
💼 Case: composition of alternative, suppose we can move from $s$ to $s'$ via either $u_1$ or $u_2$
For stochastic we get
$$ {\left . \begin{aligned} p(s,u_1,s')&=p(s,u_1) \times p(u_1,s') \\ p(s,u_2,s')&=p(s,u_2)\times p(u_2,s')
\end{aligned} \right \}p(s,s')=p(s,u_1)p(u_1,s')+p(s,u_2)p(u_2,s')} $$
For quantum we have
$$ {\left . \begin{aligned} \psi(s,u_1,s')&=\psi(s,u_1) \times \psi(u_1,s') \\ \psi(s,u_2,s')&=\psi(s,u_2)\times \psi(u_2,s')
\end{aligned} \right \}\psi(s,s')=\psi(s,u_1)\psi(u_1,s')+\psi(s,u_2)\psi(u_2,s')} $$
Now calculating the probability we get
$$ \begin{aligned} p_Q(s,s')&=|\psi(s,u_1)\psi(u_1,s')+\psi(s,u_2)\psi(u_2,s')|^2 \\ &=|\psi(s,u_1)\psi(u_1,s')|^2+|\psi(s,u_2)\psi(u_2,s')|^2 + \psi(s,u_1)\psi(u_1,s')\psi^(s,u_2)\psi^(u_2,s')+\psi^(s,u_1)\psi^(u_1,s')\psi(s,u_2)\psi(u_2,s') \end{aligned}
$$
💎 Conclusion: $p(s,s')\ne p_Q(s,s')$ ie we can get positive and destructive interference in Quantum
Stochastic:
Quantum
We start by defining a complex inner product space $H$. The following conditions must be satisfied
$$ \begin{aligned} (1)\;&|\psi\rangle, |\varphi\rangle \in H \Rightarrow |\psi\rangle + |\varphi\rangle = |\psi + \varphi\rangle = |\varphi\rangle + |\psi\rangle \in H \\ (2)\;&|\psi\rangle, |\varphi\rangle, |\zeta\rangle \in H \Rightarrow (|\psi\rangle + |\varphi\rangle) + |\zeta\rangle = |\psi\rangle + (|\varphi\rangle + |\zeta\rangle) \\ (3)\;&0 \in H \text{ and } 0 + |\psi\rangle = |\psi\rangle \in H \\ (4)\; &|\psi\rangle \in H, \lambda \in \mathbb{C} \Rightarrow |\lambda \psi\rangle = \lambda |\psi\rangle \in H \\ (5)\; &|\psi\rangle \in H, \lambda, \mu \in \mathbb{C} \Rightarrow |\lambda \mu \psi\rangle = \lambda |\mu \psi\rangle = \lambda \mu |\psi\rangle \in H \\ (6)\; &|\psi\rangle, |\varphi\rangle \in H, \lambda \in \mathbb{C} \Rightarrow \lambda (|\psi\rangle + |\varphi\rangle) = \lambda |\psi\rangle + \lambda |\varphi\rangle \in H \\ (7)\; &|\psi\rangle \in H, \lambda, \mu \in \mathbb{C} \Rightarrow (\lambda + \mu)|\psi\rangle = \lambda |\psi\rangle + \mu |\psi\rangle \in H \\ (8)\; & |\psi\rangle, |\varphi\rangle \in H \Rightarrow \langle \varphi | \psi \rangle \in \mathbb{C} \\ (9)\; & \langle \psi | \varphi \rangle = \langle \varphi | \psi \rangle^* \\ (10)\; & \langle \zeta | (|\psi\rangle + |\varphi\rangle) = \langle \zeta | \psi \rangle + \langle \zeta | \varphi \rangle \\ (11) \; & \mu \in \mathbb{C} \Rightarrow \langle \psi | \mu \varphi \rangle = \mu \langle \psi | \varphi \rangle = \langle \mu^* \psi | \varphi \rangle \\ (12) \; & \langle \psi | \psi \rangle \geq 0 \text{ and } \langle \psi | \psi \rangle = 0 \Rightarrow |\psi\rangle = 0 \end{aligned} $$
<aside> 🐧
The norm of $\ket{\psi}$ is defined as $||\psi|| =\sqrt{\braket{\psi|\psi}}$
The norm has properties:
$||\psi||\ge 0$
$||\lambda\psi||=|\lambda|\cdot||\psi||$
if $||\psi||=0$ then $\ket{\psi} =0$ </aside>
🗒️ Note: If $\braket{\psi|\varphi}=0$ then $\ket{\psi}$ and $\ket{\varphi}$ are orthogonal
