Notes - Quantum Information HT24, Rules of quantum theory
These are the rules as described by the lecture notes for Oxford Quantum Information course, I don’t think the numbering “rule 1”, “rule 2”, etc. is standard.
Flashcards
Can you state rule 1 of quantum theory, about the pure states of a quantum system?
A quantum system with $d < \infty$ perfectly distinguishable states is associated to the vector space $\mathcal H = \mathbb C^d$.
The pure states of the system are represented by unit vectors, up to global phase.
Can you state rule 2 of quantum theory, about the basic measurements?
The basic measurements on a $d$-dimensional quantum system are represented by ONBs in $\mathbb C^d$, up to global phase.
If a system is in the state $ \vert \psi\rangle$ and we perform the basic measurement with ONB $\{ \vert \psi _ n \rangle, n = 0, \cdots, d- 1\}$, then the probability we obtain the outcome $n$ is
\[p_n = |\langle \psi_n | \psi\rangle|^2\]If we obtain the outcome $n$, then the state of the system immediately after the measurement is $ \vert \psi _ n\rangle$.
Can you state rule 3 of quantum theory, about the reversible quantum processes?::
The reversible processes on a given $d$-dimensional quantum system are described by $d\times d$ unitary matrices up to a global phase.
If the system is initially in the state $ \vert \psi\rangle$ and the reversible process described by the unitary matrix $U$ is applied, then the state of the system at the end of the process is $U \vert \psi\rangle$.
Can you state rule 4 of quantum theory, about composite systems?
Let $\mathcal H _ A$ and $\mathcal H _ B$ be the Hilbert spaces of two quantum systems $A$ and $B$. Then the Hilbert space of the composiste system $AB$ is $\mathcal H _ {AB} := \mathcal H _ A \otimes \mathcal H _ B$.
If system $A$ is in the state $ \vert \alpha\rangle \in \mathcal H _ A$ and system $B$ is in the state $ \vert \beta\rangle \in \mathcal H _ B$, then the system $AB$ is in the state $ \vert \alpha\rangle \otimes \vert \beta\rangle$.
Measurements on two systems are described by $(\langle \alpha _ n \vert \otimes \langle \beta _ n \vert )( \vert \alpha\rangle \otimes \vert \beta\rangle)$.
If system $A$ undergoes reversible process $U _ A$ and system $B$ undergoes the reversible prorcess $U _ B$, then system $AB$ undergoes the reversible gate $U _ A \otimes U _ B$.