B1 - postulates

postulate 1: state

the state of an isolated quantum system is described by a unit vector in a hilbert space

| ψ ⟩ = (\begin{matrix} c_{1} \\ ⋮ \\ c_{2} \end{matrix})

where, $c_{i} \in C$ and $| ψ ⟩ \in C^{n}$

postulate 2: evolution

the evolution of a closed quantum system is described by a unitary operator

| ψ (t_{2}) ⟩ = U (t_{1}, t_{2}) | ψ (t_{1}) ⟩

where, $U^{†} U = U U^{†} = I$

postulate 3: measurement

a measurement is described by an observable $M$ , a hermitian operator on the system's hilbert space
if $M = M^{2}$ , it is a projective measurement
if $M = \sum_{m} λ_{m} P_{m}$ , the probability of getting $λ_{m}$ on the state $| ψ ⟩$ is:

p (m) = ⟨ ψ | P_{m} | ψ ⟩

the corresponding state of the system after that is:

\frac{P_{m} | ψ ⟩}{\sqrt{p (m)}}

postulate 4: composite systems

the hilbert space, $H$ , of a composite quantum system is the tensor product of the hilbert spaces, $H_{i}$ of the component quantum systems

H = H_{1} \otimes H_{2} \otimes \dots \otimes H_{n} = \otimes_{i = 1}^{n} H_{n}

a quantum state of the component system $ψ \in H$ is not necessarily a tensor product of the component systems
states that are of this form are called product states, and those that are not are called entangled states
the first three postulates are about a single quantum system, and determine the behaviour of transistors
the fourth postulate is about composite quantum systems
quantum computers rely on all of them