This follows (verbatim) the Postulates as written in \cite{Nielsen.Chuang:2010}.

Four postulates are defined to outline the space quantum mechanics exists in. 1., describes the state of a closed system. 2. describes the evolution of such system in time. 3. describes the influence of observations or information gathering from such a system and 4. defines how multiple systems can be combined.

1. Associated to any isolated physical system is a complex vector space with inner product (that is, a Hilbert space) known as the state space of the system. The system is completely described by its state vector, which is a unit vector in the system’s state space.

2. The evolution of a closed quantum system is described by a unitary transformation. That is, the state $\ket{\psi}$ of the system at time t1 is related to the state $\ket{\psi'}$ of the system at time t2 by a unitary operator U which depends only on the times t1 and t2, $\ket{\psi'} = U \ket{\psi}$

2.' The time evolution of the state of a closed quantum system is described by the Schrödinger equation, $$ i\hbar \partial_t \ket(\psi) = H \ket(\psi).$$ In this equation, $\hbar$ is a physical constant known as Planck’s constant whose value must be experimentally determined. The exact value is not important to us. In practice, it is common to absorb the factor $\hbar$ into $H$, effectively setting $\hbar = 1$. H is a fixed Hermitian operator known as the Hamiltonian of the closed system.

3. Quantum measurements are described by a collection ${M_m}$ of measurement operators. These are operators acting on the state space of the system being measured. The index $m$ refers to the measurement outcomes that may occur in the experiment. If the state of the quantum system is $\ket{\psi}$ immediately before the measurement then the probability that result $m$ occurs is given by $$p(m) = \Braket{\psi | M_m^\dag M_m | \psi}$$ and the state of the system after the measurement is $$\frac{M_m \ket{\psi}}{\sqrt{\Braket{\psi | M_m^\dag M_m | \psi}}}.$$ The measurement operators satisfy the completeness equation ¹⁾, $$\sum_m{M_m^\dag M_m} = I$$

4. The state space of a composite physical system is the tensor product of the state spaces of the component physical systems. Moreover, if we have systems numbered $1$ through $n$, and system number $i$ is prepared in the state \ket{\psi_i}, then the joint state of the total system is $\ket{\psi_i} \otimes \ket{\psi_i} \otimes \ldots \otimes \ket{\psi_i}$.