A **quantum state** is any possible state in which a quantum mechanical system can be. A fully specified quantum state can be described by a *state vector*, a wavefunction, or a complete set of quantum numbers for a specific system. A partially known quantum state, such as an ensemble with some quantum numbers fixed, can be described by a density operator.

Paul A. M. Dirac invented a powerful and intuitive mathematical notation to describe quantum states, known as bra-ket notation.

### Basis states

Any quantum state ∣*ψ*⟩ can be expressed in terms of a sum of *basis states* (also called *basis kets*), ∣*k*_{i}⟩

∣*ψ*⟩ = ∑_{i}*c*_{i}∣*k*_{i}⟩

where *c*_{i} are the coefficients representing the probability amplitude, such that the absolute square of the probability amplitude, ∣*c*_{i}∣^{2} is the probability of a measurement in terms of the basis states yielding the state ∣*k*_{i}⟩. The normalization condition mandates that the total sum of probabilities is equal to one,

∑_{i}∣*c*_{i}∣^{2} = 1.

The simplest understanding of basis states is obtained by examining the quantum harmonic oscillator. In this system, each basis state ∣*n*⟩ has an energy $E_n = \hbar \omega \left(n + {\begin{matrix}\frac{1}{2}\end{matrix}}\right)$. The set of basis states can be extracted using a construction operator *a*^{ † } and a destruction operator *a* in what is called the ladder operator method.

### Superposition of states

If a quantum mechanical state ∣*ψ*⟩ can be reached by more than one path, then ∣*ψ*⟩ is said to be a linear superposition of states. In the case of two paths, if the states after passing through path *α* and path *β* are

$|\alpha\rangle = \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix} |0\rangle + \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix} |1\rangle$, and

$|\beta\rangle = \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix} |0\rangle - \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix} |1\rangle$,

then ∣*ψ*⟩ is defined as the normalized linear sum of these two states. If the two paths are equally likely, this yields

$|\psi\rangle = \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|\alpha\rangle + \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|\beta\rangle = \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}(\begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|0\rangle + \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|1\rangle) + \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}(\begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|0\rangle - \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|1\rangle) = |0\rangle$.

Note that in the states ∣*α*⟩ and ∣*β*⟩, the two states ∣0⟩ and ∣1⟩ each have a probability of $\begin{matrix}\frac{1}{2}\end{matrix}$, as obtained by the absolute square of the probability amplitudes, which are $\begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}$ and $\begin{matrix}\pm\frac{1}{\sqrt{2}}\end{matrix}$. In a superposition, it is the probability amplitudes which add, and not the probabilities themselves. The pattern which results from a superposition is often called an interference pattern. In the above case, ∣0⟩ is said to constructively interfere, and ∣1⟩ is said to destructively interfere.

For more about superposition of states, see the double-slit experiment.

### Pure and mixed states

A *pure quantum state* is a state which can be described by a single ket vector, or as a sum of basis states. A *mixed quantum state* is a statistical distribution of pure states.

The expectation value ⟨*a*⟩ of a measurement *A* on a pure quantum state is given by

⟨*a*⟩ = ⟨*ψ*∣*A*∣*ψ*⟩ = ∑_{i}*a*_{i}⟨*ψ*∣*α*_{i}⟩⟨*α*_{i}∣*ψ*⟩ = ∑_{i}*a*_{i}∣⟨*α*_{i}∣*ψ*⟩∣^{2} = ∑_{i}*a*_{i}*P*(*α*_{i})

where ∣*α*_{i}⟩ are basis kets for the operator *A*, and *P*(*α*_{i}) is the probability of ∣*ψ*⟩ being measured in state ∣*α*_{i}⟩.

In order to describe a statistical distribution of pure states, or *mixed state*, the density operator (or density matrix), *ρ*, is used. This extends quantum mechanics to quantum statistical mechanics. The density operator is defined as

*ρ* = ∑_{s}*p*_{s}∣*ψ*_{s}⟩⟨*ψ*_{s}∣

where *p*_{s} is the fraction of each ensemble in pure state ∣*ψ*_{s}⟩. The ensemble average of a measurement *A* on a mixed state is given by

$\left [ A \right ] = \langle \overline{A} \rangle = \sum_s p_s \langle \psi_s | A | \psi_s \rangle = \sum_s \sum_i p_s a_i | \langle \alpha_i | \psi_s \rangle |^2 = tr(\rho A)$

where it is important to note that two types of averaging are occurring, one being a quantum average over the basis kets of the pure states, and the other being a statistical average over the ensemble of pure states.