A _Bloch vector,_ $\mathbf{r}$ is a $1\times 3$ [vector](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#Vectors) that describes a point either on the surface of a [Bloch sphere](Bloch%20spheres.md) when describing a [pure state](Pure%20state.md) or inside of a Bloch sphere when describing a [mixed state.](mixed%20state.md) Thus this object only models [two-level Systems](Two-Level%20Systems.md) or [ensembles of two level systems.](Ensembles%20of%20two%20level%20systems.md)  Bloch vectors have the following key properties: *  *  # Constructing two level systems with Bloch vectors All information associated with a [two-level system](Two-Level%20Systems.md) (a [qubit](Qubits.md)) is contained in its Bloch vector since it's all we need to construct [the density matrix for a two level system.](Ensembles%20of%20two%20level%20systems.md#Describing%20ensembles%20of%20two%20level%20systems%20in%20terms%20of%20Pauli%20matrices) ^9a484a    # Pure and mixed state Bloch vectors A [Bloch vector,](Bloch%20vector.md) $\mathbf{r},$ describes a [pure state](Bloch%20spheres.md#Mapping%20a%20pure%20state%20on%20a%20Bloch%20sphere) if [$|\mathbf{r}|=1$](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#Vector%20length) and describes a [mixed state](Bloch%20spheres.md#Mapping%20a%20mixed%20state%20on%20a%20Bloch%20sphere) if $|\mathbf{r}|<1.$ ([proof](Bloch%20vector.md#Proof%20that%20a%20two-level%20state%20is%20pure%20if%20and%20only%20if%20its%20Bloch%20vector%20has%20a%20length%20of%201)) ^95235c If the [Bloch vector](Bloch%20vector.md) is 0 it describes a [maximally mixed state](mixed%20state.md#Maximally%20mixed%20states) since $\hat{\rho}=\frac{\hat{\sigma}_0+\mathbf{r}\cdot\mathbf{\sigma}}{2} = \frac{\hat{\sigma}_0}{2},$ which matches the form of a [maximally mixed state density matrix](mixed%20state.md#^895809) where any [two level system](Ensembles%20of%20two%20level%20systems.md) may be expressed as [$\hat{\rho}=\frac{\hat{\sigma}_0+\mathbf{r}\cdot\mathbf{\sigma}}{2}.$](ensembles%20of%20two%20level%20systems.md#^248151) ^c59c23 --- # Proofs and examples ## Proof that a two-level state is pure if and only if its Bloch vector has a length of 1     #QuantumMechanics/QuantumInformation