The information contained in a single [qubit](Qubits.md) and by extension a [two-level system](Two-Level%20Systems.md) may be visualized as a [state vector,](State%20vector.md) $|\psi\rangle$ that [maps](Bloch%20spheres.md#Algebraic%20properties%20of%20the%20Bloch%20sphere) to a point along the surface of a _Bloch sphere._ If we were to also plot a [mixed state](Bloch%20spheres.md#Mapping%20a%20mixed%20state%20on%20a%20Bloch%20sphere) qubit, it maps to a point inside of the sphere as shown below.  The vector that defines the point in or on the Bloch sphere is referred to as a [Bloch vector.](Bloch%20vector) ^270d0d # Mapping a pure state on a Bloch sphere Given that a [Two-level systems](Two-Level%20Systems.md), $|\psi\rangle,$ may be described in terms of a [superposition,](Quantum%20superposition.md) [$|\psi\rangle = c_1|\psi_1\rangle+c_2|\psi_2\rangle,$ where $c_1, c_2 \in \mathbb{C}$ and $|c_1|+|c_2|=1,$](Two-level%20Systems.md#^3a5abd) when using a [Bloch sphere](Bloch%20spheres.md) to describe the system, we can thus express $|\psi\rangle,$ as $|\psi\rangle = e^{i\gamma}\bigg(\cos{\frac{\theta}{2}}|0\rangle+e^{i\phi}\sin{\frac{\theta}{2}}|1\rangle\bigg)$ where $e^{i\gamma}$ is a [global phase factor we may ignore when considering a single qubit.](State%20vector.md#Phases%20of%20state%20vectors) Thus we express express a qubit on a Bloch sphere as $|\psi\rangle = \bigg(\cos{\frac{\theta}{2}}|0\rangle+e^{i\phi}\sin{\frac{\theta}{2}}|1\rangle\bigg)$ where $\theta$ and $\phi$ are the angles used to define the point on the Bloch sphere. %%Note we can't cleanly generalize Bloch spheres to multple qubits, but then how do you get points inside of a Bloch sphere?%% ^d890d8 # Mapping a mixed state on a Bloch sphere In order to map a [mixed state](mixed%20state.md) onto a [Bloch sphere](Bloch%20spheres.md) we need to know its [Bloch vector](Bloch%20vector.md) in relation to its [density matrix.](density%20matrix.md) We find that:   # Algebraic properties of the Bloch sphere  #QuantumMechanics/StationaryStateQuantumSystems #QuantumMechanics/TwoLevelSystems