A two-level system is any [pure state](Pure%20state.md) [quantum system,](Quantum%20systems.md) $|\psi\rangle$ that allows for the presence of only two [states](State%20vector.md) in that system. Thus, such a system is modeled on a [two dimensional Hilbert space](Hilbert%20space%20dimension%20in%20quantum%20mechanics.md#2%20Dimensional%20Hilbert%20spaces%20in%20quantum%20mechanics) and may be thus modeled as a [quantum superposition](Quantum%20superposition.md) where [$|\psi\rangle = \alpha|\psi_1\rangle+\beta|\psi_2\rangle.$ where $\alpha, \beta \in \mathbb{C}$ and $|\alpha|^2+|\beta|^2=1.$](quantum%20superposition#^687da0) In matrix form, the [state](State%20vector.md) is expressed as $|\psi\rangle = \alpha\begin{pmatrix} 1 \\ 0 \end{pmatrix}+\beta\begin{pmatrix} 0 \\ 1\end{pmatrix}$ and operators that act on two level systems are thus also $2\times 2$ matrices. %%This conclusion in the end motivates elaborating on hilbert space dimension. in the linked parts.%% ^7590ea In terms of a [density matrix](density%20matrix.md) we may express a [pure state](Pure%20state.md) [two-level system](Two-Level%20Systems.md) as [$\hat{\rho}=\frac{\hat{\sigma}_0+\mathbf{r}\cdot\mathbf{\sigma}}{2}$](ensembles%20of%20two%20level%20systems.md#^248151) where [$|\mathbf{r}|=1$,](Bloch%20vector.md#^95235c) $\mathbf{\sigma}$ is a [vector containing the Pauli matrices,](Pauli%20Matrices.md#Vector%20of%20Pauli%20Matrices) and we denote the identity matrix as $\hat{\sigma}_0.$ ^0ed7fb # Measurements of two level systems ## Observables of two level systems    ([... see more](Observables%20of%20two%20level%20systems.md)) # Qubits Any [two-level system](Two-Level%20Systems.md) may also be described as a [qubit.](Qubits.md)  ^b07675 [(... see more)](Qubits.md) # Bloch Spheres  ([... see more](Bloch%20spheres.md)) # Ensembles of two-level systems   ([... see more](Ensembles%20of%20two%20level%20systems.md)) # Energy of a two level system %%For now this entry is justified by the fact that two level systems have special significance towards both quantum information theory and possibly also experimental quantum physics. But otherwise, this note would be redundant.%% %%In order to fully explain two level systems start with the description in Woit and then move on to the description in sakurai. Those initial descriptions will be in the note for STATIONARY two level systems and then quoted here in a block reference.%% %%Topic sources: Griffiths page 341. Here they're discussed in terms of time dependent perturbation theory.%% # Examples of two-level systems where we don't consider time dependence * [[photon polarization]] * [[ensemble of circularly polarized photons]] #QuantumMechanics/StationaryStateQuantumSystems #QuantumMechanics/TwoLevelSystems