We consider a quantum state to be a pure state if it can be expressed in terms of a single [State vector](State%20vector), $|\psi\rangle$ . This may represent a single [isolated system](Closed%20quantum%20systems.md) or an [ensemble of quantum mechanic systems](Ensembles%20of%20quantum%20systems.md) each described with the same [state vector.](State%20vector.md) The latter is referred to as a [quantum ensemble.](Ensembles%20of%20quantum%20systems.md#Quantum%20ensembles) %%Would such a quantum ensemble be necessarily a closed quantum system? Or say an ensemble of closed systems?%% ^b60077 # Density Matrix A [[Pure state]] density matrix, unlike a [mixed state density matrix](mixed%20state.md#Density%20Matrix), contains only one [state vector.](State%20vector.md) Starting from the general expression for a [density matrix](density%20matrix.md) [$\hat{\rho} = \sum_i^n p_i|\psi_i\rangle\langle\psi_i|$](density%20matrix.md#^bab548), a pure state describes the situation where there's a single [isolated system](Closed%20quantum%20systems.md) or every system in a given [ensemble](Ensembles%20of%20quantum%20systems.md) is described by the same [state vector](State%20vector.md) and the sum is reduced to a single term and $p_i=1.$ That is, if one were to pick a system from the ensemble, the probability of picking a system in state $|\psi\rangle$ is $1$. Thus a pure state density matrix is a [projection operator](Projection%20operators%20in%20quantum%20mechanics.md) on state $|\psi\rangle$ and is given as $\hat{\rho} =|\psi\rangle\langle\psi|$ ^98c745 %%Again are pure state consisting of more than one subsystem necessarily closed or open? are each of the states in that system closed necessarily?%% ## Properties In addition to the usual density matrix [properties](density%20matrix.md#Properties%20of%20density%20matrices) density matrices of pure states have the following properties 1) $\hat{\rho}^2=\hat{\rho}$ ([idempotency](Idempotent%20operators.md)) ^930e4b 2) [$\mathrm{tr}$](Trace.md)$(\hat{\rho}^2)=\mathrm{tr}(\hat{\rho})=1$ (we refer to the quantity $\mathrm{tr}(\hat{\rho}^2)$ as the [purity.](Quantum%20state%20purity.md)) ^d5926e # Measurement on a pure state ## Expectation value [expectation value](expectation%20value.md) --- # Examples * [spin ensemble with superpositions](spin%20ensemble%20with%20superpositions.md) #QuantumMechanics/QuantumStateRepresentations/StateVectors #QuantumMechanics/QuantumStateRepresentations/DensityMatrices #QuantumMechanics/MultiParticleQuantumSystems