We refer to any collection of [$N$](Quantum%20state%20population.md) [quantum systems](Quantum%20systems.md) where we can select individual subsystems as a _discrete ensemble,_ or more concisely as an _[[ensemble]]_, where the same word is also used more generally to define any collection of physical systems in a [statistical model.](Thermodynamics%20and%20Statistical%20Physics%20(Index).md#Basic%20Concepts) Given the [mathematical Formulations of quantum mechanics,](Mathematical%20Formulations%20of%20Quantum%20Mechanics.md) we are then required to define we are to [represent](Ensembles%20of%20quantum%20systems.md#Representations%20of%20ensembles%20of%20quantum%20systems) ensembles of quantum systems in terms of [state vectors,](State%20vector.md) either as a set of state vectors that each represent the quantum states present in the system or [density matrices.](density%20matrix.md) ^865b77 # Representations of ensembles of quantum systems Each [quantum subsystem](Quantum%20systems.md#Composite%20quantum%20systems) in an [ensemble](Ensembles%20of%20quantum%20systems.md), taken on its own, is represented by a [state vector.](State%20vector.md) If every subsystem in an ensemble is in the same quantum state, $|\psi\rangle,$ then this state vector also represents the whole ensemble. Such an ensemble, referred to as a [quantum ensemble,](Ensembles%20of%20quantum%20systems.md#Quantum%20ensembles) would be a [Pure state](Pure%20state.md) and thus be represented by the following [density matrix,](density%20matrix.md) [$\hat{\rho} =|\psi\rangle\langle\psi|$](Pure%20state.md#^98c745) Otherwise, an ensemble may contain $n$ different quantum states that form a set: [$\{|\psi_1\rangle...|\psi_i\rangle...|\psi_n\rangle\}.$ ](mixed%20state#^f2e78b) ^17093e with populations [$\{p_1...p_i...p_n\}$](Quantum%20state%20population.md#^0830ba)^06b999   ^9eb2ad Such an ensemble forms what's called a [mixed state,](mixed%20state.md) represented by the following density matrix [$\hat{\rho} = \sum_i^n p_i|\psi_i\rangle\langle\psi_i|$](mixed%20state.md#^8a9096) # Quantum ensembles A _quantum ensemble_ or an _ensemble of [[Pure state]]s_ is a collection of $N$ particles in the same state, $|\psi\rangle.$ %%Are these necessarily closed quantum systems%% --- # Recommended reading For a concise introduction and general introduction to modeling [Ensembles of quantum systems](Ensembles%20of%20quantum%20systems.md) with [density matrices](density%20matrix.md) see * [Nielson, M. A., Chuang, I. L. _Quantum Computation and Quantum Information_, Cambridge University Press, 2010](Nielsen,%20M.%20A.,%20Chuang,%20I.%20L.%20Quantum%20Computation%20and%20Quantum%20Information,%20Cambridge%20University%20Press,%202010.md) pgs 99-101. In the same chapter the notion of density matrices modeling ensembles of quantum particles is motivated with a discussion of quantum teleportation and quantum cryptography. * [Hughston L. P., A complete classification of quantum ensembles having a given density matrix, _Phys Lett A 183_, 1993.](Hughston%20L.%20P.,%20A%20complete%20classification%20of%20quantum%20ensembles%20having%20a%20given%20density%20matrix,%20Phys%20Lett%20A%20183.%201993..md) #QuantumMechanics/QuantumStateRepresentations/StateVectors #QuantumMechanics/QuantumStateRepresentations/Wavefunctions #QuantumMechanics/QuantumStateRepresentations/DensityMatrices #QuantumMechanics/MultiParticleQuantumSystems