A _mixed state_ is an [ensemble](Ensembles%20of%20quantum%20systems.md) of quantum systems (e.g. quantum particles) that can each be in one of $n>1$ different quantum [states](State%20vector). The possible states that any given subsystem in that ensemble could be in is represented by a [set](Sets.md) of states,[$\{|\psi_1\rangle...|\psi_i\rangle...|\psi_n\rangle\}.$](Ensembles%20of%20quantum%20systems.md#^17093e) ^f2e78b Each state in the set makes up a particular proportion of that ensemble referred to as its [_fractional population_,](Quantum%20state%20population.md#Fractional%20population) where the fractional populations form the set [$\{p_1...p_i...p_n\}$](Ensembles%20of%20quantum%20systems.md#^06b999) ^ab077e These population proportions are equivalently statistical weights on any measurement being made on the system. Thus, in an experimental setting, when measuring a subsystem from the ensemble, there is a _classical_ probability $p_i$ that one will select a state, $\psi_i$ along with the [quantum probability](Born%20rule.md) in that measurement. ^f936c1 Since the values, $p_i,$ are proportions, [$\sum_i^n p_i =1$](Ensembles%20of%20quantum%20systems.md#^9eb2ad) ^a6b2bc Rather than expressing mixed states as [sets](Sets.md) we usually use what is referred to as a [density matrix.](mixed%20state.md#Density%20Matrix) # Density Matrix Since a [mixed state](mixed%20state.md) can be represented by a set of state vectors: [$\{|\psi_1\rangle...|\psi_i\rangle...|\psi_n\rangle\}$](Ensembles%20of%20quantum%20systems.md#^17093e) with a corresponding set of proportions, [$\{p_1...p_i...p_n\}$](Ensembles%20of%20quantum%20systems.md#^06b999), we may use the elements of these sets to construct a single [density matrix](density%20matrix.md), $\hat{\rho} = \sum_i^n p_i|\psi_i\rangle\langle\psi_i|$ ^8a9096 Where there are $n$ possible states in the ensemble with probabilities $p_i$ of measuring each of these states and ^e999ac  ## Properties In addition to the usual density matrix [properties](density%20matrix.md#Properties%20of%20density%20matrices) density matrices of mixed states have the following properties 1) $0<$[tr](Trace.md)$(\hat{\rho}^2)<1$ (we refer to the quantity $\mathrm{tr}(\hat{\rho}^2)$ as the [purity.](Quantum%20state%20purity.md)) ^2cdafa 2) $\hat{\rho}=\frac{1}{n}\hat{\mathbb{1}}$ if each of the $n$ possible states for a given system exists in equal proportions. Such a state may be said to be _[maximally mixed](mixed%20state.md#Maximally%20mixed%20states)_. ^895809 # Maximally mixed states A state is _maximally [mixed](mixed%20state.md)_ if every possible state in the [ensemble](Ensembles%20of%20quantum%20systems.md) populates the ensemble in equal proportions. ^bbc797 The density matrices for maximally mixed states reduce to diagonal matrices where [$\hat{\rho}=\frac{1}{n}\hat{\mathbb{1}}$](mixed%20state.md#^895809) where is the number $n$ of allowed states in the system and therefore also the [Hilbert space dimension](Hilbert%20space%20dimension%20in%20quantum%20mechanics.md) $d$ for the system. %%This term is defined in lecture 6 of Scott Aaronson's notes on the topic of quantum information.%% ^bf5f2c # Mixed state ensembles of two level systems ## Mapping two level mixed states onto [Bloch spheres.](Bloch%20spheres.md) # Schrödinger–HJW theorem [[Schrödinger–HJW theorem]] # Measurement on a mixed state ## Expectation value [expectation value](expectation%20value.md) --- # Examples * [ensemble of circularly polarized photons](ensemble%20of%20circularly%20polarized%20photons.md) #QuantumMechanics/QuantumStateRepresentations/StateVectors #QuantumMechanics/QuantumStateRepresentations/DensityMatrices #QuantumMechanics/MultiParticleQuantumSystems