The _purity,_ of a quantum system is a numerical value ranging from $0$ to $1$ that's used to express the degree to which a [quantum system](Quantum%20systems.md) is a [mixed](Quantum%20state%20purity.md#Mixed%20States) or [Pure state.](Quantum%20state%20purity.md#Pure%20states) In terms of a system's [density matrix,](density%20matrix.md) it is defined as $\mathcal{P}=\mathrm{tr}{(\rho^2)}.$ The quantum state purity is the length, $|\mathbf{n}|,$ of a [[Bloch vector]]. %%Here at some point you should quote other ways of calculating Bloch vector length besides the trace of the squared density matrix.%% [Pure states](Quantum%20state%20purity.md#Pure%20states) have purity of [$\mathrm{tr}(\hat{\rho}^2)=\mathrm{tr}(\hat{\rho})=1$](Pure%20state.md#^d5926e) where here we note that we know a state pure if [$\hat{\rho}^2=\hat{\rho}$.](Pure%20state.md#^930e4b) ^42f737 [Mixed states](Quantum%20state%20purity.md#Mixed%20States) have a purity of [$0<\mathrm{tr}(\hat{\rho}^2)<1$](mixed%20state.md#^2cdafa) ^a8e7a3 [Maximally mixed states](Quantum%20state%20purity.md#Maximally%20mixed%20states) have a purity of $\mathrm{tr}(\hat{\rho}^2)=\frac{1}{d}$ where $d$ is the [Hilbert space dimension](Hilbert%20Spaces%20in%20Quantum%20Mechanics.md) for that particular quantum system. ^4e27c5 # Pure states   # Mixed States    ## Maximally mixed states   and thus,  --- %%Purity is defined in pg 110 of Gerry and Knight in terms of Bloch vectors and then again in 299 in terms of the trace%% #QuantumMechanics/FoundationsOfQuantumMechanics #QuantumMechanics/QuantumInformation