The asymmetry properties of pure quantum states

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A continuous function ϕ ⁡ ( g ) \phi(g) on a group G with the Haar measure d ​ g dg is called positive definite if it satisfies ∫ d ​ g ​ 𝑑 h ​ f ∗ ​ ( g ) ​ ϕ ​ ( g − 1 ​ h ) ​ f ​ ( h ) ≥ 0 . \int dgdh\ {f}^{\ast}(g)\phi(g^{-1}h)f(h)\geq\ 0. for any f ∈ L 1 ​ ( G ) f\in L^{1}(G) . For finite groups we replace the integral with summation

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Fidelity is a measure of distinguishability of two states defined as Fid ​ ( ρ , σ ) ≡ tr ​ ( ρ ​ σ ​ ρ ) \textrm{Fid}(\rho,\sigma)\equiv\textrm{tr}(\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}})

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The commutator subalgebra [ 𝔤 , 𝔤 ] [\mathfrak{g},\mathfrak{g}] (also known as the derived subalgebra) is the subalgebra spanned by [ L 1 , L 2 ] [L_{1},L_{2}] for all L 1 , L 2 ∈ 𝔤 L_{1},L_{2}\in\mathfrak{g}

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