PDF《Condensed Matter Physics, 2018,》

2 , (2.1) where ? ρ1 and ? ρ1 are density matrices of the ?rst and the second mixed states. We de?ne geometric measure of mixing of quantum states as minimal squared Hilbert-Schmidt distance from the given mixed state to a set of pure states D = min |ψ Tr ? ρ ? ? ρpure

2 , (2.2) where ? ρ is density matrix of the given mixed states, ? ρpure = |ψ ψ| (2.3) is density matrix of a pure state described by the state vector |ψ , and minimization is done over all possible pure states. Let us rewrite the geometric measure of mixing of quantum states as follows: D = min |ψ Tr ? ρ2 + Tr ? ρ2 pure ?

2 Tr ? ρ ? ρ pure . (2.4) Three terms in (2.4) can be calculated separately. For the ?rst term, we ?nd Tr ? ρ2 = i λ2 i , (2.5) where λi are eigenvalues of density matrix ? ρ. For pure state ? ρ2 pure = ? ρpure, so the second term reads Tr ? ρ2 pure = Tr ? ρpure = 1. (2.6) Trace is invariant with respect to choosing the basic vectors. To calculate the third term, we use the following orthogonal basic vectors |ψ , |ψ1 , |ψ2 where the ?rst vector is equal to the state of pure state in (2.3), ψ|ψi = 0, ψi |ψj = 0, i = 1, 2,j = 1, 2,Then, ? ρpure|ψ = |ψ , (2.7) ? ρpure|ψi = |ψ ψ|ψi = 0, i = 1, 2,2.8) As a result, for the third term we have Tr ? ρ ? ρpure = ψ| ? ρ|ψ . (2.9) Substituting (2.5), (2.6), (2.9) into (2.4), we ?nd D = min |ψ i λ2 i +

1 ?

2 ψ| ? ρ|ψ . (2.10) 33003-2 Geometric measure of mixing of quantum state This expression reaches a minimal value when |ψ is equal to the eigenvector of density matrix ? ρ with maximal eigenvalue. Thus, ?nally, for geometric measure of mixing of quantum state we have D = i λ2 i +

1 ? 2λmax = (1 ? λmax)2 + λi