Mixed State Can Be Divided Into The Necessary Conditions

In this essay we mainly consider the subset of the space D(H) of the density states on the Hilbert space H = H¹ (?) H². i.e. the space of mixed states:D^k(H) = {A:A = (T|-)^tt, T∈G gl{H), Tr(A) = 1, rank{A) = k}, (k > 1).A necessary condition is given for the separable states in the space of mixed statesin the following corollary:If A ∈ D^k(H) (k > 1) is a separable state, then the subspace of H generatedby the eigenvectors of A contains at least k separable vectors. The above result is a corollary of the following proposition: For A ∈ S(H¹ (?) H²), there exists an orthonormal basis e₁, e₂ ... e_n for H, suchthatA = λ₁e₁(e|-)₁^t + λ₂e₂(e|-)₂^t + .... +λ_ke_k(e|-)_k^t,Σ_i=1^k λ_i = 1, and λ_i > 0, i = 1,2 .... k are the eigenvalues of A. Since A is a separable state, there are f_j = a_j & b_j, a_}∈ H¹,b_j∈H², j = 1,2,... ,l, such that , and Σ_j=1¹ t_j= 1, t_j > 0, where f₁, f₂, ... , f_s is the maximum independent set of f₁, f₂, ... , f_l. Then k = s and up to an ordering e₁, e₂ ... e_k and f₁, f₂, ... , f_s can be linearly represented by each other.