Algebra 2 [Lecture notes] by Ben Smith

By Ben Smith

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8 φ is completely positive. 3, hence by the first part φ ∗ is completely positive. Then by the same proposition φ is completely positive. The converse is obvious. The above corollary can be extended to maps of C ∗ -algebras. Then it states that every k-positive map of a C ∗ -algebra A into another B is completely positive if and only if either A or B has all its irreducible representations on Hilbert spaces of dimension less than or equal to k, see [93]. We shall need to know the Choi matrix for φ ∗ when φ ∈ P (H ), the cone of positive maps of B(H ) into itself.

Thus if ωξ ◦ φ = ωη we have for a ∈ B(K) Tr φ ∗ [ξ ] a = Tr [ξ ]φ(a) = ωξ ◦ φ(a) = ωη (a) = Tr [η]a . Thus φ ∗ ([ξ ]) = [η], and φ ∗ : B(H ) → B(K) is faithful and maps 1-dimensional projections to 1-dimensional projections. Let ξ and μ be mutually orthogonal unit vectors in H . Let η and ρ be unit vectors in K such that ωξ ◦ φ = ωη , and ωμ ◦ φ = ωρ . 1 either φ([η]) = 1, in which case support φ = [η], so that φ(a) = φ([η]a[η]) = ωη (a)1, so φ is a vector state, or φ([η]) = [ξ ], φ([ρ]) = [μ].

Suppose there is no pure state ρ such that ρ ◦ φ is a pure state. 1(ii), ρ φ(e11 ) ρ φ(e22 ) > ρ φ(e12 ) 2 . Since the set of pure states on M2 is compact there exists α > 0 such that α ≤ ρ φ(e11 ) ρ φ(e22 ) − ρ φ(e12 ) 2 for all pure states ρ. Since |ρ(φ(e12 ))|2 ≤ 1 (1 ± α) ρ φ(e12 ) 2 ≤ ρ φ(e11 ) ρ φ(e22 ) . Define two maps ψ + and ψ − of M2 into itself as follows; ψ ± is linear, ψ ± (eii ) = φ(eii ), i = 1, 2, and ψ ± (e12 ) = (1 ± iδ)φ(e12 ), ψ ± (e21 ) = (1 ∓ iδ)φ(e21 ), where 0 < δ < α 1/2 , so that |1 ± iδ|2 = 1 + δ 2 < 1 + α.

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