Norms of Positive Maps
There are several different norms that can be introduced to positive maps. In this chapter we shall study some, which are closely related to mapping cones, and we show how positivity properties are reflected in norm properties of the maps. For the simples
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Norms of Positive Maps
As we saw in Chap. 1 the uniform norm of a positive map φ from an operator system A into B(H ) is defined by φ = sup φ(a), a≤1
which equals φ(1) if A is a unital C ∗ -algebra. However, there are many other norms that could be used. In this chapter we shall consider some of these norms, first for general positive maps in B(B(K), H ) and then for the so-called Werner maps of the form Tr − AdV , V : H → K.
8.1 Norms of Maps Let K and H be finite dimensional Hilbert spaces, C a mapping cone in P (H ) and PC (K) the C -positive maps in B(B(K), H ), and PC (K)◦ the dual cone of PC (K). Let in analogy with Definition 7.4.1 SC = ρ ∈ B(K ⊗ H )∗ : ρ = Tr(Cψ · ), Tr(Cψ ) = 1, ψ ∈ PC (K)◦ . Thus SC is a convex set of linear functionals. Note that if C ⊃ CP(H ), then as pointed out in the proof of Lemma 7.4.2, every map ψ ∈ PC (K)◦ is completely positive, hence the definitions of SC given in Definition 7.4.1 and above, coincide. If a ∈ B(K ⊗ H ) let aSC = sup ρ(a) : ρ ∈ SC , and if φ ∈ B(B(K), H ), let φC = sup Tr(Cφ Cψ ) : ρ = Tr(Cψ · ) ∈ SC . Then φC = Cφ SC . E. Størmer, Positive Linear Maps of Operator Algebras, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-34369-8_8, © Springer-Verlag Berlin Heidelberg 2013
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Norms of Positive Maps
Lemma 8.1.1 SC and C are norms on B(K ⊗ H ) and B(B(K), H ) respectively. Proof The norm properties λaSC = |λ|aSC and λφC = |λ|φC are clear, and the same is subadditivity, i.e. φ + ψC ≤ φC + ψC . Since the composition of a super-positive map with a positive map is super-positive by Lemma 5.1.3, the super-positive maps in B(B(K), H ) belong to the dual cone PC (K)◦ by Theorem 6.1.6. Thus SC contains all states with density operator corresponding to superpositive maps, hence all separable states by Proposition 5.1.4. By Lemma 5.1.7 and its proof, if ω(a) = 0 for all separable states then ρ(a) = 0 for all states ρ, hence a = 0. Thus SC and C are norms. Recall that if φ ∈ B(B(K), H ) is positive then Cφ is a self-adjoint operator with positive and negative parts Cφ+ and Cφ− , so Cφ = Cφ+ − Cφ− , and Cφ+ Cφ− = 0. Let φ + and φ − be the completely positive maps such that Cφ + = Cφ+ , Cφ − = Cφ− . Then we have Proposition 8.1.2 Let C be a mapping cone in P (H ) containing CP(H ). Let φ ∈ B(B(K), H ) be C -positive. Then + φ ≥ φ − , C C or equivalently, Cφ+ SC ≥ Cφ− SC . Proof As noted in Lemma 7.4.2, since C ⊃ CP(H ), PC (K)◦ is contained in the cone of completely positive maps of B(K) into B(H ). Therefore if ρ = Tr(Cψ ·) ∈ SC , then ρ is a state. Since φ ∈ PC (K), 0 ≤ Tr(Cφ Cψ ) = Tr Cφ+ Cψ − Tr Cφ− Cψ . Thus, since Cψ ≥ 0 by Theorem 4.1.8, + φ ≥ sup Tr C − Cψ = φ − . φ C C ψ
Since Cφ+ SC = φ + C , and the same for φ − , the proof is complete.
The reader should note the related result, Corollary 7.5.7, that if φ is unital then Cφ− ≤ 1 = φ. We saw in Theorem 4.1.12 that each positive map in B(B(K), H ) can be written in the fo
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