By P.H.E. Meijer, Edmond Bauer

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It follows that B A = 0 and B A = 0. 28) 0 ≤ Mv ≤ I − Mu when u = v. 4 with A = Mu , B = Mv , C = I − Mu . Thus, the formal quantum analog of classical deterministic measurements is the orthogonal resolutions of identity {E u }: E u E v = δuv E u , E u = I. u 26 Alexander Holevo The corresponding quantum measurements are called simple. Whereas a classical deterministic measurement deÀnes a decomposition of the phase space into mutually disjoint sets (u) , a simple quantum measurement furnishes a decomposition of the unitary vector space H into the direct sum of mutually orthogonal subspaces H(u) = E u (H ).

A reader can check that for the Ànite-rank operators ∗ |ϕ j )(ψ j | = j |ψ j )(ϕ j |. , (U ϕ|U ψ) = (ϕ|ψ); ϕ, ψ ∈ H. 5) the last relation can be written as U ∗ U = I, where I is the unit operator, mapping |ψ) into |ψ), ψ ∈ H. An isometric operator mapping H onto H is called unitary. The condition of unitarity is U ∗U = UU ∗ = I. , X ∗ = X . There is a polarization identity which linearly expresses a Hermitean form through its “diagonal” values for ϕ = ψ; therefore a Hermitean operator X is uniquely deÀned by the values (ψ|X ψ), ψ ∈ H.

19). Thus S(H ) is a convex set.

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