Note on non-entangling measurements

The general form of non-entangling unitary maps for measurement schemes is determined. It is shown that any POVM admits a non-entangling measurement. We prove the following. Proposition 1 Let H1,H2 be complex separable Hilbert spaces, φ0 a unit vector in H2. Assume U : H1 ⊗ H2 → H1 ⊗ H2 is a unitary map such that for all φ ∈ H1, U(φ ⊗ φ0) = φ′ ⊗ φ′ for some unit vectors φ′ ∈ H1, φ′ ∈ H2. Then U acts in one of the following two ways: (1) U(φ ⊗ φ0) = V1(φ) ⊗ φ′, where V1 is an isometry in H1 and φ′ is a fixed unit vector in H2; (2) U(φ ⊗ φ0) = φ′ ⊗W12φ, where W12 is an isometry from H1 to H2 and φ′ is a fixed unit vector in H1. Proof. Let {φn}n∈N be an orthonormal basis of H1. There are systems of unit vectors φn ∈ H1, φn ∈ H2 such that Uφn ⊗φ0 = φn ⊗φn. Due to the unitarity of U , all the vectors φn ⊗φn are mutually orthogonal. We show that one of two cases (a), (b) must hold: (a) {φn}n∈N is an orthonormal system, all φn are parallel to φ1; (b) {φn}n∈N is an orthonormal system, all φn are parallel to φ1. For two unit vectors ψ, ξ which are mutually orthogonal, 〈ψ|ξ〉 = 0, we will write ψ ⊥ ξ. From φ1 ⊥ φ2, it follows that either φ1 ⊥ φ2 or φ1 ⊥ φ2. Consider the first case. Then