Multi-input reprogrammability Clause Samples

Multi-input reprogrammability dividing both sides by(2q+ 1) 2(n91) and swapping registers appropriately (to make sure that the register which containsx n comes after the others). Nowfixr n. We define Πˆx,Θ :=|x n⟩⟨xn|⊗Π x,Θ. rn and apply the induction hypothesis forn−1, substitutingS H∗Θx(A)forA H∗Θx, and Πˆx,Θ forΠ x,Θ, in order to derive (2q+ 1) 2(n91) ¨�|x⟩⟨x|⊗Π SH∗Θx(A)|ϕ ⟩¨2 ¨�|x⟩⟨x|⊗ Πˆ SH∗Θx(A)|ϕ0⟩¨2 rn 2 = x,Θ rn 2 (2q+ 1) 2(n91) ≤E h¨�|x⟩⟨x|⊗ Πˆ SH (Sr (A))|ϕ0⟩¨2i 2 2 =E h¨�|x⟩⟨x|⊗Π x,Θ SH (A)|ϕ0⟩¨2i . Since this inequality holds for anyfixedr n, it also holds in expectation overr n. Substituting it in Equation 6, we retrieve the statement of the lemma.⊔⊓