Clean-ancilla zero-branch postselection

Let $A$ be a finite register with distinguished unit computational-basis vector $\mathrm{\mid }{0}_A⟩$ (unit by The computational basis is orthonormal), let $S$ be a finite-dimensional complex-inner-product space, and define the isometry $J:S\to A\otimes S$ by $Jx=\mathrm{\mid }{0}_A⟩\otimes x$. The product inner product makes $J$ an isometry; Finite-dimensional linear isometries satisfy the adjoint identity gives $J^{†}J=I_S$, and An isometric quantum encoding gives an orthogonal code projector makes ${\mathrm{\Pi }}_0:=J{J}^{†}$ the orthogonal projection onto its image. On elementary tensors this is $\mathrm{\mid }{0}_A⟩⟨0_A\mathrm{\mid }\otimes I_S$ by the convention of Tensor products of finitely many linear maps. For a unit vector $\varphi \in A\otimes S$, define its zero-branch vector $q:=J^{†}\varphi$, zero-branch weight $p_0:=\mathrm{\parallel }q{\mathrm{\parallel }}^2$, and complementary-branch weight $p_{\perp }:=\mathrm{\parallel }(I-{\mathrm{\Pi }}_0)\varphi {\mathrm{\parallel }}^2$. When $p_0>0$, the conditional system vector is $q\mathrm{/}\sqrt{p_0}$; when $p_0=0$, no conditional vector is assigned. This definition fixes only the clean-ancilla two-outcome branch rule used on this page; it makes no physical realization claim.