Unitary hybrids on a preserved clean-work subspace

The exact identity $U^{\ast }(JV-UJ)V^{\ast }=U^{\ast }J-J{V}^{\ast }$, together with unitary invariance, gives $\mathrm{\parallel }{U}^{\ast }J-J{V}^{\ast }{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}=\mathrm{\parallel }JV-UJ{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le \eta$. Thus reversing the actual circuit supplies the inverse clean-input bound, without applying a forward error promise to dirty work.

If $E:H\to K$ has singular values $s_j$, [F2] gives orthonormal bases in which $\mathrm{\parallel }E\psi {\mathrm{\parallel }}^2={\sum }_j{s}_j^2\mathrm{\mid }{\psi }_j{\mathrm{\mid }}^2$. Tensor those bases with any orthonormal reference basis; for $\mathrm{\Psi }={\sum }_{j,r}{c}_{j,r}{e}_j\otimes f_r$, $\mathrm{\parallel }(E\otimes I)\mathrm{\Psi }{\mathrm{\parallel }}^2={\sum }_{j,r}{s}_j^2\mathrm{\mid }{c}_{j,r}{\mathrm{\mid }}^2\le \mathrm{\parallel }E{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}^2\mathrm{\parallel }\mathrm{\Psi }{\mathrm{\parallel }}^2$. A top singular vector tensored with any unit reference vector attains equality, so $\mathrm{\parallel }E\otimes I{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}=\mathrm{\parallel }E{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$. If the reference space is zero, the amplified map is zero and its bound is immediate. For coherent control against identity, the difference between the actual and ideal clean-input maps is zero on the control-zero summand and $E$ on the orthogonal control-one summand; its norm is at most $\mathrm{\parallel }E{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$. These statements apply to step 1.1 as well as to the original forward bound.

For $s\ge 1$, expansion and cancellation give $U_s\cdots U_{1}J-J{V}_s\cdots V_1={\sum }_{j=1}^s{U}_s\cdots U_{j+1}(U_{j}J-J{V}_j)V_{j-1}\cdots V_1\mathrm{.}$ Every factor outside the local difference is unitary, so each summand has norm at most ${\eta }_j$. The triangle inequality yields the claimed sum. On the right of each local difference are only ideal system unitaries followed by $J$, so this expansion evaluates that difference on clean ideal inputs. The actual later unitaries on the left propagate the error without any cleanliness assumption. For $s=0$, both products are identity maps, so the difference is zero.