Fidelity Data Processing

Given: Nonzero finite-dimensional spaces $H,K$, a channel $\mathrm{\Phi }$, and density operators $\rho ,\sigma$ on $H$.

An isometric Stinespring dilation writes $\mathrm{\Phi }(X)={\mathrm{Tr}}_E(VX{V}^{\ast })$ for an isometry $V:H\to K\otimes E$ (Finite-dimensional quantum channel, Every finite-dimensional channel has a finite isometric dilation). The QC05 definition and theorem bodies are present as drafts.

For every auxiliary space $R$ with $\dim R\ge \dim H$, the maximum overlap of purifications of the inputs on $H\otimes R$ is $F(\rho ,\sigma )$ and is attained (Fidelity Purification Characterization). The QC06 proof is a draft, not accepted.

Tensor maps act factorwise and the wire permutation reorders factors (Tensor products of finitely many linear maps, Ordered tensor registers, their inner product, and wire permutations).

Partial trace is characterized by its trace-pairing identity (Partial trace over a finite tensor factor, The partial trace is characterized by the trace pairing and is basis-independent). The successive-trace rule follows by pairing both sides with any endomorphism $T$ on the retained factor: each pairing is $\mathrm{tr}(Z(T\otimes I\otimes I))$. For the specific local-isometry use below, write $x={\sum }_{i,a}{x}_{ia}{e}_i\otimes r_a$ and $V{e}_i={\sum }_{b,c}{V}_{bc,i}{k}_b\otimes f_c$. The $(b,b^{\mathrm{\prime }})$ entry after tracing $E\otimes R$ from $(V\otimes I_R)\mathrm{\mid }x⟩⟨x\mathrm{\mid }(V^{\ast }\otimes I_R)$ is ${\sum }_{c,a,i,i^{\mathrm{\prime }}}{V}_{bc,i}{x}_{ia}\stackrel{‾}{V_{b^{\mathrm{\prime }}c,i^{\mathrm{\prime }}}}\stackrel{‾}{x_{i^{\mathrm{\prime }}a}}$; tracing $R$ first and then $E$ from $V({\mathrm{Tr}}_R\mathrm{\mid }x⟩⟨x\mathrm{\mid })V^{\ast }$ gives the same finite sum. Thus the partial-trace identity in step 1.1 follows directly in the ordered product coordinates.