Quantum acceptance probability for a finite promise computation

Fix a length-$n$ finite promise computation from Finite quantum promise-computation model, with input $x\in \{0,1{\}}^n$, zero-initialized auxiliary register $\mathrm{\mid }{0}^{a(n)}⟩$, circuit unitary $U_{C_n}$, and ordered output-wire tuple $O_n$. Let $P_{O_n}$ be the tensor-factor permutation that moves $O_n$, in its declared order, to the front, and let $W_n$ be the ordered tuple of all remaining wires (Ordered tensor registers, their inner product, and wire permutations). Set ${\psi }_x:=U_{C_n}(\mathrm{\mid }x⟩\otimes \mathrm{\mid }{0}^{a(n)}⟩),v_x:=\mathrm{coord}(P_{O_n}{\psi }_x)\mathrm{.}$ Here coordinates are in the ordered computational basis. In this definition, $⟨\cdot ,\cdot {⟩}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}:=⟨\cdot ,\cdot {⟩}_{\otimes }$ denotes the inner product on the reordered complete tensor register from Ordered tensor registers, their inner product, and wire permutations, linear in its first argument and conjugate-linear in its second (Real and complex inner-product spaces and their induced length). For $z\in \{0,1{\}}^{\mathrm{\mid }{O}_n\mathrm{\mid }}$ and $w\in \{0,1{\}}^{\mathrm{\mid }{W}_n\mathrm{\mid }}$, write $e_{z,w}$ for the corresponding product-basis vector and define $p_{C_n}(z\mid x):={\sum }_{w\in \{0,1{\}}^{\mathrm{\mid }{W}_n\mathrm{\mid }}}{\mid ⟨P_{O_n}{\psi }_x,e_{z,w}{⟩}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}\mid }^2\mathrm{.}$ The acceptance probability is $p_{C_n}(x):={\sum }_{\begin{array}{c}z\in \{0,1{\}}^{\mathrm{\mid }{O}_n\mathrm{\mid }}\\ A_n(x,z)=1\end{array}}{p}_{C_n}(z\mid x)\mathrm{.}$ Equivalently, if $E_{A,x}$ is the diagonal projection onto the basis vectors $(z,w)$ with $A_n(x,z)=1$, then $p_{C_n}(x)=\mathrm{Tr}(E_{A,x}{v}_x{v}_x^{†})$; the matrix adjoint is conjugate transpose in an orthonormal basis (The adjoint $T^{\ast }:W\to V$ is characterised by $⟨Tv,w{⟩}_W=⟨v,T^{\ast }w{⟩}_V$, Every finite-dimensional real or complex inner product space has an orthonormal basis).

These are finite sums. The definition itself does not assert normalization, positivity of the resulting operator expression, or a probability bound; those consequences are the purpose of The partial-output acceptance law is a finite probability law. No Dirac bra convention is used here: every amplitude uses the root pairing displayed above.