The quadratic normalization

A deterministic replacement of one printed rational word by another generally changes an orthogonal classical label. Its channel distance need not be small when the numerical difference is small. A quantitative comparison must therefore specify the common encoding interface.

For each real catalogue entry $a_j$, choose a positive rational scale ${\Lambda }_j$ and a supplied rational approximation $q_j$ such that

Define

After clock outcome $j$, prepare one fresh score qubit in

using the declared finite preparation permission, and destructively read it. The score qubit is consumed. The surviving register $S$ is not measured.

The complete common-interface channel is

where

In (eq:tablechannel), the index $j$ ranges over $0,\dots ,D-1$, and $y$ ranges over $\{0,1\}$.

The maps ${\mathcal{I}}_a$ and ${\mathcal{I}}_q$ are CPTP, including arbitrary untouched references, and

Both maps return the complete conditional $S$-blocks. Discarding $R,Y$ gives ${\mathrm{Tr}}_{C}X$.

The bound concerns the declared encoded table. It is not a field-acquisition allowance or a certificate of an Einstein jet.

Proof. Kraus maps for (eq:tablechannel) are

Their squared moduli sum to $I_{CS}$.

For any input–reference state, write its positive diagonal clock blocks as ${\rho }_{jj}$. The output difference is block diagonal in $j,y$. Its half trace norm is

The standard state characterization of diamond distance for a difference of channels gives the upper bound Watrous. A deterministic clock input $P_j$, with $j$ maximizing the difference and a fixed surviving state, attains it. Equation (eq:probencode) gives (eq:tablebound). Summing the score probabilities and the clock blocks proves the discard statement.

This is an explicit probability encoding implementing the fixed-table comparison discipline of X-A-ST :table CheungXAST. Its constants refer to (eq:probencode), not to an unspecified encoding. Oai283-A is a finite-interface comparator only CheungOai283.