Theorem: Common factors at the complete public transcript

For a completed bit-output protocol satisfying Specification spec:protocol, there are a finite positive measure $\mu$ on the complete transcript space and measurable positive matrices $X_i(t),Y_j(t)$ such that Eve's unnormalized bit-branch densities are

outside a single null set. The same two local pairs work in all four branches, and

Proof. For a finite protocol, fixing the public transcript fixes all public choices. The compatible private histories sum separately in the two laboratories. Thus the final branch map is a product of two local CP maps. Pulling back the identity gives $X_i,Y_j$, and tracing the laboratory outputs in (eq:canonical) gives (eq:transcript).

We give the complete-history argument because finite-prefix factorization alone does not justify every infinite terminal readout.

Let $a=\dim A$, $b=\dim B$, and run the same protocol on

The references $A_0,B_0$ are untouched. On this auxiliary input replace a missing local bit by zero, without replacing a bit already produced by the other party.

Choose finite informationally complete POVMs on the references. They can be measured at the beginning because they commute with all laboratory operations. Conditional on $C_0$, the tested experiment has independent local sampling tapes $U,V$. Measurable sampling of standard Borel probability kernels and regular conditional laws are available in this setting Kallenberg.

Write ${\mathcal{F}}_m$ for the public prefix sigma field. Induction gives conditional independence of $U,V$ given ${\mathcal{F}}_m$. For example, after Alice's next message, ${\mathcal{F}}_{m+1}\subseteq \sigma (U)\vee {\mathcal{F}}_m$, and

Conditioning once more and multiplying by bounded functions of $U$ proves the product identity at $m+1$. The Bob step is symmetric. At a public-source step the new kernel depends only on the public prefix and does not change either conditional private marginal.

For bounded $u,v$, martingale convergence of conditional expectations in $L^2$, followed by Cauchy–Schwarz for the products, therefore gives

Countable generating algebras on the standard Borel tape spaces make this a conditional product-law identity outside one null set. In particular it applies to the defaulted local bits and the local reference-test outcomes. The supplied local terminal processes in (P6) are included in these genuine tested laws.

Next retain the reference systems instead of measuring them. By (P5), the finite-prefix reference matrix measures are consistent. Their scalar positive tests extend to the countable history space; polarization recovers the matrix entries. Positivity follows by testing a countable dense family of reference vectors and then continuity. The terminal refinements in (P6) give positive bit-conditioned reference measures $W_{ij}$ on that same complete history.

Push these measures to $T$ and dominate them by

Entrywise Radon–Nikodym derivatives yield positive densities $W_{ij}(t)$. Set

The conditional product law of the informationally complete tests gives

Since $E_r\otimes G_s$ span the Hermitian matrices,

simultaneously for all branches.

Finally apply the algebraic reference filter

Acting on the auxiliary product probes, it prepares exactly (eq:canonical). It commutes with every laboratory operation and public scalar kernel. Reference compatibility and uniqueness of the complete-history measures consequently give filtered branch laws $K{W}_{ij}{K}^{†}$.

On the actual input the event of a missing bit has probability zero. Its filtered positive matrix is therefore zero, as is the filtered matrix of every subevent. Auxiliary defaults thus make no contribution to the actual output. Taking

gives (eq:transcript). Normalization follows from completion. The filter need not be a physical operation and no inverse of $R$ is required.