Appendix A 4 October 2026

A compact proof of complete output and conditioning bounds

Jeremy Rodgers · Independent Researcher
doi:10.5281/zenodo.23131069

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A A compact proof of complete output and conditioning bounds

For normalized $u,v$ , the pure-state trace distance satisfies

D(\ket u\bra u,\ket v\bra v) =\sqrt{1-|\langle u,v\rangle|^2}\le\norm{u-v}.

Any common quantum channel, including a final position classification and an identity on a retained reference, contracts trace distance. These comparisons use the same complete retained output space. Restricting to a common event gives positive unnormalized outputs; normalization requires that the event have positive probability in both compared models.

Lemma A.1 (Conditioning on a common retained event)

Let $P,Q$ be two probability laws on the same retained output or history space, with $\TV(P,Q)\le\epsilon$ . For a common event $E$ , put $p=P(E)$ and $q=Q(E)$ . If both $p>0$ and $q>0$ , then

\TV\bigl(P(\,\cdot\mid E),Q(\,\cdot\mid E)\bigr) \le\min\{1,2\epsilon/p\}.

The sufficient condition $\epsilon<p$ ensures $q>0$ . For positive unnormalized quantum outputs $\sigma,\tau$ on the same retained space, if $\norm{\sigma-\tau}_1\le d$ , $p=\tr\sigma>0$ and $q=\tr\tau>0$ , then

\norm{\sigma/p-\tau/q}_1\le\min\{2,2d/p\}.

Here $d<p$ is sufficient for positivity of the second trace.

Proof

For a measurable set $A$ , add and subtract $Q(A\cap E)/p$ . The first difference is at most $\epsilon/p$ , and the second is at most $|p-q|/p\le\epsilon/p$ , since $Q(A\cap E)\le q$ . Taking the supremum proves the classical estimate. For the quantum estimate add and subtract $\tau/p$ , use positivity to obtain $\norm{\tau}_1=q$ , and use $|p-q|\le\norm{\sigma-\tau}_1$ :

\norm{\sigma/p-\tau/q}_1 \le d/p+|p-q|/p\le2d/p.

The upper bounds one and two are the maximal respective distances. Finally $q\ge p-\epsilon$ or $q\ge p-d$ proves the positivity claims.

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The lemma applies to complete path laws only when a joint path-law bound on that common space has actually been established. In Theorem 8.3, $E_{\rm out}$ controls complete retained quantum outputs and their physical readout probabilities, while $E_{\rm hist}$ controls the stated classical declaration-and-display histories. Neither bound asserts closeness of raw guidance trajectories or TV closeness of the ontic pair $(\Psi,Q)$ . A discarded reservoir, a missing key, or an abstract sampled path cannot be identified with an existing physical record by conditioning. A zero-probability event has no normalized conditional branch, and arbitrarily rare events have no uniform conditional guarantee.