Target-Relative Necessity of Completion: When Readout Loss Obstructs, and What a Sufficient Extension Must Retain

What every sufficient extension must retain

Jeremy RodgersSection 8 of 16Original PDF ↗Publication DOI ↗

What every sufficient extension must retain

Theorem 7.1 (Necessary separation).

Let $e\colon S\to E$ be any $Q$ -sufficient readout extension. Then $e$ separates every pair of source states with different correct answers: $a_Q(s_0)\neq a_Q(s_1) \qquad\Longrightarrow\qquad e(s_0)\neq e(s_1), \qquad s_0,s_1\in S.\tag{17}$

Proof. We prove the contrapositive. Suppose $e(s_0)=e(s_1)=:x$ . By $Q$ -sufficiency there is $b\colon E\to A_Q$ with $a_Q=b\circ e$ . Then $a_Q(s_0)=b\bigl(e(s_0)\bigr)=b(x)=b\bigl(e(s_1)\bigr)=a_Q(s_1),$ since $b(x)$ is a single element of $A_Q$ . Hence $e(s_0)=e(s_1)\implies a_Q(s_0)=a_Q(s_1)$ , which is (17). ◻

Corollary 7.2 (The critical case: fiber-mates with different answers).

Suppose $s_0,s_1\in S$ satisfy $p(s_0)=p(s_1)$ and $a_Q(s_0)\neq a_Q(s_1)$ , that is, they form an obstructing pair in the sense of (6). Then every $Q$ -sufficient readout extension $e$ satisfies $e(s_0)\neq e(s_1)$ , while $u(e(s_0))=u(e(s_1))$ . In particular $e$ is strictly finer than $p$ : the extension is not merely allowed to distinguish $s_0$ from $s_1$ , it is compelled to, and it must do so while remaining compatible with the readout value they share.

Proof. The separation $e(s_0)\neq e(s_1)$ is Theorem 7.1. The identity $u(e(s_i))=p(s_i)$ from (15), together with $p(s_0)=p(s_1)$ , gives $u(e(s_0))=u(e(s_1))$ . Since $e(s_0)\neq e(s_1)$ but $p$ does not separate $s_0,s_1$ , the partition $\sim_e$ is a strict refinement of $\sim_p$ . ◻

It is useful to restate Theorem 7.1 in partition language, where its content is transparent. Let $\sim_e$ , $\sim_p$ , and $\sim_{a_Q}$ denote the equivalence relations on $S$ induced respectively by $e$ , by $p$ , and by $a_Q$ (so $s\sim_{a_Q}s'$ iff $a_Q(s)=a_Q(s')$ ). Lemma 6.2 (3) says $\sim_e\;\subseteq\;\sim_p$ , and Theorem 7.1 says $\sim_e\;\subseteq\;\sim_{a_Q}$ . Thus:

A readout extension is sufficient for $Q$ precisely when its induced partition refines both the readout partition and the answer partition.

Indeed the converse of Theorem 7.1 holds as well and is worth recording, since together they characterize sufficiency exactly.

Proposition 7.3 (Separation characterizes sufficiency).

A readout extension $e\colon S\to E$ is $Q$ -sufficient if and only if (17) holds.

Proof. Necessity is Theorem 7.1. For sufficiency, assume (17). Its contrapositive reads $e(s)=e(s')\implies a_Q(s)=a_Q(s')$ , i.e. $a_Q$ is constant on every fiber of $e$ . By Lemma 6.5, $e$ is $Q$ -sufficient. ◻

We emphasize that Theorem 7.1 does not say that $e$ must separate every pair of distinct source states, i.e. that $e$ must be injective, i.e. that the source must be reconstructed. It says only that $e$ must separate pairs whose answers differ. Pairs with equal answers may be freely collapsed. That is the loophole that Section 8 exploits to construct a completion that is often dramatically smaller than $S$ , and it is why a target-relative theory of completion is not a disguised demand for full reconstruction.