# Question: 09-measurability

- **Subproject:** 03-ilang-space
- **Package:** 09-measurability
- **Equation tags:** (9.k)
- **Created:** 2026-10-08

## Goal

Determine how readings on the companion affect the view of a part and its witness geometry: the points, the witness angle $\alpha$ and the chain distance $\ell$. In particular, determine in which sense a distant reading, or a distant choice of how the system is cut, can or cannot change a local distance. The notation introduced below ($\Pi^{\bar A}_r$, $\Psi_r$, $p(r)$, $W_r$) is fixed for later packages.

**Setting.** A description satisfying A1–A7. A part $A$ with nonempty companion $\bar A$, at fixed $\lambda$, in an admissible state $\lvert\Psi\rangle$. A reading on the companion is a partition of the joint places of $\bar A$ into classes $S^{\bar A}_r$, lifted to the joint places of the system as in A4 with $A$ and $\bar A$ exchanged. Write $\Pi^{\bar A}_r$ for the projector onto the span of $S^{\bar A}_r$ in $\mathcal H_{\bar A}$, and $\Pi_r=\mathbb 1_A\otimes\Pi^{\bar A}_r$. The reading is label-free: each class is invariant under every swap of same-type objects of $\bar A$.

1. **Grouping of the companion.** Show that $V_A$, and hence $X_V$, $\alpha$, $\sim$ and $\ell$, depend only on $\lvert\Psi\rangle$ and on the set $A$. They do not depend on how the companion is grouped into subsets, or on the order of its factors.

2. **State update.** For a label-free reading on $\bar A$ with outcome $r$, derive:
   - the outcome probability $p(r)$;
   - the post-reading state $\lvert\Psi_r\rangle$, by (1.3);
   - its branch vectors $\lvert\psi^{(r)}_h\rangle$;
   - its view $V_A(\Psi_r)$.

   Show that $\lvert\Psi_r\rangle$ is admissible.

3. **No signalling on average.** Show that $\sum_rp(r)\,V_A(\Psi_r)=V_A(\Psi)$. Conclude that the statistics of every local reading on $A$, averaged over the outcomes of the distant reading, do not depend on whether the distant reading was made, nor on which label-free distant reading was chosen.

4. **The geometry after one outcome.** For places $h,h'$ that are present in $\Psi_r$, derive the witness data $W_r(h,h')$ of the post-reading state in terms of the branch vectors of $\Psi$. Show by an explicit example that a distant reading can change the local witness geometry for a given outcome. One example should do all of the following:
   - one outcome merges two points of $A$, so that $\alpha$ becomes $0$;
   - another outcome of the same reading separates them, so that $\alpha$ becomes $\pi/2$;
   - the view averaged over the outcomes is unchanged, as item 3 requires.

5. **When nothing changes.** Show that if the companion splits as $\bar A=\bar A_1\cup\bar A_2$ with $\lvert\Psi\rangle=\lvert\Psi_{A\bar A_1}\rangle\otimes\lvert\Omega_{\bar A_2}\rangle$, then every label-free reading on $\bar A_2$ alone leaves $V_A$, and hence $X_V$, $\alpha$ and $\ell$, unchanged for every outcome.

## Inputs

The base problem is used throughout: (1.3) reading, (1.5) view, (1.11) families, A4 local readings, A7 parts.

From 02-view-content@v1. Notation fixed there: $p_h:=\langle h\vert V_A\vert h\rangle$, $H_V:=\{h:p_h>0\}$, $\lvert\psi_h\rangle:=\bigl(\langle h\rvert\otimes\mathbb 1_{\bar A}\bigr)\lvert\Psi\rangle$, $W(h,h')=\lvert G_{hh'}\rvert^2$.

Eq. (2.1):

$$
\lvert\Psi\rangle=\sum_h\lvert h\rangle\otimes\lvert\psi_h\rangle,
\qquad
\lvert\psi_h\rangle=\bigl(\langle h\rvert\otimes\mathbb 1_{\bar A}\bigr)\lvert\Psi\rangle=\sum_r\Psi_{hr}\lvert r\rangle,
\qquad
\sum_h\lVert\psi_h\rVert^2=\lVert\Psi\rVert^2=1 .
$$

Eq. (2.2):

$$
V_A=\sum_{h,h'}\langle\psi_{h'}\vert\psi_h\rangle\,\lvert h\rangle\langle h'\rvert,
\qquad\text{i.e.}\qquad
(V_A)_{hh'}=\langle\psi_{h'}\vert\psi_h\rangle,
\qquad
p_h=\lVert\psi_h\rVert^2 .
$$

Eq. (2.5). This holds for a local reading on $A$ with classes $S^A_r$ and projectors $\Pi^A_r$.

$$
p(r)=\bigl\lVert\Pi_r\lvert\Psi\rangle\bigr\rVert^2=\Bigl\lVert\sum_{h\in S^A_r}\lvert h\rangle\otimes\lvert\psi_h\rangle\Bigr\rVert^2=\sum_{h\in S^A_r}p_h=\operatorname{Tr}\bigl(\Pi^A_rV_A\bigr).
$$

Eq. (2.11):

$$
W(h)=1,\qquad W(h,h')=\lvert G_{hh'}\rvert^2=\frac{\lvert(V_A)_{hh'}\rvert^2}{p_hp_{h'}} .
$$

From 03-witness-distance@v1. Notation fixed there: $h\approx h'$ iff $W(h,h')=1$; $X_V:=H_V/{\approx}$; $\alpha(x,x'):=\arccos\sqrt{W(x,x')}\in[0,\pi/2]$.

Eq. (3.1):

$$
\varrho_h:=\lvert E_h\rangle\langle E_h\rvert=\frac{\lvert\psi_h\rangle\langle\psi_h\rvert}{p_h},
\qquad
h\approx h'\;\Longleftrightarrow\;W(h,h')=1\;\Longleftrightarrow\;\varrho_h=\varrho_{h'} .
$$

Eq. (3.5). Here $\alpha$ is a metric on $X_V$.

$$
\cos\alpha(x,x')=\sqrt{W(x,x')}=\frac{\lvert(V_A)_{hh'}\rvert}{\sqrt{p_h\,p_{h'}}}
=\frac{\lvert\langle h\vert V_A\vert h'\rangle\rvert}{\sqrt{\langle h\vert V_A\vert h\rangle\langle h'\vert V_A\vert h'\rangle}},\qquad h\in x,\ h'\in x' .
$$

From 04-neighbours@v1. Notation fixed there: related points $x\asymp x'$ iff $W(x,x')>0$; neighbours $x\sim x'$ by (4.3), a relation defined from the values of $\alpha$ on $X_V$ alone.

Eq. (4.2):

$$
x\asymp x'\;\Longleftrightarrow\;(V_A)_{hh'}\neq0\quad\text{for one, equivalently for every, pair } h\in x,\ h'\in x' .
$$

Eq. (4.6):

$$
\ell(x,x'):=\min\Bigl\{\sum_{j=0}^{m-1}\alpha(y_j,y_{j+1})\;:\;y_0=x,\ y_m=x',\ y_j\sim y_{j+1}\Bigr\},
\qquad \ell(x,x'):=+\infty\ \text{between components.}
$$

## Assumptions

- The setting stated under "Goal", at fixed $\lambda$.
- In Ilanguage a reading yields a single value, and the state becomes the post-reading state (1.3) (Law 1). "Averaged over outcomes" in item 3 is a statement about statistics. No mixed state is introduced (A3).

## Scope

- In scope:
  - readings on the companion: state update, the averaged view, and the post-reading witness geometry;
  - the grouping of the companion;
  - the factorized case.
- Out of scope:
  - readings on $A$ itself;
  - readings followed by evolution;
  - how fast a distant reading can influence $A$ through contracts (a light cone for readings);
  - identical-type issues beyond the label-free condition;
  - any physical meaning beyond the definitions (M1, M4).

## Depth

- Item 1: short argument.
- Item 2: derive.
- Item 3: short argument.
- Item 4: derive, including the example.
- Item 5: short argument.

## Expected result

The main model expects the following. Derive each statement independently, and report any discrepancy.

- Item 2: the closed forms below, and admissibility, because $\Pi_r$ commutes with every swap.
  - $p(r)=\sum_h\langle\psi_h\vert\Pi^{\bar A}_r\vert\psi_h\rangle$;
  - $\lvert\Psi_r\rangle=\Pi_r\lvert\Psi\rangle/\sqrt{p(r)}$;
  - $\lvert\psi^{(r)}_h\rangle=\Pi^{\bar A}_r\lvert\psi_h\rangle/\sqrt{p(r)}$;
  - $\bigl(V_A(\Psi_r)\bigr)_{hh'}=\langle\psi_{h'}\vert\Pi^{\bar A}_r\vert\psi_h\rangle/p(r)$.
- Item 3: the identity, from $\sum_r\Pi^{\bar A}_r=\mathbb 1$.
- Item 4: $W_r(h,h')=\dfrac{\lvert\langle\psi_{h'}\vert\Pi^{\bar A}_r\vert\psi_h\rangle\rvert^2}{\langle\psi_h\vert\Pi^{\bar A}_r\vert\psi_h\rangle\langle\psi_{h'}\vert\Pi^{\bar A}_r\vert\psi_{h'}\rangle}$, and an explicit example. One possible example uses a companion with three places $k_1,k_2,k_3$ and the branch states $(\lvert k_1\rangle+\lvert k_2\rangle)/\sqrt2$ and $(\lvert k_1\rangle+\lvert k_3\rangle)/\sqrt2$ of two places of $A$. With the reading $\{k_1\}\,\vert\,\{k_2,k_3\}$, the outcome $k_1$ merges the two points and the other outcome separates them.
- Item 5: $V_A(\Psi_r)=V_A(\Psi)$ for every outcome.

Give every main result a tag $(9.k)$.

Consistency checks, at most three, chosen from:

- the trivial reading with a single class changes nothing;
- a reading on a companion that perfectly records the place of $A$ ($\langle\psi_{h'}\vert\psi_h\rangle=0$ for $h\neq h'$, and the reading distinguishes the records) leaves a single present place, hence a single point;
- in the example of item 4, the outcome average of the post-reading views equals the original view.

## Code

None.