Version 1 · current · External review, round 1: correct

# Neighbours and chain distance from the witness angle

- **Subproject:** 03-ilang-space
- **Package:** 03-ilang-space/04-neighbours
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-07

## Setup and assumptions

A description satisfying A1–A7 is fixed, with a part $A$ and a nonempty companion $\bar A$, at fixed $\lambda$, in the joint state $\lvert\Psi\rangle$. No further assumption and no approximation is made. All quantities are dimensionless.

Inputs, as quoted in `question.md`:

- From 02-view-content@v1: $\lvert\psi_h\rangle=(\langle h\rvert\otimes\mathbb 1_{\bar A})\lvert\Psi\rangle$ (2.1); $(V_A)_{hh'}=\langle\psi_{h'}\vert\psi_h\rangle$ and $p_h=\langle h\vert V_A\vert h\rangle=\lVert\psi_h\rVert^2$ (2.2); $H_V=\{h:p_h>0\}$; $W(h,h')=\lvert G_{hh'}\rvert^2$ with $G_{hh'}=\langle E_{h'}\vert E_h\rangle$.
- From 03-witness-distance@v1: $h\approx h'\iff W(h,h')=1\iff\varrho_h=\varrho_{h'}$ (3.1); the points $X_V=H_V/{\approx}$, and $W(x,x')=W(h,h')=\operatorname{Tr}(\varrho_x\varrho_{x'})$ for any $h\in x$, $h'\in x'$ (3.2); for a type-preserving permutation $\pi$ of the objects, the canonical bijection $[h]\mapsto[h^\pi]$ from $X_V^A$ to $X_V^{\pi(A)}$ preserves $W$. It is the identity when $\pi$ permutes only companion objects or only same-type instances inside $A$, and the points and $W$ are well defined in the sense of M6 (3.3). The witness angle $\alpha(x,x')=\arccos\sqrt{W(x,x')}\in[0,\pi/2]$ is a metric on $X_V$, with
$$
\cos\alpha(x,x')=\sqrt{W(x,x')}=\frac{\lvert(V_A)_{hh'}\rvert}{\sqrt{p_h\,p_{h'}}},\qquad h\in x,\ h'\in x'\qquad\text{(3.5)}.
$$

Facts and conventions used:
- $X_V$ is finite, because $H_V$ is a subset of the finite set of joint places of $A$ (A1).
- For distinct points, $\alpha(x,x')>0$, because $\alpha$ is a metric.
- A graph on $X_V$ is a set of unordered pairs of distinct points (its edges).
- A chain from $x$ to $x'$ is a sequence $y_0=x,\dots,y_m=x'$ in which consecutive elements are joined by edges; $m=0$ is allowed when $x=x'$.
- Components are the classes of the relation "joined by a chain".

## Derivation

### Step 1. Related points (item 1)

For distinct $x,x'\in X_V$ define
$$
x\asymp x'\;:\Longleftrightarrow\;W(x,x')>0\;\Longleftrightarrow\;\alpha(x,x')<\pi/2 .
\tag{4.1}
$$
The two forms are equivalent because $\arccos$ is strictly decreasing on $[0,1]$ with $\arccos0=\pi/2$. The relation is symmetric, because $W(x,x')=\operatorname{Tr}(\varrho_x\varrho_{x'})$ is symmetric. $\Gamma_V$ is the graph on $X_V$ whose edges are the pairs with $x\asymp x'$.

**Claim.**
$$
x\asymp x'\;\Longleftrightarrow\;(V_A)_{hh'}\neq0\quad\text{for one, equivalently for every, pair } h\in x,\ h'\in x' .
\tag{4.2}
$$
*Argument.* Take any $h\in x$ and $h'\in x'$. They lie in $H_V$, so $p_h,p_{h'}>0$. By (3.5), $\lvert(V_A)_{hh'}\rvert=\sqrt{p_hp_{h'}\,W(x,x')}$, which is nonzero iff $W(x,x')>0$. By (3.2), $W(x,x')$ does not depend on the chosen representatives. Hence a nonzero cross term for one pair of representatives is equivalent to $x\asymp x'$, which is in turn equivalent to a nonzero cross term for every pair.

### Step 2. Neighbours: symmetry and well-definedness (item 2a)

For distinct $x,x'\in X_V$ define
$$
x\sim x'\;:\Longleftrightarrow\;x\asymp x'\ \text{ and there is no } y\in X_V \text{ with } \max\bigl(\alpha(x,y),\alpha(y,x')\bigr)<\alpha(x,x') .
\tag{4.3}
$$
$N_V$ is the graph whose edges are the pairs with $x\sim x'$.

Two remarks used below:
- $y=x$ and $y=x'$ never satisfy the inequality: for $y=x$ the maximum equals $\alpha(x,x')$, since $\alpha(x,x)=0$, and similarly for $y=x'$. A violating $y$ is therefore automatically a third point.
- For a related pair, a violating $y$ has $\alpha(x,y),\alpha(y,x')<\alpha(x,x')<\pi/2$, so $x\asymp y$ and $y\asymp x'$.

*Symmetry.* $\asymp$ is symmetric (Step 1) and so is $\alpha$. For each $y$, the condition for the ordered pair $(x',x)$ reads $\max(\alpha(x',y),\alpha(y,x))<\alpha(x',x)$, which is the same as the condition for $(x,x')$. Hence $\sim$ is symmetric, $N_V$ is an undirected graph, and its edge set is contained in that of $\Gamma_V$.

*M6.* Definition (4.3) uses only the set $X_V$ and the function $\alpha$ on it. Let $\Phi:X_V\to X_V'$ be a bijection with $\alpha'(\Phi x,\Phi x')=\alpha(x,x')$. As $y$ runs over $X_V$, $\Phi y$ runs over $X_V'$, so the conditions (4.3) for $(x,x')$ and for $(\Phi x,\Phi x')$ coincide term by term. Hence $x\sim x'\iff\Phi x\sim'\Phi x'$, and the same holds for $\asymp$. Apply this to the bijection of (3.3), which preserves $W$ and therefore $\alpha$:
- a relabelling $\pi$ of the objects maps $N_V^A$ isomorphically onto $N_V^{\pi(A)}$;
- when $\pi$ permutes only companion objects or only same-type instances inside $A$, the bijection is the identity, so $N_V$ is unchanged (Law 5).

The common phase (1.2) and the phase split in (1.6) cannot enter either: $X_V$ and $W$ depend on $\Psi$ only through the $\varrho_h$ (3.1), and these are well defined in the sense of M6 (3.3). The same holds for $\Gamma_V$.

### Step 3. Only the order of the angles matters (item 2b)

Let $g$ be strictly increasing on $[0,\pi/2]$ and let $\beta=g\circ\alpha$. Define $\sim_g$ by (4.3), with $\alpha$ replaced by $\beta$ and the condition $\alpha<\pi/2$ replaced by $\beta<g(\pi/2)$. For $u,v\in[0,\pi/2]$:
- $g(u)<g(v)\iff u<v$;
- $\max(g(u),g(v))=g(\max(u,v))$.

Hence $\beta(x,x')<g(\pi/2)\iff\alpha(x,x')<\pi/2$, and for every $y$, $\max(\beta(x,y),\beta(y,x'))<\beta(x,x')\iff\max(\alpha(x,y),\alpha(y,x'))<\alpha(x,x')$. So $\sim_g=\sim$.

With $g(t)=1-\cos t$ (where $g(\pi/2)=1$) this gives the form used in item 4:
$$
x\sim x'\iff\cos\alpha(x,x')>0\ \text{ and no } y \text{ has } \min\bigl(\cos\alpha(x,y),\cos\alpha(y,x')\bigr)>\cos\alpha(x,x') .
\tag{4.4}
$$
With $g(t)=\sin^2t$, the same statement holds with $\cos\alpha$ replaced by $W$. In words: $x$ and $x'$ are neighbours iff their branch states overlap, and no third point overlaps more strongly with both.

### Step 4. $N_V$ and $\Gamma_V$ have the same components (item 2c)

**Claim.** Every related pair $x\asymp x'$ is joined by a chain in $N_V$.

*Argument.* Use strong induction over the finitely many values that $\alpha$ takes on related pairs ($X_V$ is finite). Assume the claim for every related pair with a smaller angle than $\alpha(x,x')$.
- If $x\sim x'$, the one-step chain joins them.
- Otherwise, (4.3) gives a $y$ with $\alpha(x,y)<\alpha(x,x')$ and $\alpha(y,x')<\alpha(x,x')$. By Step 2, $y\notin\{x,x'\}$, and $x\asymp y$ and $y\asymp x'$ are related pairs with smaller angles. By the hypothesis, each of them is joined by a chain in $N_V$, and concatenating the two chains joins $x$ to $x'$.

For the smallest value no such $y$ can exist, so that pair is a neighbour pair; this covers the start of the induction.

Every edge of $N_V$ is an edge of $\Gamma_V$ (Step 2), so every $N_V$-chain is a $\Gamma_V$-chain. Conversely, replacing each edge of a $\Gamma_V$-chain by an $N_V$-chain from the Claim gives an $N_V$-chain with the same ends. Hence
$$
x,x'\ \text{are joined in }\Gamma_V\iff x,x'\ \text{are joined in }N_V ,
\tag{4.5}
$$
and $N_V$ and $\Gamma_V$ have the same connected components.

### Step 5. The chain distance (item 3)

$$
\ell(x,x'):=\min\Bigl\{\sum_{j=0}^{m-1}\alpha(y_j,y_{j+1})\;:\;y_0=x,\ y_m=x',\ y_j\sim y_{j+1}\Bigr\},
\qquad \ell(x,x'):=+\infty\ \text{between components.}
\tag{4.6}
$$
By (4.5), a chain exists exactly when $x$ and $x'$ lie in the same component of $\Gamma_V$. On such a component, $\ell$ is the shortest-path distance of the finite connected graph $N_V$ with positive edge weights $\alpha(y,y')>0$. By the standard result, the minimum is attained (on a simple path) and is a metric.

The value $+\infty$ between components keeps the triangle inequality $\ell(x,z)\le\ell(x,y)+\ell(y,z)$: if $x$ and $z$ lie in different components, then $y$ lies in a different component from at least one of them, and the right side is $+\infty$. Hence $\ell$ is an extended metric on $X_V$.

*$\ell\ge\alpha$.* For any chain, the triangle inequality of $\alpha$ (3.5), applied repeatedly, gives $\alpha(x,x')\le\sum_j\alpha(y_j,y_{j+1})$. Taking the minimum gives $\ell\ge\alpha$; between components, $\ell=+\infty$.

*$\ell=\alpha$ on neighbours.* If $x\sim x'$, the chain $(x,x')$ gives $\ell(x,x')\le\alpha(x,x')$. Together with $\ell\ge\alpha$:
$$
\ell\ \text{is an extended metric on } X_V,\qquad \ell\ge\alpha,\qquad \ell(x,x')=\alpha(x,x')\ \text{ if } x\sim x' .
\tag{4.7}
$$
An example with $\ell>\pi/2$ is (4.14) in Step 10.

### Step 6. Chain example: branch data and view (item 4)

$a$ and $c$ have different types and each is the only instance of its type. No pair of objects falls under (1.11), so $\mathcal H_{\rm adm}=\mathcal H$ and $\lvert\Psi\rangle$ is admissible. Write $S_i=\{i,\dots,i+m-1\}\subseteq\{1,\dots,n+m-1\}$ for the support of $\lvert E_i\rangle$.

1. By (2.1), $\lvert\psi_{h_i}\rangle=n^{-1/2}\lvert E_i\rangle$, and $\lVert E_i\rVert^2=\lvert S_i\rvert/m=1$. So $p_{h_i}=1/n>0$ and $H_V=\{h_1,\dots,h_n\}$, and (1.6) holds with $a_{h_i}=n^{-1/2}$ and $\lvert E_{h_i}\rangle=\lvert E_i\rangle$.
2. Since the $\lvert k_s\rangle$ are orthonormal, $G_{h_ih_j}=\langle E_j\vert E_i\rangle=\lvert S_i\cap S_j\rvert/m$.
3. Let $d=\lvert i-j\rvert$. Then $S_i\cap S_j=\{\max(i,j),\dots,\min(i,j)+m-1\}$, which has $m-d$ elements if $d<m$ and is empty otherwise.
4. By (2.2), $(V_A)_{h_ih_j}=\langle\psi_{h_j}\vert\psi_{h_i}\rangle=G_{h_ih_j}/n$.

Hence
$$
G_{h_ih_j}=\frac{\max(0,\,m-\lvert i-j\rvert)}{m},\qquad (V_A)_{h_ih_j}=\frac1n\,G_{h_ih_j}.
\tag{4.8}
$$

### Step 7. Points and witness angles (all $m\ge1$)

By (4.8), $W(h_i,h_j)=G_{h_ih_j}^2$, and this equals $1$ iff $i=j$. So for every $m\ge1$ the $n$ points $x_i=[h_i]$ are distinct, $X_V=\{x_1,\dots,x_n\}$, and
$$
\cos\alpha(x_i,x_j)=\frac{\max(0,\,m-\lvert i-j\rvert)}{m},\qquad
\alpha(x_i,x_j)=\begin{cases}\arccos\bigl(1-\lvert i-j\rvert/m\bigr), & \lvert i-j\rvert<m,\\[2pt] \pi/2, & \lvert i-j\rvert\ge m.\end{cases}
\tag{4.9}
$$
By (4.1), $x_i\asymp x_j\iff1\le\lvert i-j\rvert\le m-1$. So $\Gamma_V$ is the band graph with these edges:
$$
x_i\asymp x_j\iff 1\le\lvert i-j\rvert\le m-1 .
\tag{4.10}
$$

### Step 8. The case $m=1$

Here $\lvert E_i\rangle=\lvert k_i\rangle$ are mutually orthogonal, and (4.9) gives $\alpha(x_i,x_j)=\pi/2$ for all $i\neq j$. By (4.10), no two points are related. $\Gamma_V$ and $N_V$ have no edges, every point is its own component, and by (4.6)
$$
m=1:\qquad \Gamma_V=N_V=\text{no edges},\qquad \ell(x_i,x_j)=\begin{cases}0,&i=j,\\+\infty,&i\neq j.\end{cases}
\tag{4.11}
$$

### Step 9. The neighbour graph for $m\ge2$

Let $c(e)=\max(0,m-e)/m$ for integers $e\ge0$. It is strictly decreasing on $\{0,\dots,m\}$ and zero for $e\ge m$.
1. Take a related pair, so $d=\lvert i-j\rvert\in\{1,\dots,m-1\}$ and $c(d)>0$.
2. For any integer $e\ge0$, $c(e)>c(d)\iff e<d$. Indeed, $e<d$ gives $c(e)>c(d)$ by the strict decrease. If $d\le e\le m$, then $c(e)\le c(d)$, and if $e>m$, then $c(e)=0<c(d)$.
3. By (4.4) and (4.9), $x_i\not\sim x_j$ iff some index $k$ has $\lvert i-k\rvert<d$ and $\lvert k-j\rvert<d$.
4. If $d\ge2$, the index $k=i+\operatorname{sgn}(j-i)$ gives $\lvert i-k\rvert=1<d$ and $\lvert k-j\rvert=d-1<d$, so the pair is not a neighbour pair.
5. If $d=1$, the condition $\lvert i-k\rvert<1$ forces $k=i$, and then $\lvert k-j\rvert=1\not<1$. So the pair is a neighbour pair; it is related because $m\ge2$.

$$
m\ge2:\qquad x_i\sim x_j\iff\lvert i-j\rvert=1,\qquad N_V:\ x_1\sim x_2\sim\dots\sim x_n .
\tag{4.12}
$$

### Step 10. The chain distance for $m\ge2$

By (4.9), every edge of $N_V$ has weight $\theta_m:=\alpha(x_i,x_{i+1})=\arccos(1-1/m)\in(0,\pi/2)$. Take $i<j$ without loss of generality. By (4.12), a chain from $x_i$ to $x_j$ is a walk on $\{1,\dots,n\}$ with steps $\pm1$. It must cross each edge $\{k,k+1\}$ with $i\le k<j$ at least once, because that edge separates $i$ from $j$ in the path graph. So its length is at least $(j-i)\theta_m$, and the monotone chain $x_i,x_{i+1},\dots,x_j$ attains this bound. Hence
$$
m\ge2:\qquad \ell(x_i,x_j)=\lvert i-j\rvert\,\arccos\Bigl(1-\frac1m\Bigr).
\tag{4.13}
$$
*Example with $\ell>\pi/2$ (item 3).* Take $n=3$ and $m=2$. Then $\theta_2=\arccos\frac12=\frac\pi3$, and by (4.9), with $\lvert1-3\rvert=2=m$,
$$
\ell(x_1,x_3)=\frac{2\pi}{3}>\frac{\pi}{2}=\alpha(x_1,x_3).
\tag{4.14}
$$

## Result

- **Item 1:** $x\asymp x'\iff W(x,x')>0\iff\alpha(x,x')<\pi/2$ (4.1), which holds iff $(V_A)_{hh'}\neq0$ for one, equivalently every, pair of representatives (4.2).
- **Item 2:** $\sim$ (4.3) is:
  - (a) symmetric and well defined in the sense of M6: relabellings act as graph isomorphisms of $N_V$, and as the identity for permutations inside $\bar A$ or of same-type instances inside $A$;
  - (b) invariant under $\alpha\mapsto g\circ\alpha$ for strictly increasing $g$, in particular expressible through $\cos\alpha$ or $W$ (4.4);
  - (c) $N_V$ and $\Gamma_V$ have the same components (4.5).
- **Item 3:** $\ell$ (4.6) is an extended metric on $X_V$ (a metric on each component), with $\ell\ge\alpha$ and $\ell=\alpha$ on neighbour pairs (4.7). An example with $\ell>\pi/2$ is (4.14).
- **Item 4:**
  - For every $m\ge1$, the points $x_i=[h_i]$ are distinct, with $\cos\alpha(x_i,x_j)=\max(0,m-\lvert i-j\rvert)/m$ (4.9), and $x_i\asymp x_j\iff1\le\lvert i-j\rvert\le m-1$ (4.10).
  - For $m=1$, no two points are related and $\ell=+\infty$ off the diagonal (4.11).
  - For $m\ge2$, $N_V$ is the chain $x_1\sim\dots\sim x_n$ (4.12), and $\ell(x_i,x_j)=\lvert i-j\rvert\arccos(1-1/m)$ (4.13).

## Consistency checks

1. **Product state.** Take $\lvert\Psi\rangle=\lvert\phi\rangle\otimes\lvert\chi\rangle$. Then $\lvert\psi_h\rangle=\phi_h\lvert\chi\rangle$, so all $\varrho_h=\lvert\chi\rangle\langle\chi\rvert$ coincide, and by (3.1) $X_V$ is a single point. There are no distinct pairs, so $\Gamma_V$ and $N_V$ have no edges and $\ell\equiv0$. The cross terms $(V_A)_{hh'}=\phi_h\phi_{h'}^*\neq0$ occur only between places of the same point, consistent with (4.2), which concerns distinct points only.
2. **$m=1$ via the view.** By (4.8), $V_A=\frac1n\sum_i\lvert h_i\rangle\langle h_i\rvert$ is diagonal. By (4.2) alone, no two points are related, so every component of $\Gamma_V$ is a single point. This agrees with (4.11), which was obtained from the angles, and with (4.5).
3. **$m\ge n$.** Then $\lvert i-j\rvert\le n-1\le m-1$ for all pairs, so $\Gamma_V$ is complete by (4.10), while $N_V$ is still the chain (4.12): Step 9 used only $d\le m-1$. Take $n=m=3$: $\ell(x_1,x_3)=2\arccos\frac23\approx1.682>\alpha(x_1,x_3)=\arccos\frac13\approx1.231$. This agrees with $\ell\ge\alpha$ (4.7), with strict inequality for a related pair that is not a neighbour pair.

## Open issues

- Comparison with the expected result: no discrepancy in the closed forms.
  - The points $x_i$ are distinct for every $m\ge1$, not only for $m\ge2$.
  - Formula (4.13) holds only for $m\ge2$. At $m=1$, $\ell=+\infty$ off the diagonal (4.11), not $\lvert i-j\rvert\pi/2$.
- Definition (4.3) uses a strict inequality, so ties do not remove an edge. For example, if all distinct pairs have the same $W>0$, then $N_V$ is complete. Whether this is the desired behaviour is a question about other neighbour rules, which is out of scope.
- That $\ell$ is well defined in the sense of M6 was not asked. It follows as in Step 2, since $\ell$ is built from $N_V$ and $\alpha$ only.

## Methods used

- Matrix elements of the partial trace; overlaps of branch states
- Quotient by an equivalence relation; angle metric
- Invariance under strictly increasing reparametrization
- Strong induction over a finite ordered set
- Shortest-path (extended) metrics on graphs with positive edge weights
- Counting the overlap of integer intervals