# Question: 03-witness-distance

- **Subproject:** 03-ilang-space
- **Package:** 03-witness-distance
- **Equation tags:** (3.k)
- **Created:** 2026-10-07

## Goal

Define points and a distance from the witness data in the view of a part $A$, and determine how far the choice of the distance is fixed. The notation introduced below ($\approx$, $[h]$, $X_V$, $\alpha$) is fixed for later packages. "Point" and "distance" are defined notions in the sense of M1; no further spatial meaning is attached to them.

1. **Points.** On the present places $H_V$ of $A$, define $h\approx h'$ iff $W(h,h')=1$. Show:
   - (a) $W(h,h')=1$ iff $\lvert E_h\rangle$ and $\lvert E_{h'}\rangle$ differ only by a phase, that is, iff the companion does not witness the difference between $h$ and $h'$;
   - (b) $\approx$ is an equivalence relation, and $W(h,h')$ depends only on the classes $[h]$ and $[h']$;
   - (c) the set of classes $X_V:=H_V/{\approx}$, whose elements are the points, and $W$ on $X_V$ are well defined in the sense of M6;
   - (d) if $A$ contains several instances of the same type, then present joint places of $A$ that differ only by a permutation of these instances lie in the same point.

2. **Witness angle.** For points $x=[h]$ and $x'=[h']$, define
   $$\alpha(x,x'):=\arccos\sqrt{W(h,h')}\in[0,\pi/2].$$
   State that $\alpha$ is a metric on $X_V$, and express $\cos\alpha$ through matrix elements of $V_A$.

3. **Distances from the witness data.** Consider candidate distances $d=g\circ\alpha$ with $g:[0,\pi/2]\to[0,\infty)$; equivalently, functions of the pair invariant $W$.
   - (a) **Realizability.** Show that every triple $(\alpha_1,\alpha_2,\alpha_3)\in[0,\pi/2]^3$ that satisfies the three triangle inequalities occurs, in some description, as the witness angles $\bigl(\alpha(x_1,x_2),\alpha(x_2,x_3),\alpha(x_1,x_3)\bigr)$ of the points of three present places. Give a sufficient condition on the description. An example of such a condition: the companion contains an object of a type with at least three places, and that type does not occur in $A$.
   - (b) **Characterization.** Conclude that $g\circ\alpha$ is a metric on $X_V$ for every description iff three conditions hold: $g(0)=0$; $g>0$ on $(0,\pi/2]$; and $g(\alpha_3)\le g(\alpha_1)+g(\alpha_2)$ for every such triple. Give a simple sufficient condition, for example that $g$ is nondecreasing and subadditive.
   - (c) **Candidates.** Decide for each of the following candidates whether it is a metric for every description: $\alpha$, $\sqrt{1-W}$, $\sqrt{2(1-\sqrt W)}$, $1-\sqrt W$, $1-W$, and $\sqrt{-\ln W}$ (the last one on descriptions with all $W>0$). For each candidate that fails, give a realizable triple that violates the triangle inequality. For each candidate that qualifies, state its range.

4. **Additivity.** Call three points aligned, with $x'$ between $x$ and $x''$, if $\alpha(x,x'')=\alpha(x,x')+\alpha(x',x'')$. Take $g$ as in item 3, and in addition nondecreasing or continuous. Show that $g\circ\alpha$ is additive on every aligned triple,
   $$g\bigl(\alpha(x,x'')\bigr)=g\bigl(\alpha(x,x')\bigr)+g\bigl(\alpha(x',x'')\bigr),$$
   iff $g(\alpha)=c\,\alpha$ with a constant $c>0$. The argument uses that aligned triples are realizable for every pair of angles with sum at most $\pi/2$.

## Inputs

From 02-view-content@v1.

Notation fixed there: $p_h:=\langle h\vert V_A\vert h\rangle$ and $H_V:=\{h : p_h>0\}$, where $h$ runs over the joint places of $A$; $\lvert\psi_h\rangle:=\bigl(\langle h\rvert\otimes\mathbb 1_{\bar A}\bigr)\lvert\Psi\rangle$; $G_{hh'}:=\langle E_{h'}\vert E_h\rangle$ for $h,h'\in H_V$; $W(h_1,\dots,h_k):=G_{h_1h_2}G_{h_2h_3}\cdots G_{h_{k-1}h_k}G_{h_kh_1}$.

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. (2.7), the general decomposition (1.6), with arbitrary real $\theta_h$:

$$
a_h=\sqrt{p_h}\,e^{-i\theta_h},\qquad \lvert E_h\rangle=e^{i\theta_h}\,\frac{\lvert\psi_h\rangle}{\sqrt{p_h}}\qquad(h\in H_V),\qquad a_h=0\qquad(h\notin H_V),
$$

Eq. (2.8):

$$
G_{hh'}=\langle E_{h'}\vert E_h\rangle=e^{i(\theta_h-\theta_{h'})}\,\frac{(V_A)_{hh'}}{\sqrt{p_hp_{h'}}},\qquad h,h'\in H_V .
$$

Eq. (2.11):

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

Eq. (2.16). Here $\pi$ is a permutation of the objects with $T(\pi(a))=T(a)$, and $(h^\pi)_{\pi(a)}:=h_a$. For every admissible state:

$$
p^{\pi(A)}_{h^\pi}=p^A_h,\qquad H_V^{\pi(A)}=\{h^\pi:h\in H_V^A\},\qquad W^{\pi(A)}(h_1^\pi,\dots,h_k^\pi)=W^A(h_1,\dots,h_k).
$$

## Assumptions

None beyond the base problem. The part $A$ and its companion $\bar A$ are both nonempty, and $\lambda$ is fixed.

## Scope

- In scope:
  - one part $A$ and one admissible state at fixed $\lambda$;
  - points as classes of present places;
  - distances as functions of the pair invariant $W(h,h')$.
- Out of scope:
  - neighbourhood, locality, paths and geodesics beyond item 4, and dimension;
  - the cyclic products $W(h_1,\dots,h_k)$ with $k\ge3$ and the phases they carry;
  - evolution, several parts, and comparison of the points of different parts;
  - any physical or spatial meaning beyond the definitions (M1, M4).

## Depth

- Item 1: short argument.
- Item 2: standard. $\alpha$ is the angle between the rays of $\lvert E_h\rangle$ and $\lvert E_{h'}\rangle$, and that this angle is a metric on rays may be used without proof. The expression for $\cos\alpha$ takes one line.
- Item 3 (a): derive.
- Item 3 (b): short argument.
- Item 3 (c): short argument for each candidate.
- Item 4: short argument. The solution of Cauchy's functional equation for monotone or continuous functions may be used without proof.

## Expected result

- Item 1: the equivalence relation and its classes, with statements (a)–(d).
- Item 2: $\alpha$ is a metric on $X_V$, and $\cos\alpha(x,x')=\lvert(V_A)_{hh'}\rvert/\sqrt{p_hp_{h'}}$.
- Item 3: an explicit realization of the triples; the if-and-only-if characterization with a sufficient condition; and for each candidate a verdict, with a counterexample triple or the range.
- Item 4: the if-and-only-if statement $g=c\,\alpha$.

Give every main result a tag $(3.k)$.

Consistency checks, at most three, chosen from:

- for a product state, $X_V$ consists of a single point;
- for mutually orthogonal branch states, every two distinct points are at witness angle $\pi/2$;
- for two present places, $\cos\alpha$ equals the modulus of the overlap $\langle E_1\vert E_0\rangle$ in the two-branch witness rule (base problem, Section 3).

## Code

None.