# Question: 02-view-content

- **Subproject:** 03-ilang-space
- **Package:** 02-view-content
- **Equation tags:** (2.k)
- **Created:** 2026-10-07

## Goal

Determine exactly what the view $V_A$ (1.5) of a part $A$ retains of an admissible joint state $\lvert\Psi\rangle$, and which of the retained quantities are well defined in the sense of M6. The notation introduced below ($p_h$, $H_V$, $\lvert\psi_h\rangle$, $G$, $W$) is fixed for later packages.

1. **Basic properties.** Show that $V_A$ is a density operator on $\mathcal H_A$: self-adjoint, positive semidefinite, unit trace. Define the weights and the set of present places,
   $$p_h:=\langle h\vert V_A\vert h\rangle,\qquad H_V:=\{h : p_h>0\},$$
   where $h$ runs over the joint places of $A$. Show that $\langle h\vert V_A\vert h'\rangle=0$ unless $h,h'\in H_V$. Show that for every local reading on $A$ (A4), whose classes come from a partition $\{S^A_r\}$ of the joint places of $A$, the outcome probability (1.3) equals $\operatorname{Tr}\bigl(\Pi^A_r V_A\bigr)$, with $\Pi^A_r$ the projector onto the span of $S^A_r$ in $\mathcal H_A$. This is the language's statement that the view gives the statistics of every local reading.

2. **The view as a Gram matrix.** For every joint place $h$ of $A$, define the branch vector
   $$\lvert\psi_h\rangle:=\bigl(\langle h\rvert\otimes\mathbb 1_{\bar A}\bigr)\lvert\Psi\rangle\in\mathcal H_{\bar A}.$$
   Show that $\langle h\vert V_A\vert h'\rangle=\langle\psi_{h'}\vert\psi_h\rangle$ and $\lVert\psi_h\rVert^2=p_h$. Relate this to (1.6)–(1.7): $\lvert\psi_h\rangle=a_h\lvert E_h\rangle$.

3. **What the view does not see.** Show that two admissible states $\lvert\Psi\rangle$ and $\lvert\Phi\rangle$ have the same view of $A$, $V_A(\Psi)=V_A(\Phi)$, if and only if there is a unitary $U$ on $\mathcal H_{\bar A}$ with $\lvert\varphi_h\rangle=U\lvert\psi_h\rangle$ for every joint place $h$ of $A$, that is, $\lvert\Phi\rangle=(\mathbb 1_A\otimes U)\lvert\Psi\rangle$.

4. **Well-defined companion data.** The phase split between $a_h$ and $\lvert E_h\rangle$ in (1.6) is a convention. For $h\in H_V$ it is fixed only up to $a_h\to e^{-i\theta_h}a_h$, $\lvert E_h\rangle\to e^{i\theta_h}\lvert E_h\rangle$ with arbitrary real $\theta_h$. Define the branch Gram matrix on $H_V$, the overlap of (1.7):
   $$G_{hh'}:=\langle E_{h'}\vert E_h\rangle,\qquad h,h'\in H_V .$$
   Under a change of convention it transforms by a rephasing, $G_{hh'}\to e^{i(\theta_h-\theta_{h'})}G_{hh'}$. By M6, a quantity built from $G$ is well defined only if it is invariant under every rephasing.
   - (a) Show that the cyclic products
     $$W(h_1,\dots,h_k):=G_{h_1h_2}G_{h_2h_3}\cdots G_{h_{k-1}h_k}G_{h_kh_1},\qquad k\ge1,\ h_i\in H_V,$$
     are invariant under every rephasing, and express them through matrix elements of $V_A$ alone.
   - (b) Completeness: show that two Gram matrices $G$ and $G'$ on the same $H_V$ are related by a rephasing if and only if all their cyclic products agree. Conclude that every well-defined quantity built from $G$ is a function of the cyclic products.

5. **Label-freedom.** Let $\pi$ be a permutation of the objects that maps every object to an object of the same type. Show that for every admissible state, $V_{\pi(A)}(\Psi)$ equals $V_A(\Psi)$ under the natural identification of $\mathcal H_{\pi(A)}$ with $\mathcal H_A$, which identifies the factor of object $a\in A$ with that of $\pi(a)$. Conclude that $p_h$ and the cyclic products do not change when the part is replaced by a part that differs only in which identical-type instances it contains.

## Inputs

None. The package builds only on the base problem.

## Assumptions

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

## Scope

- In scope: one part $A$ and one admissible state at fixed $\lambda$, with arbitrary types, families and contracts.
- Out of scope:
  - any distance, geometric or spatial reading of $p_h$, $G$ or $W$ (M1, M4), and the question of which entities are points;
  - evolution and dependence on $\lambda$;
  - which Gram matrices can occur under the family constraint (1.11), beyond item 5;
  - relations between views of parts that are not related by a permutation of identical-type instances.

## Depth

- Item 1: short argument.
- Item 2: derive.
- Item 3: short argument. The standard fact that two families of vectors with the same Gram matrix are related by an isometry between their spans may be used without proof.
- Item 4 (a): derive.
- Item 4 (b): derive.
- Item 5: short argument.

## Expected result

- Item 1: the listed properties, and $p(r)=\operatorname{Tr}\bigl(\Pi^A_r V_A\bigr)$.
- Item 2: the identity $\langle h\vert V_A\vert h'\rangle=\langle\psi_{h'}\vert\psi_h\rangle$.
- Item 3: an if-and-only-if statement.
- Item 4: $W$ as a ratio of products of matrix elements of $V_A$, and the completeness statement as an if-and-only-if.
- Item 5: the equality of the two views and its consequence.

Give every main result a tag $(2.k)$.

Consistency checks, at most three, chosen from:
- the two-branch witness rule of Section 3 of the base problem is recovered as a special case of item 2;
- for a product state $\lvert\Psi\rangle=\lvert\psi_A\rangle\otimes\lvert e\rangle$ every cyclic product equals $1$: the companion witnesses nothing;
- for mutually orthogonal branch states $G$ is the identity on $H_V$, so every cyclic product that contains two different places vanishes.

## Code

None.