LemmaStatement: AI-generatedProof: AI-generated

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Completion independence with an entangled reference

Statement

Under the code and noise hypotheses of support-normalized transpose recovery and completion, use its completed map $T$ and uncompleted map $T_{0}$ , and let $V : C^{d} \to H_{P}$ be an isometry onto the nonzero code. For every finite-dimensional complex inner-product reference space $K$ and every operator $W \in L (K \otimes C^{d})$ , one has

$({id}_{K} \otimes (T N {Ad}_{V})) (W) = ({id}_{K} \otimes (T_{0} N {Ad}_{V})) (W) .$

Here $L (H)$ denotes the complex-linear endomorphisms of $H$ , ${Ad}_{V} (X) = V X V^{†}$ , composition is read from right to left, and ${id}_{K} \otimes Φ$ is the linear amplification on operator spaces with the reference factor first, characterized by $A \otimes B \mapsto A \otimes Φ (B)$ . For $K = 0$ it is the unique map between zero operator spaces.

Statement references

Support-normalized transpose recovery and completion

Code restriction and decoded channel types

Ordered tensor registers, their inner product, and wire permutations

Tensor products of operator matrix units form a basis

Finite ancilla amplification is a well-defined block map

Facts & Assumptions

Given: The finite-dimensional spaces, code, noise, recovery maps and encoding in the Statement.

Goal: Prove equality of the reference-first amplifications on every operator $W$ .

[F1]

The completed and uncompleted transpose maps agree after noise on every code-supported operator $X = P X P$ , not just on states. Support-normalized transpose recovery and completion

[F2]

The encoding is ${Ad}_{V} (X) = V X V^{†}$ , with $V^{†} V = I$ and code projector $P = V V^{†}$ . The channel and recovery maps are complex-linear. Code restriction and decoded channel types Support-normalized transpose recovery and completion

[F3]

For finite-dimensional complex inner-product spaces with ordered orthonormal bases, the products of their operator matrix units form a basis of the full endomorphism space of their tensor product, including empty dimensions. Tensor products of operator matrix units form a basis

[F4]

Ordered tensor registers have their ordered product bases and product inner products; exchanging two factors is the linear map exchanging their product basis vectors, with inverse the reverse exchange. A zero-dimensional factor gives the zero tensor space. Ordered tensor registers, their inner product, and wire permutations

[F5]

For a complex-linear map $Φ : L (H) \to L (G)$ and $R = C^{r}$ , $r \geq 1$ , the ancilla-last block map $Φ^{(r)} : L (H \otimes R) \to L (G \otimes R)$ is well-defined and linear, and sends the block family $[X_{i j}]$ to $[Φ (X_{i j})]$ . Finite ancilla amplification is a well-defined block map

[F6]

Every finite-dimensional complex inner-product space admits an orthonormal basis, with the empty basis in dimension zero. Every finite-dimensional real or complex inner product space has an orthonormal basis

Proof

Given: The spaces, maps and operators in the Statement, and the conventions in [F1]–[F6].

Goal: Establish the well-defined reference-first product action and the asserted equality on all $W$ .

Proof technique: direct.

1.1F1F2algebra

For every $X \in L (C^{d})$ , $P (V X V^{†}) P = V X V^{†}$ , since $P V = V$ and $V^{†} P = V^{†}$ follow from $P = V V^{†}$ and $V^{†} V = I$ . Thus [F1] gives $(T N {Ad}_{V}) (X) = (T_{0} N {Ad}_{V}) (X)$ . Both maps are complex-linear.

1.2F4F6choose

If $K = 0$ , the input and output tensor spaces, and hence their endomorphism spaces, are zero, so their unique linear maps coincide. Otherwise put $r = \dim K \geq 1$ , choose an ordered orthonormal basis $(k_{i})_{i < r}$ of $K$ , and let $(e_{i})_{i < r}$ be the standard basis of $R = C^{r}$ . For each of the spaces $H = C^{d}$ and $G = H_{P}$ , choose an ordered orthonormal basis as well.

2.1F4F5step 1.2construct

For a complex-linear $Φ : L (H) \to L (G)$ , define $J_{H} : K \otimes H \to H \otimes R$ by $J_{H} (k_{i} \otimes x) = x \otimes e_{i}$ , and define $J_{G}$ in the same way. These maps send product bases bijectively to product bases and preserve the product inner products; their inverses send $x \otimes e_{i}$ to $k_{i} \otimes x$ . The block-map supplier therefore defines a well-defined linear map $Ψ_{Φ} (W) = J_{G}^{- 1} Φ^{(r)} (J_{H} W J_{H}^{- 1}) J_{G}$ on the full operator space.

3.1F3F5step 2.1algebra

Write $E_{u v} k_{i} = δ_{v i} k_{u}$ for the reference matrix units. For any $B \in L (H)$ , direct evaluation on $x \otimes e_{i}$ shows that $J_{H} (E_{u v} \otimes B) J_{H}^{- 1}$ has block $B$ at $(u, v)$ and zero at every other block. By [F5], its image has block $Φ (B)$ at $(u, v)$ and all other blocks zero. Thus $Ψ_{Φ} (E_{u v} \otimes B) = E_{u v} \otimes Φ (B)$ . Expanding an arbitrary reference operator $A$ in its matrix entries gives $Ψ_{Φ} (A \otimes B) = A \otimes Φ (B)$ . By [F3], this product action uniquely determines a linear map on the full operator space. Consequently the construction is independent of the chosen bases and is precisely the reference-first amplification ${id}_{K} \otimes Φ$ .

4.1F3step 1.1step 1.2step 3.1∎

Expand $W$ in the basis $E_{u v} \otimes L_{j k}$ from [F3], where $L_{j k}$ are the logical operator matrix units. By step 1.1 the two recovered maps agree on every $L_{j k}$ ; by step 3.1 their reference-first amplifications agree on every $E_{u v} \otimes L_{j k}$ . Linearity proves the claimed equality for every $W$ . Together with the zero-dimensional case in step 1.2 this covers every stated reference space. In particular, taking $K = C^{d}$ includes the maximally entangled projector.

Depends on

Used by · 1

Feasibility and range of transpose entanglement fidelity

Dependency tree · 2 levels

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