TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Tensor maps preserve composition, identities, and adjoints

Statement

Let $(T_{k} : H_{k} \to K_{k})_{k < n}$ and $(S_{k} : K_{k} \to L_{k})_{k < n}$ be finite ordered families of linear maps between finite-dimensional complex inner-product spaces. Then

$(⨂_{k < n} S_{k}) (⨂_{k < n} T_{k}) = ⨂_{k < n} (S_{k} T_{k}), ⨂_{k < n} I_{H_{k}} = I_{\otimes_{k < n} H_{k}}, {(⨂_{k < n} T_{k})}^{*} = ⨂_{k < n} T_{k}^{*} .$

In the first formula, $S_{k} T_{k}$ means $S_{k} \circ T_{k}$ . Here $I_{H_{k}}$ denotes the identity map $x \mapsto x$ . All adjoints use the repository pairing, linear in the first argument. The empty tensor-map identity is the identity map $z \mapsto z$ on $C$ .

Facts & Assumptions

Given: The finite ordered families of spaces and linear maps in the Statement.

[F1]

The tensor map is the unique linear extension of its action on elementary tensors (Tensor products of finitely many linear maps).

[F2]

Elementary tensors in the specified product basis form an orthonormal basis and span (The product basis is orthonormal and tensor dimensions multiply).

[F3]

Every linear map between finite-dimensional inner-product spaces has a unique adjoint (Every linear map between finite-dimensional inner product spaces has a unique adjoint).

[F4]

An adjoint $T^{*}$ satisfies $⟨ T x, y ⟩ = ⟨ x, T^{*} y ⟩$ (The adjoint $T^{*} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$ ).

[F5]

Each root inner product is linear in the first argument and conjugate-linear in the second (Real and complex inner-product spaces and their induced length).

[F6]

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

[F7]

Identity maps and compatible composites of linear maps are linear (Identity maps and composites of linear maps are linear).

Proof

1.1givenF1F2F7

By [F7], each $S_{k} T_{k}$ is linear and the composite of the two tensor maps is linear, so [F1] applies to both sides. On an elementary tensor $⨂_{k < n} x_{k}$ , the left side of the composition identity equals $⨂_{k < n} S_{k} (T_{k} x_{k})$ by [F1], which is the value of the right side. Since elementary tensors span by [F2], the two linear maps are equal.

1.2F2F3F4F5F6choose

By [F6], choose orthonormal bases for the finite factor spaces. For elementary tensors $x = ⨂_{k < n} x_{k}$ and $y = ⨂_{k < n} y_{k}$ , the product pairing in [F2] and the factor adjoint identities in [F4] give $⟨ (⨂_{k} T_{k}) x, y ⟩ = \prod_{k} ⟨ T_{k} x_{k}, y_{k} ⟩ = \prod_{k} ⟨ x_{k}, T_{k}^{*} y_{k} ⟩ = ⟨ x, ⨂_{k} T_{k}^{*} y_{k} ⟩$ . The pairing is extended using the first-linear convention [F5].

2.1step 1.1F1F2F7algebra

Each factor identity and the identity on the tensor space are linear by [F7]. The tensor map $⨂_{k < n} I_{H_{k}}$ fixes every elementary tensor by [F1]. The identity map does too; spanning in [F2] gives equality. If $n = 0$ , both maps are the identity on $C$ .

3.1step 1.2F2F3F4F5F6algebra∎

Sesquilinearity and spanning extend the identity in step 1.2 to all $x, y$ . By existence and uniqueness in [F3] and the defining condition [F4], $(⨂_{k} T_{k})^{*} = ⨂_{k} T_{k}^{*}$ . For $n = 0$ both sides are the identity on $C$ ; zero-dimensional factors are covered because the product bases may be empty.

Depends on

Used by

Distance corrects unlocated errors and known erasures Corollary
Partial trace over the first tensor factor Definition
A local gate has a unique global wire embedding Lemma
CNOT Pauli action Lemma
Hadamard pauli action Lemma
Phase gate pauli action Lemma
Positive Marginals Imply Joint Support Lemma
Preserving the logical marginal forces a gauge-only channel Lemma
Symmetric Power Irreducible Lemma
Trace distance is unchanged by a fixed tensor factor Lemma
A finite Kraus sum is completely positive Theorem
Binary Pauli quotient isomorphism Theorem
Channel Composition Hybrid Bound Theorem
Channel Diamond Norm One Theorem
Channel Tensor Error Bound Theorem
Diamond Norm Ancilla Dimension Bound Theorem
Fidelity range equality and products Theorem
Fingerprint swap-test error and communication cost Theorem
Finite instruments can be coherently deferred Theorem
Finite-dimensional channels compose and tensor Theorem
Pauli product and adjoint Theorem
Phase estimation on a superposition of eigenspaces Theorem
Swap-test measurement and pure-state outcome law Theorem
Symmetric Subspace Dimension And Haar Moment Theorem
The Knill–Laflamme criterion for a protected subsystem Theorem

Dependency tree · two levels

21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Watrous, The Theory of Quantum Information (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, Appendix A.6 (standard reference, not scraped)