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Finite-Dimensional Complex Inner-Product Spaces and Matrix Interfaces — Examples

1 · Prerequisites

2 · Summary

This example computes a rank-one Dirac map on $C^{2}$ in the root linear-first convention. It verifies its action, coordinate matrix, and adjoint directly, making the conjugation placement in the bra visible.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-6-luna)Open item page →

Dirac and rank-one coordinate check

Example

Let $H = K = C^{2}$ with standard ordered orthonormal basis $e_{0} = (1, 0)$ , $e_{1} = (0, 1)$ and root pairing $⟨ z, w ⟩ = z_{0} \overline{w_{0}} + z_{1} \overline{w_{1}}$ . Set $T = ∣ e_{0} ⟩ ⟨ e_{1} ∣$ . The example checks its action, coordinate matrix, and adjoint in the root linear-first convention.

Facts & Assumptions

Given: The standard basis and pairing on $C^{2}$ and the map $T = ∣ e_{0} ⟩ ⟨ e_{1} ∣$ .

[F1]

In ordered orthonormal bases, the rank-one map has action $∣ x ⟩ ⟨ y ∣ z = ⟨ z, y ⟩_{r o o t} x$ and coordinate matrix $[x] [y]^{†}$ (Dirac Notation Coordinate Identity).

[F2]

The standard complex coordinate pairing is $⟨ z, w ⟩ = \sum_{j} z_{j} \overline{w_{j}}$ (The standard formulas $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and $\sum_{k < n} x_{k} \overline{y_{k}}$ on $C^{n}$ are inner products).

[F3]

$S$ is an adjoint of $T$ exactly when $⟨ T z, w ⟩ = ⟨ z, S w ⟩$ (The adjoint $T^{*} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$ ).

[F4]

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

Verification

1.1givenF1F2algebra

For $z = (z_{0}, z_{1})$ , [F1]–[F2] give $T z = ⟨ z, e_{1} ⟩ e_{0} = z_{1} e_{0} = (z_{1}, 0)$ . The coordinate formula is linear in $z$ .

2.1F1step 1.1algebra

The matrix columns are $T e_{0} = (0, 0)$ and $T e_{1} = (1, 0)$ , so $[T] = (\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}) .$ This agrees with $[e_{0}] [e_{1}]^{†}$ in [F1].

2.2F2step 1.1constructalgebra

Define $S : C^{2} \to C^{2}$ by $S (w_{0}, w_{1}) = (0, w_{0})$ . This is linear. By [F2], for all $z, w \in C^{2}$ , $⟨ T z, w ⟩ = z_{1} \overline{w_{0}} = ⟨ z, S w ⟩$ .

3.1F3F4step 2.2algebra∎

By [F3], step 2.2 makes $S$ an adjoint of $T$ , and uniqueness [F4] gives $S = T^{*}$ . Its columns are $(0, 1)$ and $(0, 0)$ , so $[T^{*}] = (\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array}) = [T]^{†} .$

Sources

Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415v6 (2026-08-27), Appendices A.3, A.5 and A.10