DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

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Unitary conjugation map

Definition

Let $H, K$ be nonzero finite-dimensional (have finite bases by Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis) complex inner-product spaces of equal dimension. We call a surjective linear isometry $U : H \to K$ (Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces) a unitary map between these spaces. By Finite-dimensional linear isometries satisfy the adjoint identity, its adjoint exists uniquely, $U^{†} U = I_{H}$ , and $U U^{†} = I_{K}$ . Define the unitary conjugation map ${Ad}_{U} : L (H) \to L (K), {Ad}_{U} (X) = U X U^{†} .$ Here $L (H)$ and $L (K)$ are the complex-linear endomorphism spaces (Linear map between vector spaces over the same field). It is well-defined because the composition is $U \circ X \circ U^{†} : K \to K$ , and it is complex-linear in $X$ .

Depends on

Used by

Unitary conjugation is a channel Proposition
Unitary conjugation is invariant under a global phase Proposition

Dependency tree · two levels

25 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 (2018), §2.2.1 (standard reference, not scraped)