TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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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.
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Operator error bounds measured output variation

Statement

Let $U, V$ be unitary operators on a finite-dimensional Hilbert space, let $ρ$ be any density operator, and let $M$ be any finite POVM. If $∥ U - V ∥_{o p} \leq ϵ$ , then the outcome distributions obtained by measuring $U ρ U^{*}$ and $V ρ V^{*}$ have total-variation distance at most $ϵ$ .

Facts & Assumptions

Given: Finite-dimensional unitaries $U, V$ , a density operator $ρ$ , a finite POVM, and $∥ U - V ∥_{o p} \leq ϵ$ .

[F1]

For unit vectors $u, v$ , pure-state trace distance is at most $∥ u - v ∥$ . Pure-state trace distance is bounded by vector error

[F2]

Trace distance is one half the trace norm of the density-operator difference. Trace Distance

[F3]

The trace norm obeys the triangle inequality and positive homogeneity. Trace Norm Duality And Norm Properties

[F4]

A finite POVM's total-variation change is bounded by trace distance. Measurement Variation At Most Trace Distance

[F5]

Every finite-dimensional density operator has an orthonormal spectral decomposition; its eigenvalues are nonnegative and sum to one. Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely Density operator

[F6]

The operator norm is induced by the Hilbert-space norm, and a unitary preserves that vector norm. The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv|| Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces

Proof

1.1F1F2F6givenalgebra

First let $ρ = ∣ ψ ⟩ ⟨ ψ ∣$ be pure with $∥ ψ ∥ = 1$ . Then $U ψ, V ψ$ are unit vectors by [F6] and $∥ U ψ - V ψ ∥ = ∥ (U - V) ψ ∥ \leq ϵ$ by the operator-norm definition in [F6]. By [F2] and the pure-state estimate in [F1], the output trace distance is at most $ϵ$ .

2.1F2F3F5step 1.1algebra

For a general density operator, [F5] gives $ρ = \sum_{j} p_{j} ∣ ψ_{j} ⟩ ⟨ ψ_{j} ∣$ with $p_{j} \geq 0$ and $\sum_{j} p_{j} = 1$ . Unitary conjugation preserves this mixture. The trace-norm triangle inequality and homogeneity [F3], applied to the sum of output projector differences, bound the output trace distance by $\sum_{j} p_{j} ϵ = ϵ$ using step 1.1.

3.1F4step 2.1∎

Apply the POVM variation bound [F4] to the two output density operators from step 2.1. Their total variation distance is at most their trace distance, hence at most $ϵ$ .

Depends on

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Sources

John Preskill, Quantum Computation, Chapter 2: Foundations I: States and Ensembles (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes (standard reference, not scraped)