LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Modified-operator distance and residual composition

Statement

Assume the Axiom of Countable Choice, as required by the quadrant-modification construction. Let $V = C^{d \times d}$ for a positive integer $d$ , with spectral norm $∥ X ∥_{s p} = \sup_{∥ z ∥_{2} \leq 1} ∥ X z ∥_{2}$ , where $∥ z ∥_{2} = (\sum_{j} ∣ z_{j} ∣^{2})^{1 / 2}$ . Use the quadrant maps $B (t), L (t), M (t) : V \to V$ , which are complex-linear, with induced norm $∥ T ∥ = \sup_{∥ X ∥_{s p} \leq 1} ∥ T (X) ∥_{s p}$ . Fix $Γ > 0$ , $τ \geq 0$ and $R \geq 0$ . On an open real time interval $J$ , assume $M$ is differentiable in induced norm and $\dot{M} = L - B$ , $∥ M ∥ \leq Γ τ$ , $∥ L ∥ \leq Γ / 2$ . The derivative identity is a hypothesis of this composition estimate.

Let $ρ : J \to V$ be differentiable in spectral norm, with each $ρ (t)$ a density operator on $C^{d}$ , and let $ξ : J \to V$ satisfy $\dot{ρ} = B ρ + ξ$ , $∥ ξ ∥_{s p} \leq R$ and $∥ \dot{ρ} ∥_{s p} \leq Γ / 2$ . Products such as $B ρ$ denote evaluation $B (t) [ρ (t)]$ , $L M ρ = L (t) [M (t) [ρ (t)]]$ , and $I$ is the identity map on $V$ . All identities and bounds are pointwise on $J$ .

Then $σ = (I + M) ρ$ satisfies $∥ σ - ρ ∥_{s p} \leq Γ τ$ and $\dot{σ} = L σ + η$ with $η = ξ + M \dot{ρ} - L M ρ$ and $∥ η ∥_{s p} \leq R + Γ^{2} τ$ . For $R = Γ^{2} τ$ this is $2 Γ^{2} τ$ ; for the finite-preparation budget $R = Γ^{2} τ + δ \geq 0$ it is $2 Γ^{2} τ + δ$ . Positivity or exact trace normalization of $σ$ is not required for these operator estimates.

Facts & Assumptions

[F1]

The preparation-budget lemma retains the finite tail and shows only convergence of residual budgets, not of the states. Finite-preparation residual budget and its remote-boundary limit

[F2]

Under Countable Choice and their defining dominated-kernel hypotheses, the quadrant maps act on this same matrix space and obey the stated induced-norm bounds. Bounded quadrant modification and generator norms

[F3]

Differentiation is the real-time difference-quotient limit in the indicated norm. To identify it with real-coordinate differentiation, regard the matrix and map spaces as real vector spaces. Let $E_{j k}$ have entry $1$ at $(j, k)$ and zero elsewhere, for $0 \leq j, k < d$ . The matrices $E_{j k}, i E_{j k}$ form a real basis of $V$ : their coefficients are precisely the real and imaginary entries. For the complex-linear map space use $H_{j k, a b} (X) = X_{a b} E_{j k}$ and $i H_{j k, a b}$ , with $0 \leq j, k, a, b < d$ . Their coefficients are the real and imaginary entries of $T (E_{a b})$ , so they form a real basis. Each map is bounded: $∣ X_{a b} ∣ \leq ∥ X ∥_{s p}$ gives $∥ T (X) ∥_{s p} \leq \sum_{a, b} ∣ X_{a b} ∣ ∥ T (E_{a b}) ∥_{s p} \leq (\sum_{a, b} ∥ T (E_{a b}) ∥_{s p}) ∥ X ∥_{s p}$ . Thus [F6] supplies norms on these real spaces. Over the complete field $R$ , finite-dimensional norm equivalence compares each indicated norm above and below with its real-coordinate maximum norm by positive constants. Applied to difference quotients minus a candidate limit, those comparisons identify norm convergence with coordinate convergence, hence identify the derivatives with the real-coordinate definition. The assumed identity $\dot{M} = L - B$ is used directly; no continuity of $B$ or $L$ is invoked here. The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral Finite dimensional norm equivalence over a complete valued field

[F4]

A density operator is a non-negative endomorphism with trace one. Non-negative means self-adjoint with nonnegative quadratic form, and trace is computed in any ordered basis. Density operator Non-negative and positive operators The basis-independent trace of an endomorphism of a finite-dimensional vector space

[F5]

A normal endomorphism of a finite-dimensional complex inner-product space has an orthonormal eigenbasis. In particular this applies to a self-adjoint endomorphism, since it commutes with its adjoint. Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely

[F6]

The operator norm satisfies $∥ T (X) ∥_{s p} \leq ∥ T ∥ ∥ X ∥_{s p}$ and is a norm on the space of bounded maps. The spectral norm is itself the operator norm for matrices acting on Euclidean vectors. The operator norm as the least bound and as the unit-sphere or unit-ball supremum The operator norm is a norm on the space of bounded linear operators

[F7]

Matrix addition and scalar multiplication in $V = M_{d \times d} (C)$ are entrywise; addition and scalar multiplication of the maps are pointwise in their matrix outputs. The vector space M_m × n(F) := F^ m × n of m by n matrices over a field, with entrywise operations The operator norm is a norm on the space of bounded linear operators

[A1]

Assume the Axiom of Countable Choice inherited from the quadrant-modification supplier. It is used to invoke that supplier's Lebesgue-measure construction and the quadrant-map bounds in [F2]; the local product differentiation and norm estimates require no additional choice principle. The Axiom of Countable Choice (AC_ω) Bounded quadrant modification and generator norms

Proof

Given: Countable Choice, the matrix space, quadrant maps, differentiable curves, derivative identity and bounds in the Statement. Goal: Establish the modified-operator distance and residual estimates.

1.1givenF4F5F6

Fix $t \in J$ . By [F4] and [F5], $ρ (t)$ has an orthonormal eigenbasis $e_{1}, \dots, e_{d}$ with eigenvalues $λ_{j}$ . Nonnegativity gives $λ_{j} = ⟨ ρ (t) e_{j}, e_{j} ⟩ \geq 0$ , and the trace computed in that basis gives $\sum_{j} λ_{j} = 1$ . Hence $0 \leq λ_{j} \leq 1$ . For $z = \sum_{j} c_{j} e_{j}$ , orthonormality gives $∥ ρ (t) z ∥_{2}^{2} = \sum_{j} λ_{j}^{2} ∣ c_{j} ∣^{2} \leq \sum_{j} ∣ c_{j} ∣^{2} = ∥ z ∥_{2}^{2}$ . Taking the unit-ball supremum shows $∥ ρ (t) ∥_{s p} \leq 1$ .

1.2givenF3F6F7

The product differentiation needed here follows directly from difference quotients. For fixed $t$ and nonzero sufficiently small real $h$ , put $Q_{M} (h) = (M (t + h) - M (t)) / h$ and $Q_{ρ} (h) = (ρ (t + h) - ρ (t)) / h$ . By differentiability these converge in their respective norms to $\dot{M} (t)$ and $\dot{ρ} (t)$ ; in particular they are bounded for small $h$ , and $ρ (t + h) - ρ (t) = h Q_{ρ} (h) \to 0$ . The difference quotient of $M ρ$ equals $Q_{M} (h) [ρ (t + h)] + M (t) [Q_{ρ} (h)]$ . By [F6], the difference of its first term from $\dot{M} (t) [ρ (t)]$ is bounded by $∥ Q_{M} (h) ∥ ∥ ρ (t + h) - ρ (t) ∥_{s p} + ∥ Q_{M} (h) - \dot{M} (t) ∥ ∥ ρ (t) ∥_{s p}$ , which tends to zero; the second term converges to $M (t) [\dot{ρ} (t)]$ . Thus $\dot{σ} = \dot{ρ} + \dot{M} ρ + M \dot{ρ}$ . Substitution of the two assumed differential identities gives $\dot{σ} = L ρ + ξ + M \dot{ρ} = L σ + ξ + M \dot{ρ} - L M ρ$ .

2.1givenA1F2F6step 1.1

Therefore $∥ σ - ρ ∥_{s p} = ∥ M ρ ∥_{s p} \leq ∥ M ∥ ∥ ρ ∥_{s p} \leq Γ τ$ .

3.1givenA1F1F2F6step 1.1step 1.2∎

The three residual summands have spectral norms at most $R$ , $(Γ τ) (Γ / 2)$ and $(Γ / 2) (Γ τ) ∥ ρ ∥_{s p}$ , respectively. By the triangle inequality and step 1.1, $∥ η ∥_{s p} \leq R + Γ^{2} τ$ . Substituting each stated value of $R$ proves both specializations, retaining the finite-preparation tail as in [F1]. No positivity or trace claim for the modified operator entered these estimates. They bound a differential residual, rather than the distance between residual-free trajectories, and assert no convergence of preparation families.

Depends on

Used by · 1

ULE residual bound and the modified-state distinction

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.