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Finite-dimensional gapped protocol: the 1/T leakage estimate, the intertwining propagator and the Born leakage probability

Physical scope

The declared finite-dimensional adiabatic protocol: regularity of the followed cluster projection, the intertwining of the modified propagator, the operator-norm leakage estimate ||(I-P(s))U_T(s)P(0)|| <= C/T with the explicit constant, and the resulting Born leakage and survival probabilities of an initial preparation in the followed subspace. The theorem is conditional on the finite branch, a uniform positive gap and coherent following; it asserts no unbounded family, no gap closing, no laboratory ramp, no branch selection and no empirical result.

Statement

Adopt the declared adiabatic protocol of The declared adiabatic protocol model: slow parameter, followed spectral subspace, uniform gap and leakage probability in its finite branch (f): $H = C^{d}$ for some $d \geq 1$ , $H \in C^{2} ([0, 1]; Herm (C^{d}))$ , $T > 0$ and $U_{T} (s, σ)$ the protocol propagator, and let $P (s)$ be the spectral projection of $H (s)$ for a spectral cluster $σ_{P} (s)$ that is uniformly gapped with $dist (σ_{P} (s), σ (H (s)) ∖ σ_{P} (s)) \geq g > 0 (s \in [0, 1])$ and coherently followed in the sense of the protocol definition (continuity in operator norm, equivalently the local Riesz continuation of $P (0)$ ). Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Then:

(a) Regularity of the followed cluster. $P \in C^{2} ([0, 1]; End (C^{d}))$ , the commutator $K (s) : = [P^{'} (s), P (s)]$ is anti-Hermitian of class $C^{1}$ with $[K (s), P (s)] = P^{'} (s)$ , and there is a $C^{1}$ Hermitian family $X (s)$ with $[H (s), X (s)] = K (s)$ for every $s \in [0, 1]$ .

(b) Intertwining. Let $U_{A}$ solve $i \partial_{s} U_{A} (s) = (T H (s) / ℏ + i K (s)) U_{A} (s)$ , $U_{A} (0) = I$ (equivalently $U_{A}^{'} = (- i T H / ℏ + K) U_{A}$ ). Then $P (s) U_{A} (s) = U_{A} (s) P (0)$ for every $s \in [0, 1]$ .

(c) Leakage estimate. With $C : = ℏ (2 \sup_{s \in [0, 1]} ∥ X (s) ∥ + \int_{0}^{1} (∥ X^{'} (s) ∥ + ∥ K (s) ∥ ∥ X (s) ∥) d s) < \infty,$ one has, for every $s \in [0, 1]$ , $∥ (I - P (s)) U_{T} (s) P (0) ∥ \leq \frac{C}{T},$ and therefore, for every unit vector $ψ (0) \in ran P (0)$ with $ψ (s T) = U_{T} (s, 0) ψ (0)$ , $p_{l e a k} (s) = ∥ (I - P (s)) ψ (s T) ∥^{2} \leq \frac{C^{2}}{T^{2}}, ⟨ P (s) ψ (s T), ψ (s T) ⟩ \geq 1 - \frac{C^{2}}{T^{2}} .$ The same Born leakage and survival bounds hold for every supported mixed preparation $ρ_{0} = P (0) ρ_{0} P (0)$ , with $ρ (s) = U_{T} (s, 0) ρ_{0} U_{T} (s, 0)^{*}$ and $p_{l e a k} (s) = tr (ρ (s) (I - P (s)))$ .

(d) Scope. The estimate is conditional on the uniform positive gap, on coherent following and on the finite branch; the constant $C$ depends on the family and its label, not on $T$ . The theorem asserts no statement for gap-closing families, no unbounded or common-domain family (which is the subject of the companion theorem), no laboratory protocol, no branch selection or relaxation, and no empirical result.

Facts & Assumptions

Given: the declared finite-branch adiabatic protocol with family $H$ , duration $T$ , gap $g$ , followed projection $P$ and propagator $U_{T}$ ; a unit vector $ψ (0) \in ran P (0)$ and $s \in [0, 1]$ ; the Axiom of Countable Choice and the Axiom of Choice.

[F1]

Finite gapped adiabatic supplier and physical scaling. Applied to the normalized generator $A = H / ℏ$ , Finite gapped adiabatic estimate gives $P \in C^{2}$ , $K = [P^{'}, P]$ , $[K, P] = P^{'}$ and a $C^{1}$ Hermitian $X_{A}$ with $[A, X_{A}] = K$ . Its modified propagator solves $i U_{A}^{'} = (T A + i K) U_{A}$ , intertwines $P$ , and its leakage constant is $C_{A} = 2 \sup ∥ X_{A} ∥ + \int_{0}^{1} (∥ X_{A}^{'} ∥ + ∥ K ∥ ∥ X_{A} ∥)$ . Put $X = X_{A} / ℏ$ ; then $[H, X] = K$ and $C_{A} = ℏ (2 \sup ∥ X ∥ + \int_{0}^{1} (∥ X^{'} ∥ + ∥ K ∥ ∥ X ∥))$ . The normalized protocol generator $T H / ℏ$ has the unitary Dyson propagator (Norm Dyson series and uniqueness), which solves exactly the physical equation $i ℏ U_{T}^{'} = T H U_{T}$ .

[F2]

Protocol readings. The protocol declares $U_{T}$ as its propagator, $P (s)$ as the followed subspace and $p_{l e a k} (s) = ∥ (I - P (s)) ψ (s T) ∥^{2}$ as the leakage probability of a unit initial vector in $ran P (0)$ (The declared adiabatic protocol model: slow parameter, followed spectral subspace, uniform gap and leakage probability clauses (b), (c)); the probability is the Born outcome probability of the projection $I - P (s)$ in the evolved preparation, in the P2 reading (Assigned sharp observables and the outcome law of the framework clauses (a), (b); State and composition postulate P1: Hilbert spaces, preparations and composite systems; Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes; The state space: density operators, pure and mixed states, and purity; Dimensions, SI unit conventions and quantity types of the framework).

[F3]

Norm and inner-product estimates. For a bounded operator $B$ and a vector $v$ one has $∥ B v ∥ \leq ∥ B ∥ ∥ v ∥$ (A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum); for a projection $Q$ and unit $v$ the identity $∥ Q v ∥^{2} + ∥ (I - Q) v ∥^{2} = ∥ v ∥^{2} = 1$ holds in the inner product of $C^{d}$ (Real and complex inner product spaces, with the inner product linear in the first argument; Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors); and $I - P (s)$ is again a projection because $P (s)$ is an orthogonal spectral projection.

[F4]

Supported densities. Every density has a trace-norm convergent orthonormal expansion with positive eigenvalue weights $ρ_{0} = \sum_{j} w_{j} T_{u_{j}, u_{j}}$ , $\sum_{j} w_{j} = 1$ , and its projection probabilities are the weighted pure projection probabilities (Density operators and scalar spectral probability, Statement(a),(b)).

Proof

technique · direct; instantiate the finite gapped adiabatic supplier at the protocol data and convert its operator-norm leakage estimate into the Born leakage probability

Given: the data of the Statement and [F1]-[F4].

1.1F1given

[Regularity, commutator and intertwining] The family $A = H / ℏ$ is C² Hermitian with the same coherently followed spectral projections and gap $g / ℏ > 0$ . Applying [F1] and putting $X = X_{A} / ℏ$ gives $P \in C^{2}$ , the anti-Hermitian $K = [P^{'}, P]$ with $[K, P] = P^{'}$ , the $C^{1}$ Hermitian $X$ with $[H, X] = K$ , and the intertwining $P (s) U_{A} (s) = U_{A} (s) P (0)$ . These are clauses (a) and (b).

2.1F1step 1.1given

[Leakage operator estimate] [F1] gives $∥ (I - P (s)) U_{T} (s) P (0) ∥ \leq C / T$ with the displayed physical constant $C = C_{A}$ for every $s \in [0, 1]$ . This is the first display of clause (c).

3.1step 2.1F2F3algebra

[Born leakage and survival] Let $ψ (0) \in ran P (0)$ be unit, so $P (0) ψ (0) = ψ (0)$ , and let $ψ (s T) = U_{T} (s, 0) ψ (0)$ , a unit vector because $U_{T}$ is unitary [F1]. Then $(I - P (s)) ψ (s T) = (I - P (s)) U_{T} (s, 0) P (0) ψ (0),$ so $∥ (I - P (s)) ψ (s T) ∥ \leq ∥ (I - P (s)) U_{T} (s, 0) P (0) ∥ \leq C / T$ by [F3] and step 2.1; squaring gives $p_{l e a k} (s) \leq C^{2} / T^{2}$ by [F2]. Since $P (s)$ and $I - P (s)$ are complementary projections on the unit vector $ψ (s T)$ , the survival probability is $⟨ P (s) ψ (s T), ψ (s T) ⟩ = 1 - ∥ (I - P (s)) ψ (s T) ∥^{2} \geq 1 - C^{2} / T^{2}$ by [F3]. This is the second and third display of clause (c).

4.1step 3.1F4algebra

[Supported mixed preparations] Let $ρ_{0} = P (0) ρ_{0} P (0)$ . In the expansion of [F4], every vector of positive eigenvalue $w_{j}$ satisfies $u_{j} = w_{j}^{- 1} ρ_{0} u_{j} \in ran P (0)$ . Unitary transport gives the same weights and orthonormal vectors $U_{T} (s, 0) u_{j}$ in the evolved density. By [F4], its leakage projection probability is $\sum_{j} w_{j} ∥ (I - P (s)) U_{T} (s, 0) u_{j} ∥^{2}$ . The pure bound in step 3.1 holds for each such $u_{j}$ , so this sum is at most $C^{2} / T^{2}$ since $\sum_{j} w_{j} = 1$ . The complementary projection probability is $1 - p_{l e a k} (s) \geq 1 - C^{2} / T^{2}$ . All operators in the trace expression are bounded projections; no unbounded-density product or eigenvector selection within the degenerate energy level is required.

5.1step 1.1step 2.1step 3.1given∎

[Scope] The estimate used only the finite branch, the uniform gap $g > 0$ , the coherent following and the finite constant $C$ , which does not depend on $T$ ; no gap-closing, unbounded, laboratory, selection or empirical statement entered, which is clause (d).

Remarks

What the theorem replaces. The sources motivate the adiabatic behaviour of a gapped level by the slowness of the perturbation; the theorem replaces that qualitative statement by the explicit operator bound $C / T$ and the Born bound $C^{2} / T^{2}$ , valid uniformly on $[0, 1]$ and only under the stated gap and following hypotheses.
Why coherent following is a hypothesis. Without the uniform gap the Riesz projection of the initial cluster need not be defined away from $s = 0$ ; the protocol therefore declares the followed cluster together with the gap, and both enter the constant through $X$ and $K$ .

Finite-dimensional gapped protocol: the 1/T leakage estimate, the intertwining propagator and the Born leakage probability

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources