Physics theoremStatement: AI-adaptedProof: AI-generated

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Degenerate no-decay state is preserved

PDF p. 15: Eq. (S4.6) and following paragraph

Statement

Adopt the finite lifted model and all physical, spectral and choice hypotheses of the first-order fast-check channel and uniform-decay result, including the Axiom of Choice, Countable Choice and Dependent Choice. Its orthonormal basis is $∣ J, m ⟩$ for integer $J = 0, \dots, N$ and $m = - J, \dots, J$ , with $N \geq 1$ and decay-rate parameter $γ > 0$ . Set every lift $Δ (J, m)$ to zero. Let $J_{0} \geq 1$ be an available block, $P = \sum_{m = - J_{0}}^{J_{0}} ∣ J_{0}, m ⟩ ⟨ J_{0}, m ∣$ , and let $ρ_{0}$ be a density operator supported on that block: $P ρ_{0} P = ρ_{0}$ . A density operator is positive semidefinite with trace one.

Write $H$ for the model Hamiltonian, $L_{δ}$ for its downward jumps, $V = - \frac{1}{2} \sum_{δ} L_{δ}^{†} L_{δ}$ , and $K_{n u l l} = I + τ (- i H + V)$ . Then $[H, ρ_{0}] = 0$ and, as $τ ↓ 0$ at fixed model parameters, $K_{n u l l} ρ_{0} K_{n u l l}^{†} = (1 - γ τ) ρ_{0} + O (τ^{2})$ in matrix spectral norm. The normalized approximate null output equals $ρ_{0}$ whenever its trace is positive.

For the exact finite GKSL channel $N_{τ}$ , an ideal projective check with null projector $P$ has unnormalized null output $P N_{τ} (ρ_{0}) P = e^{- γ τ} ρ_{0}$ for every finite $τ \geq 0$ . Its normalized null state is therefore exactly $ρ_{0}$ . Repeating such checks after intervals of model evolution leaves the state unchanged on every all-null record; because the same time-homogeneous channel starts from the same state after each check, the checking clock can be reset. This is a conditional model prediction, not a claim of an observed measurement.

Facts & Assumptions

[F1]

In the finite model, $H = \sum_{J, m} [ω_{0} J (J + 1) + Δ (J, m)] ∣ J, m ⟩ ⟨ J, m ∣$ ; each $L_{δ}$ lowers $J$ by one and $H$ and $V$ are block diagonal. The model carries the physical, spectral and choice hypotheses stated above. First-order fast-check channel

[F2]

In the equal-rate benchmark, $V = - \frac{γ}{2} \sum_{J = 1}^{N} \sum_{m} ∣ J, m ⟩ ⟨ J, m ∣$ . Normalized jumps give a scalar total decay operator

[F3]

The ideal syndrome check has unnormalized null output $P N_{τ} (ρ_{0}) P$ and normalizes it by its trace when positive. Its first-order approximation has null numerator $K_{n u l l} ρ_{0} K_{n u l l}^{†}$ . Null-syndrome state and first-order expansion

[F4]

For the finite GKSL channel, the ordered-jump expansion has no-jump amplitude $e^{t G}$ , where $G = - i H + V$ , and every other term contains at least one jump separated by factors $e^{u G}$ . The series converges in induced matrix norm. Finite no-jump and ordered-jump expansion

[F5]

The finite GKSL channel is $N_{t} = e^{t L}$ for a time-independent generator $L$ ; matrix adjoints are conjugate transposes. Finite GKSL evolution and the QEC channel interface

Proof

Given: The finite model, its inherited hypotheses, vanishing lifts, supported density operator and ideal check in the Statement.

Goal: Prove the first-order numerator and the normalized approximate and exact null-state conclusions.

1.1givenF1F2algebra

Put $h = ω_{0} J_{0} (J_{0} + 1)$ . By [F1], $H P = h P$ because the lifts vanish. The support condition therefore gives $H ρ_{0} = ρ_{0} H = h ρ_{0}$ , so $[H, ρ_{0}] = 0$ . By [F2], $V P = - γ P / 2$ . Hence $G P = (- i h - γ / 2) P$ and $K_{n u l l} P = (1 - i h τ - γ τ / 2) P$ .

2.1step 1.1F3algebra

Consequently $K_{n u l l} ρ_{0} K_{n u l l}^{†} = ∣ 1 - i h τ - γ τ / 2 ∣^{2} ρ_{0} = (1 - γ τ) ρ_{0} + O (τ^{2})$ in spectral norm at fixed $h$ and $γ$ . Its trace is the same nonnegative scalar since $Tr ρ_{0} = 1$ . Whenever this scalar is positive, normalization gives exactly $ρ_{0}$ , as prescribed by [F3].

2.2step 1.1F1F4algebra

By [F4] and step 1.1, $e^{τ G} P = e^{(- i h - γ / 2) τ} P$ , so the no-jump term projected onto $P$ is $e^{- γ τ} ρ_{0}$ . Each propagation factor in any other ordered term preserves every $J$ block, while each jump lowers $J$ by one by [F1]. Thus any term containing a jump maps the input block strictly below $J_{0}$ and is killed by the final projection $P$ . Projection is continuous in this finite-dimensional norm, so the convergent expansion in [F4] gives $P N_{τ} (ρ_{0}) P = e^{- γ τ} ρ_{0}$ .

3.1step 2.2F3F5algebra∎

The exact null weight is $e^{- γ τ} > 0$ for finite $τ \geq 0$ , so [F3] gives conditional state $ρ_{0}$ . After each null check the state is again $ρ_{0}$ ; applying the same argument successively proves the all-null assertion. The time-independent generator in [F5] makes each subsequent interval use the same channel family from that restored state, giving the stated clock-reset interpretation.

Depends on

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF p. 15, Eq. (S4.6) and following paragraph