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Exact two-step equal-rate cascade restricted to the detection isometry

Statement

Assume AC. Work on $V_{2} \oplus V_{1} \oplus V_{0}$ with the spin bases and the isometry $W$ from the spin-two population lemma. Fix $γ > 0$ , $ω_{0} > 0$ and real $Δ$ . Take the diagonal matrix $H e_{J, m} = [ω_{0} J (J + 1) + Δ J m] e_{J, m}$ and finite lowering matrices $L_{δ}$ , whose $V_{J} \to V_{J - 1}$ blocks are $\sqrt{γ} \sqrt{(2 J + 1) / (2 J - 1)} C_{1, δ; J, m}^{J - 1, m + δ}$ for $J = 1, 2$ , and whose other blocks vanish. Let $N_{t} = e^{t L}$ be their finite GKSL channel, and let $g = e_{0, 0}$ . For every $t \geq 0$ and every $X \in L (C^{2})$ , the restricted channel $N_{t} (W X W^{†})$ has orthogonal output sectors $V_{2}, V_{1}, V_{0}$ of the form $p_{0} W_{t} X W_{t}^{†} \oplus p_{1} C_{t} (X) \oplus p_{2} Tr (X) ∣ g ⟩ ⟨ g ∣,$ where $x = γ t$ , $p_{0} = e^{- x}$ , $p_{1} = x e^{- x}$ , $p_{2} = 1 - (1 + x) e^{- x}$ , and $W_{t} = e^{- i t H ∣_{V_{2}}} W$ is an isometry. For $t > 0$ , $C_{t} : L (C^{2}) \to L (V_{1})$ is a channel with $C_{t} (I_{2}) = 2 I_{3} / 3$ and $C_{t} (Z) = diag (1 / 3, - 2 / 3, 1 / 3)$ , where $Z = diag (1, - 1)$ on the logical input and output order is $m = - 1, 0, 1$ . The family $C_{t}$ has a continuous extension to $t = 0$ . At $t = 0$ the sector formula uses this extension.

Facts & Assumptions

[F1]

The normalized $V_{2}$ -to- $V_{1}$ jump blocks $B_{δ}$ have the stated input-isometry populations and adjoint-product normalization. Spin-two downward coefficients and detection-code output populations

[F2]

Spin-block coordinates, ladder coefficients, magnetic labels and coupled phases are fixed. Finite spin blocks, spherical tensors and Clebsch–Gordan phases

[F3]

The normalized lowering components have adjoint-product sum equal to identity separately on V2 and V1. Downward dipole coefficient normalization contract

[F4]

A finite Hermitian Hamiltonian and finite jump matrices define the displayed GKSL exponential channel. Finite GKSL evolution and the QEC channel interface

[F5]

Matrix exponentials are differentiable groups and solve finite constant-coefficient linear evolution uniquely. The exponential series of a bounded operator

[F6]

Continuous finite-dimensional integrands obey the fundamental theorem of calculus under Countable Choice. Fundamental theorem of calculus for Banach-valued continuous curves

[F7]

Reorder absolutely integrable matrix entries over finite time intervals by Fubini. Fubini's theorem for L¹ functions on a sigma-finite product

[F8]

Assume full AC, carrying Countable Choice for integration. The Axiom of Choice

[F9]

Every finite-dimensional completely positive trace-preserving map admits a finite normalized Kraus family. Finite channel, Kraus, Choi, and Stinespring equivalence

Proof

Given: The three finite spin blocks and matrices in the statement.

Goal: Derive the number-of-decays sector formula and the needed one-decay population identities.

1.1givenF1F2F3F4F5algebra

Because different output spin blocks are orthogonal, [F3] gives $\sum L_{δ}^{†} L_{δ} = γ (I_{V_{2}} \oplus I_{V_{1}}) \oplus 0$ . Starting with an operator in $V_{2}$ , the block-diagonal Hamiltonian and lowering jumps preserve the direct sum of diagonal spin-block operator sectors; no off-diagonal spin block is produced. Its top sector is therefore $Z_{2} (t) = e^{- γ t} U_{2} (t) W X W^{†} U_{2} (t)^{†}$ , where $U_{J} (v) = e^{- i v H ∣_{V_{J}}}$ . This is obtained by direct differentiation of the block GKSL equation.

1.2F1F4F5F6F8F9step 1.1algebra

Variation of constants in the V1 sector gives $Z_{1} (t) = γ e^{- γ t} \int_{0}^{t} \sum_{δ} U_{1} (t - s) B_{δ} U_{2} (s) W X W^{†} U_{2} (s)^{†} B_{δ}^{†} U_{1} (t - s)^{†} d s .$ Indeed differentiating this expression gives its incoming jump term from $Z_{2}$ and the V1 no-jump generator, with zero initial value. Define $C_{t}$ as the integral in this expression divided by $t$ . Each integrand is a channel: its Kraus operators are $U_{1} (t - s) B_{δ} U_{2} (s) W$ and their adjoint-product sum is $I_{2}$ by [F1]. Its average is completely positive at every finite ancilla and trace-preserving. It is thus a finite-dimensional channel and has a finite Kraus family by [F9]. This yields $Z_{1} = p_{1} C_{t} (X)$ .

2.1F3F4F5F6F8step 1.1step 1.2algebra

The ground block is a scalar multiple of $∣ g ⟩ ⟨ g ∣$ , and its scalar derivative is $γ Tr Z_{1} (t)$ by the V1 normalization. Since $Tr Z_{1} (t) = γ t e^{- γ t} Tr X$ , integration gives $Z_{0} (t) = [1 - (1 + γ t) e^{- γ t}] Tr X ∣ g ⟩ ⟨ g ∣$ . The displayed sectors solve the entire block equation and have the correct initial data; uniqueness follows by multiplying the difference of solutions by $e^{- t L}$ and differentiating. Thus they are exactly the channel, including its two-decay sector.

3.1F1F2F6F7step 1.2algebra∎

Before a first jump, $U_{2} (s) a$ differs from $a$ only by phases on its two magnetic components and a common phase, whereas $U_{2} (s) b$ differs from $b$ only by a common phase. Thus [F1] gives diagonal one-decay output populations $(1 / 2, 0, 1 / 2)$ and $(1 / 6, 2 / 3, 1 / 6)$ for every $s$ . The subsequent diagonal $U_{1} (t - s)$ leaves them unchanged. Summing these two outputs gives $C_{t} (I_{2}) = 2 I_{3} / 3$ and subtracting them gives the stated $C_{t} (Z)$ . Rescale $s = t u$ in the definition of $C_{t}$ : it becomes an integral over $0 \leq u \leq 1$ of matrix entries continuous jointly in $(t, u)$ . Compact-uniform continuity then gives a continuous extension to $t = 0$ by integration, without dividing by a vanishing time.

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