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Normalized jumps give a scalar total decay operator

PDF p. 14: Eq. (S4.1)

Statement

Assume the Axiom of Choice, Countable Choice and Dependent Choice, and all system, bath, preparation, weak-Markov, regularity, tensor and pointwise spectral hypotheses of the finite downward-jumps result. Adopt the equal-rate benchmark for integer-spin blocks $J = 0, 1, \dots, N$ , where $N \geq 1$ and $γ > 0$ . The magnetic vectors $∣ J, m ⟩$ , $m = - J, \dots, J$ , form an orthonormal basis. For $δ \in {- 1, 0, 1}$ , write $C_{1, δ; J, m}^{J - 1, m + δ}$ for the specified ordered-coupling Clebsch–Gordan coefficient, taken to be zero when $∣ m + δ ∣ > J - 1$ .

For each $δ$ , the modeled jump satisfies $L_{δ}^{†} L_{δ} = γ \sum_{J = 1}^{N} \sum_{m = - J}^{J} \frac{2 J + 1}{2 J - 1} ∣ C_{1, δ; J, m}^{J - 1, m + δ} ∣^{2} ∣ J, m ⟩ ⟨ J, m ∣$ . Defining $V = - \frac{1}{2} \sum_{δ = - 1}^{1} L_{δ}^{†} L_{δ}$ gives $V = - \frac{γ}{2} \sum_{J = 1}^{N} \sum_{m = - J}^{J} ∣ J, m ⟩ ⟨ J, m ∣$ . Thus every excited basis state has total instantaneous downward jump rate $γ$ , while the ground block has rate zero.

The article's sums over all integer $J \geq 1$ have the same identities on finite-support linear combinations of the labeled basis vectors: each side acts by the displayed finite sum on any such vector. This algebraic statement does not assert an infinite-dimensional physical evolution.

Facts & Assumptions

[F1]

Under the stated finite emitter and bath hypotheses, the downward-jumps result gives $L_{δ} = \sqrt{γ} \sum_{J = 1}^{N} \sum_{m = - J}^{J} f (J) C_{1, δ; J, m}^{J - 1, m + δ} ∣ J - 1, m + δ ⟩ ⟨ J, m ∣$ , with out-of-range terms zero. Canonical angular-momentum downward jumps

[F2]

The benchmark chooses $f (J) = \sqrt{(2 J + 1) / (2 J - 1)}$ and $η_{J} = J (J + 1)$ for the finite integer-spin emitter; $γ$ is its positive jump-rate scale. Equal-rate rigid-rotor benchmark parameters

[F3]

The magnetic basis is orthonormal, and the coefficients use the ordered coupling and zero convention in the Statement. Angular-momentum basis and tensor labels

[F4]

For each integer $J \geq 1$ and allowed $m$ , $\sum_{δ = - 1}^{1} ∣ C_{1, δ; J, m}^{J - 1, m + δ} ∣^{2} = (2 J - 1) / (2 J + 1)$ . Clebsch–Gordan total-rate identity

[F5]

In the finite GKSL convention, the jump contribution associated with $L_{δ}$ is $L_{δ} ρ L_{δ}^{†}$ , and $†$ denotes conjugate transpose. Finite GKSL evolution and the QEC channel interface

[A1]

Assume the three stated choice principles and the physical and spectral hypotheses required by the downward-jumps result. The Axiom of Choice The Axiom of Countable Choice (AC_ω) The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain Canonical angular-momentum downward jumps

Proof

Given: The finite emitter, benchmark, choice principles and physical hypotheses in the Statement.

Goal: Calculate each $L_{δ}^{†} L_{δ}$ , their sum, and the instantaneous rates, and establish the stated finite-support interpretation.

1.1F1F2F3A1algebra

For a basis input with $J \geq 1$ , orthonormality selects one input term in [F1], giving $L_{δ} ∣ J, m ⟩ = \sqrt{γ} f (J) C_{1, δ; J, m}^{J - 1, m + δ} ∣ J - 1, m + δ ⟩$ , with zero for an inadmissible output. The ground block is annihilated. For fixed $δ$ , distinct admissible pairs $(J, m)$ give distinct output basis labels $(J - 1, m + δ)$ . Hence their output vectors are orthogonal, so $L_{δ}^{†} L_{δ}$ is diagonal with entry $γ ∣ f (J) ∣^{2} ∣ C_{1, δ; J, m}^{J - 1, m + δ} ∣^{2}$ on $∣ J, m ⟩$ and zero on the ground block. Substituting [F2] proves the first displayed identity.

2.1step 1.1F4algebra

Sum the diagonal entries over $δ$ and apply [F4]. For every excited basis vector the entry is $γ \frac{2 J + 1}{2 J - 1} \frac{2 J - 1}{2 J + 1} = γ$ , and it is zero on the ground block. Multiplication by $- 1 / 2$ gives the stated formula for $V$ .

3.1step 2.1F5algebra

Define the total instantaneous downward jump rate at a normalized basis state to be the sum of the traces of the jump contributions in [F5]. For $ρ = ∣ J, m ⟩ ⟨ J, m ∣$ this is $\sum_{δ} ∥ L_{δ} ∣ J, m ⟩ ∥^{2}$ , the corresponding diagonal entry of $\sum_{δ} L_{δ}^{†} L_{δ}$ . Step 2.1 therefore gives $γ$ on each excited basis state and zero on the ground block. This is a conditional instantaneous rate, not a claim about later evolution or an observation.

4.1step 1.1step 2.1F2F3F4algebra∎

For the unbounded label set, define the displayed jumps and their formal adjoints on finite-support vectors by the same matrix coefficients. For fixed $δ$ , each basis input and each basis output has at most one corresponding nonzero term for a given adjacent pair of blocks, so these maps preserve finite support. The calculation in steps 1.1 and 2.1 applies to every basis vector and hence by linearity to every finite-support vector. This proves the article's infinite-sum identities in that algebraic sense without assuming an infinite-dimensional physical channel.

Depends on

Used by · 14

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF page 14, Eq. (S4.1), and PDF page 15, first paragraph