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Canonical angular-momentum downward jumps

PDF p. 3: Eq. (10); PDF p. 13: Eq. (S3.20) and preceding downward-branch paragraph

Statement

Assume the Axiom of Choice, Countable Choice and Dependent Choice. Take the finite-block angular-momentum emitter model with one copy of each integer-spin block $J = 0, 1, \dots, N$ , $N \geq 1$ , and strictly increasing real level factors $η_{J}$ . Its energy is $E_{J} = ω_{0} η_{J}$ , with $ω_{0} > 0$ , and $∣ J, m ⟩$ , $m = - J, \dots, J$ , is its orthonormal magnetic basis. Adopt all system, bath, self-adjoint realization, trace-class closure, remote-past preparation, weak-Markov, kernel regularity, pointwise Fourier and principal-value hypotheses of the static ULE result, for three Hermitian Cartesian couplings $X_{x}, X_{y}, X_{z}$ of norm at most one and a centered stationary Gaussian bath prepared in vacuum. Assume the fixed-coupling thermal-family and integrable-envelope hypotheses of the vacuum-density result, and identify this bath's ordinary ULE spectral matrix with $J (x) I_{3}$ for every real $x$ , where $J$ is a nonnegative representative of that family's limiting vacuum density. This identification and all ULE applicability conditions at the vacuum are additional physical hypotheses; total-variation convergence alone does not establish them. Define $X_{+ 1} = - (X_{x} + i X_{y}) / \sqrt{2}$ , $X_{- 1} = (X_{x} - i X_{y}) / \sqrt{2}$ , $X_{0} = X_{z}$ , and assume that these components form a rank-one spherical tensor family between the retained blocks.

Use a nonnegative spectral representative $J$ that agrees with the vacuum density almost everywhere and satisfies $J (ω) = 0$ at every required nonpositive transition frequency, including zero. This pointwise condition is an additional model assumption: an almost-everywhere density alone does not prescribe its values at transition frequencies. Let $γ > 0$ be the coupling scale in the static ULE convention, and put $Ω_{J} = ω_{0} (η_{J} - η_{J - 1})$ . For each $J \geq 1$ , let $R_{J}$ be the reduced matrix-element scalar in the displayed Clebsch–Gordan convention and define $f (J) = \sqrt{2 π J (Ω_{J})} R_{J}$ . Then a valid unitary choice of the three ULE jumps is

$L_{δ} = \sqrt{γ} \sum_{J = 1}^{N} \sum_{m = - J}^{J} f (J) C_{1, δ; J, m}^{J - 1, m + δ} ∣ J - 1, m + δ ⟩ ⟨ J, m ∣, δ \in {- 1, 0, 1} .$

A term with $∣ m + δ ∣ > J - 1$ is zero. The coefficient $C_{1, δ; J, m}^{J - 1, m + δ}$ is the product-basis coefficient in the chosen normalized coupled spin- $(J - 1)$ basis for $V_{1} \otimes V_{J}$ . The complex scalar $f (J)$ is independent of $m$ and $δ$ and incorporates the reduced matrix element and the selected spectral and coupling normalization; any radial and species-specific factors included in $X$ are included in $R_{J}$ . The finite sum preserves coherent contributions between different $J$ blocks in each channel. The corresponding infinite expression is formal on finite-support vectors unless separate convergence and domain hypotheses are supplied.

Statement references

Angular-momentum emitter, photon bath, and dipole approximation

Static ULE formulas and principal-value kernel

Spherical vector components and transition-frequency convention

Finite GKSL evolution and the QEC channel interface

Vacuum limit and cubic free-space spectral density

Rank-one angular matrix elements

Finite spin blocks, spherical tensors and Clebsch–Gordan phases

Facts & Assumptions

[F1]

The emitter postulate gives the magnetic basis and energies $E_{J} = ω_{0} η_{J}$ , independent of $m$ , in the finite-block model. Angular-momentum emitter, photon bath, and dipole approximation

[F2]

Under its static-system, bath, preparation, regularity and pointwise Fourier hypotheses, the static ULE result gives Cartesian jumps $L_{i} = 2 π \sqrt{γ} \sum_{a, b, j} g_{i j} (E_{b} - E_{a}) ⟨ a ∣ X_{j} ∣ b ⟩ ∣ a ⟩ ⟨ b ∣$ , with $g (x) = [J (x) / (2 π)]^{1 / 2}$ . Static ULE formulas and principal-value kernel

[F3]

Under the vacuum-density hypotheses, the limiting continuum spectral measure is supported on nonnegative frequencies and has the stated density almost everywhere; that statement alone does not fix a representative at individual frequencies. Vacuum limit and cubic free-space spectral density

[F4]

For fixed input and output spin blocks, the rank-one Wigner–Eckart result gives a scalar independent of magnetic labels and tensor component multiplying the ordered $V_{1} \otimes V_{J}$ Clebsch–Gordan coefficient. Matrix elements vanish unless $m^{'} = m + δ$ and $J^{'}$ occurs in the rank-one coupling list. Rank-one angular matrix elements Finite spin blocks, spherical tensors and Clebsch–Gordan phases

[F5]

The displayed Cartesian-to-spherical component matrix $U$ is unitary on the three channel labels, and transition frequency is source energy minus destination energy. Spherical vector components and transition-frequency convention

[F6]

For finite matrices in units $ℏ = 1$ , the dissipative part of the GKSL generator is $D_{L} (ρ) = \sum_{i} (L_{i} ρ L_{i}^{†} - \frac{1}{2} {L_{i}^{†} L_{i}, ρ})$ . The same Hamiltonian and dissipator define the same generator. Finite GKSL evolution and the QEC channel interface

[A1]

Assume the three choice principles and the physical, regularity, tensor and pointwise spectral hypotheses stated above. 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

Proof

Given: The finite integer-spin blocks, strictly increasing energies, rank-one tensor family and compatible ULE and vacuum hypotheses in the Statement.

Goal: Derive the displayed three downward jump operators and identify the scope of their coefficients.

1.1givenF2A1algebra

The spectral-identification hypothesis gives $J (x) = J (x) I_{3}$ at every real frequency, so its positive square root is $g (x) = \sqrt{J (x) / (2 π)} I_{3}$ . Apply [F2] under the explicitly assumed static ULE hypotheses at the vacuum. Substitution gives $L_{i} = \sqrt{2 π γ} \sum_{a, b} \sqrt{J (E_{b} - E_{a})} ⟨ a ∣ X_{i} ∣ b ⟩ ∣ a ⟩ ⟨ b ∣$ . This application uses the ordinary ULE spectral matrix supplied in the hypotheses, independently of any convergence claim for finite-temperature generators.

2.1F5F6step 1.1algebra

Let $U$ be the unitary component matrix in [F5] and set $L_{δ} = \sum_{i} U_{δ i} L_{i}$ . Since $\sum_{δ} U_{δ i} \overline{U_{δ j}} = δ_{i j}$ , expansion gives $\sum_{δ} L_{δ} ρ L_{δ}^{†} = \sum_{i} L_{i} ρ L_{i}^{†}$ and $\sum_{δ} L_{δ}^{†} L_{δ} = \sum_{i} L_{i}^{†} L_{i}$ . Thus [F6] gives the same dissipator, with unchanged Hamiltonian part. Linearity of matrix elements in step 1.1 yields $L_{δ} = \sqrt{2 π γ} \sum_{a, b} \sqrt{J (E_{b} - E_{a})} ⟨ a ∣ X_{δ} ∣ b ⟩ ∣ a ⟩ ⟨ b ∣$ . This proves the spherical formula locally for the assumed vacuum bath.

3.1F1F4A1step 2.1

Insert the basis $∣ J, m ⟩$ from [F1] into the formula in step 2.1. By [F4], a nonzero matrix element can only have $m^{'} = m + δ$ and $J^{'}$ in the rank-one coupling list. For integer $J \geq 1$ this list is $J - 1, J, J + 1$ ; for $J = 0$ the only possible output spin is $1$ . Output blocks outside the finite emitter are absent.

4.1F1F3A1step 3.1

For $J^{'} = J$ , the transition frequency $E_{J} - E_{J^{'}}$ is zero. For $J^{'} = J + 1$ , strict increase of $η_{J}$ makes it negative. The pointwise assumption in [A1] makes their spectral factors zero. Thus only $J^{'} = J - 1$ , with $J \geq 1$ , remains. The vacuum measure in [F3] motivates this convention but does not by itself determine those point values.

5.1F4step 2.1step 4.1algebra∎

For each remaining pair of blocks, [F4] gives $⟨ J - 1, m + δ ∣ X_{δ} ∣ J, m ⟩ = R_{J} C_{1, δ; J, m}^{J - 1, m + δ}$ , with a single $R_{J}$ for all $m$ and $δ$ . Its frequency is $Ω_{J}$ , independent of $m$ . Substitution into the formula in step 2.1 and the definition of $f (J)$ give the displayed formula, including zero out-of-range terms. The spherical formula in step 2.1 is a sum over all transitions in each channel, so this substitution retains the cross-block coherent sum. Since the emitter contains finitely many blocks, the displayed operators are finite matrices. The infinite notation has only the stated formal meaning without further analytic hypotheses.

Source notes

The article writes the sum without a finite cutoff. The finite-block statement follows from the completed finite-dimensional ULE supplier; no convergence of the unbounded infinite model is inferred. The article's $f (J)$ can absorb convention-dependent phases and physical factors, as made explicit by the definition above.

The vacuum coefficient formula is a direct local application of the static ULE contract. Its bath realization, Gaussian preparation, scalar spectral identification and analytic regularity at the vacuum are assumed separately; the vacuum-density limit supplies the limiting measure, rather than those dynamical applicability hypotheses.

Depends on

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF page 3, Eq. (10); PDF page 13, Eqs. (S3.17)–(S3.20) and intervening text