Physics theoremStatement: AI-adaptedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Three spherical emission channels

PDF p. 13: Eq. (S3.17) and following paragraph

Statement

Assume the Axiom of Choice, Countable Choice and Dependent Choice. Take the finite-dimensional static system, three Hermitian Cartesian coupling operators $X_{x}, X_{y}, X_{z}$ , stationary Gaussian bath, preparation, weak-Markov regime, Fourier identities and kernel regularity required by the static ULE result. Also take the regulated isotropic thermal photon model and its limit under the hypotheses of the isotropic-bath result. Require that its nonnegative scalar spectral representative $J$ agrees pointwise at every required transition frequency with the ordinary spectral density in the static ULE result, and that the latter's remaining hypotheses hold. In particular $γ > 0$ is the coupling scale in that ULE convention, and all square roots below are nonnegative.

Work in the inherited units $ℏ = 1$ , so the energies in $H_{S}$ are angular frequencies. For a finite family $L = (L_{a})$ on the system space define its dissipator on every system matrix $ρ$ by $D_{L} (ρ) = \sum_{a} (L_{a} ρ L_{a}^{†} - \frac{1}{2} L_{a}^{†} L_{a} ρ - \frac{1}{2} ρ L_{a}^{†} L_{a})$ , where $†$ denotes conjugate transpose. This is the dissipative part of the finite GKSL generator. To compare units with its supplier, fix a time unit $T > 0$ seconds and an action unit $A > 0$ joule-seconds: the numerical matrices here correspond to $L_{a} / \sqrt{T}$ , physical Planck constant $A$ , and physical Hamiltonian $(A / T) H_{S}$ there. Multiplication of that physical generator by $T$ gives the convention used here. The channel rotation below leaves the Hamiltonian part fixed.

Choose an orthonormal energy eigenbasis $H_{S} ∣ n ⟩ = E_{n} ∣ n ⟩$ and put $ω_{m n} = E_{n} - E_{m}$ . Define $X_{+ 1} = - (X_{x} + i X_{y}) / \sqrt{2}$ , $X_{- 1} = (X_{x} - i X_{y}) / \sqrt{2}$ and $X_{0} = X_{z}$ . A valid unitary choice of the three ULE jump operators is

$L_{q} = \sqrt{2 π γ} \sum_{m, n} \sqrt{J (ω_{m n})} ⟨ m ∣ X_{q} ∣ n ⟩ ∣ m ⟩ ⟨ n ∣, q \in {- 1, 0, + 1} .$

The article denotes this channel label by $δ$ in its main text. For the angular-momentum interpretation, additionally suppose the system basis consists of magnetic eigenvectors and the $X_{q}$ form a rank-one spherical tensor. Specify the emission process by the following local physical assumptions. Let $R$ be a set of remainder-field records and $P$ a set of single-photon records, with real-valued maps $r : R \to R$ and $s : P \to R$ assigning angular-momentum projections on the chosen axis in units of $ℏ$ . The initial field record is $a \in R$ , with projection $r (a) = 0$ ; the final field record is $(b, p) \in R \times P$ , consisting of the remainder $b$ and exactly one emitted photon $p$ , with projection $r (b) + s (p)$ . Assume the remainder projection is unchanged, $r (b) = r (a)$ , and the transition conserves the sum of system and field projections. These record types and additive assignments are assumptions for this interpretation; they assert no general factorization of a thermal field. If the system matrix element for that transition is nonzero, the emitted photon's projection is $s (p) = - q$ . This assertion concerns the projection and the stated emission process; the formula alone does not exclude absorption at nonzero temperature.

Statement references

Static ULE formulas and principal-value kernel

Off-diagonal correlators vanish in an isotropic photon bath

Spherical vector components and transition-frequency convention

Finite spin blocks, spherical tensors and Clebsch–Gordan phases

Wigner–Eckart with explicit magnetic-label contract

Stationary correlation spectra and ULE kernel regularity

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

Finite GKSL generators and units

Facts & Assumptions

[F1]

Under its stated system, bath, preparation, weak-Markov and regularity hypotheses, the static ULE result gives $L_{i} = 2 π \sqrt{γ} \sum_{m, n, j} g_{i j} (ω_{m n}) ⟨ m ∣ X_{j} ∣ n ⟩ ∣ m ⟩ ⟨ n ∣$ , with $g = [J / (2 π)]^{1 / 2}$ . Static ULE formulas and principal-value kernel Stationary correlation spectra and ULE kernel regularity

[F2]

Under the stated isotropic photon-model hypotheses, the spectral matrix has a scalar nonnegative representative and its positive jump correlator is the corresponding scalar square root times the identity. The isotropic result requires the three stated choice assumptions. Off-diagonal correlators vanish in an isotropic photon bath 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

[F3]

The displayed spherical components are obtained from the Cartesian components by a unitary coefficient matrix $U$ ; $ω_{m n}$ is source energy minus destination energy. Spherical vector components and transition-frequency convention

[F4]

A rank-one spherical tensor satisfies $[J_{z}, X_{q}] = q X_{q}$ . Equivalently, its matrix elements between magnetic eigenvectors vanish unless the final system magnetic label equals the initial label plus $q$ . Finite spin blocks, spherical tensors and Clebsch–Gordan phases Wigner–Eckart with explicit magnetic-label contract

[F5]

The finite GKSL definition supplies the dissipator $D_{L}$ displayed in the Statement; its supplier has domain mathematics. The finite system dimension and the three finite jump matrices meet its algebraic hypotheses, and the positive units $T, A$ specified above convert its physical units to the inherited $ℏ = 1$ convention. No microscopic approximation is inferred from that definition. Finite GKSL generators and units

[A1]

Assume the choice, matching, regularity and physical hypotheses in the Statement, including the locally specified field-record, additivity, unchanged-remainder and conservation hypotheses only for the final interpretation.

Proof

Given: The system, bath, spectral matching and choice hypotheses in the Statement; for the final assertion, also the stated tensor and local field-record assumptions, including additivity, unchanged remainder projection and conservation.

Goal: Derive the three displayed jump operators and the conditional photon-projection label.

1.1givenA1F1F2algebra

At each required transition frequency, [F2] and the pointwise matching assumption give $g_{i j} (ω_{m n}) = δ_{i j} \sqrt{J (ω_{m n}) / (2 π)}$ . Substitution into [F1] gives $L_{i} = \sqrt{2 π γ} \sum_{m, n} \sqrt{J (ω_{m n})} ⟨ m ∣ X_{i} ∣ n ⟩ ∣ m ⟩ ⟨ n ∣$ for the Cartesian labels. The coefficient is real and nonnegative by the specified representative.

2.1F3F5step 1.1algebra

Let $L_{q} = \sum_{i} U_{q i} L_{i}$ , with $i = x, y, z$ . By [F3], $\sum_{q} U_{q i} \overline{U_{q j}} = δ_{i j}$ , so expansion of the finite sums gives $\sum_{q} L_{q} ρ L_{q}^{†} = \sum_{i} L_{i} ρ L_{i}^{†}$ and $\sum_{q} L_{q}^{†} L_{q} = \sum_{i} L_{i}^{†} L_{i}$ . Substituting these identities into the definition in [F5] gives $D_{(L_{q})} (ρ) = D_{(L_{i})} (ρ)$ for every system matrix $ρ$ ; the unchanged Hamiltonian part therefore gives the same ULE generator. Linearity of matrix elements and [F3] turn the Cartesian expression from step 1.1 into the displayed formula with $X_{q} = \sum_{i} U_{q i} X_{i}$ . Thus it is a valid choice of the three jump operators.

3.1givenA1F4algebra∎

Write $μ_{i n}$ and $μ_{o u t}$ for the initial and final system magnetic labels. For a nonzero matrix element, [F4] gives $μ_{o u t} - μ_{i n} = q$ . Under the local assumptions in [A1], conservation is $μ_{i n} + r (a) = μ_{o u t} + r (b) + s (p)$ . Since $r (b) = r (a)$ , subtraction yields $s (p) = μ_{i n} - μ_{o u t} = - q$ . The additive field assignment identifies this scalar as the emitted photon's projection. The remainder may contain other excitations, but its projection must be unchanged; conservation alone would only give $s (p) = - q + r (a) - r (b)$ . This proves the stated conditional interpretation.

Source notes

The article uses the same symbol for the scalar bath spectrum in its spherical formula and describes $γ$ as an effective decay scale. Here the normalization is fixed by the completed ULE and isotropic-bath interfaces, with pointwise matching explicitly required. The article's $- q$ description is retained as a projection balance under the stated tensor, conservation, additive field-record and unchanged-remainder assumptions. The field-record model is a local physical assumption, not an imported fact from the article. The source passage is attribution, not a premise for that balance.

Depends on

Used by · 1

Small-lift amplitude approximation and energy-scale scope

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF p. 13, Eq. (S3.17) and following paragraph