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Invariant finite angular-momentum cutoff

PDF p. 3: Final paragraph continuing onto p. 4

Statement

Assume the Axiom of Choice, Countable Choice and Dependent Choice. Take the finite-block angular-momentum emitter with one copy of each integer-spin block $V_{J}$ , $J = 0, 1, \dots, N$ , for $N \geq 2$ , and adopt all hypotheses of the canonical downward-jumps result, including its vacuum, strictly increasing energy, tensor, regularity and pointwise spectral-representative assumptions. In particular, the representative is zero at every required nonpositive transition frequency. Let the finite GKSL generator have those three downward jumps and a Hermitian effective Hamiltonian $H$ that preserves each $V_{J}$ . This includes the stated diagonal unperturbed emitter Hamiltonian and any added block-preserving lift, provided their sum is the effective Hamiltonian used in the generator.

Let $W = V_{0} \oplus V_{1} \oplus V_{2}$ . For every $t \geq 0$ , an operator initially supported on $W$ remains supported on $W$ , and its evolution is exactly the channel generated on $W$ by the restrictions of $H$ and the three jumps. Here supported means $X = P_{W} X P_{W}$ , where $P_{W}$ is the orthogonal projection onto $W$ ; a channel is a completely positive trace-preserving linear map. The space $W$ has dimension nine. The article uses this $J \leq 2$ restriction for its channel rank study.

Statement references

Angular-momentum emitter, photon bath, and dipole approximation

Canonical angular-momentum downward jumps

Vacuum limit and cubic free-space spectral density

Invariant finite truncation for lowering generators

Finite GKSL evolution and the QEC channel interface

Finite spin blocks, spherical tensors and Clebsch–Gordan phases

Facts & Assumptions

[F1]

Under its stated finite-emitter, vacuum and pointwise spectral hypotheses, each canonical jump maps $V_{J}$ into $V_{J - 1}$ , with zero image for $J = 0$ . Canonical angular-momentum downward jumps

[F2]

The vacuum limiting spectral measure is supported on nonnegative frequencies, but its density specified almost everywhere does not determine values at individual transition frequencies. Vacuum limit and cubic free-space spectral density

[F3]

For a finite orthogonal grading, a Hermitian block-preserving Hamiltonian and finitely many lowering jumps, the cutoff-supported operators remain supported and their evolution equals the compressed GKSL exponential. Invariant finite truncation for lowering generators

[F4]

The finite GKSL interface defines the generator $L (X) = - i [H, X] + \sum_{δ} (L_{δ} X L_{δ}^{†} - {L_{δ}^{†} L_{δ}, X} / 2)$ and the channel $N_{t} = e^{t L}$ ; $A^{†}$ is the adjoint, $[A, B] = A B - B A$ , and ${A, B} = A B + B A$ . Finite GKSL evolution and the QEC channel interface

[F5]

The integer-spin block $V_{J}$ has orthonormal magnetic basis $∣ J, m ⟩$ with $m = - J, - J + 1, \dots, J$ , hence dimension $2 J + 1$ . Finite spin blocks, spherical tensors and Clebsch–Gordan phases

[F6]

The emitter postulate specifies an unperturbed Hamiltonian diagonal in $∣ J, m ⟩$ , with real energies depending only on $J$ . Angular-momentum emitter, photon bath, and dipole approximation

[A1]

Assume the three choice principles and all physical, regularity and pointwise 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 emitter, the three jumps, the block-preserving Hermitian effective Hamiltonian, the stated hypotheses, and $t \geq 0$ .

Goal: Establish exact evolution on the $J \leq 2$ cutoff and its dimension.

1.1F1F2F5F6A1given

The basis in [F5] makes the spin blocks mutually orthogonal in the finite direct sum. By [F1], each of the three jumps lowers the block index by one, with the $J = 0$ image zero. The effective Hamiltonian preserves every block by assumption; the unperturbed Hamiltonian in [F6] is an example, and adding a block-preserving lift preserves that property. The pointwise condition in [A1] is needed for the jump conclusion: the almost-everywhere statement in [F2] alone does not supply those point values.

2.1F3F4F5step 1.1algebra∎

Apply [F3] to the generator and channel defined in [F4], with cutoff $J_{0} = 2$ . Its hypotheses hold by step 1.1 and $N \geq 2$ . Thus every $X = P_{W} X P_{W}$ remains supported on $W$ , and the full finite evolution on such $X$ agrees with the exponential of the generator formed from the restrictions to $W$ . Finally, [F5] gives $\dim W = (2 \cdot 0 + 1) + (2 \cdot 1 + 1) + (2 \cdot 2 + 1) = 9$ .

Source notes

The article identifies this cutoff with its rank-study restriction. The proof above concerns a finite ambient block cutoff $N$ and does not establish a limiting infinite-dimensional evolution. Zero temperature alone, without the stated pointwise convention and block-preserving effective Hamiltonian, is not used as a sufficient condition.

Depends on

Used by · 2

Dependency tree · 2 levels

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF pages 3–4, paragraph beginning “To see how the lift reshapes the channel” and the following Kraus-rank discussion