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Long-time atomic channel and source-resolving Kraus operators

PDF p. 4: Fig. 2 caption and first paragraph

Statement

Assume the Axiom of Choice, Countable Choice and Dependent Choice, and the finite-emitter, vacuum, regularity, tensor and pointwise spectral hypotheses of Canonical angular-momentum downward jumps and Invariant finite angular-momentum cutoff. In particular, the three jumps $L_{δ}$ lower spin by one, and the Hermitian effective Hamiltonian $H$ preserves each spin block. Restrict the resulting finite GKSL channel $N_{t}$ to $W = V_{0} \oplus V_{1} \oplus V_{2}$ , with magnetic orthonormal basis $∣ J, m ⟩$ for $J = 0, 1, 2$ and $m = - J, \dots, J$ . Put $g = ∣ 0, 0 ⟩$ and $R = \sum_{δ = - 1}^{1} L_{δ}^{†} L_{δ}$ on $W$ . Assume additionally that there is $r_{*} > 0$ such that $P_{J} R P_{J} \geq r_{*} P_{J}$ for $J = 1, 2$ , where $P_{J}$ projects onto $V_{J}$ and the inequality is in quadratic forms.

Then, as $t \to \infty$ , $N_{t}$ converges to $R (X) = Tr (X) ∣ g ⟩ ⟨ g ∣$ . Thus every initial basis state reaches the ground in the limit. One canonical Kraus representation of $R$ consists of the nine operators $∣ 0, 0 ⟩ ⟨ J, m ∣$ ; its Choi rank and minimum Kraus number are nine. At every sufficiently small positive time, every Kraus representation, including a canonical one, has an operator that acts nontrivially on at least two distinct magnetic-basis source states. In the limiting channel a source-resolving canonical choice exists, but canonical operators are not unique: their degenerate Choi eigenspace may be rotated into operators supported on several source states.

A Kraus representation of a channel is a finite family of maps $K_{a}$ with $N (X) = \sum_{a} K_{a} X K_{a}^{†}$ . Here canonical means that the vectorized operators are the square-root-weighted orthonormal positive-eigenvalue vectors of the input-first, unnormalized Choi matrix. An operator acts nontrivially on a source $∣ J, m ⟩$ when its value on that vector is nonzero.

Statement references

Canonical angular-momentum downward jumps

Invariant finite angular-momentum cutoff

Finite absorbing lowering dynamics and replacement limit

Canonical Choi eigenvectors, Kraus rank and branch weights

Finite GKSL evolution and the QEC channel interface

Finite spin blocks, spherical tensors and Clebsch–Gordan phases

Facts & Assumptions

[F1]

Under the stated emitter and pointwise spectral hypotheses, the three jumps lower $V_{J}$ to $V_{J - 1}$ and annihilate $V_{0}$ . Canonical angular-momentum downward jumps

[F2]

With a block-preserving Hermitian effective Hamiltonian, the $J \leq 2$ cutoff is invariant and its channel equals the finite GKSL exponential formed from the restricted operators. Invariant finite angular-momentum cutoff Finite GKSL evolution and the QEC channel interface

[F3]

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

[F4]

For finite orthogonal grading with one-dimensional ground block, block-preserving Hermitian Hamiltonian, lowering jumps and a strictly positive uniform excited-block exit-rate bound, the finite GKSL channel converges to ground replacement. For any orthonormal input basis, the associated ground-replacement operators form a Kraus representation and the Choi rank is the input dimension. Finite absorbing lowering dynamics and replacement limit

[F5]

A finite channel's canonical Kraus operators are obtained from its positive Choi eigenvectors; the Choi rank is the minimum Kraus number. An orthonormal basis of a repeated-eigenvalue eigenspace can be rotated. Canonical Choi eigenvectors, Kraus rank and branch weights

[A1]

Assume the three choice principles, all physical and pointwise hypotheses in the Statement, and the stated positive exit-rate bound. 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 restricted finite channel and all hypotheses in the Statement, including the positive exit-rate bound.

Goal: Establish the ground-replacement limit and the stated early- and long-time Kraus properties.

1.1F1F2F3F4A1

By [F1] and [F2], the restricted generator has lowering jumps and a block-preserving Hermitian Hamiltonian. By [F3], $V_{0} = C g$ and $\dim W = 1 + 3 + 5 = 9$ . The additional bound in [A1] supplies the excited-block rate hypothesis of [F4], which is not implied merely by the displayed lowering form. Applying [F4] gives $N_{t} (X) \to Tr (X) ∣ g ⟩ ⟨ g ∣$ and, in particular, sends each basis-state density operator to $∣ g ⟩ ⟨ g ∣$ in the limit.

2.1F3F4F5step 1.1algebra

By [F4], the nine maps $K_{J m} = ∣ g ⟩ ⟨ J, m ∣$ give the limiting channel and its Choi rank is nine. Indeed, its input-first unnormalized Choi matrix is $I_{W} \otimes ∣ g ⟩ ⟨ g ∣$ ; its nine positive eigenvalues are all one, with eigenvectors $∣ J, m ⟩ \otimes g$ . By [F5], these eigenvectors give the displayed canonical choice and nine is the minimum Kraus number. Since the positive eigenvalue is degenerate, rotating that eigenbasis gives other canonical choices; a rotated vector can involve several source basis states.

3.1F2F3F5step 2.1algebra∎

Choose two distinct magnetic-basis vectors $e$ and $f$ . By [F2], $N_{t} = e^{t L}$ for a finite linear generator, so the exponential series gives $N_{t} (∣ e ⟩ ⟨ f ∣) \to ∣ e ⟩ ⟨ f ∣ \neq 0$ as $t ↓ 0$ . Hence this image is nonzero for all sufficiently small positive $t$ . If every Kraus operator in a representation acted nontrivially on at most one source basis vector, each term $K_{a} ∣ e ⟩ ⟨ f ∣ K_{a}^{†}$ would vanish, contradicting that nonzero image. This applies in particular to any canonical representation supplied by [F5]. Together with step 2.1, this proves the early-time and limiting assertions without selecting a unique physical trajectory.

Source notes

The article's figure caption describes early multi-source operators and a source-resolving long-time picture. The argument above establishes an early interval and a canonical limiting choice under the explicit rate bound; it does not assert that every canonical basis at the degenerate limit is source-resolving or reproduce the figure's numerical thresholds.

Depends on

Used by · 1

Jump histories, distinguishability, and limits of Kraus rank

Dependency tree · 2 levels

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF page 4, Fig. 2 caption and first paragraph below the figure