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Onset of the third four-level Kraus operator

PDF p. 3: Kraus-rank paragraph; PDF p. 7: Onset paragraph and Fig. 5

Statement

Assume the four-level spontaneous-decay model, with decay rate $γ > 0$ , elapsed time $t > 0$ , and real splitting $Δ$ . The channel acts on the full four-dimensional state space with ordered orthonormal basis $∣ 0 ⟩, ∣ 1 ⟩, ∣ 2 ⟩, ∣ 3 ⟩$ . Its Kraus rank, the minimum number of operators in a Kraus representation of the channel, is two when $Δ = 0$ and three when $Δ \neq 0$ . The additional direction is the phase-adjusted difference of the two decay-path operators $∣ 2 ⟩ ⟨ 0 ∣$ and $∣ 3 ⟩ ⟨ 1 ∣$ ; its nonzero weight represents the distinction between the paths in this model.

Statement references

Four-level spontaneous-decay model

Kraus rank and Choi representation

Finite operator and ket conventions

Facts & Assumptions

Given: The model and parameters in the Statement.

Goal: Determine the minimum Kraus representation size and identify the additional direction.

[F1]

The model channel has a Kraus representation $K_{1}, K_{2}, K_{3}$ in which $K_{1}$ is diagonal and nonzero, $K_{2} = a (u F_{0} + F_{1})$ and $K_{3} = b (- u F_{0} + F_{1})$ , where $F_{0} = ∣ 2 ⟩ ⟨ 0 ∣$ , $F_{1} = ∣ 3 ⟩ ⟨ 1 ∣$ , $∣ u ∣ = 1$ , $a^{2} = e^{- γ t} f_{+} (t) / 2$ , and $b^{2} = - e^{- γ t} f_{-} (t) / 2$ . Here $f_{+} (t) = γ ∣ e^{(γ + i Δ) t} - 1 ∣ / ∣ γ + i Δ ∣ + (e^{γ t} - 1)$ . Exact four-level Kraus operators

[F2]

For $γ > 0$ and $t > 0$ , $f_{-} (t) \leq 0$ , with equality exactly when $Δ = 0$ . Degenerate four-level amplitude-damping form

[F3]

The input-first unnormalized Choi matrix of a channel is $J = \sum_{i, j} ∣ i ⟩ ⟨ j ∣ \otimes N (∣ i ⟩ ⟨ j ∣)$ , and the minimum Kraus representation size equals $rank J$ . Kraus rank and Choi representation Kraus rank equals Choi rank

[F4]

The basis outer products obey $Tr [(∣ k ⟩ ⟨ j ∣)^{†} ∣ l ⟩ ⟨ m ∣] = δ_{k l} δ_{j m}$ . Finite operator and ket conventions

Proof

1.1givenF1F2F4algebra

Since $γ t > 0$ , $e^{γ t} - 1 > 0$ , so [F1] gives $f_{+} (t) > 0$ and $a > 0$ . By [F2], $b = 0$ exactly at $Δ = 0$ , and $b > 0$ otherwise. The path operators $F_{0}, F_{1}$ are nonzero and orthogonal for the Hilbert–Schmidt inner product $⟨ A, B ⟩ = Tr (A^{†} B)$ . Thus $K_{2}$ is nonzero, and $K_{3}$ is nonzero exactly when $Δ \neq 0$ .

2.1F1F3F4step 1.1algebra

By [F4], the diagonal $K_{1}$ is orthogonal to both path operators, while $⟨ u F_{0} + F_{1}, - u F_{0} + F_{1} ⟩ = - ∣ u ∣^{2} + 1 = 0$ . Therefore the nonzero operators among $K_{1}, K_{2}, K_{3}$ are mutually orthogonal and hence linearly independent. Expanding the Choi formula in [F3] using the representation in [F1] gives $J = \sum_{k = 1}^{3} ∣ K_{k} ⟩ ⁣ ⟩ ⟨ ⁣ ⟨ K_{k} ∣$ , where $∣ K ⟩ ⁣ ⟩ = \sum_{i} ∣ i ⟩ \otimes K ∣ i ⟩$ . These vectorized operators are mutually orthogonal whenever nonzero, since their inner products are $Tr (K_{j}^{†} K_{k})$ . Consequently the rank of $J$ is the number of nonzero operators: two at zero splitting and three otherwise.

3.1F1F3F4step 1.1step 2.1algebra∎

By [F3], this Choi rank is the minimum number of Kraus operators, proving both rank values. The third operator is proportional to $- u F_{0} + F_{1}$ , the phase-adjusted difference of the decay paths; its squared Hilbert–Schmidt norm is $2 b^{2}$ , positive precisely for nonzero splitting. This identifies the additional distinguishability direction as a consequence of the model.

Depends on

Dependency tree · 2 levels

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF p. 3, Kraus-rank paragraph; PDF p. 7, onset paragraph and Fig. 5