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Exact four-level Kraus operators

PDF p. 6: Eqs. (12)–(14), continued on p. 7; PDF p. 7: Definitions following Eq. (14)

Statement

Assume the four-level model of Four-level spontaneous-decay model, with $t \geq 0$ , $γ \geq 0$ and real $ω_{0}, Δ$ . For a set $X$ of basis labels put $Π_{X} = \sum_{i \in X} ∣ i ⟩ ⟨ i ∣$ ; write $Π_{01}$ for $X = {0, 1}$ and $Π_{j}$ for $X = {j}$ . Put $Ω = γ + i Δ$ . When $γ > 0$ and $t > 0$ , define $f_{\pm} (t) = γ ∣ e^{Ω t} - 1 ∣ / ∣ Ω ∣ \pm (e^{γ t} - 1)$ and $u (t) = ∣ Ω (e^{t Ω^{*}} - 1) ∣ / [Ω (e^{t Ω^{*}} - 1)]$ . Then the channel on the entire four-dimensional operator space has Kraus representation $N_{t} (X) = \sum_{j = 1}^{3} K_{j} X K_{j}^{†}$ , where

$K_{1} = e^{- i ω_{0} t} e^{- γ t / 2 + i Δ t} Π_{01} + e^{i Δ t} Π_{2} + Π_{3},$

$K_{2} = e^{- γ t / 2} \sqrt{f_{+} (t) / 2} [u (t) ∣ 2 ⟩ ⟨ 0 ∣ + ∣ 3 ⟩ ⟨ 1 ∣],$

$K_{3} = e^{- γ t / 2} \sqrt{- f_{-} (t) / 2} [- u (t) ∣ 2 ⟩ ⟨ 0 ∣ + ∣ 3 ⟩ ⟨ 1 ∣] .$

The square roots are real and nonnegative. At $t = 0$ or $γ = 0$ , use $K_{2} = K_{3} = 0$ and the displayed $K_{1}$ ; this also specifies the channel when the printed phase or $Ω$ is undefined. In particular, for $Δ = 0$ the representation reduces to $K_{1} = e^{- i ω_{0} t} e^{- γ t / 2} Π_{01} + Π_{2} + Π_{3}$ , $K_{2} = \sqrt{1 - e^{- γ t}} (∣ 2 ⟩ ⟨ 0 ∣ + ∣ 3 ⟩ ⟨ 1 ∣)$ and $K_{3} = 0$ .

Facts & Assumptions

Given: The postulated model and parameters in the Statement.

Goal: Derive the displayed representation for every input operator, including the stated endpoints.

[F1]

The model has $H = ω_{0} Π_{01} + Δ Π_{3}$ , $L = \sqrt{γ} (∣ 2 ⟩ ⟨ 0 ∣ + ∣ 3 ⟩ ⟨ 1 ∣)$ , and generator $L (X) = - i [H, X] + L X L^{†} - {L^{†} L, X} / 2$ . Four-level spontaneous-decay model

[F2]

The ordered orthonormal basis, adjoints, and outer products obey the finite operator conventions. Finite operator and ket conventions

[F3]

For the finite GKSL generator, the channel has the convergent ordered-jump expansion with no-jump amplitude $E (t) = e^{t (- i H - L^{†} L / 2)}$ and one-jump amplitude $E (t - s) L E (s)$ . Finite no-jump and ordered-jump expansion

[F4]

A two-path branch with coefficient matrix $p (\begin{array}{cc} 1 & χ \\ \overline{χ} & 1 \end{array})$ , where $p \geq 0$ and $∣ χ ∣ \leq 1$ , has the orthogonal Kraus decomposition obtained from its two eigenvectors. Two-path coherence Gram spectrum and Kraus weights

Proof

1.1givenF1F2F3algebra

By [F1] and [F2], $L^{†} L = γ Π_{01}$ and $E (s) = e^{(- i ω_{0} - γ / 2) s} Π_{01} + Π_{2} + e^{- i Δ s} Π_{3}$ . Since $L$ maps the excited span into the decay span and annihilates the latter, $L E (r) L = 0$ for every $r \geq 0$ . Thus all terms with two or more jumps vanish in [F3]. Moreover $K_{1} = e^{i Δ t} E (t)$ , so its individual phase does not change the no-jump map.

2.1givenF2F3step 1.1algebra

Set $F_{0} = ∣ 2 ⟩ ⟨ 0 ∣$ and $F_{1} = ∣ 3 ⟩ ⟨ 1 ∣$ . The one-jump amplitude is $\sqrt{γ} e^{(- i ω_{0} - γ / 2) s} [F_{0} + e^{- i Δ (t - s)} F_{1}]$ . Hence its integral acts on every four-level operator as $p (F_{0} X F_{0}^{†} + F_{1} X F_{1}^{†}) + c F_{0} X F_{1}^{†} + \overline{c} F_{1} X F_{0}^{†}$ , with $p = 1 - e^{- γ t}$ and $c = γ \int_{0}^{t} e^{- γ s} e^{i Δ (t - s)} d s$ . This form includes inputs with coherences outside the excited span: both path operators annihilate the decay span. The triangle inequality gives $∣ c ∣ \leq p$ .

3.1givenF4step 2.1algebra

For $γ > 0$ and $t > 0$ , $p > 0$ , $Ω \neq 0$ , and integration gives $c = γ (e^{i Δ t} - e^{- γ t}) / Ω$ . Its numerator is nonzero because $e^{- γ t} < 1$ . The displayed $u$ is $c / ∣ c ∣$ : indeed $c = γ e^{- γ t} (e^{Ω t} - 1) / Ω$ , and $Ω (e^{Ω^{*} t} - 1)$ is the complex conjugate of $Ω^{*} (e^{Ω t} - 1)$ . Also $e^{- γ t} f_{+} = ∣ c ∣ + p$ and $- e^{- γ t} f_{-} = p - ∣ c ∣$ . Applying [F4] with $χ = c / p$ , then multiplying the minus eigenoperator by the harmless phase $- 1$ , gives exactly $K_{2}, K_{3}$ . The bound $∣ c ∣ \leq p$ makes both square roots nonnegative.

4.1givenF3step 1.1step 2.1step 3.1algebra∎

If $t = 0$ or $γ = 0$ , the one-jump amplitude in step 2.1 vanishes and the stated zero operators together with $K_{1}$ give the channel. If $Δ = 0$ and $γ t > 0$ , then $c = p$ , $u = 1$ , $f_{-} = 0$ and $e^{- γ t} f_{+} / 2 = p$ , giving the stated amplitude-damping form. The same form follows at the zero-decay endpoints. Combining these cases with steps 1.1, 2.1 and 3.1 proves the representation on the full operator space.

Depends on

Used by · 8

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF pp. 6–7, Eqs. (12)–(14) and definitions following Eq. (14)