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Degenerate four-level amplitude-damping form

PDF p. 7: Eqs. (15)–(16)

Statement

Assume the four-level model and Kraus representation, with $t \geq 0$ , $γ \geq 0$ and real $ω_{0}, Δ$ . In its ordered orthonormal basis $∣ 0 ⟩, \dots, ∣ 3 ⟩$ , put $Π_{01} = ∣ 0 ⟩ ⟨ 0 ∣ + ∣ 1 ⟩ ⟨ 1 ∣$ and $Π_{23} = ∣ 2 ⟩ ⟨ 2 ∣ + ∣ 3 ⟩ ⟨ 3 ∣$ . Complex integrals are componentwise Riemann integrals under $C ≅ R^{2}$ , with $\int_{0}^{0} g (s) d s = 0$ . For all these parameters define $f_{-} (t) = γ ∣ \int_{0}^{t} e^{(γ + i Δ) s} d s ∣ - (e^{γ t} - 1)$ . Where $Ω = γ + i Δ$ is nonzero, this agrees with $γ ∣ e^{Ω t} - 1 ∣ / ∣ Ω ∣ - (e^{γ t} - 1)$ .

Then $f_{-} (t) \leq 0$ . If $γ > 0$ and $t > 0$ , equality holds exactly when $Δ = 0$ . At $t = 0$ or $γ = 0$ , equality holds for every $Δ$ . For $Δ = 0$ , $f_{-} (t) = 0$ for every $t \geq 0$ , and the representation reduces to $K_{3} = 0$ , $K_{1} = e^{- i ω_{0} t} e^{- γ t / 2} Π_{01} + Π_{23}$ and $K_{2} = \sqrt{1 - e^{- γ t}} (∣ 2 ⟩ ⟨ 0 ∣ + ∣ 3 ⟩ ⟨ 1 ∣)$ .

Statement references

Exact four-level Kraus operators

Finite operator and ket conventions

The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral

exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0

Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive

Facts & Assumptions

Given: The model, representation and parameters in the Statement.

Goal: Establish the sign and equality conditions and the degenerate Kraus operators.

[F1]

The four-level result gives the stated Kraus representation, including its $t = 0$ and $γ = 0$ endpoint conventions, and defines its nonzero-parameter $f_{-}$ by the displayed exponential quotient. Exact four-level Kraus operators

[F2]

Kets, bras, outer products and the ordered orthonormal basis use the finite operator conventions. Finite operator and ket conventions

[F3]

The Riemann integral in $R^{m}$ , for $m = 1, 2$ , is componentwise and real-linear, with integral zero on a degenerate interval; identify $x + i y$ with $(x, y)$ for complex integrals. The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral

[F4]

A continuous real function on a nondegenerate compact interval is bounded and Riemann integrable. Apply this to each component for complex functions. A continuous function on [a,b] is Riemann integrable, by Heine-Cantor and Riemann's criterion

[F5]

For an integrable function $g : [a, b] \to R^{m}$ with $a \leq b$ , ${∥ \int_{a}^{b} g ∥}_{2} \leq \int_{a}^{b} ∥ g ∥_{2}$ ; complex modulus is the Euclidean norm on $R^{2}$ . For a ≤ b and f : [a,b] → ℝᵐ integrable when a<b, ‖∫ₐᵇ f‖₂ ≤ ∫ₐᵇ ‖ f‖₂; for a<b, ‖ f‖₂ is integrable

[F6]

The complex exponential is entire with derivative itself. For real $x, y$ , $e^{x + i y} = e^{x} (\cos y + i \sin y)$ and $∣ e^{x + i y} ∣ = e^{x}$ ; the real exponential is positive. The complex exponential is entire and its complex derivative is itself exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0 The exponential is positive and satisfies exp(-x)=1/exp(x)

[F7]

If a curve $G : [a, b] \to R^{m}$ , $a < b$ , is differentiable at every point, using the corresponding one-sided derivatives at the endpoints, and $G^{'}$ is integrable, then $\int_{a}^{b} G^{'} = G (b) - G (a)$ . If f : [a,b] → ℝᵐ is differentiable with integrable f' then ∫ₐᵇ f' = f(b)-f(a); and a bounded derivative makes f Lipschitz

[F8]

A continuous nonnegative real function on $[a, b]$ , $a < b$ , whose integral is zero vanishes everywhere on that interval. A continuous f ≥ 0 on [a,b] with ∫ₐᵇ f = 0 is identically 0

[F9]

Complex conjugation respects sums and products, $z \overline{z} = ∣ z ∣^{2}$ , modulus is multiplicative and nonnegative, and $∣ z ∣ = 0$ exactly when $z = 0$ . Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive

Proof

Given: The model, representation and parameters in the Statement.

Goal: Establish the sign and equality conditions and the degenerate Kraus operators.

1.1givenF3F4F6algebra

For each $c \in C$ , the real-parameter curve $s \mapsto e^{c s}$ is continuously differentiable with derivative $c e^{c s}$ . Indeed, for $c \neq 0$ its difference quotient is $c (e^{c (s + h)} - e^{c s}) / (c h)$ , which tends to $c e^{c s}$ by [F6]; for $c = 0$ the curve is constant. This also supplies the one-sided derivatives on a restricted compact interval. By [F6], $e^{(γ + i Δ) s} = e^{γ s} e^{i Δ s}$ and $∣ e^{i Δ s} ∣ = 1$ . The integrands in the Statement therefore have continuous real and imaginary components and are integrable by [F3] and [F4], including the zero-interval convention. Fixed real-linear coordinate combinations commute with integration by the componentwise formula and real-linearity in [F3].

2.1givenF1F3F4F5F6F7step 1.1algebra

Let $A = \int_{0}^{t} e^{(γ + i Δ) s} d s$ and $B = \int_{0}^{t} e^{γ s} d s$ . For $t > 0$ , apply [F7] to the curves in step 1.1 with $c = Ω$ and $c = γ$ ; their derivatives are continuous and integrable by [F4]. This gives $Ω A = e^{Ω t} - 1$ and $γ B = e^{γ t} - 1$ . The identities also hold at $t = 0$ by [F3]. Thus $A = (e^{Ω t} - 1) / Ω$ when $Ω \neq 0$ , agreeing with [F1]. By [F5] and [F6], $∣ A ∣ \leq B$ , so $f_{-} = γ (∣ A ∣ - B) \leq 0$ , including $γ = 0$ .

3.1givenF3F6F8F9step 1.1step 2.1algebra

Suppose $γ > 0$ and $t > 0$ . Step 2.1 gives $B \geq 0$ , and $B = 0$ would contradict [F8] applied to the continuous positive function $s \mapsto e^{γ s}$ , so $B > 0$ . If $f_{-} = 0$ , then $∣ A ∣ = B > 0$ ; set $u = A / ∣ A ∣$ , so $∣ u ∣ = 1$ by [F9]. The function $h (s) = e^{γ s} ∣ e^{i Δ s} - u ∣^{2}$ is continuous, real and nonnegative. Expanding its squared modulus using [F9], and integrating real and imaginary parts using [F3] and step 1.1, gives $\int_{0}^{t} h (s) d s = 2 B - 2 Re (\overline{u} A) = 2 B - 2 ∣ A ∣ = 0$ .

4.1givenF3F6F8F9step 1.1step 2.1step 3.1algebra

By [F8], $h$ in step 3.1 vanishes everywhere. Positivity of $e^{γ s}$ and definiteness of modulus imply $e^{i Δ s} = u$ for every $s \in [0, t]$ . At $s = 0$ this gives $u = 1$ . The right derivative at zero of this constant curve is zero, whereas step 1.1 computes it as $i Δ$ ; hence $Δ = 0$ . Conversely, if $Δ = 0$ then $A = B$ and step 2.1 gives $f_{-} = 0$ . At $t = 0$ the integrals vanish by [F3], and at $γ = 0$ the defining expression for $f_{-}$ vanishes. This proves all stated equality and endpoint conditions.

5.1givenF1F2step 4.1algebra∎

At $Δ = 0$ , [F1] gives the displayed $K_{1}, K_{2}, K_{3}$ for positive $γ t$ , and its endpoint convention gives the same formulas when $t = 0$ or $γ = 0$ . By [F2], $Π_{23}$ is the sum of the two ground-state basis projectors in that representation. Step 4.1 gives $f_{-} = 0$ throughout this degenerate case. This proves the conditional result.

Depends on

Used by · 1

Onset of the third four-level Kraus operator

Dependency tree · 2 levels

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF p. 7, definitions following Eq. (14) and Eqs. (15)–(16)