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Decay Kraus weights and photon overlap

PDF p. 7: Paragraph before Fig. 5 and upper right paragraph

Statement

Assume the four-level model with fixed real $ω_{0}, Δ$ , fixed $γ > 0$ , and $t > 0$ . Let $K_{2}, K_{3}$ be the model's decay Kraus operators, $p = 1 - e^{- γ t}$ , and $χ = γ (e^{i Δ t} - e^{- γ t}) / [(γ + i Δ) p]$ as its conditional decay coherence. Define the weight of an operator as $w_{j} = Tr (K_{j}^{†} K_{j})$ . Then $w_{2, 3} = p (1 \pm ∣ χ ∣)$ , with the upper sign for $K_{2}$ . The fraction of the sum of the two decay weights carried by $K_{3}$ is $(1 - ∣ χ ∣) / 2$ . For fixed $γ, Δ$ , as $t ↓ 0$ this fraction is $Δ^{2} t^{2} / 48 + O (t^{3})$ , and as $t \to \infty$ it tends to $[1 - γ / \sqrt{γ^{2} + Δ^{2}}] / 2$ .

The two operators are respectively the sum and difference of the two decay-path operators after a relative phase adjustment, as represented by the signs in Fig. 5. In an environmental realization in which the normalized emitted-photon path states have overlap $χ$ in the convention below, their overlap modulus is $∣ χ ∣$ . The reduced-system model alone does not identify a unique measured photon state. At $t = 0$ or $γ = 0$ both decay weights vanish; their normalized fraction and the conditional coherence are then undefined.

Statement references

Four-level spontaneous-decay model

Exact four-level Kraus operators

Conditional decoded state and decay coherence

Finite operator and ket conventions

The trace tr(A) as the sum of the diagonal entries

Facts & Assumptions

[F1]

The model has orthonormal basis $∣ 0 ⟩, \dots, ∣ 3 ⟩$ , with decay paths $F_{0} = ∣ 2 ⟩ ⟨ 0 ∣$ and $F_{1} = ∣ 3 ⟩ ⟨ 1 ∣$ . Adjoints, outer products and the coordinate inner product use the finite operator conventions. Four-level spontaneous-decay model Finite operator and ket conventions

[F2]

For $γ t > 0$ , the decay probability and coherence are $p = 1 - e^{- γ t}$ and $χ = γ (e^{i Δ t} - e^{- γ t}) / [(γ + i Δ) p]$ . Conditional decoded state and decay coherence

[F3]

The two decay Kraus operators are those in Exact four-level Kraus operators. With $Ω = γ + i Δ$ , that result supplies $f_{\pm} = γ ∣ e^{Ω t} - 1 ∣ / ∣ Ω ∣ \pm (e^{γ t} - 1)$ and a unit-modulus phase $u$ , with $K_{2} = e^{- γ t / 2} \sqrt{f_{+} / 2} (u F_{0} + F_{1})$ and $K_{3} = e^{- γ t / 2} \sqrt{- f_{-} / 2} (- u F_{0} + F_{1})$ . For positive $γ, t$ , its derivation gives the one-jump coefficients $p$ and $p χ$ and $u = χ / ∣ χ ∣$ . Its square roots are real and nonnegative, and both operators are zero at zero decay probability.

[F4]

For a two-path branch with coefficient matrix $p (\begin{array}{cc} 1 & χ \\ \overline{χ} & 1 \end{array})$ , $p > 0$ and $∣ χ ∣ \leq 1$ , the orthogonal Kraus weights are $p (1 \pm ∣ χ ∣)$ and the smaller fraction is $(1 - ∣ χ ∣) / 2$ . Its eigenoperators are the phase-adjusted sum and difference of the two paths. Two-path coherence Gram spectrum and Kraus weights

[F5]

For a finite list of vectors in an inner product space, its Gram matrix has entries $G_{j k} = ⟨ v_{j}, v_{k} ⟩$ . Here the coordinate inner product is linear in its first argument. The Gram matrix G(v₀,…,vᵣ₋₁)=(⟨ vᵢ,vⱼ⟩)_i,j<r and Gram determinant, with empty value 1 Finite operator and ket conventions

[F6]

Scalar exponential expansions and quotient expansions with nonzero constant denominator have controlled remainders at fixed parameters. Finite exponential and smooth quotient remainder control

[F7]

The trace of a square matrix is the sum of its diagonal entries. We use $Tr$ for this matrix trace in the fixed model basis. The trace tr(A) as the sum of the diagonal entries

Proof

Given: The model, positive rate and time, and the definitions in the Statement.

Goal: Derive the weights, fraction, limits and the stated conditional overlap interpretation.

1.1givenF1F2F3F4algebra

Put $r = ∣ χ ∣$ and $Ω = γ + i Δ$ . Here $0 < p < 1$ and $Ω \neq 0$ . The formula in [F2] gives $p r = γ e^{- γ t} ∣ e^{Ω t} - 1 ∣ / ∣ Ω ∣$ , hence $e^{- γ t} f_{+} = p (1 + r)$ and $- e^{- γ t} f_{-} = p (1 - r)$ . The real square root of $- f_{-} / 2$ supplied by [F3] implies $f_{-} \leq 0$ , so $r \leq 1$ . The two-path branch on the code therefore satisfies the hypotheses of [F4]; both path maps annihilate the orthogonal decay space. This argument invokes the already established Kraus representation, independently of the weight identities proved below.

2.1F1F3F4F7step 1.1algebra

By [F1], $F_{0}^{†} F_{1} = F_{1}^{†} F_{0} = 0$ , $F_{0}^{†} F_{0} = ∣ 0 ⟩ ⟨ 0 ∣$ and $F_{1}^{†} F_{1} = ∣ 1 ⟩ ⟨ 1 ∣$ . Write $Π_{01} = ∣ 0 ⟩ ⟨ 0 ∣ + ∣ 1 ⟩ ⟨ 1 ∣$ . Since $∣ u ∣ = 1$ , [F3] and step 1.1 give $K_{2}^{†} K_{2} = p (1 + r) Π_{01} / 2$ and $K_{3}^{†} K_{3} = p (1 - r) Π_{01} / 2$ . In the four-level basis $Π_{01}$ has diagonal $(1, 1, 0, 0)$ , so [F7] gives $w_{2, 3} = p (1 \pm r)$ . Equivalently, summing the diagonal entries of $K_{j}^{†} K_{j}$ gives the sum of the squared moduli of all entries of $K_{j}$ , its squared Hilbert–Schmidt norm. The two extra input columns outside the code are zero. The weight sum is $2 p$ , and the $K_{3}$ fraction is $(1 - r) / 2$ . The displayed operators are the phase-adjusted sum and difference directions of [F4], up to the overall sign of the minus operator; this explains the signs in Fig. 5 without asserting a unique measured trajectory.

3.1givenF2F6step 2.1algebra

Cancel the common factor $t$ in the numerator and denominator of $χ$ . By [F6], expansion at fixed $γ, Δ$ gives $χ = 1 + i Δ t / 2 + (- Δ^{2} / 6 + i γ Δ / 12) t^{2} + O (t^{3})$ . Consequently $∣ χ ∣^{2} = 1 - Δ^{2} t^{2} / 12 + O (t^{3})$ and $∣ χ ∣ = 1 - Δ^{2} t^{2} / 24 + O (t^{3})$ ; the last equality follows from $∣ χ ∣ - 1 = (∣ χ ∣^{2} - 1) / (∣ χ ∣ + 1)$ . Substitution into step 2.1 gives $Δ^{2} t^{2} / 48 + O (t^{3})$ .

3.2givenF2F3step 2.1algebra

Since $e^{- γ t} \to 0$ for fixed $γ > 0$ , [F2] gives $∣ χ ∣ = γ ∣ e^{i Δ t} - e^{- γ t} ∣ / [\sqrt{γ^{2} + Δ^{2}} (1 - e^{- γ t})] \to γ / \sqrt{γ^{2} + Δ^{2}}$ . Step 2.1 gives the asserted limiting fraction. At $t = 0$ or $γ = 0$ , [F3] gives $K_{2} = K_{3} = 0$ , and [F2] supplies no conditional decay coherence at zero probability.

4.1givenF1F5step 1.1step 2.1step 3.1step 3.2algebra∎

For an explicit normalized overlap model, take the finite coordinate space $C^{2}$ with the inner product of [F1], and put $v_{0} = (χ, \sqrt{1 - r^{2}})$ and $v_{1} = (1, 0)$ . Step 1.1 makes the square root real and nonnegative. Coordinate calculation gives $⟨ v_{0}, v_{0} ⟩ = ∣ χ ∣^{2} + 1 - r^{2} = 1$ , $⟨ v_{1}, v_{1} ⟩ = 1$ and $⟨ v_{0}, v_{1} ⟩ = χ$ . Thus [F5] identifies their Gram matrix as $(\begin{array}{cc} 1 & χ \\ \overline{χ} & 1 \end{array})$ , the same coefficient matrix as the normalized two-path branch. If a physical environmental realization identifies this normalized path-state Gram matrix with that of its emitted photons, their overlap modulus is $∣ χ ∣$ . That identification remains an additional physical interpretation, not a consequence of the reduced channel alone. Together with steps 2.1, 3.1 and 3.2 this proves the conditional assertions.

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF p. 7, paragraph before Fig. 5 and upper right paragraph