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Exact four-level transpose fidelity

PDF p. 8: Eq. (S1.7) and following equivalence

Statement

Assume the postulated four-level model with orthonormal basis $∣ 0 ⟩, ∣ 1 ⟩, ∣ 2 ⟩, ∣ 3 ⟩$ , code $C = span {∣ 0 ⟩, ∣ 1 ⟩}$ , decay rate $γ \geq 0$ , real splitting $Δ$ , and time $t \geq 0$ . Let $H_{L} = C^{2}$ have ordered orthonormal basis $∣ 0 ⟩_{L}, ∣ 1 ⟩_{L}$ , and encode by the isometry $V : H_{L} \to C^{4}$ with $V ∣ j ⟩_{L} = ∣ j ⟩$ for $j = 0, 1$ . Write $N_{t}$ for the model channel and $T_{t}$ for any support-completed transpose recovery for the code projector $P = V V^{†}$ . The resulting decoded logical channel is $E_{t} (X) = V^{†} T_{t} (N_{t} (V X V^{†})) V$ . Let $F_{t r} (t)$ be the entanglement fidelity of the resulting logical channel at the maximally mixed logical input $I / 2$ .

For $γ t > 0$ , set $p_{d e c a y} = 1 - e^{- γ t}$ and $χ_{d} (t) = γ (e^{i Δ t} - e^{- γ t}) / [(γ + i Δ) (1 - e^{- γ t})]$ . Then

$F_{t r} (t) = 1 - \frac{p_{d e c a y}}{2} (1 - ∣ χ_{d} (t) ∣^{2}) = \frac{(γ^{2} + Δ^{2}) \sinh (γ t) + γ^{2} [\cosh (γ t) - \cos (Δ t)]}{(γ^{2} + Δ^{2}) (e^{γ t} - 1)} .$

At $γ t = 0$ , the fidelity is $1$ ; the conditional decay coherence is not assigned at zero decay probability. The displayed quotient has the same limiting value $1$ as $γ t \to 0$ along the model parameter domain.

Statement references

Four-level spontaneous-decay model

Transpose (Petz) recovery on a code

Code restriction and decoded channel types

Conditional decoded state and decay coherence

Qubit coherence channels and fidelity functionals

Facts & Assumptions

[F1]

In the postulated model the code and decay spans are orthogonal, and the evolution is the specified finite-dimensional channel with $γ \geq 0$ , real $Δ$ , and $t \geq 0$ . Four-level spontaneous-decay model

[F2]

On the code, the exact channel has a no-jump branch of weight $e^{- γ t}$ and a decay branch with diagonal weight $p = 1 - e^{- γ t}$ and off-diagonal coefficient $c = γ \int_{0}^{t} e^{- γ s} e^{i Δ (t - s)} d s$ . In particular $∣ c ∣ \leq p$ , and the branch maps into the decay span. Exact four-level Kraus operators

[F3]

For $p > 0$ , identifying $∣ 2 ⟩, ∣ 3 ⟩$ with the logical basis gives the normalized decay coherence $χ_{d} = c / p$ as displayed in the statement; the normalized no-jump output is the input. Conditional decoded state and decay coherence

[F4]

The support-completed transpose recovery uses the inverse square root of the channel output on the code projector on its support, and its completion does not affect outputs of encoded inputs. Transpose (Petz) recovery on a code

[F5]

A qubit coherence map $D_{λ}$ preserves diagonal matrix entries and multiplies the $01$ entry by $λ$ and the $10$ entry by its conjugate. For $∣ λ ∣ \leq 1$ its entanglement fidelity at $I / 2$ is $(1 + Re λ) / 2$ , and fidelity is affine under a weighted sum of channels. Qubit coherence channels and fidelity functionals

[F6]

For a channel with orthogonal flagged branches $(1 - p) id$ and $p D_{χ}$ , its support-normalized transpose recovery produces the logical channel $(1 - p) id + p D_{∣ χ ∣^{2}}$ , including zero-weight branches. Transpose recovery for a flagged coherence channel

[F7]

Matrix entries, adjoints and basis identifications follow the finite orthonormal-basis conventions. Finite operator and ket conventions

[F8]

For an isometric encoding $V$ with code projector $P = V V^{†}$ and a code-supported physical recovery $T_{t}$ , the decoder is $Y \mapsto V^{†} T_{t} (Y) V$ , and composing it with $N_{t} \circ {Ad}_{V}$ gives the logical channel $E_{t}$ in the statement. Code restriction and decoded channel types

Proof

Given: The model, code, parameters and transpose-recovery procedure in the statement.

Goal: Compute the fidelity, prove the two formulas agree, and establish the zero-decay boundary.

1.1givenF1F2F3F5F7F8

Put $q = e^{- γ t}$ and $p = 1 - q$ . By [F1] and [F2], the output restricted to encoded inputs has two orthogonal flags: the code span and the decay span. Identify the code span with $H_{L}$ by $V$ and the decay span by the isometry $W ∣ j ⟩_{L} = ∣ j + 2 ⟩$ for $j = 0, 1$ . The no-jump branch is $q id$ ; the decay branch preserves diagonal entries with coefficient $p$ and multiplies the $01$ entry by $c$ . Thus, when $p > 0$ , the restricted channel is the flagged channel $q id \oplus p D_{χ}$ with $χ = c / p = χ_{d}$ . The bound in [F2] gives $∣ χ ∣ \leq 1$ . These equalities hold on all input matrices by their entries, not merely on a chosen state.

2.1givenF4F6F7F8step 1.1algebra

Let $K_{k}$ be the model-channel Kraus matrices and put $B_{k} = K_{k} V$ . The restricted noise $N_{t} \circ {Ad}_{V}$ has Kraus matrices $B_{k}$ , and $A = N_{t} (P) = \sum_{k} B_{k} B_{k}^{†}$ . By [F4] and [F8], its decoded support-normalized recovery has Kraus matrices $V^{†} P K_{k}^{†} A^{- 1 / 2} = B_{k}^{†} A^{- 1 / 2}$ , since $V^{†} P = V^{†}$ . These are exactly the transpose-recovery matrices for the restricted channel with whole input code $H_{L}$ . The completion has no effect on the restricted noise outputs by [F4]. Thus the physical recovery and decoding defining $E_{t}$ agree, under the basis identifications of step 1.1, with the recovered flagged channel in [F6].

3.1F4F5F6F8step 1.1step 2.1algebra

Applying the flagged transpose rule to step 1.1, using the decoded-recovery identification in step 2.1 and the support convention in [F4], gives $q id + p D_{∣ χ_{d} ∣^{2}}$ . The fidelity formula and affinity in [F5] therefore give $F_{t r} = q + p (1 + ∣ χ_{d} ∣^{2}) / 2 = 1 - p (1 - ∣ χ_{d} ∣^{2}) / 2$ . This proves the first expression when $p > 0$ .

4.1F3step 3.1algebra

For $p > 0$ , write $D = γ^{2} + Δ^{2} > 0$ . Directly taking the modulus in [F3] gives $∣ χ_{d} ∣^{2} = γ^{2} (1 + q^{2} - 2 q \cos (Δ t)) / (D p^{2})$ . Substitution into step 3.1 and multiplication of numerator and denominator by $e^{γ t}$ yield $F_{t r} = [D \sinh (γ t) + γ^{2} (\cosh (γ t) - \cos (Δ t))] / [D (e^{γ t} - 1)]$ , proving the printed quotient and its equivalence to the first expression.

5.1givenF2F6step 1.1step 3.1step 4.1algebra∎

If $γ t = 0$ , then $p = 0$ and [F2] gives no decay branch; the no-jump branch is the identity on the encoded logical qubit. The zero-weight case of [F6] therefore gives fidelity $1$ . For $p > 0$ , step 1.1 gives $0 \leq ∣ χ_{d} ∣^{2} \leq 1$ , so step 3.1 implies $0 \leq 1 - F_{t r} \leq p / 2$ . As $γ t \to 0$ , $p \to 0$ , establishing the stated limit of both formulas without assigning a conditional state to the zero-weight branch.

Depends on

Used by · 3

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF p. 8, Eq. (S1.7) and the sentence immediately following it