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Aydin–Barg fidelity time curves and complete-information-loss plateau

PDF p. 16: S5 introductory simulation paragraph; PDF p. 17: Fig. S5.1 and Numerical method

Statement

Consider the two-dimensional logical Aydin–Barg code in the spin- $4$ block, the equal-rate downward-decay benchmark with rate scale $γ > 0$ , and the diagonal lift of strength $Δ$ . For each nonnegative time, form the modeled channel and evaluate the entanglement fidelity $F_{t r}$ of its support-completed transpose recovery on the maximally mixed logical input.

The article reports curves of $F_{t r}$ against time for $Δ / γ = 0, 0.1, 1, 100$ , with faster fidelity decay for larger lifting strengths and a long-time plateau at $1 / 4$ , which it interprets as complete loss of logical information. These are reported numerical predictions. No numerical replication, uncertainty bound, convergence estimate or universal monotonicity claim is supplied here.

In particular, if the decoded logical channel is a replacement channel $X \mapsto Tr (X) σ$ for some fixed logical density operator $σ$ , then its entanglement fidelity is $1 / 4$ . Such a channel sends every logical input state to the same state and hence retains no information that distinguishes logical inputs. The value $1 / 4$ alone does not establish that a channel is a replacement channel.

Facts & Assumptions

[F1]

The specified spin- $4$ encoding is an isometry from a two-dimensional logical space; its two codewords are the signed Aydin–Barg vectors. J equals four Aydin–Barg encoding

[F2]

The benchmark adopts the finite downward-jump model with $f (J) = \sqrt{(2 J + 1) / (2 J - 1)}$ and rigid-rotor level factors $η_{J} = J (J + 1)$ . Under its stated physical, spectral and choice hypotheses, the normalized-jump theorem proves total instantaneous downward jump rate $γ$ for each magnetic basis state with $J \geq 1$ , and zero for the ground block. Equal-rate rigid-rotor benchmark parameters Normalized jumps give a scalar total decay operator

[F3]

The phenomenological lift adds $H_{Δ} = Δ \sum_{J, m} J m ∣ J, m ⟩ ⟨ J, m ∣$ to the finite emitter Hamiltonian. Its specification is a physical model assumption. Phenomenological angular-momentum degeneracy lift

[F4]

For finite generator matrices, the numerical-method interface forms the channel by full Lindbladian exponentiation and obtains Kraus operators from the Choi matrix. Its exact operations have no implied floating-point error bound. Lindbladian exponentiation and Choi Kraus extraction

[F5]

For an isometric finite-dimensional code and a finite Kraus channel, the QEC-matrix formula gives the entanglement fidelity of support-completed transpose recovery, including singular cases. Transpose fidelity from the QEC matrix

[F6]

For a logical qubit channel $E$ , entanglement fidelity at the maximally mixed input is $⟨ Φ ∣ (i d \otimes E) (∣ Φ ⟩ ⟨ Φ ∣) ∣ Φ ⟩$ , where $∣ Φ ⟩ = (∣ 00 ⟩ + ∣ 11 ⟩) / \sqrt{2}$ . A density operator is positive semidefinite with trace one. Entanglement fidelity on a finite logical space

[F7]

The article reports the four curves, their qualitative ordering and the plateau in the cited simulation passage and caption; this attribution is not an independently reproduced computation.

[A1]

Assume the physical, spectral and finite-model hypotheses of the benchmark and lift interfaces, together with the Axiom of Choice, Countable Choice and Dependent Choice required by those interfaces. Equal-rate rigid-rotor benchmark parameters Phenomenological angular-momentum degeneracy lift The Axiom of Choice The Axiom of Countable Choice (AC_ω) The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain

Proof

Given: The finite model, code and recovery specified above, and, for the information-loss calculation, a decoded logical replacement channel with fixed density operator $σ$ .

Goal: Identify the conditional fidelity benchmark and explain the status of the reported curves.

1.1givenF1F2F3F4F5F7A1

The encoding has logical dimension two by [F1]. The equal instantaneous rate $γ$ used in the benchmark follows from the normalized-jump theorem in [F2], applied with the same finite integer-spin blocks, coefficients and inherited hypotheses; the diagonal Hamiltonian lift in [F3] leaves the jump operators and their total rate unchanged. Under the stated physical and finite-model assumptions, [F4] specifies the channel and its Kraus representation, while [F5] identifies the resulting transpose-recovery quantity as entanglement fidelity. These operations specify what the article's reported numerical curves represent; [F7] attributes their qualitative ordering and plateau to the article, without establishing their numerical accuracy.

2.1givenF6step 1.1algebra∎

For the replacement channel, expand $∣ Φ ⟩ ⟨ Φ ∣ = \frac{1}{2} \sum_{i, j = 0}^{1} ∣ i ⟩ ⟨ j ∣ \otimes ∣ i ⟩ ⟨ j ∣$ . Since $Tr (∣ i ⟩ ⟨ j ∣) = δ_{i j}$ , applying the channel to the second factor gives $(I / 2) \otimes σ$ . Its overlap with $∣ Φ ⟩$ is $\frac{1}{4} Tr σ = 1 / 4$ . Every density input has trace one, so every input has output $σ$ . Conversely, the fidelity value alone does not prove replacement: for example, the qubit unitary channel $X \mapsto Z X Z^{†}$ with $Z = diag (e^{i π / 3}, e^{- i π / 3})$ has entanglement fidelity $∣ Tr Z ∣^{2} / 4 = 1 / 4$ , while it preserves distinguishable computational-basis states. Thus the replacement interpretation is conditional and does not follow solely from the reported plateau value.

Depends on

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF p. 16, S5 introductory simulation paragraph; PDF p. 17, Fig. S5.1 caption and Numerical method