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Minimal-code decay detection without correction

PDF p. 6: Minimal-code bullet; PDF p. 4: Code-comparison paragraph

Statement

Adopt the article's finite equal-rate, zero-temperature downward-jump benchmark, with $γ > 0$ held fixed as the lift $Δ$ varies. In this effective model the generator uses $H_{S} + H_{Δ}$ and retains the unperturbed $L_{δ}$ of Eq. (10), as explicitly specified after Eq. (11). Assume the full physical hypotheses of the unlifted downward-jump supplier and the inherited choice assumptions. Restrict the input to the single- $J = 1$ code $a = ∣ 1, - 1 ⟩$ , $b = ∣ 1, 1 ⟩$ .

A decay leaves $J = 1$ and is flagged, but both codewords decay to $g = ∣ 0, 0 ⟩$ . The article describes these paths as emission of photons of opposite polarizations; this description is retained as attributed physical model input. The decay branch cannot be corrected, and for any finite $t > 0$ the accumulated channel is not exactly correctable on this two-dimensional code. For the complementary record associated with the three Kraus maps displayed in the proof, the normalized decay-sector records of the two codewords are orthogonal for every finite $t > 0$ , including at $Δ = 0$ . For the channel-specific transpose recovery, the recovered logical channel is $E_{t} (X) = p X + (1 - p) Tr (X) I_{2} / 2$ , where $p = e^{- γ t}$ . Consequently its entanglement fidelity is $(1 + 3 e^{- γ t}) / 4$ and its checking time at any fixed target is independent of $Δ$ .

Facts & Assumptions

[F1]

The minimal code has orthonormal logical basis $a = ∣ 1, - 1 ⟩$ , $b = ∣ 1, 1 ⟩$ and encoding $V ∣ 0 ⟩ = a$ , $V ∣ 1 ⟩ = b$ . Minimal single-manifold encoding

[F2]

Under its complete finite-block, tensor, bath, preparation, regularity and spectral hypotheses, the unlifted downward jumps send $∣ J, m ⟩$ only to $∣ J - 1, m + δ ⟩$ in channel $δ$ , and annihilate the ground $g = ∣ 0, 0 ⟩$ . Canonical angular-momentum downward jumps

[F3]

The lift is diagonal: $H_{Δ} ∣ J, m ⟩ = Δ J m ∣ J, m ⟩$ . Phenomenological angular-momentum degeneracy lift

[F4]

Use the effective equal-rate benchmark with $γ > 0$ , $f (J) = \sqrt{(2 J + 1) / (2 J - 1)}$ and $η_{J} = J (J + 1)$ . Under the same finite downward-jump hypotheses, the scalar total-decay theorem gives $\sum_{δ} L_{δ}^{†} L_{δ} = γ \sum_{J = 1}^{N} \sum_{m = - J}^{J} ∣ J, m ⟩ ⟨ J, m ∣$ . Thus every excited basis state has total downward rate $γ$ , while the ground has rate zero. The parameter postulate supplies the coefficients; the theorem supplies this rate identity. Equal-rate rigid-rotor benchmark parameters Normalized jumps give a scalar total decay operator

[F5]

The finite channel is the exponential of $L (X) = - i [H, X] + \sum_{δ} (L_{δ} X L_{δ}^{†} - {L_{δ}^{†} L_{δ}, X} / 2)$ . Finite GKSL evolution and the QEC channel interface

[F6]

The code-restricted noise is $N_{t} \circ {Ad}_{V}$ , and exact correction requires a decoder whose composition with this map is the identity on all logical operators. Code restriction and decoded channel types

[F7]

For a chosen Kraus family $A_{j}$ , the abstract complementary record has entries $Tr (A_{j} X A_{k}^{†})$ in an orthonormal label basis ${∣ e_{j} ⟩}$ . The choice of labels and basis is part of this mathematical representation. Complementary error records in a Kraus basis

[F8]

A channel on a code of dimension at least two that has a nonzero rank-one Kraus map with nonzero input projection on the code is not exactly correctable. Dimensional boundary of a rank-one KL obstruction

[F9]

The support-normalized transpose recovery has Kraus maps $P K_{j}^{†} N (P)^{- 1 / 2}$ ; completion off the support does not affect recovered code inputs. Transpose (Petz) recovery on a code

[F10]

For a logical qubit channel $E$ , its entanglement fidelity is $⟨ Φ ∣ (i d \otimes E) (∣ Φ ⟩ ⟨ Φ ∣) ∣ Φ ⟩$ with $∣ Φ ⟩ = (∣ 00 ⟩ + ∣ 11 ⟩) / \sqrt{2}$ . Entanglement fidelity on a finite logical space

[F11]

At target $F_{0} \in (0, 1]$ the transpose checking time is $\inf {t \geq 0 : F_{t r} (t) < F_{0}}$ , with $\inf \emptyset = + \infty$ . Operational checking time

[F12]

The finite operator exponential is differentiable, with inverse at negative time and derivative given by the generator. The exponential series of a bounded operator

[F13]

Assume the Axiom of Choice inherited from the physical and probability suppliers. The Axiom of Choice

[F14]

Carry the Countable Choice assumption of the regulated-bath and waiting-law interfaces. The Axiom of Countable Choice (AC_ω)

[F15]

Carry the Dependent Choice assumption of the specified zero-lift atomic-jump supplier. The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain

[A1]

Fixed effective jumps under the lift are a source-stated physical model choice: use the same finite $L_{δ}$ and effective $γ$ for the generator with $H_{S} + H_{Δ}$ . The article states “We retain the unperturbed jump operators” after Eq. (11), and the Fig. 2 caption specifies the channel generated by $H_{S} + H_{Δ}$ and $L_{δ}$ . This assumption is not derived from the energy-lift postulate alone. The amplitude-correction estimate accompanying the article's choice is not an input to this deduction.

Proof

Given: The fixed-jump effective benchmark, the minimal code, all inherited physical and choice assumptions, and elapsed time $t \geq 0$ .

Goal: Establish flagged but uncorrectable decay, compute the chosen abstract complementary record and prove the transpose checking-time independence.

1.1F1F2F3F4A1algebra

From [F2], the only nonzero decay on $a$ has $δ = + 1$ , since the output magnetic label must be zero; the only nonzero decay on $b$ has $δ = - 1$ . By the total-decay identity in [F4] and these unique allowed channels, write $L_{+ 1} a = α g$ , $L_{- 1} b = β g$ , with $∣ α ∣^{2} = ∣ β ∣^{2} = γ$ ; all other actions on $a, b$ , and all jump actions on $g$ , vanish. On the invariant subspace $span {a, b, g}$ the summed jump term is $γ Tr (X) ∣ g ⟩ ⟨ g ∣$ for $X$ supported on the code, and $\sum_{δ} L_{δ}^{†} L_{δ} = γ P$ there, where $P = ∣ a ⟩ ⟨ a ∣ + ∣ b ⟩ ⟨ b ∣$ . The energies of $a, b$ are $2 ω_{0} - Δ$ and $2 ω_{0} + Δ$ . These formulas remain the modeled jump actions as $Δ$ changes by [A1].

2.1F5F6F12step 1.1algebra

On the logical basis put $U_{t} = diag (e^{i Δ t}, e^{- i Δ t})$ , $p = e^{- γ t}$ and $q = 1 - p$ . The overall phase $e^{- 2 i ω_{0} t}$ cancels on density matrices. Substitution of the actions in step 1.1 into [F5] gives the exact code-restricted channel $N_{t} (V X V^{†}) = p V U_{t} X U_{t}^{†} V^{†} + q Tr (X) ∣ g ⟩ ⟨ g ∣ .$ Indeed the code block solves $\dot{Z} = - i [H ∣_{C}, Z] - γ Z$ , its trace is $p Tr (X)$ , and the ground population has derivative $γ p Tr (X)$ and initial value zero. All code-ground coherences remain zero. These equations have the displayed solution: the difference of any two solutions of the finite linear equation is annihilated after multiplication by $e^{- t L}$ , by [F12]. Thus the formula is the specified exponential channel, rather than a first-order jump approximation.

3.1F1F6F8step 2.1algebra

A finite Kraus representation of this restricted channel $L (C^{2}) \to L (H_{P})$ is $A_{0} = \sqrt{p} V U_{t}$ , $A_{1} = \sqrt{q} ∣ g ⟩ ⟨ 0 ∣$ , $A_{2} = \sqrt{q} ∣ g ⟩ ⟨ 1 ∣$ . It is trace-preserving because $\sum_{j} A_{j}^{†} A_{j} = I_{2}$ . The ground is orthogonal to $J = 1$ , so a projection onto $J = 1$ versus the ground flags a decay. Given a decay, every input density matrix has the same output $∣ g ⟩ ⟨ g ∣$ , so no channel acting on the emitter alone can recover both distinct basis inputs. For $t > 0$ , $q > 0$ and $A_{1}$ is a nonzero rank-one Kraus operator on the two-dimensional logical input, so [F8] also rules out exact recovery of the full finite-time channel.

4.1F1F7step 2.1step 3.1algebra

For the family $A_{0}, A_{1}, A_{2}$ in step 3.1, use the orthonormal environment labels $∣ e_{0} ⟩, ∣ e_{1} ⟩, ∣ e_{2} ⟩$ of [F7]. For $t > 0$ , the two logical basis inputs have complementary records $p ∣ e_{0} ⟩ ⟨ e_{0} ∣ + q ∣ e_{1} ⟩ ⟨ e_{1} ∣$ and $p ∣ e_{0} ⟩ ⟨ e_{0} ∣ + q ∣ e_{2} ⟩ ⟨ e_{2} ∣$ , respectively: the cross terms with $A_{0}$ vanish because the code and ground are orthogonal, and the other cross terms vanish on these basis inputs. Projecting onto $Q = ∣ e_{1} ⟩ ⟨ e_{1} ∣ + ∣ e_{2} ⟩ ⟨ e_{2} ∣$ has probability $q > 0$ for either input. Dividing the projected records by $q$ gives the orthogonal pure states $∣ e_{1} ⟩ ⟨ e_{1} ∣$ and $∣ e_{2} ⟩ ⟨ e_{2} ∣$ . Thus measurement in this chosen label basis distinguishes the codewords perfectly conditional on the decay sector, including at zero lift. The shared no-decay label gives no such discrimination. These conclusions concern the abstract complementary record of [F7]; they do not identify its label measurement with a physical photon-polarization measurement.

4.2F6F9step 2.1step 3.1algebra

The transpose construction may be computed for the restricted channel using $A_{j}$ and logical code projector $I_{2}$ : this is the decoded physical transpose, since for any physical Kraus family $K_{j}$ its restriction is $K_{j} V$ and $V^{†} P K_{j}^{†} = V^{†} K_{j}^{†}$ . Put $D = \sum_{j} A_{j} A_{j}^{†} = p P + 2 q ∣ g ⟩ ⟨ g ∣$ . For $t > 0$ , its support inverse square root is $p^{- 1 / 2} P + (2 q)^{- 1 / 2} ∣ g ⟩ ⟨ g ∣$ . The recovery maps $A_{j}^{†} D^{- 1 / 2}$ are consequently $U_{t}^{†} V^{†} P$ , $∣ 0 ⟩ ⟨ g ∣ / \sqrt{2}$ , and $∣ 1 ⟩ ⟨ g ∣ / \sqrt{2}$ . Applying them to step 2.1 yields $E_{t} (X) = p X + q Tr (X) I_{2} / 2$ . At $t = 0$ the ground is outside the output support; the support convention of [F9] gives the same formula with $q = 0$ . Any off-support completion is irrelevant on the code noise output by [F9].

5.1F10F11step 4.2algebra∎

Under [F10], the identity channel contributes entanglement fidelity one, while the replacement channel $X \mapsto Tr (X) I_{2} / 2$ sends the maximally entangled projector to $I_{2} \otimes I_{2} / 4$ , whose overlap with its unit input vector is $1 / 4$ . Linearity therefore gives $F_{t r} (t) = p + q / 4 = (1 + 3 e^{- γ t}) / 4$ . This has no $Δ$ dependence when $γ$ is fixed, so its strict threshold-crossing set, and hence the checking time of [F11], is identical for every $Δ$ in this fixed effective model.

Source notes

The fixed-jump premise was missing from the escalated draft's declared model interface, but is explicit in the original article: PDF p. 3, paragraph after Eq. (11), and PDF p. 4, Fig. 2 caption. It is adopted here as a physical modeling assumption, not inferred from shifted frequencies. The calculation is confined to input in the minimal single- $J = 1$ code; it makes no corresponding statement for a superposition across different $J$ manifolds. No microscopic derivation of lift-independent jump amplitudes or quantitative accuracy bound is supplied. In particular, the article's accompanying small-lift relative-amplitude estimate is not proved or used here; results concern the stipulated effective generator. The finite-block model and inherited bath/preparation hypotheses retain their stated scope. No numerical plot, experimental outcome or species-specific approximation has been reproduced. The original article is used for attribution and physical model choice only; the channel solution, recovery calculation and noncorrection argument above are local deductions from the stated assumptions and completed suppliers.

The article's opposite-photon-polarization description on PDF p. 4 is preserved as attributed physical model input. PDF p. 3 after Eq. (10) names the dipole-emission polarizations, and PDF p. 13 after Eq. (S3.17) assigns emitted angular-momentum projection minus the spherical channel label. These descriptions do not specify a physical realization of the finite-time Kraus-label measurement used in step 4.1. That step establishes distinguishability in the chosen abstract complementary record without assuming such a realization. The photon description is not a premise of the channel, noncorrection, recovery or fidelity calculations.

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF p.6 minimal-code bullet and Fig.4; PDF p.4 code-comparison paragraph; PDF p.17 Fig.S5.2 caption; PDF p.3 text after Eq.(10); PDF p.13 text after Eq.(S3.17)
Monolithic Quantum Error Correction in the Presence of Distinguishability — fixed-jump modeling choice · PDF p.3, Eq.(10), benchmark paragraph, Eq.(11) and following paragraph; PDF p.4, Fig.2 caption; PDF p.13, final S3 paragraph