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No-jump phase tracking without decay-rate suppression

PDF p. 16: Paragraphs after Eq. (S4.12)

Statement

Adopt the finite lifted model and all its inherited physical, spectral and choice hypotheses, including the Axiom of Choice, Countable Choice and Dependent Choice. Its orthonormal basis is $∣ J, m ⟩$ with integer $J = 0, \dots, N$ and $m = - J, \dots, J$ . Let $J_{0} \geq 1$ , let $P = Π_{J_{0}}$ project onto that block, and let the initial density operator $ρ_{0}$ be supported in it. A density operator is positive semidefinite with trace one. The model has $γ > 0$ and lift $Δ (J, m) = Δ_{0} J m$ with real $Δ_{0}$ . On the host block $J_{z} ∣ J_{0}, m ⟩ = m ∣ J_{0}, m ⟩$ .

For ideal checks at nondecreasing times $0 = t_{0} \leq t_{1} \leq \dots \leq t_{n} = t$ , condition on obtaining the host-block outcome $P$ at every check. Write $U_{s} = e^{- i Δ_{0} J_{0} s J_{z}}$ on that block. The joint probability of these outcomes is $e^{- γ t}$ , and their conditional state is $U_{t} ρ_{0} U_{t}^{†}$ . Thus each null interval of length $τ$ applies the rotation $e^{- i θ J_{z}}$ with $θ = Δ_{0} J_{0} τ$ , and the conditional matrix elements satisfy $ρ_{m m^{'}} (t) = ρ_{m m^{'}} (0) e^{- i Δ_{0} J_{0} (m - m^{'}) t}$ . This is also the uninterrupted no-jump conditional evolution. The survival probability has constant hazard $γ$ ; these ideal host-block checks do not suppress it.

If the lift and elapsed interval are calibrated and the corresponding unitary control is available, applying $U_{τ}^{†}$ after each null outcome returns the state present at the beginning of that interval. In particular, starting from a code state, it returns that state after each compensated null check. Calibration supplies known phase data; it does not establish the availability or accuracy of physical control, nor reveal an unknown decay time.

For a subsequent short interval, the first-order detected-decay formula of the single-decay result applies with its input matrix elements replaced by those of the state at the beginning of that interval. Equivalently, the end-of-interval null-evolved entries can be substituted at first order for fixed parameters: their difference from the input is $O (τ)$ , so the unnormalized decay branch changes by $O (τ^{2})$ . The normalized decay formula is a first-order conditional approximation, not an exact finite-time identity.

Statement references

General bounded diagonal lift for fast-checking analysis

Normalized jumps give a scalar total decay operator

Projective code-manifold syndrome check

Matrix elements of the fast detected-decay state

Angular-momentum basis and tensor labels

Calibrated diagonal lifts and transition phase differences

Calibrated recovery cycles and accumulated-error contract

Facts & Assumptions

[F1]

The postulated finite model has diagonal Hamiltonian eigenvalues $ω_{0} J (J + 1) + Δ_{0} J m$ and jump operators that map each $J \geq 1$ block into the $J - 1$ block. General bounded diagonal lift for fast-checking analysis

[F2]

In this model $V = - \frac{1}{2} \sum_{δ} L_{δ}^{†} L_{δ} = - γ I / 2$ on every excited block. Normalized jumps give a scalar total decay operator

[F3]

The finite trajectory expansion expresses the channel as the no-jump term $E (s) X E (s)^{†}$ plus ordered terms with one or more jumps, where $E (s) = e^{s (- i H + V)}$ . Finite no-jump and ordered-jump expansion

[F4]

The ideal check uses the two projectors $P$ and $I - P$ ; an outcome's conditional state is its unnormalized output divided by its positive trace. Projective code-manifold syndrome check

[F5]

The first-order detected-decay branch is linear in its input entries, has weight $γ τ$ for a host-block density operator, and differs from the exact unnormalized checked output by $O (τ^{2})$ at fixed finite model parameters. Matrix elements of the fast detected-decay state

[F6]

The lifted null-branch supplier gives the same phase and trace through first order for a short interval. Lifted null-branch matrix elements and normalization

[F7]

The magnetic basis diagonalizes $J_{z}$ with eigenvalue $m$ ; calibrated phase data make the specified lift values and elapsed interval available to the recovery rule, without observing an unknown jump time. Angular-momentum basis and tensor labels Calibrated diagonal lifts and transition phase differences Calibrated recovery cycles and accumulated-error contract

[A1]

Assume the Axiom of Choice, Countable Choice and Dependent Choice, together with the physical and spectral assumptions inherited by the model and its stated results. 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 General bounded diagonal lift for fast-checking analysis

Proof

Given: The stipulated model, host-supported initial state, ideal checks, and assumptions in the Statement.

Goal: Establish the exact null evolution and survival probability, the conditional compensation, and the stated scope of the next decay formula.

1.1F1F2F3F7algebra

The diagonal Hamiltonian and scalar $V$ preserve each block. On the host block, $G = - i H + V = - i ω_{0} J_{0} (J_{0} + 1) I - i Δ_{0} J_{0} J_{z} - γ I / 2$ . Hence $E (s) P = e^{- γ s / 2} e^{- i ω_{0} J_{0} (J_{0} + 1) s} U_{s} P$ . Each jump lowers the block label and every intervening $E$ preserves it, so every term with at least one jump in [F3] is annihilated by projection onto $P$ on the output. Therefore the unnormalized host outcome for a host-supported input $X$ is $e^{- γ s} U_{s} X U_{s}^{†}$ .

2.1F3F4step 1.1algebra

The trace of this output for a density operator is $e^{- γ s} > 0$ , and [F4] gives conditional state $U_{s} X U_{s}^{†}$ . Applying the same expression successively to the intervals and using $U_{a} U_{b} = U_{a + b}$ gives joint weight $\prod_{k} e^{- γ (t_{k} - t_{k - 1})} = e^{- γ t}$ and conditional state $U_{t} ρ_{0} U_{t}^{†}$ . The no-jump term in [F3] has the same weight and normalized state, proving equality with uninterrupted no-jump evolution. Since $- d (\log e^{- γ t}) / d t = γ$ , its survival hazard is unchanged by these checks.

3.1F7step 2.1algebra

By the eigenvalue relation in [F7], conjugation by $U_{t}$ multiplies the $(m, m^{'})$ matrix element by $e^{- i Δ_{0} J_{0} (m - m^{'}) t}$ . If the stated unitary control is available, $U_{τ}^{†} U_{τ} = I$ , so applying it after the null outcome restores the interval's input state. The calibration convention supplies the required phase data but does not itself supply control or a jump-time observation.

4.1F5F6step 2.1algebra∎

Apply [F5] to the actual host-supported input at the start of the next interval. By step 2.1, its null-evolved state differs entrywise by $O (τ)$ at fixed finite parameters; the displayed single-decay branch is $τ$ times a fixed linear function of those entries. Substitution of the null-evolved entries therefore changes the unnormalized first-order expression by $O (τ^{2})$ . Its positive weight is $γ τ$ for $τ > 0$ , so normalization gives the corresponding first-order conditional approximation, without asserting equality at finite $τ$ . This is consistent with the short-time null expansion in [F6].

Depends on

Used by · 1

Jump-time dephasing vanishes with the checking interval

Dependency tree · 2 levels

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF page 16, Eqs. (S4.10)–(S4.12) and the two following paragraphs