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Lifted null-branch matrix elements and normalization

PDF p. 16: Eqs. (S4.10)–(S4.12)

Statement

Adopt the finite lifted model, the hypotheses of its null-projection and uniform-decay results, and the Axiom of Choice, Countable Choice and Dependent Choice inherited there. In particular, the finite physical space has orthonormal basis $∣ J, m ⟩$ with $J = 0, \dots, N$ and $m = - J, \dots, J$ ; the initial density operator $ρ_{0}$ is supported on the block $J_{0} \in {1, \dots, N}$ . A density operator is positive semidefinite with trace one. Let $H$ be the model Hamiltonian with real lift $Δ (J, m) = Δ_{0} J m$ and rate $γ > 0$ . Put $V = - \frac{1}{2} \sum_{δ} L_{δ}^{†} L_{δ}$ , $K_{n u l l} = I + τ (- i H + V)$ , $B = K_{n u l l} ρ_{0} K_{n u l l}^{†}$ , and $r_{m m^{'}} = ⟨ J_{0}, m ∣ ρ_{0} ∣ J_{0}, m^{'} ⟩$ . Write $a_{m m^{'}} = Δ_{0} J_{0} (m - m^{'})$ and $[ρ_{0}, H] = ρ_{0} H - H ρ_{0}$ .

As $τ ↓ 0$ at fixed model parameters, $i [ρ_{0}, H]_{m m^{'}} = - i a_{m m^{'}} r_{m m^{'}}$ , $B_{m m^{'}} = [1 + (- i a_{m m^{'}} - γ) τ] r_{m m^{'}} + O (τ^{2})$ , and $Tr B = 1 - γ τ + O (τ^{2})$ . For sufficiently small $τ \geq 0$ , $Tr B > 0$ and the normalized null output $ρ_{n u l l} = B / Tr B$ satisfies $[ρ_{n u l l}]_{m m^{'}} = r_{m m^{'}} [1 - i a_{m m^{'}} τ / (1 - γ τ)] + O (τ^{2}) = r_{m m^{'}} (1 - i a_{m m^{'}} τ) + O (τ^{2})$ . The displayed quotient expression is a first-order formula with a remainder, not an asserted exact finite-time identity.

The exact channel followed by the same null projection has the same displayed first-order matrix-element, trace and normalized-state expansions for sufficiently small $τ$ . Here each $O (τ^{2})$ is bounded in absolute value by a constant times $τ^{2}$ near zero, with the model and initial state fixed; no uniform estimate over varying model parameters is claimed.

Statement references

Null-syndrome state and first-order expansion

General bounded diagonal lift for fast-checking analysis

Normalized jumps give a scalar total decay operator

Calibrated diagonal lifts and transition phase differences

Facts & Assumptions

[F1]

In the lifted model, the Hamiltonian eigenvalue on $∣ J_{0}, m ⟩$ is $E_{m} = ω_{0} J_{0} (J_{0} + 1) + Δ_{0} J_{0} m$ , so $E_{m} - E_{m^{'}} = a_{m m^{'}}$ . The diagonal phase-data convention uses differences of diagonal entries. General bounded diagonal lift for fast-checking analysis Calibrated diagonal lifts and transition phase differences

[F2]

Under the benchmark and its inherited system, bath, spectral and choice hypotheses, $V = - γ I / 2$ on every excited block. Normalized jumps give a scalar total decay operator

[F3]

The null-projection result gives $B = ρ_{0} + τ (i [ρ_{0}, H] + V ρ_{0} + ρ_{0} V) + O (τ^{2})$ in spectral norm, and the exact projected channel output equals $B + O (τ^{2})$ in that norm. The conditional output is obtained by dividing by its positive trace. Null-syndrome state and first-order expansion

[F4]

Scalar expansions with nonzero constant denominator can be divided through the desired order, retaining an $O (τ^{2})$ remainder at fixed parameters. Finite exponential and smooth quotient remainder control

[A1]

Assume the Axiom of Choice, Countable Choice and Dependent Choice as required by the physical suppliers. 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 lifted model, supported density operator and checking interval in the statement, with the assumptions in [A1].

Goal: Derive the first-order unnormalized and normalized null-branch formulas for the approximate and exact channels.

1.1F1F2F3A1algebra

Since $H$ is diagonal, $[ρ_{0}, H]_{m m^{'}} = (E_{m^{'}} - E_{m}) r_{m m^{'}} = - a_{m m^{'}} r_{m m^{'}}$ . On the support of $ρ_{0}$ , [F2] gives $V ρ_{0} + ρ_{0} V = - γ ρ_{0}$ . Substitution into [F3] yields the stated matrix-element expansion. A spectral-norm $O (τ^{2})$ bound also bounds each matrix element between unit basis vectors.

2.1givenstep 1.1algebra

Summing the finitely many diagonal entries of step 1.1 gives $Tr B = (1 - γ τ) Tr ρ_{0} + O (τ^{2}) = 1 - γ τ + O (τ^{2})$ : the commutator contribution vanishes on the diagonal. Thus the trace tends to one and is positive for sufficiently small nonnegative $τ$ . The denominator $1 - γ τ$ is also nonzero there.

3.1F4step 1.1step 2.1algebra

By [F4], division by the trace gives $[ρ_{n u l l}]_{m m^{'}} = r_{m m^{'}} (1 - i a_{m m^{'}} τ) + O (τ^{2})$ . Moreover $τ / (1 - γ τ) = τ + O (τ^{2})$ , proving the equivalent displayed quotient form with its remainder. This argument does not require $r_{m m^{'}}$ to be nonzero.

4.1F3F4step 2.1step 3.1algebra∎

By [F3], the exact projected output differs from $B$ by $O (τ^{2})$ in spectral norm. In finite dimension this also changes each entry and the trace by $O (τ^{2})$ . Its trace consequently tends to one, so it is positive sufficiently near zero; applying [F4] once more gives the same normalized expansion. All constants here may depend on the fixed finite model and state.

Depends on

Used by · 1

No-jump phase tracking without decay-rate suppression

Dependency tree · 2 levels

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF p. 16, Eqs. (S4.10)–(S4.12), with the preceding commutator calculation