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Cubic short-time loss and recoverable limits

PDF p. 2: Eq. (5) and following paragraph; PDF p. 8: Eq. (S1.5)

Statement

Assume the four-level model with real $ω_{0}, Δ$ , rate $γ \geq 0$ , and the syndrome recovery $R_{4}$ . The code has orthonormal basis $∣ 0 ⟩, ∣ 1 ⟩$ . For a normalized code state $∣ ψ ⟩ = a ∣ 0 ⟩ + b ∣ 1 ⟩$ , let $F_{ψ} (t)$ be the branch-averaged decoded state fidelity: the sum of each recovery outcome probability times its conditional output's overlap with $∣ ψ ⟩$ , with a zero-probability outcome contributing zero. At fixed parameters, as $t \to 0$ through nonnegative times,

$1 - F_{ψ} (t) = c_{ψ} γ Δ^{2} t^{3} + O (t^{4}), c_{ψ} = \frac{∣ a ∣^{2} ∣ b ∣^{2}}{3} \leq \frac{1}{12} .$

Here $O (t^{4})$ means a quantity bounded in absolute value by $C t^{4}$ for sufficiently small nonnegative $t$ , for a constant $C$ that may depend on the fixed parameters. For positive $γ$ , one power of time comes from the decay probability and two from the relative phase mismatch between decay paths. If $γ = 0$ or $Δ = 0$ , the recovered output equals the input for every code density operator at every finite $t \geq 0$ .

Statement references

Four-level spontaneous-decay model

Four-level syndrome recovery

Branch-averaged decoded state fidelity

Conditional decoded state and decay coherence

Finite operator and ket conventions

Finite exponential and smooth quotient remainder control

Density operators have an orthogonal spectral ensemble

Facts & Assumptions

[F1]

The postulated model has real $ω_{0}, Δ$ , rate $γ \geq 0$ , and the stated orthonormal code basis; its evolution is a linear channel. Four-level spontaneous-decay model

[F2]

The recovery has null operation $X \mapsto P X P$ and decay operation $X \mapsto U X U^{†}$ , with $P = ∣ 0 ⟩ ⟨ 0 ∣ + ∣ 1 ⟩ ⟨ 1 ∣$ and $U = ∣ 0 ⟩ ⟨ 2 ∣ + ∣ 1 ⟩ ⟨ 3 ∣$ ; discarding the outcome sums these operations. Four-level syndrome recovery

[F3]

For $γ t > 0$ , put $p = 1 - e^{- γ t}$ and $χ = γ (e^{i Δ t} - e^{- γ t}) / [(γ + i Δ) p]$ . The fidelity is $F_{ψ} = 1 - 2 ∣ a ∣^{2} ∣ b ∣^{2} p (1 - Re χ)$ . At $γ t = 0$ , it is one. Branch-averaged decoded state fidelity

[F4]

For $γ t > 0$ , the normalized null output is $∣ ψ ⟩ ⟨ ψ ∣$ and the normalized decay output has diagonal entries $∣ a ∣^{2}, ∣ b ∣^{2}$ and off-diagonal entries $a b^{*} χ$ and its conjugate. At $γ t = 0$ the decay branch has zero probability. Conditional decoded state and decay coherence

[F5]

Outer products and adjoints use the finite orthonormal-basis conventions. Finite operator and ket conventions

[F6]

The finite exponential series has a controlled remainder, and products and quotients with nonzero limiting denominator have the stated fixed-parameter expansions. Finite exponential and smooth quotient remainder control

[F7]

On a nonzero finite-dimensional complex inner-product space, every density operator has a finite spectral ensemble $ρ = \sum_{j = 1}^{m} λ_{j} ∣ e_{j} ⟩ ⟨ e_{j} ∣$ , where the $e_{j}$ are orthonormal unit vectors, $λ_{j} > 0$ , and $\sum_{j} λ_{j} = 1$ . Density operators have an orthogonal spectral ensemble

[F8]

For the normalized code input and every $γ \geq 0$ , $t \geq 0$ , the exact evolved operator is $ρ (t) = e^{- γ t} ∣ ψ ⟩ ⟨ ψ ∣ + (1 - e^{- γ t}) (∣ a ∣^{2} ∣ 2 ⟩ ⟨ 2 ∣ + ∣ b ∣^{2} ∣ 3 ⟩ ⟨ 3 ∣) + [a b^{*} c (t) ∣ 2 ⟩ ⟨ 3 ∣ + h . c .]$ , where $c (t) = γ (e^{i Δ t} - e^{- γ t}) / (γ + i Δ)$ when $(γ, Δ) \neq (0, 0)$ and $c (t) = 0$ at $(0, 0)$ ; h.c. denotes the Hermitian conjugate of the preceding term. Exact state evolution from a four-level codeword

Proof

Given: The model, recovery, normalized code state, and fixed parameters in the statement.

Goal: Establish the short-time expansion and exact recovered outputs in the two parameter limits.

1.1givenF3F6algebra

Suppose $γ > 0$ and $t > 0$ . Set $c = γ (e^{i Δ t} - e^{- γ t}) / (γ + i Δ)$ , so $p χ = c$ . Applying the exponential expansion in [F6] to the scalar exponentials, and using $γ + i Δ \neq 0$ , gives $p = γ t - γ^{2} t^{2} / 2 + γ^{3} t^{3} / 6 + O (t^{4})$ and $c = γ t + γ (i Δ - γ) t^{2} / 2 + γ (γ^{2} - i γ Δ - Δ^{2}) t^{3} / 6 + O (t^{4})$ . Consequently $p - Re c = γ Δ^{2} t^{3} / 6 + O (t^{4})$ .

1.2givenF2F3F4F5F8algebra

If $γ = 0$ and $Δ \neq 0$ , the expression for $c (t)$ in [F8] has zero numerator and nonzero denominator, so $c (t) = 0$ ; if also $Δ = 0$ , its specified boundary value is zero. Thus [F8] gives $ρ (t) = ∣ ψ ⟩ ⟨ ψ ∣$ at $γ = 0$ . By [F2] and [F5], $P ∣ ψ ⟩ = ∣ ψ ⟩$ and $U ∣ ψ ⟩ = 0$ , so the unnormalized recovered null output is the input projector and the decay output is zero. The loss is identically zero and satisfies the expansion. If $Δ = 0$ and $γ t > 0$ , [F3] gives $χ = 1$ ; the two conditional outputs in [F4] then both equal $∣ ψ ⟩ ⟨ ψ ∣$ . At $t = 0$ , [F8] gives $c (0) = 0$ and $ρ (0) = ∣ ψ ⟩ ⟨ ψ ∣$ , and the same recovery calculation applies. Summing outcome-weighted outputs therefore restores every pure code state in either limit.

2.1givenF3F6step 1.1algebra

By [F3], $1 - F_{ψ} = 2 ∣ a ∣^{2} ∣ b ∣^{2} (p - Re c)$ , which gives the displayed cubic coefficient. Normalization gives $∣ a ∣^{2} + ∣ b ∣^{2} = 1$ , so $(∣ a ∣^{2} - ∣ b ∣^{2})^{2} \geq 0$ implies $∣ a ∣^{2} ∣ b ∣^{2} \leq 1 / 4$ . Moreover $p = γ t + O (t^{2})$ , and division after cancelling $t$ gives $1 - Re χ = Δ^{2} t^{2} / 6 + O (t^{3})$ . Thus the two factors have the stated time orders.

3.1F1F2F5F7step 1.2step 2.1algebra∎

The evolved-and-recovered map is linear by [F1] and [F2]. Apply [F7] on the code space $C = span {∣ 0 ⟩, ∣ 1 ⟩}$ , a nonzero two-dimensional complex inner-product space by [F1] and [F5]. Every code density operator therefore has an ensemble $ρ = \sum_{j = 1}^{m} λ_{j} ∣ e_{j} ⟩ ⟨ e_{j} ∣$ with unit code vectors $e_{j}$ and positive weights summing to one. In either parameter limit, step 1.2 restores each projector, so linearity gives $R_{4} (N_{t} (ρ)) = \sum_{j} λ_{j} ∣ e_{j} ⟩ ⟨ e_{j} ∣ = ρ$ . Together with step 2.1, this proves the exact limits and the expansion for all $γ \geq 0$ .

Depends on

Used by · 2

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF p. 2, Eq. (5) and following paragraph; PDF p. 8, Eq. (S1.5)