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Gradual which-path leakage and inverse splitting timescale

PDF p. 1: Final paragraph of Introduction

Statement

Assume the four-level model Four-level spontaneous-decay model, with decay rate $γ \geq 0$ , real splitting $Δ$ , code basis $∣ 0 ⟩, ∣ 1 ⟩$ , and the syndrome recovery $R_{4}$ of Four-level syndrome recovery. For a normalized input $∣ ψ ⟩ = a ∣ 0 ⟩ + b ∣ 1 ⟩$ , write $F_{ψ} (t)$ for the decoded state fidelity averaged over recovery outcomes: each outcome contributes its probability times the overlap of its normalized decoded output with the input, and an outcome of probability zero contributes zero.

When $γ t > 0$ , let $χ_{d} (t)$ be the multiplier of the input off-diagonal entry $a b^{*}$ in the normalized decoded decay output. With $u = ∣ Δ ∣ t$ , it obeys $∣ 1 - χ_{d} (t) ∣ \leq u e^{u}$ . Thus the relative phase accumulated over a checking interval is uniformly small when $∣ Δ ∣ t ≪ 1$ ; for nonzero splitting, $1 / ∣ Δ ∣$ is the associated phase scale. This is a statement about the model's conditional coherence, not a claim that a fixed amount of information has leaked at that time.

At fixed parameters, the loss under $R_{4}$ satisfies $1 - F_{ψ} (t) = ∣ a ∣^{2} ∣ b ∣^{2} γ Δ^{2} t^{3} / 3 + O (t^{4})$ as $t ↓ 0$ . The leading coefficient grows with $∣ Δ ∣$ at fixed positive $γ$ and fixed input with $a b \neq 0$ ; this does not assert global monotonicity at arbitrary time. In particular the fidelity tends to one as the interval tends to zero. If $γ = 0$ or $Δ = 0$ , this recovery restores every code density operator exactly at every finite time.

For the model's support-completed transpose recovery, let $F_{t r} (t)$ denote recovered entanglement fidelity at the maximally mixed logical input, and define $τ (F_{0}) = \inf {t \geq 0 : F_{t r} (t) < F_{0}}$ , taking the infimum of the empty set as $+ \infty$ . For fixed $γ Δ^{2} > 0$ , $τ (F_{0}) \sim [24 (1 - F_{0}) / (γ Δ^{2})]^{1 / 3}$ as $F_{0} ↑ 1$ ; the ratio tends to one. If $γ = 0$ or $Δ = 0$ , this checking time is infinite for $F_{0} \in (0, 1]$ . This first-crossing scale is distinct from the inverse-splitting phase scale.

Statement references

Transpose (Petz) recovery on a code

Entanglement fidelity on a finite logical space

Code restriction and decoded channel types

Facts & Assumptions

[F1]

The postulated model has the stated code, real splitting and nonnegative decay rate; it evolves by the specified finite GKSL channel. Four-level spontaneous-decay model

[F2]

The specified syndrome recovery has a null branch and a decay branch, the latter mapping the two decay basis states back to their corresponding code basis states. Four-level syndrome recovery

[F3]

For $γ t > 0$ , the decoded decay coherence multiplier is $χ_{d} = γ (e^{i Δ t} - e^{- γ t}) / [(γ + i Δ) (1 - e^{- γ t})]$ ; the normalized decay output has off-diagonal entry $a b^{*} χ_{d}$ . Conditional decoded state and decay coherence

[F4]

At fixed parameters the stated syndrome recovery has loss $∣ a ∣^{2} ∣ b ∣^{2} γ Δ^{2} t^{3} / 3 + O (t^{4})$ and exactly restores every code density operator when $γ = 0$ or $Δ = 0$ . Cubic short-time loss and recoverable limits

[F5]

The support-completed transpose recovery has the stated strict first-crossing asymptotic and the stated infinite checking times in the zero-parameter cases. Four-level high-fidelity checking time

[F6]

The exponential series has a controlled remainder; in particular, for real $x$ , $∣ e^{i x} - 1 ∣ \leq ∣ x ∣ e^{∣ x ∣}$ . Finite exponential and smooth quotient remainder control

[F7]

For a finite-dimensional channel and a nonzero code, the transpose recovery uses the support inverse square root of the channel output of the code projector, with an arbitrary code-state completion on the orthogonal complement. This completion does not change recovery after noise on code inputs. Transpose (Petz) recovery on a code

[F8]

The complex exponential is differentiable everywhere with derivative itself, obeys $e^{z + w} = e^{z} e^{w}$ , and agrees with the real exponential on real arguments. The complex exponential is entire and its complex derivative is itself exp(z+w)=exp z exp w, and the complex exponential extends the real exponential

[F9]

A continuous real function on a finite closed interval has the Darboux (oriented Riemann) integral; this integral is monotone in continuous integrands and linear under real scalar combinations. Continuous functions have an oriented Riemann integral The oriented Riemann integral is linear in continuous integrands

[F10]

If a real primitive is differentiable at every point of a closed nondegenerate interval, using one-sided derivatives at its endpoints, and its derivative is Darboux integrable, the integral of the derivative is the endpoint difference. The second fundamental theorem: if G is differentiable on [a,b] with G' = f and f is integrable, then ∫ₐᵇ f = G(b)-G(a)

[F11]

For a quantum channel $E$ on $C^{d}$ , its entanglement fidelity at the maximally mixed input $I / d$ is $F_{e} (E) = ⟨ Φ_{d} ∣ (i d \otimes E) (∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣) ∣ Φ_{d} ⟩$ , where $∣ Φ_{d} ⟩ = d^{- 1 / 2} \sum_{j = 0}^{d - 1} ∣ j ⟩ \otimes ∣ j ⟩$ . Entanglement fidelity on a finite logical space

[F12]

For an isometric encoding $V$ and a code-supported physical recovery channel $T$ , the decoded logical channel is $E (X) = V^{*} T (N (V X V^{*})) V$ . This is a channel on the logical space. Code restriction and decoded channel types

Proof

Given: The model parameters, input, recovery and time in the statement.

Goal: Establish the phase bound and the stated fidelity and checking-time consequences.

1.1givenF8F9algebra

Suppose $γ t > 0$ , so $γ > 0$ and $t > 0$ , and put $p = 1 - e^{- γ t} > 0$ and $z = γ + i Δ \neq 0$ . Throughout this proof, the integral of a continuous complex function $f$ on $[0, t]$ means $\int_{0}^{t} Re f (s) d s + i \int_{0}^{t} Im f (s) d s$ , with both real integrals taken in the Darboux convention of [F9]. It exists by [F9], and its complex linearity follows by expanding real and imaginary parts and using real linearity. The primitive $Φ (s) = - γ e^{i Δ t - z s} / z$ has derivative $Φ^{'} (s) = γ e^{- γ s} e^{i Δ (t - s)}$ : the derivative in [F8] on increments $- z h$ gives the factor $- z$ , and its addition law separates the exponentials. The primitive and its derivative are continuous on $[0, t]$ , and the same calculation gives the required one-sided endpoint derivatives.

1.2givenF1F2F4algebra

Apply [F4] to the normalized input and the recovery in [F1] and [F2]. Its fixed-parameter remainder means the loss tends to zero as $t ↓ 0$ ; its leading coefficient is proportional to $Δ^{2}$ and is strictly increasing in $∣ Δ ∣$ under the stated positivity conditions. The same supplier gives exact restoration in either zero-parameter case.

2.1F3F8F9F10step 1.1algebra

Apply [F10] separately to the real and imaginary components of $Φ$ ; their derivatives are continuous and hence integrable by [F9]. This gives $\int_{0}^{t} γ e^{- γ s} e^{i Δ (t - s)} d s = Φ (t) - Φ (0) = γ (e^{i Δ t} - e^{- γ t}) / z = p χ_{d}$ by [F3]. Applying the same real theorem to the primitive $- e^{- γ s}$ gives $\int_{0}^{t} γ e^{- γ s} d s = p$ . Thus $w (s) = γ e^{- γ s} / p$ is continuous and nonnegative, $\int_{0}^{t} w (s) d s = 1$ , and $χ_{d} = \int_{0}^{t} w (s) e^{i Δ (t - s)} d s$ .

3.1F6F9step 1.1step 2.1algebra

Put $u = ∣ Δ ∣ t$ . By [F6], $∣ e^{i Δ (t - s)} - 1 ∣ \leq ∣ Δ ∣ (t - s) e^{∣ Δ ∣ (t - s)} \leq u e^{u}$ for $0 \leq s \leq t$ . The continuous function $h (s) = w (s) (e^{i Δ (t - s)} - 1)$ therefore satisfies $∣ h (s) ∣ \leq w (s) u e^{u}$ , and step 2.1 and complex linearity give $J : = \int_{0}^{t} h (s) d s = χ_{d} - 1$ . If $J = 0$ the desired bound is immediate. Otherwise set $α = \overline{J} / ∣ J ∣$ , so $∣ α ∣ = 1$ and $Re (α J) = ∣ J ∣$ . The continuous real function $Re (α h (s))$ is at most $∣ h (s) ∣ \leq w (s) u e^{u}$ . Real monotonicity and linearity in [F9], together with the componentwise convention, give $∣ J ∣ = \int_{0}^{t} Re (α h (s)) d s \leq u e^{u} \int_{0}^{t} w (s) d s = u e^{u}$ . Hence $∣ 1 - χ_{d} ∣ \leq u e^{u}$ . This tends to zero whenever $u \to 0$ , independently of the positive value of $γ t$ . For $Δ \neq 0$ , $u$ becomes order one on the scale $1 / ∣ Δ ∣$ .

4.1givenF1F5F7F11F12step 3.1step 1.2∎

The model in [F1] has a finite-dimensional channel and a nonzero code, so the recovery defined in [F7] applies. Identify the two code basis vectors with the standard basis of $C^{2}$ and apply [F12] to the model channel and the code-supported recovery in [F7]. The resulting logical channel $E_{t}$ satisfies the hypotheses of [F11] with $d = 2$ , so $F_{t r} (t) = F_{e} (E_{t})$ at input $I_{2} / 2$ . Apply [F5] with those fixed parameters, that recovery, and the strict-crossing definition in the statement. It gives the cube-root asymptotic when $γ Δ^{2} > 0$ and an empty crossing set, hence infinite checking time, in either zero-parameter case. Its high-fidelity limit is a separate limit from the phase comparison in step 3.1. These conclusions establish the stated conditional version of gradual leakage and fast correction.

Source notes

Article attribution only: The article states that which-path information leaks gradually on the inverse-splitting timescale, that recoverable-fidelity loss grows with the shift, and that sufficiently fast correction restores near-perfect protection.

The article describes which-path information leaking gradually on an inverse-splitting scale and proposes faster correction as protection. The quantitative assertions above are restricted to the displayed model and recoveries; the phase bound alone does not quantify information in an environment.

Depends on

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF p. 1, final paragraph of Introduction