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Four-level high-fidelity checking time

PDF p. 3: Eq. (8)

Statement

Assume the postulated four-level model with decay rate $γ \geq 0$ , real degeneracy-lifting parameter $Δ$ , code $span {∣ 0 ⟩, ∣ 1 ⟩}$ , and its support-completed transpose recovery at each elapsed time. Let $F_{t r} (t)$ be the recovered entanglement fidelity at the maximally mixed logical input, and define the checking time $τ (F_{0}) = \inf {t \geq 0 : F_{t r} (t) < F_{0}}$ , with $\inf \emptyset = + \infty$ .

For fixed parameters with $γ Δ^{2} > 0$ , as $F_{0} ↑ 1$ through values below $1$ ,

$τ (F_{0}) \sim {[\frac{24 (1 - F_{0})}{γ Δ^{2}}]}^{1 / 3} .$

The symbol $\sim$ means that the ratio of the two sides tends to $1$ . For fixed $F_{0} < 1$ , the displayed leading approximation decreases as $γ$ or $∣ Δ ∣$ increases while the other is positive. This comparison is of leading approximations at fixed parameters; it does not assert global monotonicity of the exact checking time. If $γ = 0$ or $Δ = 0$ , then $F_{t r} (t) = 1$ for every $t \geq 0$ , and $τ (F_{0}) = + \infty$ for every $F_{0} \in (0, 1]$ .

Statement references

Four-level spontaneous-decay model

Transpose (Petz) recovery on a code

Operational checking time

Cubic transpose-fidelity loss

Exact four-level transpose fidelity

The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents

Continuity and derivatives of positive-base real powers

On an interval I, for f continuous on I and differentiable at every interior point: f' ≥ 0 throughout gives f nondecreasing, f' > 0 gives f increasing, f' ≤ 0 and f' < 0 give the two decreasing forms; conversely a nondecreasing f has f' ≥ 0 and a nonincreasing f has f' ≤ 0 wherever it is differentiable, and no strict converse is claimed

Facts & Assumptions

[F1]

The model specifies a finite four-level channel with $γ \geq 0$ , real $Δ$ , the stated code, and the transpose recovery is applied to that channel and code. Four-level spontaneous-decay model Transpose (Petz) recovery on a code

[F2]

The checking time is the infimum of the strict fidelity-crossing set, with value $+ \infty$ when that set is empty. Operational checking time

[F3]

At fixed model parameters, $F_{t r} (t) = 1 - γ Δ^{2} t^{3} / 24 + O (t^{4})$ as $t ↓ 0$ , where the remainder is bounded in absolute value by a parameter-dependent constant times $t^{4}$ near zero. Cubic transpose-fidelity loss

[F4]

If $γ t > 0$ , put $p = 1 - e^{- γ t}$ and $χ = γ (e^{i Δ t} - e^{- γ t}) / [(γ + i Δ) p]$ . Then $F_{t r} = 1 - p (1 - ∣ χ ∣^{2}) / 2$ . If $γ t = 0$ , the fidelity is $1$ , and the formula has limit $1$ as $γ t \to 0$ . Exact four-level transpose fidelity

[F5]

If a continuous $F : [0, \infty) \to R$ satisfies $F (0) = 1$ and $1 - F (t) = a t^{p} + O (t^{p + 1})$ for fixed $a, p > 0$ , its strict first-crossing time at $1 - ϵ$ is asymptotic to $(ϵ / a)^{1 / p}$ as $ϵ ↓ 0$ . lem-qec-prereq-first-crossing-asymptotics

[F6]

For a bounded complex operator $A$ , the exponential $e^{t A}$ is continuous in $t$ in operator norm. In particular this applies to scalar complex exponentials. The exponential series of a bounded operator

[F7]

For positive bases and real exponents, the product, quotient and iterated-power laws hold: $(a b)^{r} = a^{r} b^{r}$ , $(a / b)^{r} = a^{r} / b^{r}$ and $(a^{r})^{s} = a^{r s}$ . The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents

[F8]

For every real $α$ , the function $x \mapsto x^{α}$ is continuous and differentiable on $(0, \infty)$ , with derivative $α x^{α - 1}$ . Continuity and derivatives of positive-base real powers

[F9]

A real function continuous on an interval and differentiable at every interior point is strictly decreasing there if its derivative is strictly negative throughout the interior. On an interval I, for f continuous on I and differentiable at every interior point: f' ≥ 0 throughout gives f nondecreasing, f' > 0 gives f increasing, f' ≤ 0 and f' < 0 give the two decreasing forms; conversely a nondecreasing f has f' ≥ 0 and a nonincreasing f has f' ≤ 0 wherever it is differentiable, and no strict converse is claimed

Proof

Given: The fixed model parameters, code, recovery and target described in the statement.

Goal: Establish the first-crossing asymptotic, its leading parameter dependence, and the zero-parameter cases.

1.1givenF4F6algebra

If $γ > 0$ , the exponentials in [F4] are continuous by [F6]. For $t > 0$ , $p > 0$ and $γ + i Δ \neq 0$ , so the displayed algebraic expression is continuous there. Its value at zero is $1$ and its right limit is $1$ by [F4]. If $γ = 0$ , [F4] gives the constant function $1$ . Thus $F_{t r}$ is continuous on $[0, \infty)$ and $F_{t r} (0) = 1$ .

2.1givenF2F3F5step 1.1algebra

Suppose $γ Δ^{2} > 0$ and put $a = γ Δ^{2} / 24$ and $ϵ = 1 - F_{0}$ . By [F3], $1 - F_{t r} (t) = a t^{3} + O (t^{4})$ with $a > 0$ . Applying [F5] to the continuous function from step 1.1 and the crossing convention in [F2] yields $τ (1 - ϵ) \sim (ϵ / a)^{1 / 3}$ as $ϵ ↓ 0$ , which is the displayed formula.

3.1givenF7F8F9step 2.1algebra

For fixed $ϵ > 0$ , put $d = ∣ Δ ∣ > 0$ . By [F7], the leading approximation from step 2.1 is $(24 ϵ / (γ d^{2}))^{1 / 3} = (24 ϵ)^{1 / 3} γ^{- 1 / 3} d^{- 2 / 3}$ for $γ, d > 0$ . By [F8], $x \mapsto x^{- 1 / 3}$ and $x \mapsto x^{- 2 / 3}$ are continuous and differentiable on $(0, \infty)$ with derivatives $- x^{- 4 / 3} / 3 < 0$ and $- 2 x^{- 5 / 3} / 3 < 0$ , respectively. By [F9] both functions are strictly decreasing on that interval. Holding either parameter fixed leaves a positive constant times the corresponding strictly decreasing function, so the leading approximation strictly decreases as either positive $γ$ or positive $d$ increases. The asymptotic in step 2.1 is pointwise in the parameters, so this calculation makes no global monotonicity assertion about the exact crossing time.

4.1givenF2F4algebra∎

If $γ = 0$ , [F4] gives $F_{t r} = 1$ at every time. If $Δ = 0$ and $γ > 0$ , then for $t > 0$ the numerator of $χ$ in [F4] is $γ (1 - e^{- γ t}) = γ p$ , so $χ = 1$ and the same fidelity equals $1$ ; the value at $t = 0$ is also $1$ . For any $F_{0} \in (0, 1]$ the strict crossing set is therefore empty, and [F2] gives $τ (F_{0}) = + \infty$ .

Depends on

Used by · 1

Gradual which-path leakage and inverse splitting timescale

Dependency tree · 2 levels

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF p. 3, Eq. (8) and surrounding discussion