Physics theoremStatement: AI-adaptedProof: AI-generated

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Minimal-code closed fidelity and checking time

PDF p. 17: Fig. S5.2 caption

Statement

Adopt the finite equal-rate, zero-temperature downward-jump benchmark with $γ > 0$ , at zero lift, and all physical, spectral and choice hypotheses of Minimal-code decay detection without correction. The minimal code is the two-dimensional span of $∣ 1, - 1 ⟩$ and $∣ 1, 1 ⟩$ . For its channel-specific transpose recovery, let $F_{t r} (t)$ be the entanglement fidelity at the maximally mixed logical input. Then, for $t \geq 0$ , $1 - F_{t r} (t) = 3 p_{d e c a y} (t) / 4$ , where $p_{d e c a y} (t) = 1 - e^{- γ t}$ .

For a target $F_{0} \in (0, 1]$ , define the checking time as $τ (F_{0}) = \inf {t \geq 0 : F_{t r} (t) < F_{0}}$ , with $\inf \emptyset = + \infty$ . If $1 / 4 < F_{0} \leq 1$ , then $τ (F_{0}) = - γ^{- 1} \log [1 - \frac{4}{3} (1 - F_{0})]$ . If $0 < F_{0} \leq 1 / 4$ , then $τ (F_{0}) = + \infty$ . As $F_{0} ↑ 1$ through values below $1$ , $τ (F_{0}) \sim 4 (1 - F_{0}) / (3 γ)$ .

Facts & Assumptions

[F1]

Under the stated finite benchmark and the full hypotheses of the minimal-code result, including its fixed effective jumps and inherited choice assumptions, the recovered logical channel has entanglement fidelity $F_{t r} (t) = (1 + 3 e^{- γ t}) / 4$ . Minimal-code decay detection without correction Equal-rate rigid-rotor benchmark parameters

[F2]

The minimal code uses the orthonormal logical basis $∣ 1, - 1 ⟩, ∣ 1, 1 ⟩$ ; entanglement fidelity is the maximally entangled-state overlap for the recovered logical channel. Minimal single-manifold encoding Entanglement fidelity on a finite logical space

[F3]

The checking time is the infimum of the strict below-target set, with the empty-set convention in the Statement. Operational checking time

[F4]

The exponential is strictly increasing, positive and tends to zero at minus infinity. The natural logarithm is its inverse on the positive reals. The exponential function is strictly increasing The exponential tends to +∞ at +∞ and to 0 at -∞ The natural logarithm as the inverse of the exponential function

[F5]

The logarithm is differentiable at every positive real $x$ , with derivative $1 / x$ . The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t

[A1]

Assume the Axiom of Choice, Countable Choice and Dependent Choice carried by the minimal-code supplier. 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 zero-lift benchmark and the hypotheses and target in the Statement.

Goal: Establish the fidelity, all stated checking-time cases and the high-fidelity asymptotic.

1.1givenF1F2algebra

The minimal-code result applies at zero lift. Subtracting its fidelity from one yields $1 - F_{t r} (t) = 3 (1 - e^{- γ t}) / 4 = 3 p_{d e c a y} (t) / 4$ . The fidelity and code are those specified in the Statement.

2.1givenF3F4step 1.1algebra

Put $q = (4 F_{0} - 1) / 3$ . If $1 / 4 < F_{0} < 1$ , then $0 < q < 1$ , and the strict crossing condition is $e^{- γ t} < q$ . By the inverse and strict monotonicity in [F4], this is equivalent to $t > - \log (q) / γ$ . Thus the crossing set has that boundary as its infimum. If $F_{0} = 1$ , the crossing set is $(0, \infty)$ and its infimum is zero, which is also the displayed logarithmic expression. If $0 < F_{0} \leq 1 / 4$ , positivity of the exponential makes $F_{t r} (t) > 1 / 4$ at every finite $t$ ; the crossing set is empty and its infimum is $+ \infty$ .

3.1F4F5step 2.1algebra∎

Set $ϵ = 1 - F_{0}$ . As $ϵ ↓ 0$ , the logarithm's derivative at $1$ is $1$ and $\log 1 = 0$ . Consequently $\log (1 - 4 ϵ / 3) / (- 4 ϵ / 3) \to 1$ . Substitution into step 2.1 proves $τ (1 - ϵ) \sim 4 ϵ / (3 γ)$ .

Depends on

Dependency tree · 2 levels

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF p. 2, paragraph following Eq. (4), for the decay probability; PDF p. 17, Fig. S5.2 caption, for the minimal-code formulas