Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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Amplitude damping with operational time units

Physical scope

The exact finite-dimensional qubit time-homogeneous GKSL model with one amplitude-damping jump operator. The formulas are conditional model predictions; no vacuum continuum, reservoir limit, detector record, or empirical decay rate is asserted.

Statement

For the separately selected qubit GKSL model, let $H = ℏ ω ∣ 1 ⟩ ⟨ 1 ∣$ with $ω \in R$ in $s^{- 1}$ and $ℏ > 0$ in joule-seconds, and let $L = \sqrt{γ} ∣ 0 ⟩ ⟨ 1 ∣$ with $γ \geq 0$ in $s^{- 1}$ , so $L$ has units $s^{- 1 / 2}$ . For any initial density matrix $ρ (0)$ in the ordered basis $(∣ 0 ⟩, ∣ 1 ⟩)$ , the solution $ρ (t) = e^{t L} ρ (0)$ for $t \geq 0$ satisfies $ρ_{11} (t) = e^{- γ t} ρ_{11} (0)$ , $ρ_{00} (t) = ρ_{00} (0) + (1 - e^{- γ t}) ρ_{11} (0)$ , $ρ_{01} (t) = e^{(i ω - γ / 2) t} ρ_{01} (0)$ , and $ρ_{10} (t) = e^{(- i ω - γ / 2) t} ρ_{10} (0)$ .

Statement references

Finite system-environment and Markov model assignments

Finite GKSL generators and units

Finite GKSL generates CPTP semigroups

Every finite-dimensional normed space is Banach

Finite matrix trace-norm ideal inequalities

The exponential series of a bounded operator

The complex exponential by its power series

$\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential

Facts & Assumptions

Given: The selected finite GKSL model, $ω \in R$ , $γ \geq 0$ , and a qubit density matrix $ρ (0)$ .

[A1]

Finite system-environment and Markov model assignments: the time-homogeneous finite GKSL equation is a separately adopted approximation, not inferred from the finite system-environment reduction; its time parameter $t$ is elapsed physical time measured in seconds, so $t L$ is dimensionless.

[F1]

Finite GKSL generators and units: $L (X) = - \frac{i}{ℏ} [H, X] + L X L^{*} - \frac{1}{2} {L^{*} L, X}$ for the single-jump case.

[F2]

Every finite-dimensional normed space is Banach and The exponential series of a bounded operator: the finite matrix space is complete and the bounded-operator exponential is its norm-convergent power series.

[F3]

The complex exponential by its power series: $\exp z = \sum_{n = 0}^{\infty} z^{n} / n!$ for $z \in C$ .

[F4]

Finite matrix trace-norm ideal inequalities: $∥ A X B ∥_{1} \leq ∥ A ∥ ∥ X ∥_{1} ∥ B ∥$ for finite matrices.

[F5]

Finite GKSL generates CPTP semigroups: $e^{t L}$ is a finite channel for every $t \geq 0$ .

[F6]

Finite completely positive trace-preserving maps: a finite channel is completely positive and trace-preserving on all matrix inputs.

[F7]

$\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential: on real arguments, the complex exponential agrees with the real exponential $e^{x}$ .

Proof

technique · compute powers of the finite GKSL generator on the four matrix units, then sum its defining exponential series

1.1givenA1F1algebra

Write $E_{j k} = ∣ j ⟩ ⟨ k ∣$ , $j, k \in {0, 1}$ , so $I = E_{00} + E_{11}$ . In this basis, $H = diag (0, ℏ ω)$ , $L = \sqrt{γ} E_{01}$ , and $L^{*} L = γ E_{11}$ . Direct multiplication in [F1] gives $L (E_{00}) = 0$ , $L (E_{11}) = γ (E_{00} - E_{11})$ , $L (E_{01}) = (i ω - γ / 2) E_{01}$ , and $L (E_{10}) = (- i ω - γ / 2) E_{10}$ .

2.1F1F2F3F4F7step 1.1algebra

The matrix space is finite-dimensional and complete by [F2]. The trace-ideal estimate [F4] applied to the finitely many left and right multiplications in [F1] bounds $∥ L (X) ∥_{1}$ by a finite constant times $∥ X ∥_{1}$ , so $L$ is bounded; [F2] therefore gives $e^{t L} = \sum_{n \geq 0} t^{n} L^{n} / n!$ . Put $D = E_{00} - E_{11}$ . Step 1.1 gives $L (D) = - γ D$ and $L (E_{11}) = γ D$ ; induction therefore gives $L^{n} (E_{11}) = γ (- γ)^{n - 1} D$ for $n \geq 1$ . Summing this scalar series using [F3] and the real-extension clause [F7] yields $e^{t L} (E_{11}) = E_{11} + (1 - e^{- γ t}) D$ , including $γ = 0$ because both the added coefficient and every positive power term then vanish. Also step 1.1 gives $L^{n} (E_{00}) = 0$ for $n \geq 1$ , while the two coherence matrix units are eigenvectors, so [F3] gives $e^{t L} (E_{01}) = e^{(i ω - γ / 2) t} E_{01}$ and $e^{t L} (E_{10}) = e^{(- i ω - γ / 2) t} E_{10}$ . Applying these four identities linearly to $ρ (0) = \sum_{j, k} ρ_{j k} (0) E_{j k}$ gives all four formulas in the Statement.

3.1A1F5F6step 2.1∎

By [F5] and [F6], the full evolution map is completely positive and trace-preserving for $t \geq 0$ , so it carries each initial density matrix to a density matrix. These formulas are deductions within the selected approximation and assert no microscopic bath limit.

Remarks

The parameter $γ$ is a model rate. Its numerical value and the regime in which this generator approximates a physical system must be supplied by a specific application; no measurement or statistical estimate is asserted here.

Depends on

Dependency tree · 2 levels

Sources

Lin Lin and Nathan Wiebe, Quantum Algorithms for Scientific Computation, Chapter 21, §21.1