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Specified energy-jump instrument realizes a finite chain

Definition

Use the finite energy basis, energies $E_{a}$ , rates $w_{a b}$ , Hamiltonian $H_{s}$ , and jump matrices $L_{a b} = \sqrt{w_{a b}} ∣ b ⟩ ⟨ a ∣$ of Connected Gibbs jump generators satisfy detailed balance. Define $R : = \sum_{a \neq b} L_{a b}^{*} L_{a b} = \sum_{a} r_{a} ∣ a ⟩ ⟨ a ∣$ and the no-click matrix $K : = - i H_{s} / ℏ - R / 2$ . The specified energy-resolved instrument has the following outcomes:

If the current state is the energy eigenstate $∣ a ⟩$ , then during a no-click interval of length $t$ the unnormalized state is $e^{t K} ∣ a ⟩$ .
In an infinitesimal interval $d t$ , the outcome labelled $a \to b$ has jump operation $\sqrt{d t} L_{a b}$ and probability $d t ⟨ a ∣ L_{a b}^{*} L_{a b} ∣ a ⟩ + o (d t)$ . For $w_{a b} > 0$ , its normalized post-click state is $L_{a b} ∣ a ⟩ / ∥ L_{a b} ∣ a ⟩ ∥ = ∣ b ⟩$ . If $w_{a b} = 0$ , this branch is the zero operation and has exactly zero probability; no conditional state is assigned.
The no-click operation over $d t$ is $M_{0} (d t) = I + K d t + o (d t)$ . Since $K^{*} + K = - R$ and $\sum_{a \neq b} L_{a b}^{*} L_{a b} = R$ , the no-click and jump effects sum to $I + o (d t)$ .
After each recorded jump, repeat the same rule from the new energy label.

The output path records the current label $a (t) \in E$ . This is one selected instrument model; the one-time GKSL semigroup alone does not specify a unique record process.

Facts & Assumptions

Given: The Axiom of Choice, the connected finite Gibbs jump data, the jump operators $L_{a b}$ , and the finite-state chain supplier.

[A1]

The Axiom of Choice: every family of nonempty sets has a choice function. The full axiom is retained because the supplied chain construction uses Ionescu–Tulcea.

[F1]

Connected Gibbs jump generators satisfy detailed balance: $E$ is finite, the positive-rate graph is connected, $r_{a} = \sum_{b \neq a} w_{a b} > 0$ , and the energy-resolved jump matrix is $\sqrt{w_{a b}} ∣ b ⟩ ⟨ a ∣$ .

[F2]

Finite GKSL generators and units: the Schrödinger generator for Hamiltonian $H_{s}$ and jumps $L_{a b}$ is $L (X) = - i [H_{s}, X] / ℏ + \sum_{a \neq b} (L_{a b} X L_{a b}^{*} - \frac{1}{2} {L_{a b}^{*} L_{a b}, X})$ .

[F3]

Nonexplosive finite continuous-time chain and reversible spectral gap: under full AC, the finite connected reversible rates construct a nonexplosive continuous-time Markov chain with waiting rate $r_{a}$ , destination probabilities $w_{a b} / r_{a}$ , and transition matrix $e^{t W}$ .

Proof

1.1givenF1constructalgebra

The jump matrices give $L_{a b}^{*} L_{a b} = w_{a b} ∣ a ⟩ ⟨ a ∣$ , so $R = \sum_{a} r_{a} ∣ a ⟩ ⟨ a ∣$ and $K^{*} + K = - R$ . Hence $M_{0} (d t)^{*} M_{0} (d t) = I - R d t + o (d t)$ , while the sum of jump effects is $R d t$ ; the total effect is $I + o (d t)$ as required for a normalized infinitesimal instrument. Also $K ∣ a ⟩ = (- i E_{a} / ℏ - r_{a} / 2) ∣ a ⟩$ , so $∥ e^{t K} ∣ a ⟩ ∥^{2} = e^{- r_{a} t}$ . The no-click survival probability is exactly the exponential holding-time law of rate $r_{a}$ .

1.2givenF1algebra

For a short interval $d t$ , the instrument outcome $a \to b$ has probability $d t ⟨ a ∣ L_{a b}^{*} L_{a b} ∣ a ⟩ + o (d t) = w_{a b} d t + o (d t)$ . Summing over $b \neq a$ gives total jump rate $r_{a}$ , and conditioning on a jump gives destination probability $w_{a b} / r_{a}$ . For $w_{a b} > 0$ the post-jump normalized state is $∣ b ⟩$ ; for $w_{a b} = 0$ the jump operation is zero, so that outcome never occurs and is not normalized.

2.1F1F2step 1.1algebra

For an arbitrary matrix $X$ , the no-click branch contributes $M_{0} (d t) X M_{0} (d t)^{*} = X + d t (K X + X K^{*}) + o (d t)$ , and the jump branches contribute $d t \sum_{a \neq b} L_{a b} X L_{a b}^{*} + o (d t)$ . Since $K = - i H_{s} / ℏ - R / 2$ and $R = \sum_{a \neq b} L_{a b}^{*} L_{a b}$ , their sum is $X + d t L (X) + o (d t)$ with exactly the generator in [F2]. Thus the selected infinitesimal outcomes have the specified GKSL generator.

3.1A1F3step 1.1step 1.2∎

The holding-time and destination laws in steps 1.1–1.2 are exactly those of the chain in [F3]. Full AC [A1] and [F3] supply its joint path law and nonexplosion; hence the specified instrument's recorded path is the finite continuous-time chain with generator $W$ , not a law inferred from noncommuting one-time correlations.

Remarks

The record path is a mathematical model of a chosen energy-resolved monitoring scheme. Other unravelings of the same GKSL evolution can produce different records; no unique trajectory follows from the one-time channel family.

Specified energy-jump instrument realizes a finite chain

Definition

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources