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Finite no-jump and ordered-jump expansion

Statement

Take the finite-dimensional GKSL data $d, H, L_{1}, \dots, L_{m}$ and channel $N_{t} = e^{t L}$ from the finite GKSL interface, with $t \geq 0$ . The matrix spectral norm means $∥ X ∥ = \sup_{∥ v ∥_{2} \leq 1} ∥ X v ∥_{2}$ on $C^{d}$ , where $∥ v ∥_{2} = (\sum_{i} ∣ v_{i} ∣^{2})^{1 / 2}$ ; map norms are induced by this matrix norm. Put $G = - i H - \frac{1}{2} \sum_{a} L_{a}^{†} L_{a}$ and $E (u) = e^{u G}$ . For $n \geq 1$ , indices $a_{1}, \dots, a_{n} \in {1, \dots, m}$ and ordered times $0 \leq s_{1} \leq \dots \leq s_{n} \leq t$ , define $K_{a_{1} \dots a_{n}} (s_{1}, \dots, s_{n}) = E (t - s_{n}) L_{a_{n}} E (s_{n} - s_{n - 1}) \dots L_{a_{1}} E (s_{1})$ ; for $n = 0$ put $K_{\emptyset} = E (t)$ . All matrix-valued integrals are taken entrywise. For $n \geq 2$ , the ordered-integral notation means the nested one-dimensional Riemann integral

$\int_{0 \leq s_{1} \leq \dots \leq s_{n} \leq t} f (s) d s_{1} \dots d s_{n} : = \int_{0}^{t} \int_{0}^{s_{n}} \dots \int_{0}^{s_{2}} f (s_{1}, \dots, s_{n}) d s_{1} \dots d s_{n} .$

For $n = 1$ it means $\int_{0}^{t} f (s_{1}) d s_{1}$ . Then

$N_{t} (X) = E (t) X E (t)^{†} + \sum_{n = 1}^{\infty} \sum_{a_{1}, \dots, a_{n}} \int_{0 \leq s_{1} \leq \dots \leq s_{n} \leq t} K_{a_{1} \dots a_{n}} (s) X K_{a_{1} \dots a_{n}} (s)^{†} d s_{1} \dots d s_{n} .$

The series converges absolutely in the operator norm induced by the matrix spectral norm. Thus $E (t)$ is the no-jump amplitude and $E (t - s) L_{a} E (s)$ is the single-jump amplitude at time $s$ ; these names describe terms of this mathematical expansion. If $g = ∥ G ∥$ and $j = \sum_{a} ∥ L_{a} ∥^{2}$ , the norm of the sum of all terms with at least $N$ jumps is at most $e^{2 g t} \sum_{n = N}^{\infty} (j t)^{n} / n!$ for $N \geq 1$ .

As $t ↓ 0$ , $E (t) = I + t G + O (t^{2})$ in matrix norm and $N_{t} (X) = E (t) X E (t)^{†} + t \sum_{a} L_{a} X L_{a}^{†} + O (t^{2}) ∥ X ∥$ uniformly in $X$ . Consequently the finite amplitudes $E (t)$ and $\sqrt{t} L_{a}$ represent the channel through first order, with an $O (t^{2})$ induced-norm error; they are not asserted to give the exact channel at finite time.

Facts & Assumptions

[F1]

The finite GKSL interface defines $L (X) = - i [H, X] + \sum_{a} (L_{a} X L_{a}^{†} - {L_{a}^{†} L_{a}, X} / 2)$ and $N_{t} = e^{t L}$ for finite matrices, Hermitian $H$ , a finite family of $L_{a}$ , and $t \geq 0$ . Finite GKSL evolution and the QEC channel interface

[F2]

On a complex Banach space, a strongly continuous bounded operator family with a continuous norm majorant has a unique solution given by its uniformly convergent iterated Volterra series. Banach-space Volterra and Dyson series with a norm remainder

[F3]

Every finite-dimensional complex normed space is complete. Every finite-dimensional normed space is Banach

[F4]

A bounded operator exponential is norm differentiable, obeys the group law and the bound $∥ e^{u A} ∥ \leq e^{∣ u ∣ ∥ A ∥}$ , and has quadratic remainder at zero. The exponential series of a bounded operator

[F5]

On $C^{d}$ with its Euclidean norm, the matrix operator norm in the statement makes $M_{d} (C)$ a complete normed matrix algebra with $∥ A B ∥ \leq ∥ A ∥ ∥ B ∥$ , $∥ A^{†} ∥ = ∥ A ∥$ , and $∥ I ∥ = 1$ . Finite matrix algebras with the operator norm are C star algebras

[F6]

A continuous curve in a complete normed real space has a norm-Riemann integral, linear in the curve and commuting with bounded real-linear maps; its indefinite integral has derivative equal to the curve. The integral is bounded by interval length times the supremum norm. Complex matrix spaces can be used as real spaces here. Riemann integration of continuous curves in a complete normed space, with the fundamental theorem of calculus and coefficient functionals

Proof

Given: The finite matrices and nonnegative time in the statement. Matrix norms below are spectral norms, and map norms are their induced operator norms.

Goal: Establish the ordered expansion, its tail bound, and the stated short-time approximation.

1.1givenF1F2F3F4F5algebra

Define $A (X) = G X + X G^{†}$ and $J (X) = \sum_{a} L_{a} X L_{a}^{†}$ . Direct expansion of [F1] gives $L = A + J$ . By [F5], $∥ A ∥ \leq 2 g$ and $∥ J ∥ \leq j$ , so $L$ is bounded too. The matrix space is Banach by [F5], while $C^{d}$ and the finite-dimensional space of matrix-space linear maps with the induced norm are Banach by [F3]. The entry functional obeys $∣ X_{i j} ∣ \leq ∥ X ∥$ : the $j$ th column has Euclidean norm at most $∥ X ∥$ and each entry has modulus at most that column norm. Conversely $∥ X ∥ \leq \sum_{i, j} ∣ X_{i j} ∣$ , by the triangle inequality applied to $X = \sum_{i, j} X_{i j} E_{i j}$ and $∥ E_{i j} ∥ = 1$ . These inequalities show that the exponential series in the finite GKSL interface has the same entrywise sum as the spectral-norm exponential here. By [F4], $e^{u A} (X) = E (u) X E (u)^{†}$ : for $u \geq 0$ both sides solve the same linear differential equation with initial value $X$ , by differentiating the right side and using uniqueness in [F2]. For $u < 0$ apply the same argument to $- A$ and $- G$ . In particular $∥ e^{u A} ∥ \leq e^{2 ∣ u ∣ g}$ by [F4] and [F5].

2.1givenF2F4F6step 1.1algebra

For fixed $X$ set $Y (t) = e^{- t A} e^{t L} X$ . Differentiation using [F4] yields $Y^{'} (t) = B (t) Y (t)$ , $Y (0) = X$ , where $B (t) = e^{- t A} J e^{t A}$ . On every finite interval $[0, T]$ , $B$ is norm-continuous and $∥ B (t) ∥ \leq j e^{4 T g}$ , so [F2] applies. To obtain the series in map norm as well, apply [F2] on the Banach space of matrix-space linear maps to the coefficient $Φ \mapsto B (s) Φ$ , with initial map the identity; its norm is bounded by $j e^{4 T g}$ . The recursion in [F2] nests the earliest time innermost, exactly as in the ordered-integral convention of the statement. By [F6], a fixed bounded linear map commutes with each integral. Thus its iterated integrals, multiplied on the left by $e^{t A}$ , have integrand $e^{(t - s_{n}) A} J e^{(s_{n} - s_{n - 1}) A} \dots J e^{s_{1} A} X$ on $0 \leq s_{1} \leq \dots \leq s_{n} \leq t$ ; the exponential group law cancels adjacent factors. Expanding the finite sums in $J$ and using step 1.1 gives exactly the displayed amplitudes and integrals.

3.1F2F6step 1.1step 2.1algebra

The integrals in step 2.1 exist in norm by the Volterra recursion [F2]; moving the fixed linear factors through them and expanding the finite sums uses [F6]. For each matrix entry, the real and imaginary parts of $X \mapsto X_{i j}$ are bounded real-linear functionals by step 1.1, so [F6] identifies these norm integrals with entrywise integrals. The same conclusion holds for the map-valued integrands: apply the bounded evaluation maps $Φ \mapsto Φ (X)$ and then matrix-entry functionals. The inequality $∥ \int f ∥ \leq \int ∥ f ∥$ follows by the triangle inequality on each Riemann sum and passage to the limits; $∥ f ∥$ is continuous. Finally, the nested integral of the constant $1$ is $V_{n} (t) = t^{n} / n!$ : start with $V_{0} (t) = 1$ and use $V_{n} (t) = \int_{0}^{t} V_{n - 1} (u) d u$ and the polynomial antiderivative, by the fundamental theorem in [F6]. This is the ordered integration weight used below.

4.1givenF5step 1.1step 2.1step 3.1algebra

For ordered times the nonnegative propagation intervals sum to $t$ . Therefore each $n$ -jump integrand has induced norm at most $e^{2 g t} \prod_{k = 1}^{n} ∥ L_{a_{k}} ∥^{2}$ . Summing indices gives $e^{2 g t} j^{n}$ . Applying the integral norm bound successively and using $V_{n} (t) = t^{n} / n!$ from step 3.1 gives the factorial factor. Thus the $n$ -jump term has norm at most $e^{2 g t} (j t)^{n} / n!$ . The exponential majorant proves absolute induced-norm convergence and the stated tail bound.

5.1givenF4step 1.1step 2.1step 3.1step 4.1algebra∎

By [F4], $∥ E (t) - I - t G ∥ \leq t^{2} g^{2} e^{t g} / 2$ . For the one-jump term, write its integrand as $e^{(t - s) A} J e^{s A}$ ; subtracting $J$ and using $∥ e^{u A} - I ∥ \leq e^{2 u g} - 1$ for $u \geq 0$ (from the series in [F4] and $∥ A ∥ \leq 2 g$ ) bounds the difference by $j (e^{2 t g} - 1)$ for each $s \in [0, t]$ . Hence its integral differs from $t J$ by at most $t j (e^{2 t g} - 1)$ . Step 4.1 bounds all terms with at least two jumps by $e^{2 g t} (j t)^{2} e^{j t} / 2$ . Both bounds are $O (t^{2})$ as $t ↓ 0$ , and $t J (X) = \sum_{a} (\sqrt{t} L_{a}) X (\sqrt{t} L_{a})^{†}$ . This proves the approximation and its uniform induced-norm error.

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