Wiener KL Divergence in Quantum Control

• The paper defines Wiener KL as a path-space Kullback–Leibler divergence that penalizes measurement-record drifts relative to a zero-drift Wiener process in quantum control.
• It derives the divergence via Girsanov’s theorem, establishing a closed-form expectation of quadratic drift deviations akin to Cameron–Martin energy in Gaussian measures.
• The approach demonstrates practical improvements in quantum fidelity and robustness by steering trajectories toward decoherence-free regions under open quantum system dynamics.

to=arxiv_search.search 天天中彩票中大奖 彩神争霸官网 天天中彩票买json {"query":"KL_W Wiener KL QMaxCal path-space regularization open quantum control arXiv (Moody et al., 18 Jun 2026)", "max_results": 5} to=arxiv_search.search ахәыҷി 北京赛车女json {"query":"Information Projection on Banach spaces with Applications to State Independent KL-Weighted Optimal Control Wiener space arXiv (Selk et al., 2020)", "max_results": 5} to=arxiv_search.search 福利彩票天天彩պեսjson {"query":"A representation for the Kantorovich--Rubinstein distance on the abstract Wiener space arXiv (Riabov, 2016)", "max_results": 5} Wiener KL, usually written $KL_W$, denotes in the cited arXiv literature a Kullback–Leibler divergence defined on path space with a Wiener reference measure. In the exact notation $KL_W$ used by QMaxCal, it is the KL divergence between the measurement-record distribution induced by a controlled stochastic Schrödinger evolution and standard Wiener measure, so it penalizes the observable trajectory-level consequences of control on decoherence channels rather than the control waveform itself (Moody et al., 18 Jun 2026). In a broader Wiener-space context, this usage belongs to the standard relative-entropy framework for Gaussian and path measures; in particular, for Gaussian shift measures on classical Wiener space, the KL divergence reduces to a quadratic Cameron–Martin energy (Selk et al., 2020).

1. Definition in open quantum control

In QMaxCal, an open quantum system is continuously monitored under the diffusive or homodyne unravelling. Each decoherence channel $k$ produces a measurement record

$$dI_k(t)=\alpha_k(t)\,dt+dW_k(t), \qquad \alpha_k(t)=\langle \psi(t)\mid (L_k+L_k^\dagger)\mid \psi(t)\rangle,$$

where the $dW_k$ are independent Wiener increments. The drift $\alpha_k(t)$ is the observable signature of the state’s exposure to channel $k$ (Moody et al., 18 Jun 2026).

With $P_\theta$ denoting the path measure induced by the controlled stochastic Schrödinger equation and $P_W$ denoting standard Wiener measure, the Wiener KL is defined by

$$\mathrm{KL}_W:=\mathrm{KL}(P_\theta\|P_W) = \frac12\sum_{k=1}^K \mathbb{E}_{P_\theta}\!\left[\int_0^T \alpha_k^{(\theta)}(t)^2\,dt\right].$$

Here $KL_W$0 is the zero-drift reference process, $KL_W$1. Accordingly, $KL_W$2 is literally the KL divergence between the control-induced record distribution and pure Brownian noise (Moody et al., 18 Jun 2026).

This definition makes the regularizer state- and trajectory-sensitive. It does not act on $KL_W$3 directly, but on the path-space distribution generated after the controls, Hamiltonian, Lindblad operators, and monitoring scheme have been fixed. The paper’s stated interpretation is that minimizing $KL_W$4 encourages the controlled trajectory to move into regions where decoherence has little or no effect, most importantly into a joint kernel or decoherence-free region when such a region is reachable (Moody et al., 18 Jun 2026).

2. Derivation from Girsanov’s theorem

The central derivation uses Girsanov’s theorem for diffusions with identical noise structure and different drifts. If two path measures $KL_W$5 and $KL_W$6 have drift difference $KL_W$7, then the paper states the scalar change-of-measure formula as

$KL_W$8

and the $KL_W$9-channel quantum measurement-record analogue as

$k$0

Choosing the reference drift to be zero immediately yields the Wiener KL formula above (Moody et al., 18 Jun 2026).

The derivation depends on a specific structural condition: the measurement record must be an Itô diffusion with unit diffusion coefficient independent of control. In the QMaxCal construction, two quantum evolutions share the same Lindblad operators $k$1, so the path measures differ only in the drift of the measurement record. Under exactly that hypothesis, the KL functional becomes a closed-form, differentiable expectation of a quadratic drift functional, and the paper emphasizes that it is straightforward to estimate by Monte Carlo over stochastic Schrödinger trajectories (Moody et al., 18 Jun 2026).

A plausible implication is that $k$2 inherits the computational advantages typical of quadratic-energy path-space penalties while remaining tied to an explicitly observable signal, namely the monitored record itself rather than an auxiliary latent quantity.

3. Relation to Wiener-space KL on classical Wiener space

The QMaxCal construction is not an isolated use of KL on Wiener path space. On classical Wiener space $k$3, the Cameron–Martin theorem gives explicit Radon–Nikodym derivatives for Gaussian shift measures $k$4, and the associated KL divergence is

$k$5

If $k$6 and $k$7 with $k$8, then

$k$9

The same paper formulates an information projection problem over shift measures and shows that, in this Wiener-space setting, KL projection, KL-weighted optimal control, and minimization of an Onsager–Machlup function are equivalent formulations (Selk et al., 2020).

This places $$dI_k(t)=\alpha_k(t)\,dt+dW_k(t), \qquad \alpha_k(t)=\langle \psi(t)\mid (L_k+L_k^\dagger)\mid \psi(t)\rangle,$$0 in a larger measure-theoretic lineage. The object is not a new divergence axiomatically; it is the ordinary KL divergence specialized to a Wiener reference structure. In the QMaxCal case the reference is standard Wiener measure on the monitored record; in the Banach- and Wiener-space control formulation, the reference is the law of standard Brownian motion and admissible measures are Cameron–Martin shifts (Selk et al., 2020).

A nearby but distinct result is the representation of the Kantorovich–Rubinstein distance on abstract Wiener space. There the relevant functional is not KL divergence but the $$dI_k(t)=\alpha_k(t)\,dt+dW_k(t), \qquad \alpha_k(t)=\langle \psi(t)\mid (L_k+L_k^\dagger)\mid \psi(t)\rangle,$$1-Wasserstein distance $$dI_k(t)=\alpha_k(t)\,dt+dW_k(t), \qquad \alpha_k(t)=\langle \psi(t)\mid (L_k+L_k^\dagger)\mid \psi(t)\rangle,$$2, represented via the divergence or extended stochastic integral operator: $$dI_k(t)=\alpha_k(t)\,dt+dW_k(t), \qquad \alpha_k(t)=\langle \psi(t)\mid (L_k+L_k^\dagger)\mid \psi(t)\rangle,$$3 This is conceptually adjacent because it also turns a path-space measure discrepancy into a variational norm minimization, but it is not a Wiener KL (Riabov, 2016).

4. Role inside QMaxCal and comparison with $$dI_k(t)=\alpha_k(t)\,dt+dW_k(t), \qquad \alpha_k(t)=\langle \psi(t)\mid (L_k+L_k^\dagger)\mid \psi(t)\rangle,$$4

QMaxCal incorporates the Wiener KL into a fidelity-plus-regularization objective

$$dI_k(t)=\alpha_k(t)\,dt+dW_k(t), \qquad \alpha_k(t)=\langle \psi(t)\mid (L_k+L_k^\dagger)\mid \psi(t)\rangle,$$5

where

$$dI_k(t)=\alpha_k(t)\,dt+dW_k(t), \qquad \alpha_k(t)=\langle \psi(t)\mid (L_k+L_k^\dagger)\mid \psi(t)\rangle,$$6

The paper notes that experiments often use either $$dI_k(t)=\alpha_k(t)\,dt+dW_k(t), \qquad \alpha_k(t)=\langle \psi(t)\mid (L_k+L_k^\dagger)\mid \psi(t)\rangle,$$7 or $$dI_k(t)=\alpha_k(t)\,dt+dW_k(t), \qquad \alpha_k(t)=\langle \psi(t)\mid (L_k+L_k^\dagger)\mid \psi(t)\rangle,$$8, not both simultaneously, depending on the benchmark (Moody et al., 18 Jun 2026).

The drift-variance regularizer $dI_k(t)=\alpha_k(t)\,dt+dW_k(t), \qquad \alpha_k(t)=\langle \psi(t)\mid (L_k+L_k^\dagger)\mid \psi(t)\rangle,$9 uses a different reference class. Instead of zero-drift Wiener measure, it chooses the closest constant-drift process: $dW_k$0 and obtains

$dW_k$1

with

$dW_k$2

The paper’s distinction is explicit: $dW_k$3 penalizes drift magnitude relative to zero drift, whereas $dW_k$4 penalizes fluctuations around a constant drift regardless of whether that constant is zero (Moody et al., 18 Jun 2026).

This leads to different inductive biases. $dW_k$5 has a stronger bias toward the joint kernel $dW_k$6, where drift vanishes. By contrast, $dW_k$7 vanishes on any decoherence-free subspace where the drift is constant in time and across realizations, even if that constant is nonzero. The paper therefore treats $dW_k$8 as the stronger but more specialized regularizer, and $dW_k$9 as the more generally applicable one (Moody et al., 18 Jun 2026).

5. Benchmark behavior and reported performance

Across the reported single- and multi-qubit benchmarks, as well as a multi-qubit chain calibrated to a published snapshot of the IBM Kingston processor, the regularizers outperform unregularized gradient-based and reinforcement-learning baselines along final-state fidelity, robustness to mismatch in the assumed noise model, and occupation of forbidden states. The abstract reports gains growing from $\alpha_k(t)$0 percentage points at training noise to $\alpha_k(t)$1 percentage points under $\alpha_k(t)$2 noise mismatch, reduction of infidelity by up to $\alpha_k(t)$3, and approximately $\alpha_k(t)$4 gains on the calibrated IBM Kingston chain (Moody et al., 18 Jun 2026).

Benchmark	Reported $\alpha_k(t)$5 outcome	Structural interpretation
Single-qubit amplitude damping	At $\alpha_k(t)$6, baseline fidelity $\alpha_k(t)$7, Wiener KL $\alpha_k(t)$8, $\alpha_k(t)$9 $k$0, PPO $k$1	Ground state is the kernel
STIRAP	At $k$2, fidelity stays essentially unchanged ($k$3 baseline, $k$4 Wiener KL), but peak $k$5 population drops from $k$6 to $k$7	Suppresses occupancy of the lossy intermediate state
Diamond system	At $k$8, baseline $k$9, Wiener KL with $P_\theta$0 gives $P_\theta$1, with $P_\theta$2 gives $P_\theta$3, PPO gives $P_\theta$4	Routes population through a safe kernel state

The amplitude-damping example is the clearest demonstration of the mechanism. The Lindblad operator is $P_\theta$5, so the kernel is the ground state $P_\theta$6. The paper reports that the trajectory-variance collapse is large: the time-integrated population variance falls from $P_\theta$7 to $P_\theta$8, approximately a $P_\theta$9 reduction, and the drift-squared integral $P_W$0 drops from $P_W$1 to $P_W$2 at $P_W$3 (Moody et al., 18 Jun 2026).

The STIRAP benchmark illustrates a different regime. Here Wiener KL does not materially improve the final fidelity, but it reduces occupation of the lossy excited state. The paper gives a $P_W$4 reduction in peak $P_W$5 population and a $P_W$6 reduction in time-integrated $P_W$7 exposure at $P_W$8 (Moody et al., 18 Jun 2026). This suggests that $P_W$9 can improve protocol quality even when the terminal fidelity is already near saturation.

The diamond-system robustness test shows the strongest mismatch effect. Trained at $$\mathrm{KL}_W:=\mathrm{KL}(P_\theta\|P_W) = \frac12\sum_{k=1}^K \mathbb{E}_{P_\theta}\!\left[\int_0^T \alpha_k^{(\theta)}(t)^2\,dt\right].$$0 and evaluated at $$\mathrm{KL}_W:=\mathrm{KL}(P_\theta\|P_W) = \frac12\sum_{k=1}^K \mathbb{E}_{P_\theta}\!\left[\int_0^T \alpha_k^{(\theta)}(t)^2\,dt\right].$$1, the reported fidelities are $$\mathrm{KL}_W:=\mathrm{KL}(P_\theta\|P_W) = \frac12\sum_{k=1}^K \mathbb{E}_{P_\theta}\!\left[\int_0^T \alpha_k^{(\theta)}(t)^2\,dt\right].$$2 for the baseline, $$\mathrm{KL}_W:=\mathrm{KL}(P_\theta\|P_W) = \frac12\sum_{k=1}^K \mathbb{E}_{P_\theta}\!\left[\int_0^T \alpha_k^{(\theta)}(t)^2\,dt\right].$$3 for Wiener KL, and $$\mathrm{KL}_W:=\mathrm{KL}(P_\theta\|P_W) = \frac12\sum_{k=1}^K \mathbb{E}_{P_\theta}\!\left[\int_0^T \alpha_k^{(\theta)}(t)^2\,dt\right].$$4 for PPO, with the paper noting that the gain grows monotonically with test-noise strength and reaches $$\mathrm{KL}_W:=\mathrm{KL}(P_\theta\|P_W) = \frac12\sum_{k=1}^K \mathbb{E}_{P_\theta}\!\left[\int_0^T \alpha_k^{(\theta)}(t)^2\,dt\right].$$5 percentage points over the baseline at the hardest mismatch setting (Moody et al., 18 Jun 2026).

6. Terminological scope and ambiguity of the notation

The notation $$\mathrm{KL}_W:=\mathrm{KL}(P_\theta\|P_W) = \frac12\sum_{k=1}^K \mathbb{E}_{P_\theta}\!\left[\int_0^T \alpha_k^{(\theta)}(t)^2\,dt\right].$$6 is not uniform across arXiv literatures. In the QMaxCal usage, it denotes the Wiener KL regularizer just described (Moody et al., 18 Jun 2026). In work on classical Wiener space, the operative object is still the ordinary KL divergence $$\mathrm{KL}_W:=\mathrm{KL}(P_\theta\|P_W) = \frac12\sum_{k=1}^K \mathbb{E}_{P_\theta}\!\left[\int_0^T \alpha_k^{(\theta)}(t)^2\,dt\right].$$7 on Gaussian shift measures rather than a separately named $$\mathrm{KL}_W:=\mathrm{KL}(P_\theta\|P_W) = \frac12\sum_{k=1}^K \mathbb{E}_{P_\theta}\!\left[\int_0^T \alpha_k^{(\theta)}(t)^2\,dt\right].$$8 invariant (Selk et al., 2020). In abstract Wiener-space optimal transport, the relevant discrepancy is $$\mathrm{KL}_W:=\mathrm{KL}(P_\theta\|P_W) = \frac12\sum_{k=1}^K \mathbb{E}_{P_\theta}\!\left[\int_0^T \alpha_k^{(\theta)}(t)^2\,dt\right].$$9, not KL divergence (Riabov, 2016).

The term “Wiener” is also heavily overloaded outside probability and control. In graph theory it usually denotes the Wiener index

$KL_W$00

or related invariants such as the terminal Wiener index or Wiener complexity $KL_W$01. Those papers explicitly do not define a KL-type quantity $KL_W$02 [(Chen et al., 2013); (Dobrynin et al., 2019); (Dolžan, 2023)]. Separately, the shorthand $KL_W$03 may also refer to the KL divergence between two Weibull distributions,

$KL_W$04

which is unrelated to Wiener measure despite the similar subscript notation (Bauckhage, 2013).

Accordingly, the expression “Wiener KL” is context-dependent. As an exact named object, it refers most directly to the QMaxCal path-space regularizer

$KL_W$05

while its broader mathematical background is the standard KL divergence on Gaussian and Wiener path spaces rather than a separate divergence family (Moody et al., 18 Jun 2026).