Quantum Doob Transform

• Quantum Doob transform is a framework that extends the classical Doob method to construct trace-preserving, completely positive dynamics for optimizing transport in quantum networks.
• It utilizes a tilted generator and principal eigenpair analysis to modify coherent and dissipative mechanisms, thereby enhancing observables like excitonic current.
• The approach enables practical engineering of quantum systems under constraints such as fixed dissipators, offering insights for improved quantum transport efficiency.

The quantum Doob transform is a framework for constructing trace-preserving, completely positive dynamics that optimize selected transport observables in Markovian quantum systems, particularly within the single-excitation manifold of quantum networks. By extending the classical Doob transform to quantum stochastic dynamics governed by the GKSL (Gorini-Kossakowski-Sudarshan-Lindblad) equation, it enables the systematic enhancement of transport efficiency, such as excitonic current or activity, by appropriately modifying both coherent (Hamiltonian) and incoherent (dissipative) mechanisms (Esteve et al., 6 Aug 2025).

1. Formal Construction

The starting point is an open quantum system with density matrix $\rho(t)$ evolving under a GKSL generator,

$$\dot\rho(t) = \mathcal{L}[\rho(t)] = -i[H, \rho(t)] + \sum_{jk}\Bigl(L_{jk} \rho(t) L_{jk}^\dagger - \tfrac{1}{2}\{L_{jk}^\dagger L_{jk}, \rho(t)\}\Bigr),$$

where $H$ is the Hamiltonian and $L_{jk}$ are Lindblad jump operators for transitions between network sites.

A time-integrated additive observable, such as the total number of incoherent transfers $|N\rangle \to |1\rangle$, is defined by

$$\mathcal{O} = \sum_{j\to k} O_{jk}, \qquad O_{jk} \in \{0, 1\}.$$

To bias the statistics of $\mathcal{O}$, a counting field $s$ is introduced, yielding the tilted generator,

$$\mathcal{L}_s[\cdot] = -i[H, \cdot] + \sum_{jk}\Bigl(e^{sO_{jk}} L_{jk} \cdot L_{jk}^\dagger - \tfrac{1}{2}\{L_{jk}^\dagger L_{jk}, \cdot\}\Bigr).$$

The leading eigenpair $(\theta(s), r_s)$ of $\mathcal{L}_s$ yields the scaled cumulant generating function (SCGF) $\theta(s)$ and the associated right eigenmatrix $r_s$; the left eigenmatrix $\ell_s$ solves $\mathcal{L}_s^\dagger[\ell_s] = \theta(s)\ell_s$.

The quantum Doob transform is defined as

$$\mathcal{L}^D_s[X] = \ell_s^{1/2} \, \mathcal{L}_s[\ell_s^{-1/2} X \ell_s^{-1/2}] \, \ell_s^{1/2} - \theta(s) X,$$

ensuring complete positivity and trace preservation. The resulting generator can be rewritten as GKSL with explicitly constructed Doob Hamiltonian $H^D_s$ and jump operators $L^D_{jk,s}$: $$\begin{aligned} H^D_s &= \tfrac{1}{2}\ell_s^{1/2}\Bigl(H - \tfrac{i}{2}\sum_{jk}L_{jk}^\dagger L_{jk}\Bigr)\ell_s^{-1/2} + \mathrm{H.c.}, \ L^D_{jk,s} &= e^{sO_{jk}/2} \ell_s^{1/2} L_{jk} \ell_s^{-1/2}. \end{aligned}$$

2. Numerical Procedure and Computational Cost

Optimizing transport via the quantum Doob transform requires a single diagonalization of the tilted generator $\mathcal{L}_s$ (or its adjoint) for a fixed $s$. The relevant task is to compute the principal eigenvalue $\theta(s)$ and the left eigenmatrix $\ell_s$. From these, $H^D_s$ and $L^D_{jk,s}$ are assembled analytically.

The generator $\mathcal{L}_s$ acts on density operators in a Hilbert space of dimension $N$, so it is represented as an $N^2 \times N^2$ matrix. The worst-case computational cost of diagonalization is $O(N^6)$. In practice, sparse iterative Krylov methods are applied due to the sparsity of $\mathcal{L}_s$ (Esteve et al., 6 Aug 2025).

3. Transport Observables and Large Deviation Theory

Within this framework, the probability of observing an atypical event rate $\mathcal{O}/t$ under the original dynamics ($s=0$) decays exponentially with time: $P_t(\mathcal{O}) \sim \exp[-tI(\mathcal{O}/t)],$ where $I$ is the rate function, and $\theta(s)$ is its Legendre dual SCGF. The first and second derivatives yield the average current and activity: $J(s) = \theta'(s), \qquad A(s) = \theta''(s).$ Under Doob-transformed dynamics, the rare fluctuations of the original system become typical steady-state values: $$\langle\mathcal{O}\rangle_{\rm Doob}/t = J(s),\qquad \mathrm{Var}[\mathcal{O}]_{\rm Doob}/t = A(s).$$ This enables direct realization of steady states that optimize target transport observables.

4. Physical Constraints and Structure Preservation

It is possible to incorporate restrictions so particular physical features of the original system are preserved during optimization. Two notable scenarios are:

• Fixed dissipator: The optimization uses the transformed Hamiltonian $H^D_s$ but keeps the original $L_{jk}$, discarding $L^D_{jk,s}$.
• Fixed input–output coupling: After constructing $H^D_s$, the $(1,N)$ matrix element is reset to its original value, maintaining prescribed input–output interaction structure.

These constraints are implemented post hoc and the impact on stationary currents is evaluated numerically. Empirical results indicate that substantial transport enhancements are typically retained even under such constraints for a large fraction of random network instances (Esteve et al., 6 Aug 2025).

5. Centrosymmetry and Transport Optimization

Centrosymmetry in an $N$-dimensional Hermitian Hamiltonian is quantified by

$$\varepsilon(H) = \frac{1}{N} \min_{S\in \mathrm{Perm}(2,\dots,N-1)} \|\; H - A^{-1}HA \;\|_{\mathrm{HS}},$$

with $A_{ij} = \delta_{i,N-j+1}$ and $\|\cdot\|_{\mathrm{HS}}$ the Hilbert–Schmidt norm, permitting permutations across the intermediate sites. Enhanced centrosymmetry, as induced by the quantum Doob transform, is positively correlated with increased transport current $J(s)$. Notably, systems with initially low centrosymmetry benefit most under optimization. This aligns with known results relating centrosymmetry to robust quantum transport efficiency (Esteve et al., 6 Aug 2025).

6. Scope, Implementation, and Limitations

The quantum Doob transform applies to weak-coupling, Markovian, single-excitation GKSL dynamics. The methodology as introduced focuses on single-jump observables, but the formalism generalizes to any linear combination of jump counts.

Numerical studies utilize ensembles of $10^4$ fully connected $7\times7$ random Hermitians (often using the Fenna–Matthews–Olson motif with minimized $(1,N)$ coupling). The practical range of tilts $s \lesssim 3.5$ is chosen to prevent numerical overflow issues, as matrix elements can become exponentially large or small for greater $|s|$. The main computational overhead is in eigenproblem solution for the $N^2\times N^2$ sparse matrix, efficiently addressed by iterative techniques (Esteve et al., 6 Aug 2025).

A plausible implication is that the quantum Doob transform provides a computationally efficient and controllable means for engineering quantum dynamics that realize rare, optimal transport fluctuations as stable, steady-state operation, with potential relevance for quantum technology and open quantum systems optimization.