A generic map from non-Lindblad to Lindblad master equations

Abstract: Many current problems of interest in quantum non-equilibrium are described by time-local master equations (TLMEs) for the density matrix that are not of the Lindblad form, that is, that are not strictly probability conserving and/or Markovian. Here we describe an generic approach by which the system of interest that obeys the TLME is coupled to an ancilla, such that the dynamics of the combined system-plus-ancilla is Markovian and thus described by a Lindblad equation. This in turn allows us to recover the properties of the original TLME dynamics from a physical unravelling of this associated Lindblad dynamics. We discuss applications of this generic mapping in two areas of current interest. The first one is that of "thermodynamics of trajectories", where non-Lindblad master equations encode the large-deviation properties of the dynamics, and we show that the relevant large-deviation functions (i.e. dynamical free-energies) can be recovered from appropriate observables of the ancilla. The second one is that of quantum filters, where we show tracking a quantum system undergoing a continuous homodyne measurement with another quantum system of the same size will inherently be inefficient in our framework.

• The paper introduces a generic mapping that transforms non-Lindblad time-local master equations into Lindblad master equations using an ancillary system.
• The methodology bridges non-Markovian dynamics with a Markovian framework, enabling improved quantum control, feedback, and thermodynamic analysis.
• The approach supports potential experimental implementation of TLME dynamics, while highlighting challenges in efficient sampling and state estimation.

A Generic Map from Non-Lindblad to Lindblad Master Equations

Introduction

The framework governing the evolution of quantum systems interacting with their environments is often described by quantum master equations (QMEs). Most notably, Lindblad master equations provide a robust theoretical foundation for describing Markovian quantum dynamics. However, many real-world quantum systems, particularly in non-equilibrium scenarios, are governed by time-local master equations (TLMEs) that do not adhere to the Lindblad form. This divergence presents challenges as such equations may not conserve probability and can lack Markovian characteristics.

This paper presents a novel and generic mapping approach to transform systems governed by non-Lindblad TLMEs into systems with dynamics described by Lindblad master equations by coupling with an ancillary system. This strategic coupling allows for the recovery of TLME dynamics through a Markovian framework, facilitating a deeper understanding and potential experimental implementation of non-Lindblad dynamics.

Theory and Methodology

Lindblad Formulation

A system governed by a Lindblad master equation has its density matrix $\rho(t)$ evolve according to:

$$\dot{\rho} = -i[H, \rho] + \sum_{i=1}^{N} J_i \rho J_i^\dagger - \frac{1}{2} \{J_i^\dagger J_i, \rho\},$$

where $H$ is the Hamiltonian representing coherent dynamics, and $J_i$ are bounded jump operators encoding incoherent interactions. Such equations guarantee positivity, probability conservation, and the Markov property of the density matrix.

Mapping to Lindblad Equations

The non-Lindblad TLMEs can be structured as:

$$\dot\rho_{\text{sys}} = \mathcal{L}_{\text{sys}}(\rho_{\text{sys}}) + B\rho_{\text{sys}} + \rho_{\text{sys}}C + \sum_{j=1}^{M} D_j \rho_{\text{sys}} E_j^\dagger,$$

where $\mathcal{L}_{\text{sys}}$ is Lindbladian, and $B$, $C$, $D_j$, $E_j$ are arbitrary operators. The paper derives a framework where such a TLME can be mapped onto a Lindblad master equation for a system combined with an ancilla. The density matrix $\rho_{\text{sys}}$ of the original system is recovered from the composite system-plus-ancilla using a quantum weighting technique:

$\rho_{\text{sys}} = \text{Tr}_{a}[w_t \rho],$

with $w_t$ being an operator on the ancillary space determined through the mapping.

Applications and Implications

Thermodynamics of Trajectories

One compelling application of this mapping is in the domain of the thermodynamics of trajectories. This involves exploring the large-deviation properties of time-integrated observables, which are crucial in identifying dynamical phases of quantum systems. The introduction of a counting field, $s$, biases the quantum trajectories, and the dynamical free energy, $\theta(s)$, can be extracted from the evolution of the coupled system-plus-ancilla using Lindblad dynamics (Figure 1).

Figure 1: Mapping a TLME system to Markovian evolution using a system-ancilla pairing, demonstrating recovery of non-Lindblad dynamics.

Quantum Feedback and Control

The framework also extends to quantum feedback mechanisms like quantum filters. Typically, quantum filters track a system undergoing measurement by an associated estimation process. By realizing such dynamics through the Markovian master equations, insights into optimal quantum state estimation can be gained, though an inherent inefficiency in sampling persists when translating TLME dynamics to a physically realized quantum setting.

Conclusion

This paper offers a pathway for integrating TLMEs into the established Lindblad formalism by introducing ancilla-assisted mappings. The approach facilitates a deeper understanding and potential experimental realization of non-Markovian dynamics in quantum systems. While efficient sampling in experimental contexts remains challenging, the framework paves the way for further explorations in quantum control, feedback, and the thermodynamics of quantum systems, potentially leveraging quantum hardware for simulations that are computationally prohibitive with classical systems. This research invites ongoing development and refinement of methodologies to effectively simulate and understand complex quantum dynamics across diverse physical systems.