Memory Boosts System Performance Beyond Standard Limits

Scientists have long sought to understand the subtle differences between Markovian and non-Markovian dynamics in open quantum systems, crucial for advancements in both technology and state preparation. Jihong Cai and Advith Govindarajan, from the Department of Mathematics at the University of Illinois Urbana-Champaign, alongside Marius Junge and colleagues, demonstrate a surprising connection between these seemingly disparate evolutionary pathways. Their research reveals that, under certain conditions, every trajectory generated by a non-Markovian evolution can also be replicated using a family of time-dependent Lindbladian operators, effectively suggesting that non-Markovianity becomes undetectable when considering individual system trajectories. This finding challenges conventional understanding and highlights the hidden nature of non-Markovian processes, potentially reshaping how we characterise and utilise quantum memory effects.

Zero memory retention, the defining limit of standard quantum systems, is overcome by harnessing non-Markovian evolution, allowing for enhanced performance. This discovery suggests that seemingly complex behaviours can, in fact, be replicated by simpler, conventional models. Scientists have long sought to understand how quantum systems interact with their surroundings. A field known as open quantum frameworks theory, vital for developing quantum technologies.

Dynamics governed by time-dependent Lindbladians, mathematical descriptions of how quantum states evolve, are typically Markovian, leading to decoherence and a loss of quantum information. However, Non-Markovian evolutions, retaining a ‘memory’ of past states, can outperform Markovian ones. Recent work directly compares the paths taken by both systems starting from the same initial condition. Revealing that any trajectory observed in a Non-Markovian system can also be generated by a carefully constructed family of time-dependent Lindbladians.

This suggests that Non-Markovianity, as a property, may be undetectable when considering only individual trajectories. Foundational work by researchers such as Gorini, Kossakowski, Sudarshan. Lindblad has identified the general form of the generator for Markovian dynamics, represented by the GKSL master equation. Many experimental quantum devices, including nitrogen-vacancy centres, photonic systems. Superconducting processors, exhibit memory effects, motivating the search for ways to characterise dynamics beyond the Markovian regime.

Multiple definitions of quantum Non-Markovianity exist, yet no single definition has gained universal acceptance. These approaches generally focus on the reduced dynamical map or the multi-time quantum process itself. But experimental observation is limited to trajectories, the sequence of density matrices describing the system’s state over time. From these trajectories, ensemble descriptions can be inferred, prompting the question of whether Non-Markovianity can be discerned from trajectory data alone.

New findings demonstrate that, given mild assumptions regarding trajectory smoothness, the answer is no — the project proves that Non-Markovianity is not an inherent property of individual quantum trajectories, nor even of limited sets of them. Here, this is illustrated using the pure dephasing evolution, where the principle extends to more general cases. These results do not invalidate existing definitions of Non-Markovianity. But rather highlight a fundamental limitation in identifying Markovianity solely through trajectory-preserving descriptions, requiring continuous changes in density matrix eigenspaces and excluding pathological spectral behaviour.

Recreating quantum paths with Markovian operators for Non-Markovian systems

A central technique in this effort involves reconstructing quantum trajectories from dynamical maps, focusing on the paths density matrices trace out over time. Once a trajectory is defined as a continuous map describing the evolution of density matrices, to generate equivalent trajectories using time-dependent Lindbladians. Such Lindbladians, representing Markovian dynamics, are constructed to mimic the behaviour observed in potentially Non-Markovian systems. Challenging the ability to distinguish between the two regimes based solely on observed trajectories.

The core methodological approach centres on establishing a connection between Non-Markovian and Markovian descriptions at the level of individual trajectories. The team imposed two regularity conditions on the density operator paths: continuous dependence of eigenspaces on time and the absence of accumulation points in rank changes, ensuring well-behaved spectral properties.

By constructing a ‘Lindbladian lift’, a time-dependent generator satisfying the equation of motion for the observed trajectory, forms a key step in the process. At the heart of this procedure lies the equation ρt = Ltρt, where Lt is the constructed Lindbladian. Meanwhile, the project shifts the focus to the more accessible area of trajectory reconstruction. Rather than relying on complex definitions of Non-Markovianity at the level of dynamical maps.

For instance, The team applied this approach to the pure dephasing evolution, demonstrating how a Markovian trajectory could effectively simulate a Non-Markovian one — beyond single trajectories, The effort also considered finite ensembles of paths. Since even collections of trajectories may remain indistinguishable without additional criteria, The project highlights a fundamental limitation in identifying Non-Markovianity from trajectory-based observations. By framing the problem in terms of trajectory reconstruction. At the same time, the project provides a new perspective on the long-standing debate surrounding the definition and detection of quantum Non-Markovianity.

Reconstruction of quantum trajectories via continuous Lindbladian lifting

Under mild assumptions, any trajectory of quantum states can be reproduced by a corresponding Markovian evolution. That non-Markovianity is not detectable from observing a single path. Work presented details how trajectories generated by non-Markovian dynamics are always replicable using suitably chosen Markovian evolutions. Here, this finding challenges the conventional understanding of identifying Markovianity through trajectory analysis.

Specifically, for any smooth path of density matrices, a continuous Lindbladian lift exists, effectively providing a memoryless explanation for the observed trajectory. To ensure well-behaved spectral properties, two regularity conditions were imposed: continuous dependence of eigenspaces on time and the absence of accumulation points in the times where the density matrix changes rank.

Such conditions, satisfied by many physical evolutions, allow for the construction of a time-local Lindbladian generator that accurately reproduces the observed path. Consider the example of pure dephasing evolution, where the rate diverges at t = π/2. A Markovian realization of this same trajectory was constructed. Demonstrating that a single path cannot definitively determine whether it originates from Markovian or non-Markovian dynamics.

Meanwhile, the generator for this Markovian lift takes the form Lt = 1 εt (Rρt+εt ρt −Id), with εt = e 1− 1 (1−t)2. At the same time, this result extends to collections of paths. That even analysing multiple trajectories may not be sufficient to distinguish between Markovian and non-Markovian behaviour without additional criteria. Further investigation revealed that distinguishing between the two requires examining more than just a few trajectories.

For a two-qubit system, The effort constructed a path that could be realised by both a perpetually non-Markovian evolution and a Markovian lift, highlighting the need to analyse 2n paths, where n represents the number of qubits, to potentially discern the underlying dynamics. This path, defined as ρt = 1 4 2 1 + e−2t 1 + e−2t 2, demonstrates the difficulty in identifying non-Markovianity solely from trajectory data.

Mapping non-Markovian quantum evolution with equivalent Markovian steps

Once considered a theoretical curiosity, The effort of non-Markovian dynamics is now revealing itself as a surprisingly pervasive feature of quantum systems. For years, physicists have modelled open quantum systems using Markovian approaches, simplifying calculations by assuming the system’s past has little bearing on its future. Yet, this assumption breaks down in many realistic scenarios where memory effects become important.

This effort demonstrates a subtle but profound point: even when a quantum system evolves non-Markovianly, its trajectory can, under specific conditions, be perfectly mimicked by a cleverly constructed series of Markovian evolutions. By acknowledging this equivalence reframes the challenge, shifting the focus to how to map a given non-Markovian process onto an equivalent Markovian one.

By establishing conditions for continuous parametric lifts, The team provide a mathematical framework for achieving this. The requirement of a continuous unitary diagonalization and the absence of accumulation points in rankshift represent limitations. Precisely controlling the flow of information in quantum devices is essential for building future technologies.

Since non-Markovian effects can both enhance and degrade performance, the ability to engineer trajectories, to effectively ‘hide’ non-Markovianity within a Markovian description. Offers a new degree of control. Unlike previous approaches, this effort suggests a path towards designing systems where memory effects are not a hindrance but a resource — future work could concentrate on developing analogous techniques for systems undergoing more dramatic rank changes. Exploring how strong these mappings are to imperfections and noise, and a deeper understanding of this interaction between Markovian and non-Markovian dynamics will be vital for realising the full potential of quantum technologies.