The theory of open quantum systems served as a tool to prepare entanglement at the beginning stage of quantum technology and more recently provides an important tool for state preparation. Dynamics given by time dependent Lindbladians are Markovian and lead to decoherence, decay of correlation and convergence to equilibrium. In contrast Non-Markovian evolutions can outperform their Markovian counterparts by enhancing memory. In this letter we compare the trajectories of Markovian and Non-Markovian evolutions starting from a fixed initial value. It turns out that under mild assumptions every trajectory can be obtained from a family of time dependent Lindbladians. Hence Non-Markovianity is invisible if single trajectories are concerned.
The theory of open quantum systems describes the evolution of quantum devices including interactions with an environment. Open quantum systems have been studied extensively in quantum information [1,2], control [3][4][5], and thermodynamics [6][7][8]. For completely positive and trace-preserving (CPTP) Markovian dynamics, Gorini, Kossakowski, Sudarshan [9], and Lindblad [10] identified the general form of the time-local generator of the GKSL master equation
where L t is a Lindbladian operator. Independently, the same form is derived from the Born-Markov approximation [1,11]. Data processing inequality, decoherenence, and decay of correlation are imminent to the Markovian regime.
Many experimentally relevant processes exhibit memory effects, such as information backflow [12], nonmonotonic decoherence [13], or revival of entanglement [14]. These observations have motivated sustained efforts to characterize quantum dynamics beyond the Markovian regime. Such effect can be demonstrated in many quantum devices, such as nitrogen-vacancy centers [15][16][17], photonic systems [12,14,18], nuclear magnetic resonance [19][20][21], trapped ions [22,23], and on superconducting processors [24,25].
Multiple definitions of quantum Non-Markovianity have been proposed, reflecting different structural and operational perspectives. Foundational work by Wolf and Cirac [26] and Wolf, Eisert, Cubitt, and Cirac [27] initiated a systematic analysis of Markovianity at the level of quantum dynamical maps. Despite significant progress, no single definition has achieved universal acceptance (see Refs. [28,29] for reviews): CP-divisibility [30], information backflow [31], mutual information [32], channel capacities [33], decay-rate negativity [34], trajectorybased formulations [35], tensor network [36,37], and process tensors [38]. Ongoing work continues to clarify relations between different notions [39]. While differing in formulation, these approaches consistently regard Markovianity as a property of the reduced dynamical map or, more generally, of the associated multi-time quantum process.
From a naïve standpoint, only trajectories can be experimentally observed. More precisely, given a CPTP evolution T t , observed trajectories are given by ρ t = T t ρ 0 for an initial state ρ 0 . Such a trajectory (ρ t ) t0≤t<t1 is a prototype of a path of density matrices (mixed states). From individual realizations of dynamics in the form of trajectories ensemble descriptions can be inferred. This has motivated extensive use of trajectory-based methods, including stochastic unravelings and continuousmeasurement descriptions [3,[40][41][42][43][44][45]. This leads to the following question:
Can Non-Markovianity be detected from the trajectory?
In this Letter we show that under mild assumptions, the answer is a resounding No. We prove that Non-Markovianity is not an invariant property of individual quantum trajectories, not even finite ensembles thereof. This is demonstrated on a trajectory induced by the pure dephasing evolution, such as
More general classes of such example will be discussed later in the Letter. These results do not alter existing definitions of quantum Non-Markovianity formulated at the level of dynamical maps or multi-time processes. Rather, they establish the fundamental non-identifiability of Markovianity under trajectory-preserving descriptions.
Preliminaries. Let H be a finite-dimensional Hilbert space and denote
We call an evolution T t Markovian if it is given by time-dependent Lindbladians. More precisely, it satisfies a time-local master equation of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form Eq. ( 1) where the arXiv:2602.17631v1 [quant-ph] 19 Feb 2026 generator L t is a Lindbladian. As shown in the seminal works of Gorini et al. and Lindblad [9,10], the most general form of a time-local generator compatible with CPTP maps is
where H t is a (possibly time-dependent) Hamiltonian, a i (t) are jump operators, and the rates γ i (t) ≥ 0. Timelocal Lindbladians generate Markovian evolutions T t via the time-ordered exponential T t = T exp t 0 L s ds (equivalently solving the differential equation).
Main Results. We show that trajectories generated by Non-Markovian dynamics can always be reproduced by suitably chosen Markovian evolutions. Under mild assumptions, we prove that any differentiable path of density matrices ρ t can be realized as the trajectory of a Markovian dynamical map T t . We apply this result to several classes of trajectories naturally arising from Non-Markovian evolutions, demonstrating that Markovianity cannot be inferred from a single trajectory. We further show that even collections of trajectories may remain indistinguishable in the absence of additional selection criteria.
We impose two mild regularity assumptions on the path of density operators, ensuring well-behaved spectral properties:
(1) The eigenspaces of ρ t depend continuously on t (2) The set of times at which ρ t chan
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