What Non-Markovian Means

Non-Markovian dynamics concerns memory in the evolution of a system whose environment has been left out of the description. The intuitive question is whether earlier events can affect the future beyond what the present system state tells us. Quantum theory offers several tests of this idea, and they do not all answer the same question.

Two widely used tests examine a family of reduced-state maps: failure of CP divisibility and revival of state distinguishability. A stronger operational question asks how the system responds to a sequence of measurements and preparations. One-time maps alone do not determine these multi-time responses, as explained by Li, Hall, and Wiseman and by Pollock and collaborators in the references.

A nonexponential decay, a time-dependent equation, or a structured bath can motivate a closer look, but none alone proves failure of every Markovian criterion. Always state which test is being used.

Markovian Ideal

In a classical Markov process, the future is conditionally independent of the past once the present state is known. In open quantum systems, a common time-homogeneous ideal is a quantum dynamical semigroup:

$$\Phi_0=\mathrm{id}, \qquad \Phi_{t+s}=\Phi_t\Phi_s, \qquad t,s\ge0.$$

In finite dimensions, a norm-continuous completely positive trace-preserving semigroup has a Lindblad–GKSL generator:

$$\Phi_t=e^{t\mathcal L}, \qquad \frac{d\rho}{dt} = \mathcal L(\rho).$$

This ideal combines several assumptions:

• no relevant memory variable outside the reduced state;
• no explicit dependence on the starting time;
• completely positive trace-preserving maps for all elapsed times;
• a generator whose rates do not require knowledge of the past trajectory.

Failure of time homogeneity alone is not sufficient to establish memory: an externally driven process can have different channels at different times while remaining CP-divisible. Moreover, a semigroup family of unperturbed one-time maps does not by itself specify all statistics after intermediate interventions.

Exact Reduced Dynamics

Start from a system $S$ and environment $E$:

$$\rho_S(t) = \operatorname{Tr}_E \left[ U(t,t_0)\rho_{SE}(t_0)U^\dagger(t,t_0) \right].$$

This formula is exact if $S+E$ is closed. The reduced state $\rho_S(t)$ may fail to contain all information needed for a closed equation because information can be stored in:

• the environment state;
• correlations between $S$ and $E$;
• delayed fields or finite propagation times;
• structured modes that exchange excitation with $S$;
• preparation-dependent correlations present at $t_0$.

Thus non-Markovianity is not a mysterious violation of quantum mechanics. It is usually a statement that the chosen reduced description has thrown away variables that still matter.

Meanings Side by Side

Meaning	Mathematical question	Typical page
Memory kernel	Does $\dot\rho(t)$ depend explicitly on $\rho(s)$ for $s\lt t$?	Memory Kernels
Nonsemigroup	Is there no time-independent $\mathcal L$ with $\Phi_t=e^{t\mathcal L}$?	Quantum Dynamical Semigroups
CP indivisibility	Is some intermediate map $\Phi_{t,s}$ not CPTP?	CP Divisibility
Information backflow	Does trace distance increase for some pair of states?	Information Backflow
Correlation memory	Do system–environment correlations affect later dynamics?	Initial Correlations
Structured environment	Should a bath mode be promoted into the system?	Reaction-Coordinate Mapping

The table includes both diagnostics and modeling clues. A memory kernel depends on the representation chosen, and a structured environment need not produce backflow in the regime measured. These notions often agree in simple examples, but not in general. That is why the broad chapter overview is Non-Markovian Dynamics, while the specialized pages treat each diagnostic separately. Quantitative comparisons belong in Non-Markovianity Measures.

Memory-Kernel Meaning

A time-nonlocal master equation has the schematic form

$$\frac{d}{dt}\rho_S(t) = \int_{t_0}^{t}ds\, \mathcal K(t,s)\rho_S(s) + \mathcal J(t).$$

The kernel $\mathcal K(t,s)$ explicitly refers to earlier reduced states. The inhomogeneous term $\mathcal J(t)$ can appear when initial correlations or projection choices leave relevant information outside $\rho_S(t_0)$.

This meaning is closest to the everyday word “memory.” It says the present derivative depends on the past. Projection-operator derivations, delayed feedback, finite reservoirs, and structured baths naturally produce this language.

The form of the equation is not itself a complete memory test. The same reduced evolution can often be written in time-local or memory-kernel form. Conversely, a time-dependent generator with nonnegative canonical rates is CP-divisible.

Time-Local but Memoryful

If the reduced map $\Phi_{t,t_0}$ is invertible, one can formally write

$$\frac{d\rho}{dt} = \mathcal K_{\mathrm{TCL}}(t)\rho(t), \qquad \mathcal K_{\mathrm{TCL}}(t) = \dot\Phi_{t,t_0}\Phi_{t,t_0}^{-1}.$$

This equation is local in time: it uses $\rho(t)$, not an explicit integral over $\rho(s)$. But the coefficients can remember the environment through their time dependence, singularities, or temporarily negative canonical rates.

Therefore:

time-local equation does not imply Markovian physics

For the detailed representation, see Time-Convolutionless Master Equations.

Divisibility Meaning

Given maps $\Phi_{t,0}$, divisibility asks whether the evolution from $0$ to $t$ can be split through an intermediate time $s$:

$$\Phi_{t,0} = \Phi_{t,s}\Phi_{s,0}, \qquad t\ge s\ge0.$$

If $\Phi_{t,s}$ is completely positive and trace preserving for every interval, the evolution is CP-divisible. This is a factorization of the unperturbed map family. It does not guarantee that the same channel predicts what happens after an arbitrary intervention at $s$; such an intervention can change system–environment correlations. Multi-time memory tests ask that additional question.

Failure of CP divisibility is a strong channel-level sense of non-Markovianity. It does not necessarily mean the finite-time map $\Phi_{t,0}$ is unphysical. It means the intermediate-channel interpretation fails for at least one interval.

For differentiable invertible finite-dimensional maps, CP divisibility is equivalent, under standard regularity assumptions, to a time-local generator with instantaneous GKSL form and nonnegative canonical rates.

Information-Backflow Meaning

For two system states, trace distance

$$D(\rho,\sigma) = \frac12 \lVert\rho-\sigma\rVert_1$$

measures optimal distinguishability. A quantum channel cannot increase it:

$$D(\Phi(\rho),\Phi(\sigma)) \le D(\rho,\sigma).$$

If an open-system evolution makes

$$\frac{d}{dt} D(\rho_1(t),\rho_2(t)) > 0$$

for some pair of initial states, distinguishability has returned to the system. This is interpreted as information backflow from the environment or from system–environment correlations.

This diagnostic is operational and experimentally intuitive. But it detects positive divisibility failure, not every possible failure of CP divisibility. It is a witness, not a universal definition.

Microscopic Meaning

In microscopic modeling, non-Markovianity often means that a bath cannot be replaced by a rapidly forgetting reservoir. Common sources include:

• bath correlation times comparable to system dynamics;
• narrow spectral peaks or band edges;
• finite environments and revivals;
• strong coupling and dressed system–environment eigenstates;
• slow classical noise;
• delayed coherent feedback;
• correlated ancillas in collision models;
• initially correlated preparations.

The modeling response is not always “use a more complicated master equation.” Sometimes the best move is to enlarge the system boundary. Pseudomode Methods and Reaction-Coordinate Mapping make important environmental degrees of freedom explicit, leaving a shorter-memory residual bath.

Simple Dephasing Example

Consider a qubit dephasing channel

$$\rho_{01}(t) = \eta(t)\rho_{01}(0), \qquad \rho_{00}(t)=\rho_{00}(0).$$

A time-homogeneous dephasing semigroup has

$$\eta(t) = e^{-\Gamma t} e^{-i\omega t}, \qquad \Gamma\ge0.$$

The semigroup law is

$$\eta(t+s)=\eta(t)\eta(s).$$

If $\eta(t)$ is not exponential, the dynamics is not this simple semigroup. If $|\eta(t)|$ increases over an interval, trace distance between suitable phase-superposition states increases, giving information backflow. When $\eta(s)\ne0$, if the intermediate factor

$$\eta(t,s) = \frac{\eta(t)}{\eta(s)}$$

has magnitude greater than $1$, the intermediate dephasing map is not completely positive.

If $\eta(s)=0$, use the factorization condition rather than dividing by zero. A later revival cannot be CP-divisible: two initial states that have become identical at $s$ cannot be separated again by a channel acting only on that identical state.

Choosing the Right Meaning

Use the definition that matches the question.

If the question is…	Start with…
Is a Lindblad semigroup adequate?	semigroup and approximation checks
Can every intermediate interval be a channel?	CP divisibility
Can an experiment witness memory through distinguishability revivals?	information backflow
Does the equation explicitly depend on the past?	memory kernels
Did a weak-coupling derivation discard relevant bath dynamics?	bath correlations and time-scale checks
Is a structured mode responsible for memory?	pseudomode or reaction-coordinate methods
Are there preparation-dependent effects?	initial correlations and assignment maps

For practical modeling, it is often best to state both the mathematical diagnostic and the physical source of memory.

Markovian and Non-Markovian Noise explains how channel data and interventions can test memory in quantum-information experiments.

What Non-Markovian Does Not Mean

Non-Markovian does not automatically mean:

• the model is more accurate;
• the map is unphysical;
• the master equation has an explicit integral kernel;
• a rate is negative for all conventions;
• the environment literally sends a classical signal back;
• every Markov approximation is invalid;
• memory effects are large enough to matter for the observable being measured.

It means the chosen Markovian ideal is not adequate under the diagnostic being used.

For a compact warning list aimed at modeling and data analysis, see Pitfalls in Non-Markovian Modeling.

Exercises

1. Semigroup test. A dephasing coherence factor is $\eta(t)=e^{-\Gamma t^2}$ for $t\ge0$ and $\Gamma>0$ (with units of inverse time squared). Is the family a time-homogeneous semigroup? Is it CP-divisible?

Solution

No. A semigroup coherence factor must satisfy

$$\eta(t+s)=\eta(t)\eta(s).$$

Here

$$\eta(t+s) = e^{-\Gamma(t+s)^2},$$

while

$$\eta(t)\eta(s) = e^{-\Gamma(t^2+s^2)}.$$

These are not equal unless $2\Gamma ts=0$. Nevertheless, every map is CPTP because $0<\eta(t)\le1$, and the family is CP-divisible because $\eta(t)/\eta(s)=e^{-\Gamma(t^2-s^2)}\le1$ for $t\ge s$. Its coherence-decay rate is $2\Gamma t\ge0$. This is an explicit example of nonsemigroup evolution without CP indivisibility.

2. Backflow versus finite-time positivity. If a trace-distance increase is observed for some pair of system states, what does it prove?

Solution

It proves that the intermediate evolution over that interval cannot be positive and trace preserving for all system states. Therefore it also cannot be CP-divisible. It does not by itself prove that the finite-time map from the initial time to the final time is unphysical.

3. Time local but not memoryless. Explain why a time-local generator $\mathcal K(t)$ does not automatically imply Markovian dynamics.

Solution

If the reduced map is invertible, one can formally write

$$\mathcal K(t)=\dot\Phi(t)\Phi^{-1}(t).$$

The resulting equation uses only $\rho(t)$, but the time dependence of $\mathcal K(t)$ can encode earlier exchange with the environment. Memory has been compressed into the coefficients rather than removed.

4. Boundary change. A qubit strongly exchanges excitation with a single lossy cavity mode. Why might including the cavity mode in the system make the remaining problem more Markovian?

Solution

If the cavity mode stores excitation and later returns it to the qubit, treating the cavity as part of the environment gives memory in the qubit-only dynamics. By enlarging the system to qubit plus cavity, that exchange becomes explicit system dynamics. The residual external modes may then be broad and short-memory enough to model with a Markovian loss term.

References

• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).
• H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, “Colloquium: Non-Markovian dynamics in open quantum systems,” Reviews of Modern Physics 88, 021002 (2016).
• D. Chruściński, A. Kossakowski, and Á. Rivas, “Measures of non-Markovianity: Divisibility versus backflow of information,” Physical Review A 83, 052128 (2011).
• I. de Vega and D. Alonso, “Dynamics of non-Markovian open quantum systems,” Reviews of Modern Physics 89, 015001 (2017).
• L. Li, M. J. W. Hall, and H. M. Wiseman, “Concepts of quantum non-Markovianity: A hierarchy,” Physics Reports 759, 1–51 (2018), doi:10.1016/j.physrep.2018.07.001.
• F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, “Operational Markov Condition for Quantum Processes,” Physical Review Letters 120, 040405 (2018), doi:10.1103/PhysRevLett.120.040405.
• Á. Rivas, S. F. Huelga, and M. B. Plenio, “Quantum non-Markovianity: characterization, quantification and detection,” Reports on Progress in Physics 77, 094001 (2014).