Non-Markovianity Measures

A non-Markovianity measure tries to turn memory effects into a number. That is useful for comparing models, experiments, and approximations, but the number is never definition-free. It depends on which meaning of non-Markovianity is being quantified.

The safest reading is:

measure = diagnostic + time interval + optimization domain + normalization

Trace-distance measures quantify recoveries of distinguishability. Divisibility measures quantify failures of intermediate maps to be physical channels. Entanglement and correlation measures quantify revivals of resources that local channels cannot create. These measures often agree in simple qubit examples, but they are not equivalent in general.

For the broader terminology, start with What Non-Markovian Means. For the two central diagnostics, see CP Divisibility and Information Backflow.

Defining a Non-Markovianity Measure

Before quoting a number, specify five choices.

Choice	Why it matters
Dynamical object	A family $\Phi_{t,0}$, a time-local generator, a memory kernel, or experimental state data can support different tests.
Time window	A process may be Markovian on one interval and non-Markovian on another.
Diagnostic	Backflow, CP indivisibility, entanglement revival, and kernel memory answer different questions.
Optimization set	Some measures maximize over state pairs, ancillas, or input ensembles.
Normalization	Different papers use different prefactors, logarithms, and integration conventions.

A measure is most meaningful when all five choices match the physical question. A large trace-distance backflow measure may be irrelevant for an observable that is insensitive to the revived degree of freedom. A small CP-indivisibility measure may still matter if a control protocol relies on entangled references.

Trace-Distance Backflow

The Breuer–Laine–Piilo family of measures starts from the trace distance between two reduced system states:

$$D(t) = D\!\left(\rho_1(t),\rho_2(t)\right) = \frac12 \left\lVert \rho_1(t)-\rho_2(t) \right\rVert_1 .$$

For a positive trace-preserving intermediate map, trace distance cannot increase. A positive derivative

$$\sigma(t;\rho_1,\rho_2) = \frac{d}{dt} D\!\left(\Phi_{t,0}\rho_1,\Phi_{t,0}\rho_2\right) > 0$$

therefore witnesses failure of positive divisibility. The corresponding backflow measure is

$$\mathcal N_{\mathrm{BLP}}[\Phi] = \max_{\rho_1,\rho_2} \int_{\sigma>0}dt\, \sigma(t;\rho_1,\rho_2).$$

The integral adds up all intervals where distinguishability returns to the system. The maximization asks for the pair of initial states that makes the revival largest.

This measure is operational: $D(\rho_1,\rho_2)$ is directly related to optimal single-shot state discrimination. For the norm itself, see Trace Distance.

Strengths of Backflow Measures

Trace-distance measures are attractive because they need only system states. One can compute or reconstruct $\rho_1(t)$ and $\rho_2(t)$, then look for statistically significant increases in distinguishability.

They are especially transparent for:

• dephasing channels, where transverse Bloch states reveal coherence revivals;
• amplitude-damping models, where population return can increase distinguishability;
• experiments where full dynamical-map tomography is harder than preparing state pairs;
• finite reservoirs and structured modes with visible revivals.

For a qubit dephasing map

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

the states $\lvert+\rangle$ and $\lvert-\rangle$ give

$$D(t)=|\eta(t)|.$$

Thus

$$\mathcal N_{\mathrm{BLP}} = \int_{\frac{d}{dt}|\eta(t)|>0} dt\, \frac{d}{dt}|\eta(t)|$$

for that optimal dephasing pair. In this special example, the measure is literally the total amount of coherence revival.

Limitations of Backflow Measures

Trace-distance backflow is a witness, not a universal definition.

Important limitations:

• absence of observed backflow does not prove CP divisibility;
• optimizing over all state pairs can be difficult in high dimension;
• a nonoptimal pair can miss memory that another pair detects;
• noise in tomography can create small artificial increases;
• the measure is sensitive to the chosen time grid in numerical work;
• it detects loss of positive divisibility, while CP divisibility is a stronger condition.

The last point is the main conceptual caution:

$$\text{CP divisible} \Longrightarrow \text{P divisible} \Longrightarrow \text{no trace-distance backflow}.$$

The reverse implications need not hold.

Divisibility-Based Measures

Divisibility-based measures ask how much the intermediate propagator

$$\Phi_{t+\epsilon,t}$$

fails to be a completely positive trace-preserving map. In finite dimensions, this is often tested using a Choi matrix or an ancilla extension.

A schematic Rivas–Huelga–Plenio-style infinitesimal witness is

$$g(t) = \lim_{\epsilon\to0^+} \frac{ \left\lVert \left( \Phi_{t+\epsilon,t} \otimes \operatorname{id}_A \right) \left( \lvert\Omega\rangle\langle\Omega\rvert \right) \right\rVert_1 -1 }{\epsilon},$$

where $\lvert\Omega\rangle=d^{-1/2}\sum_{j=1}^d\lvert j,j\rangle$ is normalized. The resulting Choi operator is $J/d$ in the unnormalized convention of Choi Matrix. For a trace-preserving Hermiticity-preserving map its trace is one; its trace norm equals one exactly when it is positive. If the intermediate map is CPTP, the numerator therefore vanishes. Positive $g(t)$ signals an infinitesimal failure of complete positivity. A measure integrates the positive violation:

$$\mathcal N_{\mathrm{RHP}} = \int_0^T dt\,g(t).$$

The details depend on convention, finite-dimensional regularity, and invertibility of $\Phi_{t,0}$. The conceptual target is clear: quantify how far the process is from being CP-divisible.

Rate-Negativity Measures

When the dynamics has a time-local generator in canonical instantaneous GKSL form,

$$\frac{d\rho}{dt} = - \frac{i}{\hbar} [H(t),\rho] + \sum_\alpha \gamma_\alpha(t) \left( L_\alpha(t)\rho L_\alpha^\dagger(t) - \frac12 \{L_\alpha^\dagger(t)L_\alpha(t),\rho\} \right),$$

CP divisibility is equivalent, under the standard finite-dimensional regularity assumptions, to

$$\gamma_\alpha(t)\ge0$$

for all canonical rates. A practical measure is therefore the integrated negative part:

$$\mathcal N_{\mathrm{rate}} = \int_0^T dt\, \sum_\alpha \max\{0,-\gamma_\alpha(t)\}.$$

Fix traceless Hilbert–Schmidt orthonormal canonical operators, $\operatorname{Tr}(L_\alpha^\dagger L_\beta)=\delta_{\alpha\beta}$, before assigning a numerical value to the rates. Otherwise a rescaling of an $L_\alpha$ changes the reported measure without changing the generator. The integral is dimensionless because each rate has units of inverse time.

This is easy to compute once the canonical rates are known. The word “canonical” matters: arbitrary dissipator representations can move terms between operators and rates. Rate-negativity measures should be built from a diagonal or otherwise invariant instantaneous dissipator representation, not from an arbitrary basis.

Entanglement-Based Measures

If a system $S$ evolves by a CP-divisible family and an ancilla $A$ is left isolated, then entanglement between $S$ and $A$ cannot increase under the local intermediate channel:

$$\rho_{SA}(t) = \left( \Phi_{t,0}\otimes\operatorname{id}_A \right) \rho_{SA}(0).$$

For an entanglement monotone $E$,

$$E(t) = E\!\left(\rho_{SA}(t)\right)$$

must be nonincreasing for every initial $\rho_{SA}(0)$ if the process is CP-divisible. A revival with

$$\frac{dE(t)}{dt}>0$$

therefore witnesses non-Markovianity in a CP-divisibility sense.

A schematic entanglement-based measure is

$$\mathcal N_E = \max_{\rho_{SA}(0)} \int_{\dot E>0}dt\,\dot E(t).$$

This family of measures is conceptually close to the Choi test: complete positivity is precisely the condition that the map remains positive when extended by an arbitrary reference. It is also experimentally more demanding, because one must prepare and characterize system-ancilla states.

Correlation-Based Variants

One can replace entanglement by other correlation quantities, such as quantum mutual information between the system and an isolated ancilla:

$$I(S:A) = S(\rho_S)+S(\rho_A)-S(\rho_{SA}).$$

For a CP-divisible process, local channels on $S$ cannot increase suitable distinguishability or correlation monotones with an untouched reference. A revival of such a quantity can therefore serve as a witness.

Correlation-based witnesses are useful when entanglement is too fragile or absent, but they inherit the same caution: the chosen correlation quantity defines the operational task. A mutual-information revival, an entanglement revival, and a trace-distance revival need not have the same numerical size or appear over exactly the same intervals.

Geometric and Distance-to-Markovian Sets

Another strategy asks how far a process is from a chosen set of Markovian processes. Schematically,

$$\mathcal N_{\mathrm{geom}}(\Phi) = \inf_{\Psi\in\mathcal M} d(\Phi,\Psi),$$

where $\mathcal M$ is a selected set of Markovian or CP-divisible processes and $d$ is a distance between process families.

This is attractive in principle because it resembles “distance to the nearest memoryless model.” In practice it is technically delicate:

• the set $\mathcal M$ must be specified;
• the distance $d$ may be hard to compute;
• finite time sampling changes the optimization;
• the result depends on whether one compares generators, maps, process tensors, or observed data.

Geometric measures are best used when the optimization problem and operational distance are both explicit.

Memory-Kernel and Process-Tensor Views

Measures based on memory kernels ask how strongly the present derivative depends on earlier states:

$$\frac{d}{dt}\rho(t) = \int_0^t ds\, \mathcal K(t,s)\rho(s).$$

One might quantify the norm, duration, or spectral content of $\mathcal K(t,s)$. This can be useful in model reduction, but it is representation-dependent: different projection choices or enlarged system boundaries can move memory between kernels, auxiliary modes, and time-local coefficients.

More operationally complete frameworks use multi-time process tensors. They test whether interventions at earlier times influence later statistics beyond what is mediated by the current system state. Such measures are powerful for control and experiments with interventions, but they require more data than a two-time dynamical map.

For the single-time map diagnostics used in this chapter, the canonical pages remain CP Divisibility and Information Backflow.

Comparing Common Measures

Measure family	Detects	Main input	Main limitation
Trace-distance backflow	Increase of state distinguishability	Evolved pairs of system states	Can miss CP-indivisible but P-divisible dynamics
CP-divisibility	Failure of intermediate maps to be CPTP	Dynamical map or time-local generator	Requires map reconstruction or generator control
Rate negativity	Negative canonical instantaneous rates	Canonical time-local generator	Sensitive to regularity and representation choices
Entanglement revival	Increase of system-ancilla entanglement	Ancilla-assisted evolution	Requires ancilla preparation and chosen monotone
Correlation revival	Increase of system-ancilla correlations	Reference-assisted data	Operational meaning depends on correlation measure
Geometric distance	Distance from a chosen Markovian set	Process family plus metric	Often computationally expensive and convention-dependent
Kernel memory	Explicit history dependence	Memory-kernel equation	Representation and boundary dependent

No single row dominates the others. Each row answers a different question.

Operational Cautions

A measure is not an explanation

A positive value says that a diagnostic was triggered. It does not by itself identify the physical source of memory. The source might be a structured spectral density, a finite bath, strong coupling, initial correlations, classical noise, or a poor system-boundary choice.

Zero does not always mean memoryless

A zero BLP value on sampled state pairs does not prove the process is Markovian in every sense. It may only mean that the tested pairs did not reveal backflow, or that the dynamics is CP-indivisible without trace-distance increase.

Larger is not always more important

A measure can be large for degrees of freedom irrelevant to the observable of interest. Conversely, a small memory effect can be important in precision measurements, error correction, or feedback control.

Experimental measures need uncertainty

Non-Markovianity measures often involve derivatives, optimizations, or small eigenvalue tests. In experiments, quote confidence intervals or robustness checks. In numerics, quote time-step, truncation, and positivity tolerances.

The system boundary matters

Promoting a pseudomode or reaction coordinate into the system can reduce or remove non-Markovianity in the enlarged description while preserving the same total physics. A measure is always a property of the chosen reduced description.

Exercises

BLP measure for one revival

A qubit dephasing model has $|\eta(0)|=1$, decreases to $0.3$, increases to $0.6$, and then decreases monotonically to $0.1$. What is the BLP contribution from the transverse dephasing pair?

Solution

For the transverse pair,

$$D(t)=|\eta(t)|.$$

The only positive interval increases from $0.3$ to $0.6$, so the backflow contribution is

$$0.6-0.3=0.3.$$

The later decrease contributes nothing because the BLP integral keeps only intervals where $\dot D(t)>0$.

Negative rate integral

Suppose a canonical time-local dephasing rate is

$$\gamma(t) = \begin{cases} 1, & 0\le t\le1,\\ -0.2, & 1\le t\le3,\\ 0.5, & 3\le t\le4. \end{cases}$$

Compute the integrated negative-rate measure over $0\le t\le4$.

Solution

Only the interval from $1$ to $3$ contributes:

$$\mathcal N_{\mathrm{rate}} = \int_1^3 dt\,0.2 = 0.4.$$

The positive-rate intervals contribute zero to the negative part.

Backflow versus CP indivisibility

Why can a trace-distance measure fail to detect some CP-indivisible dynamics?

Solution

Trace-distance monotonicity is tied to positive divisibility. CP divisibility is stronger because it requires complete positivity of the intermediate map, meaning positivity even when the system is entangled with a reference. A process can fail complete positivity of an intermediate map while remaining positive on all system states. In that case trace distances between system states need not increase, even though CP divisibility fails.

Choice of system boundary

A qubit plus a narrow cavity mode evolves Markovianly when the cavity is included in the system, but the qubit alone shows excitation revivals. For vacuum amplitude damping, compare the BLP witness for the qubit with a measure that vanishes on the enlarged CP-divisible dynamics.

Solution

The qubit has a positive backflow contribution when its survival probability revives. The enlarged CP-divisible family has zero for any diagnostic defined to vanish on all such families. The cavity stores excitation and later returns it to the qubit after being traced out. In the enlarged qubit-plus-cavity description, that exchange is explicit system dynamics, and only the residual bath is treated as environment. The total physics is not changed; the reduced boundary is.

References

• 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).
• H.-P. Breuer, E.-M. Laine, and J. Piilo, “Measure for the degree of non-Markovian behavior of quantum processes in open systems,” Physical Review Letters 103, 210401 (2009).
• D. Chruściński, A. Kossakowski, and Á. Rivas, “Measures of non-Markovianity: Divisibility versus backflow of information,” Physical Review A 83, 052128 (2011).
• S. Luo, S. Fu, and H. Song, “Quantifying non-Markovianity via correlations,” Physical Review A 86, 044101 (2012).
• 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, “Entanglement and non-Markovianity of quantum evolutions,” Physical Review Letters 105, 050403 (2010).
• Á. Rivas, S. F. Huelga, and M. B. Plenio, “Quantum non-Markovianity: characterization, quantification and detection,” Reports on Progress in Physics 77, 094001 (2014).
• M. M. Wolf, J. Eisert, T. S. Cubitt, and J. I. Cirac, “Assessing Non-Markovian Quantum Dynamics,” Physical Review Letters 101, 150402 (2008).