Markovianity and non-Markovianity in quantum and classical systems

May combine two claims.

For the convolution of two different exponential distributions with p > s^2/8, q(t) oscillates between positive and negative values, N(Φ) assumes a finite value, and I(Φ) is infinite.

“When p > s 2 / 8, which includes the case λ 1 = λ 2 , q ( t ) again oscillates between positive and negative values, so that one has a similar behavior as before, with N (Φ) assuming a finite value and I (Φ) ≖ ∞ .”

May combine two claims.

For the convolution of two equal exponential waiting time distributions, γ(t) takes on both positive and negative values and diverges for λt=(3/4)π mod π.

“which indeed takes on both positive and negative values, diverging for λt = (3 / 4) π mod π .”

May combine two claims.

For the dephasing map, the Rivas-Huelga-Plenio requirement for non-Markovianity is satisfied at the same time as the growth of the trace distance, but the effect is quantified differently.

“While for this model this requirement for non-Markovianity is satisfied at the same time as the growth of the trace distance, the effect is quantified in a different way.”

May combine two claims.

For the dissipative model, N(Φ) becomes positive as soon as P-divisibility is broken, depending only on the sign of γ(t), while I(Φ) becomes positive even when only CP-divisibility fails, depending also on the sign of δ(t).

“The measure N (Φ) becomes positive as soon as P-divisibility is broken, which depends on the sign of γ ( t ) only, while I (Φ) becomes positive even when only CP-divisibility does not hold, which also depends on the sign of the function δ ( t ) appearing in front of the purely quantum channel L z , which determines the dynamics of the coherences.”

May combine two claims.

For the dephasing map, P-divisibility holds if the matrices obtained as in Eq. (3) represent a positive map Φ(t,s) for any t≥s≥0, which is the case provided condition (58).

“In particular the map Φ ( t, 0) will turn out to be P-divisible if the matrices”

• Also stated in: § 1. Divisibility · p. 9 · plain

  “obtained as in Eq. (3) represent a positive map Φ ( t, s ) for any t ⩾ s ⩾ 0, which is the case provided”

May combine two claims. The quote only partly states this claim.

For the dissipative CPT map E±, the time-convolutionless master equation includes channels Lz, L+, and L−, with time-dependent rates γ(t) and δ(t), where δ(t) is given by a difference involving the always-positive hazard rate h(t).

“The time dependent rate γ ( t ) is still given by Eq. (56), while the function δ ( t ) is given by the difference”

• Also stated in: § D. Dissipative dynamics and time-local equation · p. 12 · plain

  “where h ( t ) is the so called hazard rate function introduced in Eq. (77), which is always positive.”

May combine two claims.

For the dephasing map, the P-divisibility condition is satisfied if |q(t)| is a monotonic decreasing function, and then the time-dependent rate γ(t) is always positive.

“This condition is satisfied if | q ( t ) | is a monotonic decreasing function, and therefore the time dependent rate γ ( t ) is always positive.”

May combine two claims.

For the dephasing map E_z, the time-convolutionless master equation has a single quantum channel and a time-dependent rate γ(t).

“where we have a single quantum channel given by”

• Also stated in: § B. Dephasing dynamics and time-local equation · p. 8 · plain

  “and the time dependent rate γ ( t ) is”

May combine two claims.

For the dephasing map, the region Ω+ corresponds to the time intervals where |q(t)| increases, and the maximum of the integrated trace-distance derivative is obtained for initial states with Δp(0)=0 and Δc(0)=1.

“The region Ω + now corresponds to the time intervals where | q ( t ) | increases, and the maximum is obtained for initial states such that ∆ p (0) = 0 and ∆ c (0) = 1”

May combine two claims.

For a convex mixture of two exponential distributions, the trajectories never cross, and the matrices T(t,s) always are well-defined stochastic matrices.

“For the case of a convex mixture of two exponential distributions on the contrary the trajectories never cross, and the matrices T ( t, s ) always are well-defined stochastic matrices.”

• AssumptionRests on an assumption: A semi-Markov process is Markovian if and only if its waiting time distributions are exponential.

May combine two claims.

For a memoryless waiting time distribution, q(t)=exp(-2λt), γ(t) is the positive constant λ, and each non-Markovianity measure is zero.

“For the case of a memoryless waiting time distribution of the form Eq. (22), so that k ( t ) is actually a delta function and q ( t ) = exp( -2 λt ), according to Eq. (56) the function γ ( t ) is simply given by the positive constant λ . Each non-Markovianity measure is easily assessed to be zero.”

May combine two claims.

For the projection CPT map corresponding to pure dephasing, the survival probability is positive and monotonically decreasing, and the hazard rate function is always positive.

“Note that the survival probability is a positive monotonously decreasing function, and the hazard rate function is always positive.”

May combine two claims.

For the dissipative map E±, P-divisibility and CP-divisibility are not necessarily violated together; it can happen that γ(t) stays positive while δ(t) is negative.

“Note however that in this case, due to the presence of different quantum channels, P-divisibility and CP-divisibility are not necessarily violated together, since it can well happen that γ ( t ) stays positive, but δ ( t ) takes on negative values.”

May combine two claims.

For the convolution of two equal exponential waiting time distributions, the Breuer-Laine-Piilo measure is finite and independent of λ.

“The measure proposed by Breuer, Laine and Piilo according to Eq. (64) can now be exactly calculated and is given by”

• Also stated in: § 3. Explicit example · p. 10 · plain

  “which is finite and independent on λ .”

May combine two claims.

In all these situations the process is non-Markovian, but P-divisibility of the time evolution Λ(t,0) still holds in some cases.

“It should be noticed that in all these situations the process is non-Markovian, but P-divisibility of the time evolution Λ ( t, 0) in the sense of Eq. (5) still holds in some cases.”

• AssumptionRests on an assumption: A semi-Markov process is Markovian if and only if its waiting time distributions are exponential.

May combine two claims.

For convolution of two different exponential waiting time distributions, q(t) shows oscillatory behaviour if χ becomes imaginary (for 3-2√2 ≤ λ1/λ2 ≤ 1/(3-2√2)), and is positive monotonic otherwise.

“The expression given by Eq. (38) shows an oscillatory behaviour if χ becomes imaginary, for 3 -2 √ 2 ⩽ λ 1 /λ 2 ⩽ 1 / ( 3 -2 √ 2 ) , while it is a positive monotonic function of t otherwise.”

• AssumptionRests on an assumption: A semi-Markov process is Markovian if and only if its waiting time distributions are exponential.

May combine two claims.

For π=1/2, the Kolmogorov distance is a monotonic contraction for any waiting time distribution; for π=1, the distance can show revivals depending on q(t).

“Thus, while for π = 1 / 2 the Kolmogorov distance is a monotonic contraction for any waiting time distribution, thanks to the fact that g ( t ) is a survival probability, for π = 1 the distance among distributions can show revivals depending on the explicit expression of q ( t ), as can be seen from Fig. 1 for the case of the convolution of two exponential waiting time distributions with the same”

• AssumptionRests on an assumption: A semi-Markov process is Markovian if and only if its waiting time distributions are exponential.

May combine two claims.

For the convolution of two equal exponential distributions in the dissipative model, both γ(t) and δ(t) oscillate in sign and diverge when cotg(λt) = -1, and both measures are positive.

“so that both γ ( t ) and δ ( t ) oscillate in sign and diverge when cotg ( λt ) takes on the value minus one, as shown in Fig. 4. In this case both measures are positive”

May combine two claims.

For the dissipative map E±, the only condition for P-divisibility is still Eq. (58), so the map is P-divisible whenever |q(t)| is a monotonic decreasing function.

“so that thanks to the property of the survival probability the only condition for P-divisibility is still given by Eq. (58). Therefore the map is P-divisible whenever | q ( t ) | is a monotonic decreasing function.”

May combine two claims.

For the dissipative map E±, the trace distance derivative can grow in the region Ω+ where q(t) and qdot(t) have the same sign, and the growth is maximal for Δc(0)=0 and Δp(0)=1, i.e., for projectors on ground and excited state.

“and can grow in the region Ω + where q ( t ) and ˙ q ( t ) have the same sign. In this region | q ( t ) | does increase. The growth is maximal for ∆ c (0) = 0 and ∆ p (0) = 1, so that the couple of states which maximize it is given by the projectors on ground and excited state.”

The quote only partly states this claim.

For the convolution of two equal exponential waiting time distributions, the Rivas-Huelga-Plenio measure is infinite because q(t) goes through zero and γ(t) diverges.

“The measure proposed by Rivas, Huelga and Plenio instead is equal to infinity I (Φ) ≖ ∞ , due to the fact that q ( t ) goes through zero and therefore γ ( t ) diverges.”

May combine two claims.

For the dissipative map E±, coherences and populations decouple, and the populations obey the same equation as the transition probabilities of the classical semi-Markov processes considered in Sect. III for π=1 and arbitrary waiting time distribution.

“For the dynamics described by Eq. (49), with the CPT map given by E ± as in Eq. (78), coherences and populations decouple, and the populations obey the same equation as the transition probabilities of the classical semiMarkov processes considered in Sect. III for π = 1 and arbitrary waiting time distribution.”

May combine two claims.

For the dephasing map, the growth of |q(t)| determines both the breaking of CP-divisibility and the growth of the trace distance, so both approaches detect non-Markovianity at the same time, even if they quantify it differently.

“For this model the growth of | q ( t ) | determines both the breaking of CP-divisibility as well as the growth of the trace distance, so that both approaches detect nonMarkovianity at the same time, even if they quantify it in different ways.”

May combine two claims.

For a semi-Markov process with π=1/2, P-divisibility always holds; the trajectories never cross and |q(t)| never grows.

“For a semi-Markov process with semi-Markov matrix as in Eq. (12) and π = 1 / 2, P-divisibility always holds, in particular as shown in the examples the trajectories never cross and | q ( t ) | never grows.”

• AssumptionRests on an assumption: A semi-Markov process is Markovian if and only if its waiting time distributions are exponential.

May combine two claims.

For π=1, the behaviour depends on the waiting time distribution, which determines whether q(t) shows oscillating behavior, implying trajectories cross.

“For π = 1 on the contrary the behaviour depends on the waiting time distribution, which determines whether or not the quantity q ( t ) shows an oscillating behavior, implying that the trajectories start getting closer till they cross and then get apart once again.”

• AssumptionRests on an assumption: A semi-Markov process is Markovian if and only if its waiting time distributions are exponential.

May combine two claims.

For the convolution of two different exponential distributions in the dissipative model, if λ1/λ2 is far enough from one, γ(t) stays always positive, so P-divisibility holds and N(Φ)=0, while δ(t) is negative, so CP-divisibility is violated and I(Φ)>0.

“If the ratio λ 1 /λ 2 is far enough from one γ ( t ) given by Eq. (71) as discussed above stays always positive, so that one has P-divisibility and the measure N (Φ) is equal to zero. On the contrary the function δ ( t ) is negative, so that the coefficient in front of the quantum channel is always negative and CP-divisibility is violated , thus determining a positive measure I (Φ).”

May combine two claims.

For an exponential waiting time distribution, δ(t) is zero, so the pure quantum channel is not available in the proper Markovian case.

“To consider the behavior of the measures for this model we thus have to consider also the quantity δ ( t ), which is simply equal to zero for an exponential waiting time distribution, so that in the proper Markovian case this pure quantum channel is not available.”

May combine two claims.

For a convex mixture of two memoryless distributions, γ(t) always takes on positive values, and both measures are zero.

“which according to the definitions of 〈 λ 〉 and λ given in Eq. (43) and Eq. (44) can be checked to always take on positive values.”

• Also stated in: § 3. Explicit example · p. 11 · plain

  “In this situation both measures are equal to zero.”

May combine two claims.

For the projection CPT map corresponding to pure dephasing, CP-divisibility always holds, so both non-Markovianity measures are zero, whatever the waiting time distribution is.

“As a result CP-divisibility always holds, so that both non-Markovianity measures are equal to zero, whatever the waiting time distribution is.”

May combine two claims.

The measure based on CP-divisibility [13] gives the same infinite value to quite different time evolutions, whereas the measure based on the dynamics of the trace distance [12] assigns different weights to them.

“In particular, as shown by Eq. (64) and Eq. (67) and discussed thoroughly in Sect. IV B 3, we put into evidence that the measure based on CP-divisibility [13] gives the same infinite value to quite different time evolutions, at variance with the measure based on the dynamics of the trace distance [12], which assigns to them different weights.”

• cites [13] sourceNot checked yet

  Not checked in the free run.

• cites [12] sourceNot checked yet

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May combine two claims.

There are several fundamental distinctions between the classical and quantum notion of non-Markovianity, as well as between the various quantum measures for non-Markovianity.

“Our results clearly demonstrate several fundamental distinctions between the classical and the quantum notion of non-Markovianity, as well as between the various quantum measures for non-Markovianity.”

Probably not a substantial claim: may not pass S2 (checkable).

Markovianity or non-Markovianity should be a property of the map or equivalently of the time evolved states, not of the equations admitting such states as solutions.

“Note that Markovianity or non-Markovianity should be a property of the map or equivalently of the time evolved states, not of the equations admitting such states as solutions.”

May combine two claims.

In classical semi-Markov processes, Markovianity or non-Markovianity depends only on the choice of f(t), while P-divisibility and behavior of the Kolmogorov distance depend on both f(t) and Π.

“In Sect. III we considered semiMarkov processes with a semi-Markov matrix of the form Eq. (12), with arbitrary f ( t ) and Π a bistochastic matrix, and where Markovianity or non-Markovianity of the process depended only on the choice of f ( t ), while Pdivisibility and behavior of the Kolmogorov distance did depend on both f ( t ) and Π.”

• AssumptionRests on an assumption: A semi-Markov process is Markovian if and only if its waiting time distributions are exponential.

May combine two claims.

The one-point probabilities of the considered non-Markovian semi-Markov process can be equally well obtained from a Markov process with conditional transition probability p1|1 given by T, but the n-point probabilities would differ.

“This result implies that the one-point probabilities of the considered non-Markovian semi-Markov process can be equally well obtained from a Markov process with conditional transition probability p 1 | 1 given by T , whose n -point probabilities can be obtained as in Eq. (7). The latter would however differ from those of the considered semi-Markov process.”

• AssumptionRests on an assumption: A semi-Markov process is Markovian if and only if its waiting time distributions are exponential.

Classified as probably not a substantial claim.

Markovianity or non-Markovianity is a property of the solution ρ(t), rather than of the equation.

“while Markovianity or non-Markovianity is a property of the solution ρ ( t ), rather then of the equation, it is quite important to read the signatures of a non-Markovian behavior from the equations themselves, and this task turns out to be much easier when the equations are written in time-local form.”

May combine two claims.

The lack of P-divisibility and monotonic decrease of Kolmogorov distance can be interpreted as a signature of non-Markovianity and used to quantify it.

“The lack of these properties can thus be interpreted as signature of non-Markovianity, and used to quantify it.”

Probably not a substantial claim: may not pass S3 (load-bearing).

For bistochastic CPT maps, Φ(t,0) is bistochastic, preserving both trace and identity.

“In the quantum setting we also leave f ( t ) arbitrary and consider bistochastic CPT maps, in the sense that E [ BD ] = BD , so that also Φ ( t, 0) is bistochastic, preserving both the trace and the identity.”

May combine two claims.

Knowledge of a non-Markovian classical process requires information on all conditional probability densities, whereas studying an open system's dynamics only considers the evolution of the state.

“While knowledge of a non-Markovian classical process requires information on all the conditional probability densities, when studying the dynamics of an open system one only considers the evolution of the state, expressed by a probability vector in the classical case and a statistical operator for a quantum system.”

May combine two claims.

For a single quantum channel, positivity of the time-dependent rate ensures CP-divisibility of the time evolution, which is violated if γ(t) becomes negative at some point.

“Thus, as discussed in [16], for the case of a single quantum channel positivity of the time dependent rate ensures CP-divisibility of the time evolution, which is violated if γ ( t ) becomes negative at some point.”

• cites [16] sourceNot checked yet

  Not checked in the free run.

May combine two claims.

The map Φ(t,0) is completely positive by construction and can be interpreted as the repeated action of the CPT map E, distributed in time according to f(t).

“The map Φ( t, 0) is thus completely positive by construction and can be interpreted as the repeated action of the CPT map E , implementing a quantum operation, distributed in time according to the waiting time distribution f ( t ).”

May combine two claims.

For an exponential waiting time distribution, the memory kernel is a delta function and one goes back to a semigroup dynamics.

“For the case of an exponential waiting time distribution the memory kernel is a delta function and one goes back to a semigroup dynamics.”

May combine two claims.

For a Markovian process, P-divisibility is always granted and the Kolmogorov distance is a monotone contraction; non-Markovianity can spoil these features, but neither the lack of P-divisibility nor the growth of the Kolmogorov distance can be taken as necessary signatures of non-Markovianity.

“In particular, we have seen that while for a Markovian process P-divisibility is always granted and the Kolmogorov distance is a monotone contraction, non-Markovianity can spoil these features, even though neither the lack of P-divisibility nor the growth of the Kolmogorov distance can be taken as necessary signatures of non-Markovianity.”

May combine two claims.

The master equations of the form given generate by construction completely positive dynamics, which provide a quantum counterpart of classical semi-Markov processes.

“Such master equations generate by construction completely positive dynamics, which provide a quantum counterpart of classical semi-Markov processes [5, 30].”

• cites [5] sourceNot checked yet

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• cites [30] Feller W, 1971, An Introduction to Probability Theory and its ApplicationsNot checked yet

  Not checked in the free run.

Classified as probably not a substantial claim.

The two criteria of CP-divisibility and trace distance are at the basis of two recently introduced measures of non-Markovianity for open quantum systems [12,13].

“These two criteria are at the basis of two recently introduced measures of non-Markovianity for open quantum systems [12, 13], which we here compare considering a quantum counterpart of classical semiMarkov processes.”

• cites [12] sourceNot checked yet

  Not checked in the free run.

• cites [13] sourceNot checked yet

  Not checked in the free run.