Characterizing non-Markovianity via quantum coherence based on Kirkwood-Dirac quasiprobability

Abdallah Slaoui; Rachid Ahl Laamara; Yassine Dakir

REVIEW 4 major objections 4 minor 65 references

Characterizing non-Markovianity via quantum coherence based on Kirkwood-Dirac quasiprobability

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proposes a non-Markovianity measure $N_{\rm CKD}$ built from the imaginary part of Kirkwood-Dirac quasiprobability coherence, and shows in four models that it detects memory backflow at least as well as the $\ell_1$-norm…

desk verdict Routine substitution of KD coherence into the standard positive-slope construction; single-qubit examples are clean, but the monotonicity proof is invalid and the two-qubit formula contradicts the central claim. read the letter →

arxiv 2506.21691 v1 pith:L4WNTMVZ submitted 2025-06-26 quant-ph math-phmath.MP

Yassine Dakir , Abdallah Slaoui , Rachid Ahl Laamara This is my paper · ORCID

classification quant-phmath-phmath.MP MSC 81P4581S22 PACS 03.65.Yz03.67.-a

keywords non-MarkovianityKirkwood-Diracquasiprobabilityquantumcoherenceincoherentoperationsinformationbackflowdephasingchannelamplitudedampingopensystems

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes a new detector of memory effects in open quantum systems: watch the time evolution of a coherence measure built from the imaginary part of the Kirkwood-Dirac quasiprobability, a complex-valued representation of a quantum state. The claim is that this coherence, $C_{\rm KD}$, can only decrease under incoherent completely positive trace-preserving maps, so any growth of $C_{\rm KD}$ during a dynamics signals backflow of information from the environment. The authors define a non-Markovianity measure $N_{\rm CKD}(\Phi_t)$ by integrating the positive part of $dC_{\rm KD}/dt$, and test it on single- and two-qubit dephasing and dissipative channels. In all four models the new measure becomes nonzero in exactly the parameter regimes where the established $\ell_1$-norm coherence measure detects non-Markovianity, and it requires no auxiliary system, which makes the measure potentially simpler to realize in experiments than ancilla-based witnesses.

What carries the argument

The load-bearing object is the KD coherence quantifier $C_{\rm KD}[\varrho;\{X_\mu\}] = \max_{\{|\nu\rangle\}} \sum_{\mu,\nu} |\operatorname{Im} \operatorname{Tr}(X_\nu X_\mu \varrho)|$, the $\ell_1$-norm of the imaginary part of the Kirkwood-Dirac quasiprobability $\operatorname{Tr}(X_\nu X_\mu \varrho)$, maximized over all second bases $\{|\nu\rangle\}$. The imaginary part encodes the commutator between the state and the incoherent reference basis $\{X_\mu = |\mu\rangle\langle\mu|\}$, turning non-commutativity into a real, faithful, convex coherence measure. The non-Markovianity measure $N_{\rm CKD}$ of Eq. (21) integrates the positive time-derivative of $C_{\rm KD}$ along the dynamical map, so every upward swing of the KD coherence counts as information backflow.

What would settle it

Recompute the monotonicity proof's key inequality for a single-qubit random-unitary channel with $U_k$ a rotation by angle $\theta$ about an axis not parallel to the reference-basis projectors $X_\mu$, and compare $C_{\rm KD}[\Phi(\varrho); \{X_\mu\}]$ with $C_{\rm KD}[\varrho; \{X_\mu\}]$. The paper's inequality replaces $U_k^\dagger X_\mu U_k$ with $X_\mu$; if any such channel yields a strict increase in $C_{\rm KD}$, the monotonicity premise (A5) — and with it the interpretation of a rising $C_{\rm KD}$ as memory backflow — fails. Alternatively, a direct scan of a Markovian dephasing dynamics with $\gamma(t) \ge 0$ at all times must show $dC_{\rm KD}/dt \le 0$; any positive excursion is a counterexample to the detection criterion.

Watch

Extended reading notes

Core claim

The central claim is that a coherence quantifier based on the Kirkwood-Dirac quasiprobability — the $\ell_1$-norm of its imaginary part, maximized over the second basis — can serve as a witness for non-Markovianity. Because $C_{\rm KD}$ is argued to be monotonic under incoherent completely positive trace-preserving maps, a Markovian (divisible) evolution must have $dC_{\rm KD}/dt \le 0$ at all times; the measure $N_{\rm CKD}(\Phi_t) = \max_{\varrho(0)} \int_{\sigma_C > 0} \sigma_C(t)\,dt$ collects the positive growth. The paper works out explicit formulas for four models — single-qubit dephasing, single-qubit amplitude damping, two-qubit dephasing in a global reservoir, and two-qubit amplitude damping — and finds that $N_{\rm CKD}$ is nonzero precisely in the non-Markovian regimes, vanishing where all decay rates $\gamma(t)$ stay non-negative. In the single-qubit dephasing case $N_{\rm CKD} = \tfrac{1}{2} N_{C_{\ell_1}}$, so the two measures agree on detection while assigning different magnitudes.

Load-bearing premise

The load-bearing premise is that the KD coherence $C_{\rm KD}$ never increases under any incoherent completely positive trace-preserving map (property A5); the supplied proof covers only random-unitary channels and at one step replaces $U_k^\dagger X_\mu U_k$ with $X_\mu$, shifting the reference basis. If that monotonicity fails for some incoherent map, then a rising $C_{\rm KD}$ would not reliably signal memory effects.

Editorial extensions

If this is right

In all four models examined, $N_{\rm CKD}$ is nonzero in exactly the regimes where $\gamma(t)$ takes negative values, so the KD-coherence measure flags the same Markovian-to-non-Markovian transitions as the $\ell_1$-norm coherence measure.
For single-qubit dephasing, $N_{\rm CKD} = \tfrac{1}{2} N_{C_{\ell_1}}$, meaning the two measures detect the same memory effect but the KD-based measure assigns half the magnitude.
Because $C_{\rm KD}$ is obtained from the KD quasiprobability of the state itself, the measure needs no auxiliary qubit and is within reach of existing schemes for measuring KD quasiprobabilities.
The paper explicitly notes that growth of $C_{\rm KD}$ is necessary but not sufficient for non-Markovianity, so $N_{\rm CKD}$ is a detector of memory effects rather than a complete characterization of the dynamics.
The predicted regime boundaries — ohmicity $s > 2.2$ for single-qubit dephasing, $s < 3$ for two-qubit dephasing, $\kappa/\gamma_0 \in [0.05, 1]$ for single-qubit dissipation, and $\kappa/\gamma_0 \in [0.1, 0.35]$ for two-qubit amplitude damping — are concrete thresholds where memory effects should appear and could be checked in experiments.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The exact factor $1/2$ relating $N_{\rm CKD}$ and $N_{C_{\ell_1}}$ in the single-qubit dephasing model suggests the imaginary-part KD coherence and the $\ell_1$-norm coherence may be proportional for pure dephasing, but the paper only establishes the relation in that one model; where a general proportionality holds is an open question the formulas invite.
Because property (A5) is proved only for random-unitary channels, the detection criterion is on firmest ground precisely for the computed models; a rigorous monotonicity proof for general incoherent maps, or an explicit counterexample, would determine how far the measure extends beyond qubit channels.
The KD nonclassicality $N_c$ shows revivals in the same non-Markovian regimes as $C_{\rm KD}$, so a single KD-tomography experiment could simultaneously track both quantities and test whether nonclassicality revivals are a one-to-one signature of information backflow.
The maximization over the second basis $\{|\nu\rangle\}$ in $C_{\rm KD}$ means the measure is optimized over measurement choices; the paper's 'simplified' closed forms fix that basis, so the full maximization should be checked in regimes where the two could disagree, such as the two-qubit dephasing case.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the paper substitutes Kirkwood-Dirac coherence into the standard “integrate the positive slope of a monotone” construction for non-Markovianity, and the single-qubit examples are computed honestly. But the general claim—that a positive time derivative of CKD signals non-Markovianity and that N_CKD is zero iff Markovian—rests on a monotonicity proof that is invalid, and the paper’s own two-qubit dephasing formula produces a counterexample. I would still send it to a referee, but the central measure as stated is not established.

What is new: using the imaginary part of the KD quasiprobability, maximized over a second basis, as the coherence monotone in this construction. I don’t know of a prior non-Markovianity measure built from CKD, so the definition is a real addition to the catalog. The single-qubit dephasing case is clean: CKD = (1/2)R(t), and N_CKD = -∫_{γ<0} γR dt, which is exactly proportional to the l1-norm measure and shares the γ<0 detection condition. The dissipative case gives N = (1/2)∫ d|B|/dt over positive increments, again a plausible and correctly computed quantity.

Now the soft spots. Property (A5), monotonicity under all incoherent CPTP maps, is the load-bearing wall. The proof only treats random unitary channels, and the step from Eq. (12) to Eq. (13) replaces U_k† X_μ U_k with X_μ. That is not true for a general incoherent unitary, which may permute the basis projectors. The maximization over {|ν⟩} is also performed before the replacement, so the triangle inequality does not control the optimized quantity. Without (A5), Eq. (20) and the “zero iff Markovian” conclusion have no basis.

The two-qubit dephasing section makes the problem concrete. Eq. (58) gives CKD = (1/4)|R(t)^4 sin((h1+h2)t)|. For constant γ>0, R = e^{-2γt}, and this oscillates, so its derivative is positive on intervals whenever the oscillation frequency beats the decay. That directly contradicts the claimed Markovian monotonicity. The resolution is that the calculation did not maximize over all second bases; it fixed α1=α2=π/2, β1=β2=0 and used a product basis. So either Eq. (58) is not the CKD defined in Eq. (5), or (A5) is false. Both readings kill the general claim. The two-qubit amplitude damping formula is similarly computed with a fixed basis.

Citation pattern: they cite the relevant coherence-based measures (Chanda et al., He et al.) and the KD coherence source, so there is no hiding of antecedents.

Who this is for: someone cataloging coherence-based non-Markovianity witnesses might use the single-qubit formulas, but not the general measure. The paper needs either a correct proof of A5 for a restricted class of maps or a redefinition as a witness with a stated domain. My recommendation: engage with it—send it to peer review, because the mistake is technical and the single-qubit derivations are worth preserving. But in your own work, do not cite N_CKD as a reliable quantifier until the monotonicity question is solved.

Referee Report

4 major / 4 minor

Summary. The paper proposes a non-Markovianity measure based on the imaginary part of the Kirkwood-Dirac (KD) quasiprobability. It defines a coherence quantifier CKD in Eq. (5), claims that CKD is monotonic under all incoherent completely positive trace-preserving maps (property A5), and then defines N_CKD in Eq. (21) as the integral of the positive derivative of CKD along the time evolution. The measure is applied to single-qubit dephasing and dissipative channels and to two-qubit dephasing and amplitude-damping channels, and the results are compared with the l1-norm-coherence-based measure. The single-qubit calculations are explicit and reproduce the known l1-norm behaviour up to a factor, but the central theoretical claim rests on property A5, whose proof is invalid, and the two-qubit examples do not implement the required optimization over the second basis.

Significance. If correct, the proposal would offer an experimentally accessible non-Markovianity witness based on a quasiprobability representation, complementing existing measures based on trace distance, Fisher information, or l1-norm coherence. The paper gives credit for deriving closed-form expressions in tractable models and for explicitly connecting the positive-derivative condition with negative decay rates in the single-qubit dephasing case. However, the central monotonicity property A5 is not proved: the proof in Section II is limited to random unitary channels and contains an unjustified substitution, and the two-qubit dephasing formula Eq. (58) appears to contradict monotonicity even for a Markovian constant-rate channel. The two-qubit examples also fix a particular product second basis rather than maximizing over all second bases as required by Eq. (5). These are load-bearing gaps: without a valid proof of A5 and a correct implementation of the maximization, the claim that N_CKD vanishes if and only if the process is Markovian is unsupported. The potential value of the KD-coherence approach is not enough to offset these issues as the manuscript stands.

major comments (4)

[Section II, Eq. (12)-(13)] The proof of property A5 is invalid. In going from Eq. (12) to Eq. (13), the authors replace U_k^\dagger X_\mu U_k by X_\mu inside the trace. For an incoherent unitary, this replacement is not generally correct: an incoherent unitary can permute the reference projectors, yielding U_k^\dagger X_\mu U_k = X_{\pi(\mu)} rather than X_\mu. Additionally, the proof begins by assuming a random unitary channel and never returns to the general incoherent CPTP case; random unitary channels are a strict subset of incoherent CPTP maps. The derivation of Eq. (20), which concludes dCKD/dt \le 0 in the Markovian regime, relies entirely on A5, so this gap is load-bearing for the central claim of the paper.
[Section IV.B.1, Eq. (58)] Equation (58) states CKD[\rho_AB(t)] = (1/4)|R(t)^4 \sin((h_1+h_2)t)|. For a constant positive decay rate \gamma(t)=\gamma>0, the dephasing channel is Markovian, and R(t)=e^{-2\gamma t}; then CKD has intervals of positive time derivative whenever the oscillation frequency exceeds the decay rate. This directly contradicts the claimed monotonicity of CKD under Markovian dynamics. The only way out is that Eq. (58) is not the true CKD because the calculation uses one fixed second basis instead of the maximization over all second bases required by Eq. (5). In either case, the paper's central claim that positive derivative signals non-Markovianity is not supported, and the two-qubit dephasing results cannot be interpreted as stated.
[Section IV.B, Eq. (53)-(54)] The two-qubit calculations restrict the second basis to tensor products of single-qubit bases of the form |\nu_{1\pm}\rangle \otimes |\nu_{2\pm}\rangle and then fix the angles to \alpha_1=\alpha_2=\pi/2, \beta_1=\beta_2=0 (Eq. (53) and the text after Eq. (55)). No argument is given that the maximum over all second bases in Eq. (5) is attained at this particular product basis. All subsequent two-qubit expressions, including Eq. (58) and Eq. (64), are therefore computed for a fixed basis and are not the CKD defined in Eq. (5). The false-positive behavior in the Markovian regime described above is a direct consequence of this omission. The authors must either perform the maximization explicitly or restrict the definition of the measure to a fixed-basis quantity and then justify why such a quantity should detect non-Markovianity.
[Section III, Eq. (21)] The non-Markovianity measure N_CKD is defined with a maximization over all coherent initial states \varrho(0) \in IC, but every application in Section IV evaluates the derivative only for a single fixed initial state, such as |\phi\rangle = (|0\rangle+|1\rangle)/\sqrt{2} in Eq. (26) or (|00\rangle+|11\rangle)/\sqrt{2} in Eq. (55). The reported numerical values are therefore only lower bounds on the maximized measure, and the statement that N_CKD(\Phi_t)=0 only if the process is Markovian is not established for the maximized quantity. The authors should either carry out the maximization or state clearly that the examples illustrate a lower bound rather than the full measure.

minor comments (4)

[Section II, Eq. (5)] The notation in Eq. (5) shows the maximization over {|\mu\rangle}, but the surrounding text says that the maximization is over the second basis {|\nu\rangle}; the reference basis {|\mu\rangle} is fixed. This is confusing and should be corrected.
[Section IV.A.2 and Fig. 2] The text in Section IV.A.2 and the caption of Fig. 2 use inconsistent notation for the reservoir parameters: the text mentions \gamma_0/\lambda while the figure axis and the equation refer to \kappa/\gamma_0. The symbol \lambda is not defined in this context.
[Section II, Eq. (14)] The quantity N_c defined in Eq. (14) is introduced but is not used in any proof or derivation; it appears only in the application figures. The authors should either connect it to the main argument or remove it.
[General] The manuscript contains numerous typographical and grammatical errors, including inconsistent use of \varrho and \rho, missing subscripts, and incomplete sentences (for example, 'This constant is the inverse of the relaxation time ... and is linked to the Markovian decay'). A careful editing pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proposed measure is an explicit functional of the CKD coherence monotone, and the worked examples are independent evaluations rather than fitted predictions.

full rationale

The derivation chain is not circular. Eq. (5) defines CKD as a maximized imaginary-part l1 norm of the Kirkwood-Dirac quasiprobability. Eq. (21) then defines N_CKD as the positive variation of CKD, and Eq. (20) derives dCKD/dt <= 0 for Markovian dynamics from the assumed monotonicity property A5. This is the standard witness construction used for BLP trace-distance and l1-coherence measures, not a fit or a renamed input. The explicit formulas, e.g. Eqs. (29)-(31), (47)-(48), (58) and (64), are closed-form evaluations for stated initial states and bases, and the comparison with N_Cl1 uses published independent formulas ([26]) rather than parameters fitted to the target. The paper's self-citations are contextual references to the authors' earlier work and do not carry the load-bearing monotonicity or uniqueness argument. There are serious correctness concerns in the paper, but they are not circularity: the proof of A5 in Eqs. (12)-(13) unjustifiably replaces U_k^dagger X_mu U_k by X_mu, and Eq. (58) appears inconsistent with A5 for Markovian dephasing if the maximization in Eq. (5) is enforced. These flaws undermine the soundness of the derivation but do not make the claimed prediction equivalent to the input by construction. Therefore no circular step is identified.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the assumed monotonicity of CKD under all incoherent CPTP maps, which is only partially proved, and on a restriction to product bases in the two-qubit examples. No new entities or fitted parameters are introduced.

assumptions (4)

ad hoc to paper Property A5: CKD is monotonic under all incoherent completely positive trace-preserving maps.
The proof in Section II only handles random unitary channels and contains an unjustified replacement of U_k-dagger X_mu U_k by X_mu; the general case is assumed. This property is load-bearing because it turns positive derivative of CKD into a non-Markovianity signal.
domain assumption The dynamics are governed by the time-local master equation with possibly negative rates, and Markovianity is equivalent to all rates being non-negative.
Standard in open quantum systems and used in Eqs. (15)-(18) to connect the sign of gamma(t) to the behavior of CKD.
ad hoc to paper The second basis in the two-qubit optimization can be restricted to product bases without loss of generality.
Eqs. (53)-(54) parameterize only tensor products of local Bloch-sphere bases, whereas Eq. (5) maximizes over all orthonormal bases. Entangled second bases can change the CKD value and would remove the phase oscillations seen in Eq. (58).
standard math The Kirkwood-Dirac quasiprobability representation and the CKD coherence quantifier of Budiyono and Dipojono are valid as used.
The paper builds directly on Ref. [50] for the definition and properties of CKD; it does not re-derive the validity of the quantifier.

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Pith. "Pith review of Characterizing non-Markovianity via quantum coherence based on Kirkwood-Dirac quasiprobability." pith.science (2026). https://pith.science/paper/L4WNTMVZ

@misc{pith2026250621691,
  author       = {Pith},
  title        = {Pith review of: Characterizing non-Markovianity via quantum coherence based on Kirkwood-Dirac quasiprobability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L4WNTMVZ}},
  note         = {Machine review of arXiv:2506.21691}
}

abstract

We present a new measure of non-Markovianity based on the property of nonincreasing quantum coherence via Kirkwood-Dirac (KD) quasiprobability under incoherent completely positive trace-preserving maps. Quantum coherence via the KD quasiprobability is defined as the imaginary part of the KD quasiprobability, which is maximised over all possible second bases and evaluated using an incoherent reference basis. A measure non-Markovianity based on KD quasiprobability coherence would capture memory effects via the time evolution of the imaginary part of the KD quasiprobability, providing an experimentally accessible and physically intuitive alternative to traditional measures relying on quantum Fisher information or trace distance. This approach is applied to the study of dissipation and dephasing dynamics in single- and two-qubit systems. The results obtained show that, in the cases studied, our measure based on coherence via Kirkwood-Dirac quasiprobability performs at least as well as $\ell_{1}$-norm coherence in detecting non-Markovianity, this provides a novel perspective on the analysis of non-Markovian dynamics.

Figures

Figures reproduced from arXiv: 2506.21691 by the authors.

**Figure 2.** FIG. 2. In Fig. (a) we plot coherence KD quasiprobability measurement of the non-Markovianity [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3. In Fig (a) we simplified quantum coherence based on KD quasiprobability of the non-Markovianity [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Fig. (a): Measurement of non-Markovianity for two-qubit amplitude damping channel, with the Lorentzian spectral density given [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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N CKD = 0 , while in the non-Markovian region κ/γ0 ∈ [0.05, 1], N CKD (Φt) takes non-zero values. Fig. (b): Time evolution of the coherence via KD quasiprobability CKD and of nonclassicality Nc (small plot) in the Markovian ( κ/γ0 = 0 .01 and κ/γ0 > 1) and non-Markovian (κ/γ0 = 0 .2) regimes. In the non-Markovian regime, a nonmonotonic behavior of nonclas...

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Dephasing channel with global reservoir for two qubit The Hamiltonian that describes two qubits interacting with each other and connected to a shared thermal reservoir is ex- pressed as follows H = HS + X j ωja† jaj + sz 2 X j (gjaj + g∗ j a† j) (49) The Hamiltonian of the system HS, is expressed as follows HS = 2X i=1 hi 2 σz i + λσz 1σz 2, (50) with sz ...

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Dephasing channel for a single qubit Here, we focus on a single-qubit system subjected to a de- phasing channel. The Hamiltonian that describes the interac- tion with a thermal reservoir [1], is written as follows H = ω0σz + X k ωka† kak + X k gkσzak + g∗ kσza† k , (22) For this model, the qubit’s transition frequency is denoted by ω0, the Pauli-z operato...

work page

[2] [2]

Single-qubit dissipative channel Now, we will examine the dissipative dynamics of a single qubit, using the following Hamiltonian, which is given by the formula H = ω0σ+σ− + X k ωkb† kbk + X k gkbkσ+ + g∗ i b† kσ− (35) where σ+ (σ−) represents the raising (lowering) Pauli opera- tor, and bk and b† k is the creation and annihilation operators. It’s associa...

work page

[3] [3]

N CKD = 0 , while in the non-Markovian region κ/γ0 ∈ [0.05, 1], N CKD (Φt) takes non-zero values. Fig. (b): Time evolution of the coherence via KD quasiprobability CKD and of nonclassicality Nc (small plot) in the Markovian ( κ/γ0 = 0 .01 and κ/γ0 > 1) and non-Markovian (κ/γ0 = 0 .2) regimes. In the non-Markovian regime, a nonmonotonic behavior of nonclas...

work page

[4] [4]

Dephasing channel with global reservoir for two qubit The Hamiltonian that describes two qubits interacting with each other and connected to a shared thermal reservoir is ex- pressed as follows H = HS + X j ωja† jaj + sz 2 X j (gjaj + g∗ j a† j) (49) The Hamiltonian of the system HS, is expressed as follows HS = 2X i=1 hi 2 σz i + λσz 1σz 2, (50) with sz ...

work page

[5] [5]

representing the total spin operator along the z axis, and λ being the strength coupling parameter be- tween two-qubit. The dynamics of the system is characterized by the master equation ˙ϱ (t) = −i[HS, ϱ(t)] + γ(t)(szϱ(t)sz − 1 2 [s2 z, ϱ(t)]+, (51) where γ(t) denotes the time-dependent dephasing rate deter- mined from the spectral density function Sd(ω)...

work page

[6] [6]

ka† kσB −) X k (gkakσB + + g

Amplitude damping channel for a two qubit For two-qubit amplitude damping channel, we consider the Hamiltonian H = X i=A,B ω0σi z + X k ωka† kak + X k (gkakσA + + g. ka† kσB −) X k (gkakσB + + g. ka† kσA −) (59) After the system has undergone amplitude damping, the initial state ϱAB(0) results in the diagonal and off-diagonal elements of the amplitude dam...

work page

[7] [7]

H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002)

2002

[8] [8]

Banerjee, Open Quantum Systems: Dynamics of Nonclassi- cal Evolution (Springer, Berlin, 2018), V ol

S. Banerjee, Open Quantum Systems: Dynamics of Nonclassi- cal Evolution (Springer, Berlin, 2018), V ol. 20

work page 2018