The Nakajima–Zwanzig memory-kernel master equation describes non-Markovian open-system dynamics by making the instantaneous rate of change depend on the entire history of the state, but it is notoriously difficult to decide which kernels generate a legitimate (completely positive, trace-preserving) evolution. Chruściński and Kossakowski [Phys. Rev. A 94, 020103(R) (2016)] introduced a sufficient-condition framework: a 'legitimate pair' {N(t), Q(t)} — a completely positive monotonic quantum operation N(t) playing the role of a quantum survival map, and a completely positive jump map Q(t) with a trace-normalization condition — whose convolution-renewal series always defines a legitimate dynamical map and covers Markovian semigroups, quantum semi-Markov evolution, and collision models. Three linked questions arising from that paper remain open in general: (i) which structural properties of the pair, especially N(t), encode Markovianity (CP-divisibility and related information-flow notions) of the induced dynamical map; (ii) what are necessary and sufficient conditions for a memory kernel to be admissible, given that the pair conditions are provably only sufficient; and (iii) how can the associated homogeneous Nakajima–Zwanzig kernel — available in the Laplace domain but generically not invertible to the time domain — be systematically computed or approximated. Later work resolved these questions only in restricted classes (generalized Pauli/diagonal kernels, phase-covariant qubits, commutative dynamics, renewal and semi-Markov processes, Gaussian collisional models), so the general characterization problem stands open.

Background

An open quantum system is a finite-dimensional quantum system coupled to an environment; its reduced state evolves under a family of dynamical maps Λ(t)\Lambda(t) sending the initial state ρ0\rho_0 to ρ(t)\rho(t). Physical consistency requires each Λ(t)\Lambda(t) to be completely positive (CP: the map stays positive even when the system is entangled with an arbitrary auxiliary system) and trace-preserving (TP), with Λ(0)\Lambda(0) the identity. When Λ(t)=etL\Lambda(t)=e^{tL} is a one-parameter semigroup, the generator LL has the canonical Gorini–Kossakowski–Sudarshan–Lindblad (GKLS) form with non-negative decoherence rates. Non-Markovian evolutions are described either by a time-local equation ρ˙(t)=L(t)ρ(t)\dot{\rho}(t)=L(t)\rho(t) or by the Nakajima–Zwanzig (NZ) memory-kernel equation ρ˙(t)=∫0tKNZ(t−τ) ρ(τ) dτ\dot{\rho}(t)=\int_0^t K_{\mathrm{NZ}}(t-\tau)\,\rho(\tau)\,d\tau, whose kernel KNZK_{\mathrm{NZ}} encodes memory of the whole history. For time-local equations the GKLS theorem settles exactly when CP is guaranteed, and Markovianity is standardly operationalized as CP-divisibility: the two-parameter propagators V(t,s)V(t,s) defined by Λ(t)=V(t,s)Λ(s)\Lambda(t)=V(t,s)\Lambda(s) are CP for all t≥st\ge s (equivalently, all instantaneous GKLS rates are non-negative; related information-flow measures such as trace-distance revival — a temporary re-increase of the trace distance between two evolving states, signalling information flowing back from the environment — define closely related but distinct notions). For memory kernels no analogous characterization is known: phenomenological kernels such as K(t)=k(t)LK(t)=k(t)L, or the post-Markovian kernel of Shabani and Lidar, can generate non-CP evolutions depending on the memory function and the generator, while constructions like the quantum semi-Markov processes of Breuer and Vacchini guarantee CP by design through classical waiting-time distributions.

Chruściński and Kossakowski (2016) parametrized a large class of admissible kernels by a legitimate pair {N(t),Q(t)}\{N(t),Q(t)\}: N(t)N(t) is a monotonic quantum operation — a completely positive, trace-non-increasing map with N(0)=1N(0)=\mathbb{1}, the quantum analogue of a survival map (in the semi-Markov case N(t)=g(t) 1N(t)=g(t)\,\mathbb{1} with gg the survival probability of a waiting-time distribution, and monotonicity of N(t)N(t) means the Heisenberg-picture quantity N(t)‡(1)N(t)^{\ddagger}(\mathbb{1}) is non-increasing in tt; throughout, M‡M^{\ddagger} denotes the adjoint (Heisenberg-picture) superoperator of a map MM, defined by Tr⁡[A M(B)]=Tr⁡[M‡(A) B]\operatorname{Tr}[A\,M(B)]=\operatorname{Tr}[M^{\ddagger}(A)\,B] for all operators AA and BB) — and Q(t)Q(t) is a completely positive jump map satisfying the normalization Q(t)‡(1)=−ddtN(t)‡(1)Q(t)^{\ddagger}(\mathbb{1})=-\tfrac{d}{dt}N(t)^{\ddagger}(\mathbb{1}). The pair generates the dynamical map through the renewal series

Λ(t)=N(t)+(N∗Q)(t)+(N∗Q∗Q)(t)+⋯ ,\Lambda(t) = N(t) + (N*Q)(t) + (N*Q*Q)(t) + \cdots,

where ∗* denotes time convolution; this Λ(t)\Lambda(t) is CPTP by construction and solves the NZ equation with a kernel expressed through the Laplace transforms of NN and QQ. The class contains Markovian semigroups, quantum semi-Markov evolution and collision models (stroboscopic dynamics in which the system sequentially interacts with fresh, uncorrelated environmental units), and is closed under convex combinations, reduction, and gauge transformations (compositions N′(t)=F(t)N(t)N'(t)=F(t)N(t), Q′(t)=F(t)Q(t)Q'(t)=F(t)Q(t) with a dynamical map F(t)F(t), which take a legitimate pair to another legitimate pair, though generically not to the same dynamical map) — but it is not exhaustive: the same Λ(t)\Lambda(t) can arise from pairs outside the sufficient class (e.g. a projection channel EE with E2=EE^2=E gives Λ(t)=g(t)1+[1−g(t)]E\Lambda(t)=g(t)\mathbb{1}+[1-g(t)]E), so the pair conditions are sufficient but not necessary. Moreover, the homogeneous NZ kernel associated with a general pair is given by a Laplace-domain formula that, in the authors' words, is 'in general ... not feasible to invert ... to the time domain', while the associated inhomogeneous kernel K~(s)=sQ~(s)\widetilde{K}(s)=s\widetilde{Q}(s) is simple. The authors close by noting that it would be interesting to link the properties of the pair with well-studied properties of the dynamical map such as Markovianity, and that the essential features of the evolution might already be encoded in N(t)N(t) — an open problem they explicitly leave for further analysis.

Problem Statement

For a finite-dimensional open quantum system, consider the Nakajima–Zwanzig memory-kernel master equation and the legitimate-pair parametrization of Chruściński and Kossakowski: pairs {N(t),Q(t)}\{N(t),Q(t)\} with N(t)N(t) a completely positive trace-non-increasing map with N(0)=1N(0)=\mathbb{1} and Q(t)Q(t) completely positive satisfying Q(t)‡(1)=−ddtN(t)‡(1)Q(t)^{\ddagger}(\mathbb{1})=-\tfrac{d}{dt}N(t)^{\ddagger}(\mathbb{1}), generating the dynamical map by the renewal series Λ(t)=N(t)+(N∗Q)(t)+(N∗Q∗Q)(t)+⋯\Lambda(t)=N(t)+(N*Q)(t)+(N*Q*Q)(t)+\cdots. The common objective is a structural characterization of this parametrization that makes admissibility, Markovianity and practical kernel construction decidable from the pair. Concretely, three linked questions, all open in general and all stated or demonstrated as limitations in the source paper:

(1) Encoding of Markovianity. Characterize, in terms of structural properties of the legitimate pair — especially monotonicity and decay properties of N(t)N(t), and the effect of the pair's gauge transformations on the induced dynamics — when the induced dynamical map Λ(t)\Lambda(t) is Markovian, i.e. CP-divisible (and how this relates to other standard non-Markovianity notions such as trace-distance information backflow). In particular, determine whether Markovianity in the sense of CP-divisibility of Λ(t)\Lambda(t) is determined by the survival map N(t)N(t) alone — that is, whether any two legitimate pairs {N(t),Q1(t)}\{N(t),Q_1(t)\} and {N(t),Q2(t)}\{N(t),Q_2(t)\} that share the same N(t)N(t) generate dynamical maps Λ1(t)\Lambda_1(t) and Λ2(t)\Lambda_2(t) that are simultaneously CP-divisible or simultaneously not CP-divisible — proving this where it holds or exhibiting an explicit finite-dimensional counterexample where it fails.

(2) Necessary and sufficient admissibility. Determine necessary and sufficient conditions on a memory-kernel superoperator K(t)K(t) for the induced one-parameter family to be a legitimate (completely positive, trace-preserving) dynamical map with Λ(0)=1\Lambda(0)=\mathbb{1}, going beyond the legitimate-pair conditions, which are provably only sufficient (the same Λ(t)\Lambda(t) can arise from pairs outside the sufficient class). The characterization should delimit exactly which kernels are admissible in the general finite-dimensional case, not only for kernels diagonal in the generalized Pauli (Weyl) operator basis, which generalizes the qubit Pauli matrices to dd-level systems, and should make transparent how it quotients over the non-uniqueness of the kernel-to-map relation.

(3) Time-domain reconstruction. Provide a systematic method — exact, algorithmic, or with certified error control — to compute or approximate in the time domain the homogeneous Nakajima–Zwanzig kernel associated with a general legitimate pair, whose Laplace-domain expression is generally not invertible in closed form; the source paper obtains closed-form kernels only in special cases such as semi-Markov evolution and collision models.

References

  1. D. Chruściński and A. Kossakowski, "Sufficient conditions for a memory-kernel master equation", Phys. Rev. A 94, 020103 (2016). DOI: 10.1103/PhysRevA.94.020103; arXiv:1602.01642, https://arxiv.org/abs/1602.01642
  2. A. Shabani and D. A. Lidar, "Completely positive post-Markovian master equation via a measurement approach", Phys. Rev. A 71, 020101 (2005). DOI: 10.1103/PhysRevA.71.020101; arXiv:quant-ph/0404077, https://arxiv.org/abs/quant-ph/0404077
  3. H.-P. Breuer and B. Vacchini, "Quantum semi-Markov processes", Phys. Rev. Lett. 101, 140402 (2008). DOI: 10.1103/PhysRevLett.101.140402; arXiv:0809.1501, https://arxiv.org/abs/0809.1501
  4. B. Vacchini, "Generalized master equations leading to completely positive dynamics", Phys. Rev. Lett. 117, 230401 (2016). DOI: 10.1103/PhysRevLett.117.230401; arXiv:1611.01150, https://arxiv.org/abs/1611.01150
  5. D. Chruściński and A. Kossakowski, "Generalized semi-Markov quantum evolution", Phys. Rev. A 95, 042131 (2017). DOI: 10.1103/PhysRevA.95.042131; arXiv:1701.06534, https://arxiv.org/abs/1701.06534
  6. K. Siudzińska and D. Chruściński, "Memory kernel approach to generalized Pauli channels: Markovian, semi-Markov, and beyond", Phys. Rev. A 96, 022129 (2017). DOI: 10.1103/PhysRevA.96.022129; arXiv:1706.10041, https://arxiv.org/abs/1706.10041
  7. K. Siudzińska and D. Chruściński, "Engineering fidelity of the generalized Pauli channels via legitimate memory kernels", Phys. Rev. A 100, 012303 (2019). DOI: 10.1103/PhysRevA.100.012303; arXiv:1811.07589, https://arxiv.org/abs/1811.07589
  8. D. Chruściński, "Conditions for legitimate memory kernel master equation", in Classical and Quantum Physics, Springer Proceedings in Physics, pp. 147–162 (2019). DOI: 10.1007/978-3-030-24748-5_8
  9. S. N. Filippov, A. N. Glinov and L. Leppäjärvi, "Phase covariant qubit dynamics and divisibility", Lobachevskii Journal of Mathematics 41, 617–630 (2020). DOI: 10.1134/S1995080220040095; arXiv:1911.09468, https://arxiv.org/abs/1911.09468
  10. N. Megier, A. Smirne and B. Vacchini, "The interplay between local and non-local master equations: exact and approximated dynamics", New J. Phys. 22, 083011 (2020). DOI: 10.1088/1367-2630/ab9f6b; arXiv:2001.11948, https://arxiv.org/abs/2001.11948
  11. N. Megier, A. Smirne and B. Vacchini, "Evolution equations for quantum semi-Markov dynamics", Entropy 22, 796 (2020). DOI: 10.3390/e22070796; arXiv:2007.11276, https://arxiv.org/abs/2007.11276
  12. B. Vacchini, "Quantum renewal processes", Scientific Reports 10, 5592 (2020). DOI: 10.1038/s41598-020-62260-z; arXiv:1906.00693, https://arxiv.org/abs/1906.00693
  13. R. Ramírez Camasca and G. T. Landi, "Memory kernel and divisibility of Gaussian collisional models", Phys. Rev. A 103, 022202 (2021). DOI: 10.1103/PhysRevA.103.022202; arXiv:2008.00765, https://arxiv.org/abs/2008.00765
  14. K. Nestmann, V. Bruch and M. R. Wegewijs, "How quantum evolution with memory is generated in a time-local way", Phys. Rev. X 11, 021041 (2021). DOI: 10.1103/PhysRevX.11.021041; arXiv:2002.07232, https://arxiv.org/abs/2002.07232
  15. S. Utagi, S. Banerjee and R. Srikanth, "On the non-Markovianity of quantum semi-Markov processes", Quantum Information Processing 20, 399 (2021). DOI: 10.1007/s11128-021-03302-x; arXiv:2012.11219, https://arxiv.org/abs/2012.11219
  16. N. Megier, M. Ponzi, A. Smirne and B. Vacchini, "Memory effects in quantum dynamics modelled by quantum renewal processes", Entropy 23, 905 (2021). DOI: 10.3390/e23070905; arXiv:2106.07697, https://arxiv.org/abs/2106.07697
  17. D. Chruściński, "Dynamical maps beyond Markovian regime", Physics Reports 992, 1–85 (2022). DOI: 10.1016/j.physrep.2022.09.003; arXiv:2209.14902, https://arxiv.org/abs/2209.14902
  18. T. Saha, Sahil, K. P. Athulya and S. Ghosh, "Post-Markovian master equation à la microscopic collisional model", arXiv:2411.16878 (2024), https://arxiv.org/abs/2411.16878
  19. A. Watanabe, T. Suzuki, M. Unoki and H. Nakazato, "CP conditions for GKSL-like master equations", arXiv:2406.11205 (2024), https://arxiv.org/abs/2406.11205