Adding dynamical generators in quantum master equations Jan Ko lody´nski,1 Jonatan Bohr Brask,2 Mart´ıPerarnau-Llobet,1, 3 and Bogna Bylicka1 1ICFO{Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain 2Group of Applied Physics, University of Geneva, 1211 Geneva, Switzerland 3Max-Planck-Institut f¨urQuantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany The quantum master equation is a widespread approach to describing open quantum system dynamics. In this approach, the effect of the environment on the system evolution is entirely captured by the dynamical generator, providing a compact and versatile description. However, care needs to be taken when several noise processes act simultaneously or the Hamiltonian evolution of the system is modified. Here, we show that generators can be added at the master equation level without compromising physicality only under restrictive conditions. Moreover, even when adding generators results in legitimate dynamics, this does not generally correspond to the true evolution of the system. We establish a general condition under which direct addition of dynamical generators is justified, showing that it is ensured under weak coupling and for settings where the free system Hamiltonian and all system-environment interactions commute. In all other cases, we demonstrate by counterexamples that the exact evolution derived microscopically cannot be guaranteed to coincide with the dynamics naively obtained by adding the generators. I. INTRODUCTION In this work, we address the questions of when: (i) The naive addition of generators yields physically It is generally impossible to completely isolate a small valid dynamics. system of interest from the surrounding environment. (ii) The corresponding evolution coincides with the true Thus, dissipative effects caused by the environment are system dynamics derived from the underlying micro- important in almost every quantum experiment, ranging scopic model. from highly controlled settings, where much effort is in- First, we show that (i) is satisfied for generators which vested in minimising them, to areas where the dissipation are commutative, semigroup-simulable (can be inter- is the key object of interest. In many cases, exact mod- preted as a fictitious semigroup at each time instance), elling of the environment is not practical and its effect and preserve commutativity of the dynamics under addi- is instead accounted for by employing effective models tion. These reach beyond the case of Markovian genera- describing the induced noise. Different approaches ex- tors for which (i) naturally holds. Outside of this class, ist, e.g., quantum Langevin and stochastic Schr¨odinger we find examples of simple qubit QMEs which lead to equations [1,2], quantum jump and state-diffusion mod- unphysical dynamics. We observe that (ii) holds if and els [3,4], or Hilbert-space averaging methods [5]. only if the cross-correlations between distinct environ- Arguably, the most widely applied approach is to use ments can be ignored within a QME. We show this to be the quantum master equation (QME) description [1,2]. the case in the weak-coupling regime, extending previous In this approach, the system evolution is given by a time- results in this direction [7, 12, 13]. We also provide a suf- local differential equation, where the effect of the envi- ficient condition for (ii) dictated by the commutativity of ronment is captured by the dynamical generator. A mas- Hamiltonians at the microscopic level. We combine these ter equation can be derived from a microscopic model of generic considerations with a detailed study of a specific the system and environment, and their interaction, by open system, namely a qubit interacting simultaneously tracing over the environment and applying appropriate with multiple spin baths, for which we provide examples approximations [1,2]. However, QMEs are also often ap- where (ii) is not satisfied, while choosing the microscopic arXiv:1704.08702v3 [quant-ph] 19 May 2018 plied directly, without explicit reference to an underlying Hamiltonians to fulfil particular commutation relations. model. In that case, care needs to be taken when several Our results are of relevance to areas of quantum noise processes act in parallel, as simultaneous coupling physics where careful description of dissipative dynam- to multiple baths in a microscopic model does not gen- ics plays a key role, e.g., in dissipative quantum state erally correspond to simple addition of noise generators. engineering [14{17], dissipative coupling in optomechan- Moreover, when the Hamiltonian evolution of the sys- ics [18], or in dissipation-enhanced quantum transport tem is modified, e.g., when controlling system dynamics scenarios [11, 19], including biological processes [20]. In by coherent driving [6], the form of noise generators in a particular, they are of importance to situations in which QME may significantly change. Additivity of noise at the QMEs are routinely employed to account for multiple QME level has been discussed recently for qubits when sources of dissipation, e.g., in quantum thermodynamics analysing dynamical effects of interference between dif- [21{24] when dealing with multiple heat baths [8, 25{27] ferent baths [7,8], non-additivity of relaxation rates in or in quantum metrology [28{30] where the relation be- multipartite systems [9, 10], as well as in the context of tween dissipation and Hamiltonian dynamics, encoding charge (excitation) transport [11]. the estimated parameter, is crucial [31{34]. 2 The manuscript is structured as follows. In Sec.II, we A. Physicality of dynamical generators discuss QMEs at an abstract level|as defined by families of dynamical generators whose important properties we For the QME (1) to be physically valid, it must yield summarise in Sec.IIA. We specify in Sec.IIB conditions dynamics that is consistent with quantum theory. In par- under which the addition of physically valid generators ticular, upon integration the QME must lead to a family is guaranteed to yield legitimate dynamics. We demon- of (dynamical) maps Λt (parametrised by t) that satisfy strate by explicit examples that even mild violation of ρS(t) = Λt[ρS(0)] for any t ≥ 0 and initial ρS(0), with these conditions may lead to unphysical evolutions. each Λt being completely positive and trace preserving In Sec. III, we view the validity of QMEs from the mi- (CPTP) [38, 39]. croscopic perspective. In particular, we briefly review in On the other hand, any QME (1) is unambiguously Sec. IIIA the canonical derivation of a QME based on specified by the family of (dynamical) generators Lt ap- an underlying microscopic model, in order to discuss the pearing in Eq. (1). However, as discussed in App.A, effect of changing the system Hamiltonian on the QME, although the CPTP condition can be straightforwardly as well as the generalisation to interactions with multi- checked for maps Λt, it does not directly translate onto ple environments. We then formulate a general criterion the generators Lt. As a result, for a generic QME for the validity of generator addition in Sec. IIIB, which its physicality cannot be easily inferred at the level of we explicitly show to be ensured in the weak coupling Eq. (1), unless its explicit integration is possible. Never- regime, or when particular commutation relations of the theless, we formally call a family of dynamical generators microscopic Hamiltonians are fulfilled. Lt physical if the family of maps it generates consists In Sec.IV, we develop an exactly solvable model of a only of CPTP transformations. In what follows (see also qubit interacting with multiple spin baths, which allows App.A1), we describe properties of dynamical genera- us to explicitly construct counterexamples that disprove tors that ensure their physicality. the microscopic validity of generator addition in all the Any family of dynamical generators, whether physical regimes in which the aforementioned commutation rela- or not, can be uniquely decomposed as [37] tions do not hold. Finally, we conclude in Sec.V. d2−1 X 1 L [ρ] = −i[H(t); ρ] + D (t) F ρF y − fF yF ; ρg ; t ij i j 2 j i II. TIME-LOCAL QUANTUM MASTER i;j=1 EQUATIONS (2) d2 where d is the Hilbert space dimension and fFigi=1 is any QMEs constitute a standard tool to describe reduced y orthonormal operator basis with TrfFi Fjg = δij and all dynamics of open quantum systems. They provide a com- p F traceless except F 2 = 11= d. The Hamiltonian part, pact way of defining the effective system evolution at the i d Ht, of the generator in Eq. (1) is then determined by level of its density matrix, ρS(t), without need for ex- H(t) of Eq. (2), while the dissipative part Dt is defined plicit specification neither of environmental interactions by the Hermitian matrix D(t). Although general criteria nor the nature of the noise. Although a QME may be for physicality of dynamical-generator families are not expressed in a generalised form as an integro-differential known, two natural classes of physical dynamics can be equation involving time-convolution [35], its equivalent identified based on the above decomposition. (c.f. [36]) and more transparent time-local formulation is In particular, when D(t) is positive semidefinite, Lt typically favoured, providing a more direct connection to is said to be of Gorini-Kossakowski-Sudarshan-Lindblad the underlying physical mechanisms responsible for the (GKSL) form [37, 40]. If this is the case for all t ≥ 0, dissipation [1,2]. Given a time-local QME: then the corresponding evolution is not only physical but d also CP-divisible, i.e., the corresponding family of maps ρS(t) = Lt[ρS(t)] = Ht[ρS(t)] + Dt[ρS(t)]; (1) can be decomposed as Λt = Λ~ t;sΛs, where Λ~ t;s is CPTP dt for all 0 ≤ s ≤ t. This property is typically associated all the information about the system evolution is con- with Markovianity of the evolution [41{43].