Density-operator evolution: Complete positivity and the Keldysh real-time expansion

Abstract

We study the reduced time-evolution of general open quantum systems by combining insights from quantum-information and statistical field theory. Inspired by prior work [Eur. Phys. Lett. 102, 60001 (2013) and Phys. Rev. Lett. 111, 050402 (2013)] we establish the explicit structure guaranteeing the complete positivity (CP) and trace-preservation (TP) of the real-time evolution expansion in terms of the microscopic system-environment coupling.This reveals a fundamental two-stage structure of the coupling expansion: Whereas the first stage naturally defines the dissipative timescales of the system -before having integrated out the environment completely-the second stage sums up elementary physical processes, each described by a CP superoperator. This allows us to establish the highly nontrivial functional relation between the (Nakajima-Zwanzig) memory-kernel superoperator for the reduced density operator and novel memory-kernel operators that generate the Kraus operators of an operator-sum. We illustrate the physically different roles of the two emerging coupling-expansion parameters for a simple solvable model. Importantly, this operational approach can be implemented in the existing Keldysh real-time technique and allows approximations for general time-nonlocal quantum master equations to be systematically compared and developed while keeping the CP and TP structure explicit.Our considerations build on the result that a Kraus operator for a physical measurement process on the environment can be obtained by 'cutting' a group of Keldysh real-time diagrams 'in half '. This naturally leads to Kraus operators lifted to the system plus environment which have a diagrammatic expansion in terms of time-nonlocal memory-kernel operators. These lifted Kraus operators obey coupled time-evolution equations which constitute an unraveling of the original Schrödinger equation for system plus environment. Whereas both equations lead to the same reduced dynamics, only the former explicitly encodes the operator-sum structure of the coupling expansion. arXiv:1808.09395v3 [cond-mat.stat-mech] 25 Jul 2019 42 E.2 Pasting rules 43 F Functional relation of self-energies Σ m and σ m 43 References 44

Fermionic duality: General symmetry of open systems with strong dissipation and memory

“…For semigroup dynamics this coefficient is time-constant, but in general it is time-dependent, even in the resonant level model, see π 0 (t) in Table 2. The result (33) implies that the expectation value of a system observable A can be decomposed into an instantaneous expectation value in the time-dependent fixed-point state plus corrections:…”

“…Thus, also its analytical structure proves clearly that the time-dependence of the dual propagatorΠ(t) is not physical, complementing the discussion of the algebraic, operational constraints of CP and TP [Sec. 3.2.3] which are independent of time, see also [33].…”

“…( 57), the approximate evolution ( 77) is both CP and TP. This is in general very difficult to achieve for nonperturbative approximations [29][30][31][32][33]. This class includes dynamics which is not a trivial semigroup described by a Lindblad equation, which is already the case for the resonant level model (except for T = ∞ or = µ, see Sec.…”

“…The same applies to the so-called Choi-Jamiołkowski state, which is closely related to the measurement-operator sum by a well-known isomorphism [26][27][28]. The microscopic calculation of Kraus operators has received recent attention [29][30][31][32][33] but remains very difficult, motivating our search for analytic simplifications.…”

“…This follows from Π(t) (−1) N † = 1 PΠ(t) †(23) = e −Γ t 1 Π (t)P = e −Γ t (−1) N 8. The CP property is very difficult to maintain when performing approximations, see Ref [33]. for a discussion and references.…”