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

by V. Reimer, M. R. Wegewijs

This Submission thread is now published as SciPost Phys. 7, 012 (2019)

Submission summary

As Contributors: Viktor Reimer · Maarten Wegewijs
Arxiv Link: (pdf)
Date accepted: 2019-06-19
Date submitted: 2019-06-07 02:00
Submitted by: Reimer, Viktor
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Mathematical Physics
  • Quantum Physics
Approach: Theoretical


We study the reduced time-evolution of open quantum systems by combining quantum-information and statistical field theory. Inspired by prior work [EPL 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 defines the dissipative timescales of the system --before having integrated out the environment completely-- the second stage sums up elementary physical processes described by CP superoperators. This allows us to establish the nontrivial relation between the (Nakajima-Zwanzig) memory-kernel superoperator for the density operator and novel memory-kernel operators that generate the Kraus operators of an operator-sum. 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 Schr\"odinger 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.

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Schrödinger equation

Published as SciPost Phys. 7, 012 (2019)

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Submission & Refereeing History

Published as SciPost Phys. 7, 012 (2019)

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Resubmission 1808.09395v2 on 7 June 2019

Reports on this Submission

Anonymous Report 1 on 2019-6-12 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1808.09395v2, delivered 2019-06-12, doi: 10.21468/SciPost.Report.1014


The authors have undertaken a very careful revision of the manuscript and addressed all my points of concern in a satisfactory and balanced way. It has to be expected that their thorough analysis of CP and TP properties together with the relation to quantum entanglement and expansions in terms of physical processes will bring the two communities working with quantum information based or quantum field theoretical methods closer together. It will sharpen the mind of these two communities w.r.t. the advantages and disadvantages of their methods and motivates the study of further approximation schemes bringing together the various aspects. In this respect I regard the manuscript to be a unique piece of work with a lot of potential of high impact in the field of open quantum systems. Therefore, I strongly recommend publication in SciPost physics.

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