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arXiv:2209.02288 · Open Systems & Information Dynamics

On the Use of Total State Decompositions for the Study of Reduced Dynamics

Andrea Smirne, Nina Megier, Bassano Vacchini

Published Jun 1, 2022• 1 citation
arXivHTMLPDF

Abstract

The description of the dynamics of an open quantum system in the presence of initial correlations with the environment needs different mathematical tools than the standard approach to reduced dynamics, which is based on the use of a time-dependent completely positive trace preserving (CPTP) map. Here, we take into account an approach that is based on a decomposition of any possibly correlated bipartite state as a conical combination involving statistical operators on the environment and general linear operators on the system, which allows one to fix the reduced-system evolution via a finite set of time-dependent CPTP maps. In particular, we show that such a decomposition always exists, also for infinite dimensional Hilbert spaces, and that the number of resulting CPTP maps is bounded by the Schmidt rank of the initial global state. We further investigate the case where the CPTP maps are semigroups with generators in the Gorini-Kossakowski-Lindblad-Sudarshan form; for two simple qubit models, we identify the positivity domain defined by the initial states that are mapped into proper states at any time of the evolution fixed by the CPTP semigroups.

DOI
10.1142/s1230161222500081

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The description of the dynamics of an open quantum system in the presence of initial correlations with the environment needs different mathematical tools than the standard approach to reduced dynamics, which is based on the use of a time-dependent completely positive trace preserving (CPTP) map. Here, we take into account an approach that is based on a decomposition of any possibly correlated bipartite state as a conical combination involving statistical operators on the environment and general linear operators on the system, which allows one to fix the reduced-system evolution via a finite set of time-dependent CPTP maps. In particular, we show that such a decomposition always exists, also for infinite dimensional Hilbert spaces, and that the number of resulting CPTP maps is bounded by the Schmidt rank of the initial global state. We further investigate the case where the CPTP maps are semigroups with generators in the Gorini-Kossakowski-Lindblad-Sudarshan form; for two simple qubit models, we identify the positivity domain defined by the initial states that are mapped into proper states at any time of the evolution fixed by the CPTP semigroups.
Given an open quantum system SSS with environment EEE, we establish a one-to-one correspondence between completely positive trace-preserving (CPTP) maps from SSS to EEE and assignment maps initializing the composite system SESESE in a possibly correlated state prior to unitary evolution. We show that every possible joint state ρSEρ_{SE}ρSE​ can be realized by such assignment maps, which are inherently Hermiticity-preserving and trace-preserving (HPTP). While these maps sacrifice global positivity, we derive a simple necessary and sufficient condition for any local state of SSS to belong to their domain of validity. Furthermore, we prove that the geometry of this domain is governed by the joint state assigned to the maximally mixed input, ensuring that a non-zero volume of admissible local states can readily coexist with initial system-environment entanglement. Finally, we provide an operational protocol for witnessing such limits of validity.
This paper develops the open-system companion to the unitary BV-clock quantum dynamics paper. We consider density matrices evolving over a finite-horizon sigma clock with bounded Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) continuous generators and locally finite completely positive trace-preserving (CPTP) atom maps. Under bounded measurable coefficients with integrable generator norm, the sigma-chronological product exists as a CPTP channel evolution, composes through admissible cuts, preserves trace, positivity, and complete positivity, and inserts atom maps in clock order. We also prove that averaging over a probability distribution of admissible sigma clocks preserves CPTP structure by convexity. This random-clock result is deliberately not stated as a semigroup or divisibility theorem: mixtures of admissible channels need not inherit Markovian divisibility without additional hypotheses. A finite open-qubit benchmark checks a GKSL flow with amplitude damping, excitation, dephasing, reset/depolarizing/bit-flip atoms, and random timing averages using trace, Hermiticity, state eigenvalue, Choi eigenvalue, and trace-preservation ledgers.
Abstract We investigate the dynamics of open quantum systems which are initially correlated with their environment. The strategy of our approach is to analyze how given, fixed initial correlations modify the evolution of the open system with respect to the corresponding uncorrelated dynamical behavior with the same fixed initial environmental state, described by a completely positive dynamical map. We show that, for any predetermined initial correlations, one can introduce a linear dynamical map on the space of operators of the open system which acts like the proper dynamical map on the set of physical states and represents its unique linear extension. Furthermore, we demonstrate that this construction leads to a linear, time-local quantum master equation with generalized Lindblad structure involving time-dependent, possibly negative transition rates. Thus, the general non-Markovian dynamics of an open quantum system can be described by means of a time-local master equation even in the case of arbitrary, fixed initial system–environment correlations. We present some illustrative examples and explain the relation of our approach to several other approaches proposed in the literature.
These notes are a short introduction to the mathematical theory of open quantum systems. They are meant to serve as an entry point into a broad research area which has applications across the quantum sciences dealing with systems subjected to external noise. The guiding idea is to let the key structures of the theory emerge from a concrete model. By working through the dissipative Jaynes-Cummings model the reader will dis- cover explicitly how irreversible dynamics arises from a unitary system-reservoir evolution. The notions of the continuous mode limit, correlation functions, spectral density appear in a natural manner and lead to the evolution equation of the open system in form of a master equation. This sets the stage for the more general analysis of completely positive, trace preserving (CPTP) maps and the study of quantum dynamical semigroups. We motivate and prove the Kraus representation theorem, the dilation theorem and the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) theorem. Working through the exercises (for which full solutions are supplied) will reinforce the ideas introduced in the main text.
We investigate the evolution of open quantum systems in the presence of initial correlations with an environment. Here the standard formalism of describing evolution by completely positive trace preserving (CPTP) quantum operations can fail and non-completely positive (non-CP) maps may be observed. A new classification of correlations between a system and environment using quantum discord is explored. However, we find quantum discord is not a symmetric quantity between exchange of systems and this leads to ambiguity in classifications - states which are both quantum and classically correlated depending on the order of the two systems. State preparation in quantum process tomography is investigated with regard to non-CP maps. In SQPT the preparation procedure can influence the complete-positivity of the reconstructed quantum operation if our system is initially correlated with an environment. We examine a recently proposed preparation procedures using projective measurements, and propose our own protocol that uses a single measurement followed by unitary rotations. The former can give rise to non-CP evolution while the later will always give rise to a CP map. State preparation in AAPT was found always to give rise to CP evolution. We examine the effect of statistical noise in process tomography and find it can result in the identification of a non-CP when the evolution should be CP. The variance of the distribution for reconstructed processes is found to be inversely proportional to the number of copies of a state used to perform tomography. Finally, we detail an experiment using currently available linear optics QC devices to demonstrate non-CP maps arising in SQPT.
We derive continuous-time quantum dynamical semigroups from the local geometric structure of discrete completely positive trace-preserving (CPTP) maps on finite-dimensional Hilbert spaces. Rather than assuming a continuous time parameter and differentiability from the outset, we analyze the convex structure of the CPTP set in a neighborhood of the identity channel, viewed as the neutral element of composition. By characterizing the tangent cone at this boundary point under the trace-preserving constraint, we show that admissible first-order perturbations coincide with generators of Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) form. In this sense, the GKSL structure emerges from the intrinsic convex geometry of the CPTP set. Furthermore, we establish a quantitative scaling result: under an explicit operator-norm error estimate, iterated discrete channels converge uniformly on finite intervals to the semigroup generated by the associated GKSL operator. This provides a precise mathematical link between discrete completely positive dynamics and continuous-time Lindblad evolution, clarifying their structural relationship within the foundations of open quantum dynamics.