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arXiv:quant-ph/0309081 · Physical Review A

Depolarizing channel as a completely positive map with memory

Sonja Daffer, Krzysztof Wódkiewicz, James D. Cresser, John K. McIver

Published Jul 21, 2004• 198 citations
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Abstract

The prevailing description for dissipative quantum dynamics is given by the Lindblad form of a Markovian master equation, used under the assumption that memory effects are negligible. However, in certain physical situations, the master equation is essentially of a non-Markovian nature. In this paper we examine master equations that possess a memory kernel, leading to a replacement of white noise by colored noise. The conditions under which this leads to a completely positive, trace-preserving map are discussed for an exponential memory kernel.

DOI
10.1103/physreva.70.010304

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The prevailing description for dissipative quantum dynamics is given by the Lindblad form of a Markovian master equation, used under the assumption that memory effects are negligible. However, in certain physical situations, the master equation is essentially of a non-Markovian nature. In this paper we examine master equations that possess a memory kernel, leading to a replacement of white noise by colored noise. The conditions under which this leads to a completely positive, trace-preserving map are discussed for an exponential memory kernel.
We derive sufficient conditions for the memory-kernel governing nonlocal master equation which guarantee a legitimate (completely positive and trace-preserving) dynamical map. It turns out that these conditions provide natural parametrizations of the dynamical map being a generalization of the Markovian semigroup. This parametrization is defined by the so-called legitimate pair---monotonic quantum operation and completely positive map---and it is shown that such a class of maps covers almost all known examples from the Markovian semigroup, the semi-Markov evolution, up to collision models and their generalization.
We construct a large class of completely positive and trace-preserving non-Markovian dynamical maps for an open quantum system. These maps arise from a piecewise dynamics for statistical operators characterized by a continuous time evolution interrupted by jumps, randomly distributed in time and described by a quantum channel. The state of the open system is shown to obey a closed evolution equation, given by a master equation with a memory kernel and an inhomogeneous term. The non-Markovianity of the obtained dynamics is explicitly assessed studying the behavior of the distinguishability of two different initial system's states with elapsing time.
Semi-Markov processes represent a well-known and widely used class of random processes in classical probability theory. Here, we develop an extension of this type of non-Markovian dynamics to the quantum regime. This extension is demonstrated to yield quantum master equations with memory kernels which allow the formulation of explicit conditions for the complete positivity of the corresponding quantum dynamical maps, thus leading to important insights into the structural characterization of the non-Markovian quantum dynamics of open systems. Explicit examples are analyzed in detail.
We begin with an examination of the assumptions made in the theory of passive quantum error correction. Physical motivation is discussed for the use of completely positive trace-preserving maps as a description of the evolution of quantum systems. The Lindblad master equation is presented as an example of the dynamics of a system thusly described, in the case of a Markovian system. We review noiseless subsystems and decoherence-free subspaces. Building on the generalization of the noiseless subsystem approach, we discuss recent results that provide a robust procedure for finding noiseless subsystems, given a specific, not necessarily unital, channel. Unitarily correctable subsystems are introduced, which form a type of active quantum error correction that does not require a measurement. The key result for these systems is that we can apply our procedure for finding noiseless subsystems to find unitarily correctable subsystems, in the case of a unital noise model.
We provide a general construction of quantum generalized master equations with a memory kernel leading to well-defined, that is, completely positive and trace-preserving, time evolutions. The approach builds on an operator generalization of memory kernels appearing in the description of non-Markovian classical processes and puts into evidence the nonuniqueness of the relationship arising due to the typical quantum issue of operator ordering. The approach provides a physical interpretation of the structure of the kernels, and its connection with the classical viewpoint allows for a trajectory description of the dynamics. Previous apparently unrelated results are now connected in a unified framework, which further allows us to phenomenologically construct a large class of non-Markovian evolutions taking as the starting point collections of time-dependent maps and instantaneous transformations describing the microscopic interaction dynamics.