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Physical Review A

Evolution of any finite open quantum system always admits a Kraus-type representation, although it is not always completely positive

David Cienfuegos Salgado, J. L. Sánchez‐Gómez, Miguel Ferrero

Published Nov 22, 2004• 14 citations
Publisher

Abstract

We argue that a full account of the set of quantum states prevents one from applying the rigorous definition of complete positivity. However, we give three equivalent proofs that any quantum evolution of a finite system always admits a Kraus-type decomposition, i.e., a Kraus decomposition but with Kraus matrices dependent on the initial state upon which they apply.

DOI
10.1103/physreva.70.054102

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We argue that a full account of the set of quantum states prevents one from applying the rigorous definition of complete positivity. However, we give three equivalent proofs that any quantum evolution of a finite system always admits a Kraus-type decomposition, i.e., a Kraus decomposition but with Kraus matrices dependent on the initial state upon which they apply.
For any master equation which is local in time, whether Markovian, non-Markovian, of Lindblad form or not, a general procedure is given for constructing the corresponding linear map from the initial state to the state at time t, including its Kraus-type representations. Formally, this is equivalent to solving the master equation. For an N-dimensional Hilbert space it requires (i) solving a first order N 2×N 2 matrix time evolution (to obtain the completely positive map), and (ii) diagonalizing a related N 2×N 2 matrix (to obtain a Kraus-type representation). Conversely, for a given time-dependent linear map, a necessary and sufficient condition is given for the existence of a corresponding master equation, where the (not necessarily unique) form of this equation is explicitly determined. It is shown that a “best possible” master equation may always be defined, for approximating the evolution in the case that no exact master equation exists. Examples involving qubits are given.
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.
We investigate completely positive maps for an open system interacting with its environment. The families of the initial states for which the reduced dynamics can be described by a completely positive map are identified within the framework of direct-sum decomposition of state space. They include not only separable states with vanishing or nonvanishing quantum discord but also entangled states. A general expression of the families as well as the Kraus operators for the completely positive maps are explicitly given. It significantly extends the previous results.