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arXiv:1407.2051 · Physical Review A

Completely positive maps within the framework of direct-sum decomposition of state space

Longjiang Liu, D. M. Tong

Published Jul 7, 2014• 19 citations
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Abstract

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.

DOI
10.1103/physreva.90.012305

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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.
The description of the dynamics of a system that may be correlated with its environment is only meaningful within the context of a specific framework. Different frameworks rely upon different assumptions about the initial system-environment state. We reexamine the connections between complete positivity and quantum discord within two different sets of assumptions about the relevant family of initial states. We present an example of a system-environment state with nonvanishing quantum discord that leads to a completely positive map. This invalidates an earlier claim about the necessity of vanishing quantum discord for completely positive maps. In our final remarks, we discuss the physical validity of each approach.
We introduce a framework for the construction of completely positive dynamical evolutions in the presence of system-environment initial correlations. The construction relies upon commutativity of the compatibility domain obtained by considering the marginals with respect to the environmental degrees of freedom of the considered class of correlated states, as well as basic properties of completely positive maps. Our approach allows to consider states that can have finite discord, though it does not include entangled states, and it explicitly shows the non-uniqueness of the completely positive extensions of the obtained dynamical map outside the compatibility domain. The possible relevance of such maps for the treatment of open quantum system dynamics is critically discussed, together with the connection to previous literature.
Two long-standing open problems in quantum theory are to characterize the class of initial system-bath states for which quantum dynamics is equivalent to (i) a map between the initial and final system states, and (ii) a completely positive (CP) map. The CP map problem is especially important, due to the widespread use of such maps in quantum information processing and open quantum systems theory. Here we settle both these questions by showing that the answer to the first is "all", with the resulting map being Hermitian, and that the answer to the second is that CP maps arise exclusively from the class of separable states with vanishing quantum discord.
We expand the set of initial states of a system and its environment that are known to guarantee completely positive reduced dynamics for the system when the combined state evolves unitarily. We characterize the correlations in the initial state in terms of its quantum discord [1]. We prove that initial states that have only classical correlations lead to completely positive reduced dynamics. The induced maps can be not completely positive when quantum correlations including, but not limited to, entanglement are present.
In the present thesis, we examine two subjects in the theory of open quantum systems. The first subject deals with the description of the dynamics of open systems initially correlated with the environment. In the theory of open quantum systems, maps characterizing the dynamics of a quantum system in contact with the environment are usually thought to be completely positive. However, this is not necessarily true if the system and its environment are initially correlated, unless we restrict the domain on which the map acts, in other words, only a subset of the set of states of the system gets mapped to other states by the dynamical map. We present a framework for the construction of reduced dynamical maps for subsystems of indistinguishable fermionic particles. We show that in this scenario, a reduced map in the Kraus representation is possible for some sets of states where the only non-classical correlation present is exchange. The second subject studied is related to the characterization of non-Markovian dynamics with the divisibility and entanglement criteria. We obtain the analytical expression for the Kraus decomposition of the quantum map of an environment modeled by an arbitrary quadratic fermionic Hamiltonian acting on one or two qubits, we derive simple functions to examine the non positivity of the intermediate map. In the particular case of an environment represented by the Ising Hamiltonian, we discuss the two sources of non-Markovianity in the model, one due to the finite size of the lattice, and another due to the kind of interactions.
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.