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arXiv:1807.07175 · Physical Review A

Completely positive maps for reduced states of indistinguishable particles

Leonardo da Silva Souza, Tiago Debarba, Diego L. B. Ferreira, Fernando Iemini, Reinaldo O. Vianna

Published Nov 28, 2018• 5 citations
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Abstract

We introduce a framework for the construction of completely positive maps for subsystems of indistinguishable fermionic particles. In this scenario, the initial global state is always correlated, and it is not possible to tell system and environment apart. Nonetheless, a reduced map in the operator sum representation is possible for some sets of states where the only nonclassical correlation present is exchange.

DOI
10.1103/physreva.98.052135

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We introduce a framework for the construction of completely positive maps for subsystems of indistinguishable fermionic particles. In this scenario, the initial global state is always correlated, and it is not possible to tell system and environment apart. Nonetheless, a reduced map in the operator sum representation is possible for some sets of states where the only nonclassical correlation present is exchange.
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.
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.
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.
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.
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 construct a broad class of completely positive maps and Gorini-Kossakowski-Sudarshan-Lindblad generators for fermionic systems induced by linear transformations of system and environment modes. For these maps, we derive explicit Heisenberg-picture formulas for arbitrary normally ordered monomials in terms of minors of the underlying mode-transformation matrices and environment correlation tensors. We show that for even environment states the linear span of monomials up to any fixed order is invariant, which yields closed equations for low-order moments and makes their computation efficient. We also discuss the relation of this construction to second quantization of non-Hermitian one-particle contractions and extend the formalism to completely positive maps arising from post-selection.