Criteria for system-environment entanglement generation for systems of any size in pure-dephasing evolutions

Abstract

An evolution between a system and its environment which leads to pure dephasing of the system may either be a result of entanglement building up between the system and the environment or not (the second option is only possible for initially mixed environmental states). We find a way of distinguishing between an entangling and non-entangling evolutions for systems which are larger than a single qubit. The generalization of the single qubit separability criterion to larger systems is not sufficient to make this distinction (it constitutes a necessary condition of separability). A set of additional conditions for the operators describing the evolution of the environment depending on the state of the system is required. We find that the commutation of these environmental operators with the initial state of the environment does not guarantee separability, products of the operators need to commute among themselves for a pure dephasing evolution not to be accompanied by system-environment entanglement generation. This is a qualitative difference with respect to the single-qubit case, since it allows for a system to entangle with an initially completely mixed environment.PACS numbers:

Entanglement and objectivity in pure dephasing models

“…The measurement Hamiltonian which is used to study the emergence of spectrum broadcast structure states is, in fact a special case of the general Hamiltonian which leads to pure dephasing evolutions [6][7][8][9][10][11][12][13] in a systemenvironment scenario [30,31]. The Hamiltonian is given byĤ…”

How to detect qubit environment entanglement in pure dephasing evolutions

“…It has been shown previously [10,11] that for this class of interactions and initial states, the if and only if criterion of QE separability at time t is (it is also the criterion for zero-discord [16,17]…”

“…Entanglement, which in this case is directly linked to system decoherence, describes the amount information about the state of the system which can be extracted from the environment [8,9]. It has been recently shown for mixed states, that without such information transfer, pure dephasing is not accompanied by entanglement generation [10,11].…”

“…The measure has a straightforward physical interpretation in contrast to all other measures of mixed state entanglement, directly linking the amount of information about the qubit state which is contained in the environment to the amount of QEE. This means that the pure state interpretation of entanglement for the class of interactions studied [8,9] can be directly extended to mixed states, even though the link between entanglement and decoherence cannot [10][11][12].…”

“…Pure state Fidelity is given by F (|ψ 1 , |ψ 2 ) = | ψ 1 |ψ 2 | 2 , so, since the conditional evolution of the pure initial state of the environment yields |R i (t) =ŵ i (t)|R (corresponding to the density matricesR ii (t) = |R i (t) R i (t)|), the Fidelity between the two environmental states is given by F R 00 (t),R 11…”