Characterization and measurement of qubit-environment-entanglement generation during pure dephasing

Abstract

We consider the coupling of a qubit in a pure state to an environment in an arbitrary state, and characterize the possibility of qubit-environment entanglement generation during the evolution of the joint system, that leads to pure dephasing of the qubit. We give a simple necessary and sufficient condition on the initial density matrix of the environment together with the properties of the interaction, for appearance of qubit-environment entanglement. Any entanglement created turns out to be detectable by the Peres-Horodecki criterion. Furthermore, we show that for a large family of initial environmental states, the appearance of nonzero entanglement with the environment is necessarily accompanied by a change in the state of the environment (i.e. by the back-action of the qubit). When a quantum system, a qubit (Q) in the context of this paper, is coupled to an environment (E), an initial pure state of the qubit evolves into a mixed state. It is widely recognized that there is an intimate relation between this process of decoherence [1, 2] and creation of qubit-environment entanglement (QEE). A precise statement can be made for E initialized in a pure state: the establishment of QEE is then equivalent to the reduction of purity of the reduced density matrix of Q. However, in the case of an initial mixed state of E, the situation is more complicated: the state of Q can lose its purity while no QEE is established [3]. This should not be surprising, since QEE should be associated with decoherence, understood as a process in which information is transferred from Q to E (i.e. E is in some sense "measuring" Q), not with the bare fact that the state of Q is becoming mixed. The latter can happen when there is no influence of Q on E. For example, when the self-HamiltonianĤ E of E commutes with the qubit-environment interactionV QE , and when the initial state of E fulfillsρ E (0) = f (Ĥ E ) (e.g. it is a thermal state determined byĤ E ), there is no back-action of Q on E, while E is simply a source of random, classical, and quasi-static fields acting on Q [4][5][6][7]. In this case of so-called random unitary (RU) evolution, the purity of Q decays while no QEE is established.A natural question to ask is whether the RU case is the only one for which QEE does not accompany the loss of purity of Q. While specific kinds of environments and qubit-environment couplings were investigated in this context (e.g. quantum Brownian motion [3,8] or pure dephasing due to a bath of noninteracting bosons [9]), the general answer to this question seems to be lacking. This is to a large degree caused by the fact that quantification (or even checking for presence) of QEE is a very hard problem when mixed states of the total system are considered, and when the dimension of Hilbert space of E is larger than 3 [10-13]. The existence of bound entanglemet [14,15] (which is not detected by the Peres-Horodecki criterion [16,17] of negativity of a partially transposed matrix of the total system) severely hampers the task of general understanding of Q...

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…”

“…Hence, the study of mixed initial environmental states is necessary to assess if SBS states are likely to occur. For mixed states the correlation between system-environment entanglement buildup and system decoherence is less direct; although such entanglement is always accompanied by decoherence, decoherence is also possible without said entanglement [27][28][29][30][31]. In the following we use the methods devised in Refs [30,31] to study the interrelations between system-environment entanglement generation and the possibility of the emergence of SBS states.…”

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].…”