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arXiv:1704.08180 · Physical Review A

Entanglement generation between a charge qubit and its bosonic environment during pure dephasing: Dependence on the environment size

Tymoteusz Salamon, Katarzyna Roszak

Published Sep 22, 2017• 13 citations
arXivHTMLPDF

Abstract

We study entanglement generated between a charge qubit and a bosonic bath due to their joint evolution which leads to pure dephasing of the qubit. We tune the parameters of the interaction, so that the decoherence is quantitatively independent of the number of bosonic modes taken into account and investigate how the entanglement generated depends on the size of the environment. A second parameter of interest is the mixedness of the initial state of the environment which is controlled by temperature. We show analytically that for a pure initial state of the environment, entanglement does not depend on environment size. For mixed initial states of the environment, the generated entanglement decreases with the increase of environment size. This effect is stronger for larger temperatures, when the environment is initially more mixed, but in the limit of an infinitely large environment, no entanglement is created at any finite temperature.

DOI
10.1103/physreva.96.032333

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We study entanglement generated between a charge qubit and a bosonic bath due to their joint evolution which leads to pure dephasing of the qubit. We tune the parameters of the interaction, so that the decoherence is quantitatively independent of the number of bosonic modes taken into account and investigate how the entanglement generated depends on the size of the environment. A second parameter of interest is the mixedness of the initial state of the environment which is controlled by temperature. We show analytically that for a pure initial state of the environment, entanglement does not depend on environment size. For mixed initial states of the environment, the generated entanglement decreases with the increase of environment size. This effect is stronger for larger temperatures, when the environment is initially more mixed, but in the limit of an infinitely large environment, no entanglement is created at any finite temperature.
We study entanglement generated between a qubit and a super-ohmic bosonic bath due to their joint evolution which leads to pure dephasing of the qubit. We tune the parameters of the interaction, so that the decoherence is quantitatively independent of the number of bosonic modes taken into account (and corresponds to realistic decoherence seen in small excitonic quantum dots coupled to an environment of phonons) and investigate, how the entanglement generated depends on the size of the environment. A second parameter of interest is the mixedness of the initial state of the environment which is controlled by temperature. We show analytically that for a pure initial state of the environment (zero temperature here), entanglement does not depend on environment size. At finite temperatures (mixed initial state of the environment), the generated entanglement decreases with the increase of environment size. This effect is stronger for larger temperatures, when the environment is initially more mixed, but in the limit of an infinitely large environment, no entanglement is created at any finite temperature.
Decoherence of qubits is one main limitation in quantum information processing, where the qubits themselves can be mixed because of decoherence and additional decoherence may generated through interactions with a bath. Here, we present a complete study of qubit-environment entanglement in the case of interactions that lead to pure dephasing of the qubit by allowing the initial state of the qubit to be mixed, exceeding previous restrictions. We find that the criteria for separability that hold for pure initial states of the qubit are reduced to witness status. Yet, the evolution of the witnesses reflects the array of features that entanglement can exhibit in the mixed case but are forbidden in the pure-qubit case. Most notably, this includes sudden birth of entanglement as the initial phase of the evolution is typically separable. Moreover, there is a stark numerical advantage to the study of entanglement for large environments when using the witness rather than the traditional measures available. We include the study of entanglement generated between a spin qubit and a sparse spin environment to illustrate the peculiarities of the evolution of entanglement in this case, as well as to emphasize the level of reliability of the entanglement witness.
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 the 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 backaction of the qubit).
The fact that in the course of an open evolution there is, in a vast majority of cases, the creation of system-environment entanglement out of the initial coherence, has been known for many years. However, how much entanglement can be created under general assumptions has not been analyzed yet. In this work, we investigate system-environment entanglement generation under a broad and important class of interactions known as pure dephasing interactions and for arbitrary initial conditions. Our main results are the following bounds on the relative entropy of entanglement Er(σSE)E_r(σ_{SE})Er​(σSE​) after evolution: $ C_r(ρ_S) - H(\mathcal{I}|\mathcal{M}) \leq E_r(σ_{SE}) \leq C_r(ρ_S) - C_r(σ_S) + H(\mathcal{I})$, connecting it on one side to the initial relative entropy of coherence, Cr(ρS)C_r(ρ_S)Cr​(ρS​), and information extractable from the environment, H(I∣M)H(\mathcal{I}|\mathcal{M})H(I∣M), and on the other side to the decoherence strength, $ C_r(ρ_S) - C_r(σ_S)$, making quantitatively precise the statement that larger entanglement leads to more decoherence. As an illustration, we apply our results to the spin-boson model, where a qubit interacts with a bosonic bath, showing time evolution of the entanglement and the Markovian/non-Markovian crossover. Our results help better understand and quantify the complicated correlations, produced during an open evolution, which will hopefully lead to better and new decoherence mitigation techniques.
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 entangling and nonentangling 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.
Abstract We propose a scheme for the detection of qubit–environment entanglement which requires only operations and measurements on the qubit, all within reach of current experimental state-of-the-art. The scheme works for any type of interaction which leads to pure dephasing of the qubit as long as the initial qubit state is pure. The scheme is direct in the sense that it allows the detection of entanglement present in the system at time τ\tau τ τ after the initialization of the qubit in a superposition state. It requires a measurement on the qubit at time τ\tau τ τ and a comparison of the post-measurement evolution to the evolution obtained by a modified scheme. It becomes particularly simple when one of the qubit states is neutral with respect to the environment, such as in case of the most common choice of the NV center spin qubit or for excitonic charge qubits, when the environment is initially at thermal equilibrium. In this case, the post-measurement evolution needs to be compared only to the standard decoherence which is obtained without any qubit manipulation after the preparation of the initial state.