How to detect qubit-environment entanglement in pure dephasing evolutions

The problem of detecting entanglement between a qubit and its environment is known to be complicated [1]. To simplify the issue, we study the class of Hamiltonians that describe the interacting system in such a way that the resulting evolution of the qubit alone is of pure dephasing type. Although this leads to some loss of generality, the pure dephasing Hamiltonian describes the dominant decohering mechanism for many types of qubits. When both the qubit and the environment are initially in a pure state, their interaction leading to qubit dephasing always leads to the creation of entanglement between the two [2]. It is often assumed that such a dephasing mechanism must induce entanglement between the qubit and environment also when the environment is initially in a mixed state. We have shown that while the creation of qubit-environment entanglement in the pure dephasing case is possible when the environment is initially in a mixed state, its occurrence is by no means guaranteed [3]. We have also shown that the evolution of the environment conditional on the qubit state is qualitatively different in entangling and non-entangling scenarios [3]. This serves as a basis for possible detection of qubit-environment entanglement via measurements on only one of these subsystems. Obviously, such entanglement could be straightforwardly determined by measurements on the environment, but such measurements are rarely accessible. Here, we propose a scheme for the detection of qubit-environment entanglement which requires operations and measurements on the qubit subsystem alone [4]. It relies on the fact that only for entangling evolutions does the environment behave differently in the presence of different qubit states. Hence, only if an evolution is entangling can there be a difference in the evolution of qubit coherence when the environment was allowed to relax in the presence of either qubit pointer states prior to the excitation of a superposition state. The scheme is in fact an entanglement witness. If a difference in the decay of coherence of this superposition is detected then the interaction with the environment is entangling. If not, then either there is no entanglement or the conditional evolution operators of the environment commute. We illustrate the concept with a calculation performed for a nitrogen-vacancy center in diamond, a spin qubit coupled to a nuclear spin environment that is widely used for noise spectroscopy [5].

References
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[2] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
[3] K. Roszak and L. Cywiński, Phys. Rev. A 92, 032310 (2015).
[4] K. Roszak, D. Kwiatkowski and Ł. Cywiński, Phys. Rev. A 100, 022318 (2019).
[5] L. Degen, F. Reinhard, and P. Cappellaro, Rev. Mod. Phys. 89, 035002 (2017).