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arXiv:1911.08867 · Physical Review A

Qubit-environment negativity versus fidelity of conditional environmental states for a nitrogen-vacancy-center spin qubit interacting with a nuclear environment

Małgorzata Strzałka, Damian Kwiatkowski, Łukasz Cywiński, Katarzyna Roszak

Published Oct 8, 2020• 10 citations
arXivHTMLPDF

Abstract

We study the evolution of qubit-environment entanglement, quantified using negativity, for nitrogen-vacancy-center spin qubits interacting with an environment of partially polarized nuclear spins in the diamond lattice. We compare it with the evolution of the fidelity of environmental states conditional on the pointer states of the qubit, which can serve as a tool to distinguish between entangling and nonentangling decoherence in the pure-dephasing scenario considered here. The two quantities show remarkable agreement during the evolution in a wide range of system parameters, leading to the conclusion that the amount of entanglement generated between the qubit and the environment is likely to be proportional to the trace that the joint evolution leaves on the environment.

DOI
10.1103/physreva.102.042602

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We study the evolution of qubit-environment entanglement, quantified using negativity, for nitrogen-vacancy-center spin qubits interacting with an environment of partially polarized nuclear spins in the diamond lattice. We compare it with the evolution of the fidelity of environmental states conditional on the pointer states of the qubit, which can serve as a tool to distinguish between entangling and nonentangling decoherence in the pure-dephasing scenario considered here. The two quantities show remarkable agreement during the evolution in a wide range of system parameters, leading to the conclusion that the amount of entanglement generated between the qubit and the environment is likely to be proportional to the trace that the joint evolution leaves on the environment.
We generalize the scheme for detection of qubit-environment entanglement to qudit-environment systems. This is of relevance for many-qubit systems and the quantification of the operation of quantum algorithms under the influence of external noise, since only decoherence that is not entangling in its nature can be effectively described by quantum channels and similar methods in more complicated scenarios. The generalization involves an increase in the class of entangled states which are not detected by the scheme, but the type of entanglement which cannot be detected is also least likely to qualitatively influence decoherence. We exemplify the operation of the scheme on a realistically modeled nitrogen-vacancy-center spin qutrit interacting with an environment of nuclear spins.
We propose a qubit-environment entanglement measure which is tailored for evolutions that lead to pure dephasing of the qubit, such as are abundant in solid-state scenarios.The measure can be calculated directly form the density matrix without minimization of any kind.In fact, it does not require knowledge of the full density matrix and it is enough to know the initial qubit state and the states of the environment conditional on qubit pointer states.This yields a computational advantage over standard entanglement measures, which becomes large when there are no correlations between environmental components in the conditional environmental states.In contrast to all other measures of mixed-state entanglement, the measure has a straightforward physical interpretation directly linking the amount of information about the qubit state which is contained in the environment to the amount of qubit-environment entanglement.This allows for a direct extension of the pure-state interpretation of entanglement generated during pure dephasing to mixed states, even though pure-state conclusions about qubit decoherence are not transferable.
Direct access to the initial polarization of 13C{}^{13}C13C nuclei by measuring coherence evolution of an nitrogen-vacancy center spin qubit
We introduce a method for the measurement of the lower bound on the initial polarization of spinful nuclei in a diamond by following the coherence evolution of an NV center spin qubit after a simple scheme is operated on the qubit to facilitate the transfer of information from the environment into the qubit state. Current polarization measurement techniques are challenging to implement due to the need for direct access to the environment. In our method, information is obtained by measuring the difference of the evolution of the qubit coherence resulting from preparation phase when the environment evolution is conditional on the qubit pointer state. We find that the method does not depend strongly on the applied magnetic field, but rather on the number of spinfull nuclei that lead to decoherence, and gives a reasonable estimate if the environment is polarized. The key advantage of this approach is its simplicity and minimal experimental requirements, allowing the inference of initial nuclear polarizations without direct access to the environment. We demonstrate the efficacy of this method using a simulated environment of up to fifteen randomly placed nuclear spins.
We propose a straightforward experimental protocol to test whether qubit-environment entanglement is generated during pure dephasing of a qubit. The protocol is implemented using only measurements and operations on the qubit---it does not involve the measurement of the system-environment state of interest, but the preparation and measurement of the qubit in two simple variations. A difference in the time dependencies of qubit coherence between the two cases testifies to the presence of entanglement in the state of interest. Furthermore, it signifies that the environment-induced noise experienced by the qubit cannot be modeled as a classical stochastic process independent of the qubit state. We demonstrate the operation of this protocol on a realistically modeled nitrogen-vacancy center spin qubit in diamond interacting with a nuclear spin environment, and show that the generation of entanglement should be easily observable in this case.
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.
We consider pure dephasing of Bell states of electron spin qubits interacting with a sparse bath of nuclear spins. Using the newly developed two-qubit generalization of cluster-correlation expansion method, we calculate the spin-echo decay of $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$ and $|\mathrm{\ensuremath{\Phi}}\ensuremath{\rangle}$ states for various interqubit distances. Comparing the results with calculations in which dephasing of each qubit is treated independently, we identify signatures of influence of common part of the bath on the two qubits. At large interqubit distances, this common part consists of many nuclei weakly coupled to both qubits, so that decoherence caused by it can be modeled by considering multiple uncorrelated sources of noise (clusters of nuclei), each of them weakly affecting the qubits. Consequently, the resulting genuinely two-qubit contribution to decoherence can be described as being caused by classical Gaussian noise. On the other hand, for small interqubit distances the common part of the environment contains clusters of spins that are strongly coupled to both qubits, and their contribution to two-qubit dephasing has visibly non-Gaussian character. We show that one can easily obtain information about non-Gaussianity of environmental noise affecting the qubits from the comparison of dephasing of $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$ and $|\mathrm{\ensuremath{\Phi}}\ensuremath{\rangle}$ Bell states. Numerical results are obtained for two nitrogen vacancy centers interacting with a bath of 13C^{13}\mathrm{C}13C nuclei of natural concentration, for which we obtain that Gaussian description of a correlated part of environmental noise starts to hold for centers separated by about 3 nm.