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arXiv:2002.10901 · Quantum Information Processing

A scheme for direct detection of qubit–environment entanglement generated during qubit pure dephasing

Bartosz Rzepkowski, Katarzyna Roszak

Published Dec 18, 2020• 143 citations
arXivHTMLPDF

Abstract

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 $$\tau $$ τ after the initialization of the qubit in a superposition state. It requires a measurement on the qubit at time $$\tau $$ 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.

DOI
10.1007/s11128-020-02935-8

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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.
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.
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 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.
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 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).