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arXiv:1804.09320 · Physical Review A

Experimental classification of entanglement in arbitrary three-qubit pure states on an NMR quantum information processor

Amandeep Singh, Harpreet Singh, Kavita Dorai, Arvind

Published Sep 4, 2018• 24 citations
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Abstract

We undertake experimental detection of the entanglement present in arbitrary three-qubit pure quantum states on an NMR quantum information processor. Measurements of only four observables suffice to experimentally differentiate between the six classes of states which are inequivalent under stochastic local operation and classical communication. The experimental realization is achieved by mapping the desired observables onto Pauli $z$ operators of a single qubit, which is directly amenable to measurement. The detection scheme is applied to known entangled states as well as to states randomly generated using a generic scheme that can construct all possible three-qubit states. The results are substantiated via direct full quantum state tomography as well as via negativity calculations and the comparison suggests that the protocol is indeed successful in detecting tripartite entanglement without requiring any a priori information about the states.

DOI
10.1103/physreva.98.032301

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We undertake experimental detection of the entanglement present in arbitrary three-qubit pure quantum states on an NMR quantum information processor. Measurements of only four observables suffice to experimentally differentiate between the six classes of states which are inequivalent under stochastic local operation and classical communication. The experimental realization is achieved by mapping the desired observables onto Pauli zzz operators of a single qubit, which is directly amenable to measurement. The detection scheme is applied to known entangled states as well as to states randomly generated using a generic scheme that can construct all possible three-qubit states. The results are substantiated via direct full quantum state tomography as well as via negativity calculations and the comparison suggests that the protocol is indeed successful in detecting tripartite entanglement without requiring any a priori information about the states.
Random local measurements have recently been proposed to construct entanglement witnesses and thereby detect the presence of bipartite entanglement. We experimentally demonstrate the efficacy of one such scheme on a two-qubit NMR quantum-information processor. We show that a set of three random local measurements suffices to detect the entanglement of a general two-qubit state. We experimentally generate states with different amounts of entanglement and show that the scheme is able to clearly witness entanglement. We perform complete quantum state tomography for each state and compute state fidelity to validate our results. Further, we extend previous results and perform a simulation using random local measurements to optimally detect bipartite entanglement in a hybrid system of $2\ensuremath{\bigotimes}3$ dimensionality.
We experimentally explore the state space of three qubits on a nuclear magnetic resonance (NMR) quantum-information processor. We construct a scheme to experimentally realize a canonical form for general three-qubit states up to single-qubit unitaries. This form involves a nontrivial combination of Greenberger-Horne-Zeilinger (GHZ) and WWW-type maximally entangled states of three qubits. The general circuit that we have constructed for the generic state reduces to those for GHZ and WWW states as special cases. The experimental construction of a generic state is carried out for a nontrivial set of parameters and the good fidelity of preparation is confirmed by complete state tomography. The GHZ and WWW states are constructed as special cases of the general experimental scheme. Further, we experimentally demonstrate a curious fact about three-qubit states, where for almost all pure states, the two-qubit reduced states can be used to reconstruct the full three-qubit state. For the case of a generic state and for the WWW state, we demonstrate this method of reconstruction by comparing it to the directly tomographed three-qubit state.
Employing the Pauli matrices, we have constructed a few operators, which can be used to distinguish six inequivalent classes of entanglement under SLOCC (stochastic local operation and classical communication) for three-qubit pure states. Motivated by the form of Mermin's operator, these operators have very simple structure and most importantly they can be implemented in an experiment to distinguish the types of entanglement present in a state. We have shown that the measurement of only one operator is sufficient to show that whether the tangle of the state in question is zero or not zero. Thus we are able to detect whether the given state is of GHZ-type or not. Also it is shown that it is possible to detect and classify W-type states or biseparable states or separable states by performing a small number of measurements. Furthermore, we discuss the teleportation scheme of Lee et.al. \cite{lee} and have shown that the partial tangle and hence teleportation fidelity can be computed experimentally. We have also shown that the partial tangle and hence teleportation fidelity can be used to classify genuine entangled state, biseparable state and separable state.
Quantum entanglement is the fundamental hallmark of quantum mechanics and a core resource for realizing long-distance quantum communication and scalable linear quantum computing. Accordingly, the precise detection and quantitative quantification of entanglement constitute a foundational and critical problem in quantum information theory. To date, researchers have proposed numerous sufficient conditions for entanglement detection as well as a variety of entanglement measures to characterize the entanglement strength of quantum states; nevertheless, efficient and direct measurement schemes for core entanglement parameters remain underdeveloped. Based on unitary transformations and auxiliary measurements, this paper proposes a set of quantum circuit schemes capable of directly measuring the bipartite concurrence and the tripartite 3-tangle entanglement measure. By introducing auxiliary qubits and constructing specific controlled unitary operations, the proposed scheme maps the analytical expressions of the two entanglement measures onto the measurement probabilities of output states from quantum circuits. It enables efficient and direct quantitative measurement of bipartite and tripartite entanglement without performing full quantum state tomography. This work provides a feasible technical route for the experimental characterization of entanglement properties and lays a groundwork for the practical deployment of multipartite entanglement resources in quantum information processing.
This thesis focuses on the experimental creation and detection of a variety of quantum correlations using nuclear magnetic resonance hardware. Quantum entanglement, being most common and counter-intuitive, is one of the main type considered in this thesis. Quantum correlations play a major role in achieving, the much talked, computational speedup. Creation and detection of such correlations experimentally is a major thrust area in experimental quantum information processing field. Main goals of the studies undertaken in this thesis were to come up with strategies to detect the entanglement in a 'state-independent' way and with low experimental resources. The core of all the detection protocols is based on our own developed method which enables us to measure any observable with high accuracy. The experimental protocol has been successfully implemented to detect the entanglement of random two-qubit pure states. Further, the schemes for the experimental detection as well as classification of generic and general three-qubit pure states have also been devised and implemented successfully. Quantum correlation possessed by mixed and///or separable states e.g. non-classical, bound-entanglement and of non-local nature were also investigated. In all the investigations, the results were verified by one or more alternative ways e.g. full quantum state tomography.
We study entanglement properties of generic three-qubit pure states. First, we obtain the distributions of both the coefficients and the only phase in the five-term decomposition of Acín et al. for an ensemble of random pure states generated by the Haar measure on U ( 8 ) . Furthermore, we analyze the probability distributions of two sets of polynomial invariants. One of these sets allows us to classify three-qubit pure states into four classes. Entanglement in each class is characterized using the minimal Rényi-Ingarden-Urbanik entropy. Besides, the fidelity of a three-qubit random state with the closest state in each entanglement class is investigated. We also present a characterization of these classes in terms of the corresponding entanglement polytope. The entanglement classes related to stochastic local operations and classical communication (SLOCC) are analyzed as well from this geometric perspective. The numerical findings suggest some conjectures relating some of those invariants with entanglement properties to be ground in future analytical work.