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arXiv:1902.04803 · Physical Review A

Optimal measurement strategies for fast entanglement detection

Nadia Milazzo, Daniel Braun, O. Giraud

Published Jul 17, 2019• 7 citations
arXivHTMLPDF

Abstract

With the advance of quantum information technology, the question of how to most efficiently test quantum circuits is becoming of increasing relevance. Here we introduce the statistics of lengths of measurement sequences that allows one to certify entanglement across a given bipartition of a multiqubit system over the possible sequence of measurements of random unknown states and identify the best measurement strategies in the sense of the (on average) shortest measurement sequence of (multiqubit) Pauli measurements. The approach is based on the algorithm of truncated moment sequences, which allows one to deal naturally with incomplete information, i.e., information that does not fully specify the quantum state. We find that the set of measurements corresponding to diagonal matrix elements of the moment matrix of the state are particularly efficient. For symmetric states their number increases only like the third power of the number $N$ of qubits. Their efficiency increases rapidly with $N$, leaving already for $N=4$ less than a fraction ${10}^{\ensuremath{-}6}$ of randomly chosen entangled states undetected.

DOI
10.1103/physreva.100.012328

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With the advance of quantum information technology, the question of how to most efficiently test quantum circuits is becoming of increasing relevance. Here we introduce the statistics of lengths of measurement sequences that allows one to certify entanglement across a given bipartition of a multiqubit system over the possible sequence of measurements of random unknown states and identify the best measurement strategies in the sense of the (on average) shortest measurement sequence of (multiqubit) Pauli measurements. The approach is based on the algorithm of truncated moment sequences, which allows one to deal naturally with incomplete information, i.e., information that does not fully specify the quantum state. We find that the set of measurements corresponding to diagonal matrix elements of the moment matrix of the state are particularly efficient. For symmetric states their number increases only like the third power of the number NNN of qubits. Their efficiency increases rapidly with NNN, leaving already for N=4N=4N=4 less than a fraction 10\ensuremath−6{10}^{\ensuremath{-}6}10\ensuremath−6 of randomly chosen entangled states undetected.
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If only limited control over a multiparticle quantum system is available, a viable method to characterize correlations is to perform random measurements and consider the moments of the resulting probability distribution. We present systematic methods to analyze the different forms of entanglement with these moments in an optimized manner. First, we find the optimal criteria for different forms of multiparticle entanglement in three-qubit systems using the second moments of randomized measurements. Second, we present the optimal inequalities if entanglement in a bipartition of a multiqubit system shall be analyzed in terms of these moments. Finally, for higher-dimensional two-particle systems and higher moments, we provide criteria that are able to characterize various examples of bound entangled states, showing that detection of such states is possible in this framework.
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Abstract Entanglement is one of the key resources of quantum information science which makes identification of entangled states essential to a wide range of quantum technologies and phenomena. This problem is however both computationally and experimentally challenging. Here we use autoencoder neural networks to find semi-optimal set of incomplete measurements that are most informative for the detection of entangled states . We show that it is possible to find high-performance entanglement detectors with as few as three measurements. Also, with the complete information of the state, we develop a neural network that can identify all two-qubits entangled states almost perfectly. This result paves the way for automatic development of efficient entanglement witnesses and entanglement detection using machine learning techniques.
In this work, we present a practical and efficient framework for verifying entangled states when only a tomographically incomplete measurement setting is available-specifically, when access to observables is severely limited. We show how the experimental estimation of a small number of observables can be directly exploited to construct a large family of entanglement witnesses, enabling the efficient identification of entangled states. Moreover, we introduce an optimization approach, formulated as a semidefinite program, that systematically searches for those witnesses best suited to reveal entanglement under the given measurement constraints. We demonstrate the practicality of the approach in a proof-of-principle experiment with photon-polarization qubits, where entanglement is certified using only a fraction of the full measurement data. These results reveal the maximal usefulness of incomplete measurement settings for entanglement verification in realistic scenarios.