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arXiv:1602.02099

Entanglement Detection with Fewer Measurements based on the Geometric Criterion

Bingjie Wang, STEPHEN D. BRIERLEY

Published Feb 5, 2016• 0 citations
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Abstract

We present a new technique to reduce the expected number of measurements to declare an unknown quantum state as entangled. Our method is based on the geometric criterion and so requires only local Pauli measurements. Using concentration of measure, we provide a heuristic which allows us to reinterpret a previous decision tree algorithm due to Laskowski et al and that forms the basis for our new algorithm. Numerical simulations show that for three to six qubits we use fewer measurements than either the trivial algorithm or the decision tree algorithm. In addition, our method is easy to construct, assumes no prior knowledge of the system and works for any number of qubits.

DOI
10.48550/arxiv.1602.02099

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We present a new technique to reduce the expected number of measurements to declare an unknown quantum state as entangled. Our method is based on the geometric criterion and so requires only local Pauli measurements. Using concentration of measure, we provide a heuristic which allows us to reinterpret a previous decision tree algorithm due to Laskowski et al and that forms the basis for our new algorithm. Numerical simulations show that for three to six qubits we use fewer measurements than either the trivial algorithm or the decision tree algorithm. In addition, our method is easy to construct, assumes no prior knowledge of the system and works for any number of qubits.
Experimental procedures are presented for the rapid detection of entanglement of unknown arbitrary quantum states. The methods are based on the entanglement criterion using accessible correlations and the principle of correlation complementarity. Our first scheme essentially establishes the Schmidt decomposition for pure states, with few measurements only and without the need for shared reference frames. The second scheme employs a decision tree to speed up entanglement detection. We analyze the performance of the methods using numerical simulations and verify them experimentally for various states of two, three, and four qubits.
We introduce an experimental procedure for the detection of quantum entanglement of an unknown quantum state with a small number of measurements. The method requires neither a priori knowledge of the state nor a shared reference frame between the observers and can thus be regarded as a perfectly state-independent entanglement witness. The scheme starts with local measurements, possibly supplemented with suitable filtering, which essentially establishes the Schmidt decomposition for pure states. Alternatively we develop a decision tree that reveals entanglement within few steps. These methods are illustrated and verified experimentally for various entangled states of two and three qubits.
Classifying states as entangled or separable is a fundamental, but expensive task. This paper presents a method, the forest algorithm, to improve the amount of resources needed to detect entanglement. Starting from 'optimized' methods for using geometric criterion to detect entanglement, specific steps are replaced with machine learning models. Tests using numerical simulations indicate that the model is able to declare a state as entangled in fewer steps compared to existing methods. This improvement is achieved without affecting the correctness of the original algorithm.
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We propose a new method for detecting entanglement of two qubits and discuss its relation with the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality. Without the need for full quantum tomography for the density matrix we can experimentally detect the entanglement by measuring less than 9 local observables for any given state. We show that this test is stronger than the CHSH-Bell inequality and also gives an estimation for the degree of entanglement. If prior knowledge is available we can further greatly reduce the number of required local observables. The test is convenient and feasible with present experimental technology.