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What is the current record number of entagled qubits and how has this number been increased?

The latest result on stack exchange, which is 3 years old, reports 14 via this post: How many stabilised qubits have been achieved in Quantum Computing?

in accord with this paper: http://arxiv.org/abs/1009.6126?

For the purposes of this post, I'd prefer to leave the D-Wave computers out of the answers.

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This is not an answer to either of your four specific questions, but this recent paper arxiv.org/abs/1402.4848 achieved, if I am not mistaken, record single-qubit and two-qubit gate fidelities. –  Juan Miguel Arrazola Mar 7 at 19:35

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