Quantum entanglement is essential to the development of quantum computation, communications, and technology. The controlled SWAP test, widely used for state comparison, can be adapted to an efficient and useful test for entanglement of a pure state. Here we show that the test can evidence the presence of entanglement (and further, genuine n-qubit entanglement), can distinguish entanglement classes, and that the concurrence of a two-qubit state is related to the test's output probabilities. We also propose a multipartite measure of entanglement that acts similarly for n-qubit states. The average number of copies of the test state required to detect entanglement decreases for larger systems, to four on average for many (n 8) qubits for maximally entangled states. For non-maximally entangled states, the number of copies required to detect entanglement increases with decreasing entanglement. Furthermore, the results are robust to second order when typical small errors are introduced to the state under investigation.

Driven by the value of quantum entanglement as a resource in quantum information, great interest has been shown in methods of detecting and measuring entanglement. An efficient test for entanglement in pure qubit states has recently been explored [Foulds et al, QST 6 035002 (2021)] based on an adaptation of the widely used controlled SWAP test for equivalence. We provide a range of extensions to both the controlled SWAP test for equivalence and the controlled SWAP test for entanglement, including an extension to optical states and qudits, as well as a modification to test for bipartite entanglement exclusively. We compare results from these cases with those from Foulds et al. and so show the test for equivalence can be successfully applied to these states and the test for entanglement can be applied in certain cases. We conclude that, while the test in qudits is successful in general, the extension to optical states has limited applicability, relevant only for entangled coherent states and similar systems. The test for bipartite entanglement yields useful results for bipartite cuts across a multi-qubit state, but fails to correctly detect entanglement between two qubits in a larger system, a failure which points to the limitations of the test when dealing with mixed states.

We show how to estimate a broad class of multipartite entanglement measures from Bell basis measurement data. In addition to lowering the experimental requirements relative to previously known methods of estimating these measures, our proposed scheme also enables a simpler analysis of the number of measurement repetitions required to achieve an -close approximation of the measures, which we provide for each. We focus our analysis on the recently introduced Concentratable Entanglements [ Beckey et al. Phys. Rev. Lett. 127, 140501 (2021)] because many other well-known multipartite entanglement measures are recovered as special cases of this family of measures. We extend the definition of the Concentratable Entanglements to mixed states and show how to construct lower bounds on the mixed state Concentratable Entanglements that can also be estimated using only Bell basis measurement data. Finally, we demonstrate the feasibility of our methods by realistically simulating their implementation on a Rydberg atom quantum computer.

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