Extending the controlled SWAP test to higher dimensions

Oliver Prove, Steph Foulds, Viv Kendon

Published 2021-12-08, updated 2022-02-07Version 3

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.