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The paper presents an analysis of entangling and non-local operations in a quantum nondemolition (QND) interaction between multimode light with orbital angular momentum and an atomic ensemble. A protocol consisting of two QND operations with rotations of quadratures of atomic spin coherence and light between them provides a wide range of two-qubit operations, while the multimode nature of the chosen degrees of freedom allows the implementation of parallel operations over multiple two-qubit systems. We have used the formalism of equivalence classes and local invariants to evaluate the properties of two-qubit transformations. It is shown that, when selecting suitable values of the governing parameters - the duration of each of the two QND interactions and the rotation angles of the qubits - the protocol allows to realise a deterministic non-local SWAP operation and entangling square-root-SWAP operation with probability 1/3.
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Vashukevich et al. (2024) studied this question.
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