  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 consists of two QND
operations with rotations of quadratures of atomic spin coherence and light
between them. This protocol 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, such as 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
$\sqrt{SWAP}$ operation with probability 1/3.
