Realizing quantum advantage without entanglement in single-photon states

Correlations allow us to measure, and quantitatively study, the properties of physical systems, their evolution and their interactions. Quantum discord expresses quantum correlations beyond those associated with entanglement.\footnote{K Modi, {\em et al., Rev. Mod. Phys. } {\bf 84}, 1655 (2012)}. However, discord has not yet been adopted as a standard subject of study by the experimental community. Here we propose a feasible optical setup to generate symmetric two-qubit $X$-states with controllable coherences, where the two qubits correspond to the spin and path of a photon. With these states we show how a classical random variable $K$ can be encoded by Alice and decoded by Bob. Using our previous results \footnote{A. Maldonado-Trapp, {\em et al., Quantum Inf. Process} {\bf 14} 1947 (2015)} we study the correlations between the spin and path qubits and its relation with the information about $K$ that can be decoded by Bob using local measurements with or without two-qubit gate operations.\footnote{M. Gu, {\em et al., Nature Phys. }{\bf 8}, 671 (2012)} Discord is the mutual information contained in the coherences of the system, and it is possible to exploit it for quantum advantage even in the absence of entanglement.